Recent zbMATH articles in MSC 47https://zbmath.org/atom/cc/472023-03-23T18:28:47.107421ZWerkzeugOn the spectrum of dense random geometric graphshttps://zbmath.org/1503.051062023-03-23T18:28:47.107421Z"Adhikari, Kartick"https://zbmath.org/authors/?q=ai:adhikari.kartick"Adler, Robert J."https://zbmath.org/authors/?q=ai:adler.robert-joseph"Bobrowski, Omer"https://zbmath.org/authors/?q=ai:bobrowski.omer"Rosenthal, Ron"https://zbmath.org/authors/?q=ai:rosenthal.ronLet \(G\) be an \(n\)-vertex undirected graph, and let \(A\) be its adjacency matrix. The degree of the vertex \(i\) is then \(d_i=|N_G(i)|\), the number of neighbours of vertex \(i\) in \(G\), and the Laplacian of \(G\) is defined as \(L:=D-A\), where \(D\) is the diagonal matrix of vertex degrees. The normalized graph Laplacian of \(G\) is defined as
\[
\mathcal{L}:=D^{-\frac{1}{2}}LD^{-\frac{1}{2}}.
\]
Graph Laplacians and their spectra contain important information about the connectivity structure of graphs and the behavior of random walks on them. Graph spectra and harmonics also play key roles in various applications such as network analysis and machine learning.
The random geometric graph \(G(n,r)\) is defined as the undirected graph with vertex set \(\{1,2,\ldots,n\}\), where \(i\) is adjacent to \(j\), if, and only if, \(\|X_i - X_j\|\leq r\), where \(\|\cdot\|\) denotes the norm in \(\mathbb{R}^d\).
In this paper, the authors study the spectrum of the random geometric graph \(G(n,r)\), in a regime where the graph is dense and highly connected. In the Erdös-Rényi random graph \(G(n,p)\), it is well known that upon connectivity the spectrum of the normalized graph Laplacian is concentrated around 1. The authors show that such concentration does not occur in the \(G(n,r)\) case, even when the graph is dense and almost complete. They also show that the limiting spectral gap is strictly smaller than \(1\). In the special case where the vertices are distributed uniformly in the unit cube and \(r = 1\), they prove that for all \(k\in [0,d]\) there exists at least \(\binom{d}{k}\) eigenvalues near \(1-2^{-k}\), and the limiting spectral gap is exactly 1/2. They confirm that the corresponding eigenfunctions in this case are tightly related to the geometric configuration of the points.
The study of this paper has a nice motivation and the results obtained in this paper are interesting.
Reviewer: Shuchao Li (Wuhan)A note on the equation \(A X B = B = B X A\)https://zbmath.org/1503.150132023-03-23T18:28:47.107421Z"Deng, Chunyuan"https://zbmath.org/authors/?q=ai:deng.chunyuan"Deng, Xiaoli"https://zbmath.org/authors/?q=ai:deng.xiaoli"Lin, Chujian"https://zbmath.org/authors/?q=ai:lin.chujianSummary: In this note, we give the necessary and sufficient conditions for the existence of the solution and obtain the general solution to the equation \(A X B = B = B X A\). The results improve the recent results by \textit{M. Vosough} and \textit{M. S. Moslehian} [Electron. J. Linear Algebra 32, 172--183 (2017; Zbl 1375.15028)].Singular value inequalities for convex functions of positive semidefinite matriceshttps://zbmath.org/1503.150222023-03-23T18:28:47.107421Z"Al-Natoor, Ahmad"https://zbmath.org/authors/?q=ai:al-natoor.ahmad"Hirzallah, Omar"https://zbmath.org/authors/?q=ai:hirzallah.omar"Kittaneh, Fuad"https://zbmath.org/authors/?q=ai:kittaneh.fuadSummary: In this paper, we give new singular value inequalities for matrices. It is shown that if \(A, B, X\) are \(n\times n\) matrices such that \(X\) is positive semidefinite, and if \(f:[0,\infty)\rightarrow \mathbb{R}\) is an increasing nonnegative convex function, then
\[
\begin{aligned} s_j\left( f\left( \frac{\left| AXB^\ast\right|}{\left\| X\right\|}\right) \right) \leq \frac{\left\| f\left( \frac{A^\ast A+B^\ast B}{2}\right) \right\|}{\left\| X\right\|}s_j\left( X\right) \end{aligned}
\]
and
\[
\begin{aligned} s_j\left( AXB^\ast \right) \leq \frac{1}{2}\left\| \frac{A^\ast A}{ \left\| A\right\|^2}+\frac{B^\ast B}{\left\| B\right\|^2} \right\| \left\| A\right\| \left\| B\right\| s_j\left( X\right) \end{aligned}
\]
for \(j=1,2,\dots,n\). Some of our inequalities present refinements of some known singular value inequalities.New proofs on two recent inequalities for unitarily invariant normshttps://zbmath.org/1503.150262023-03-23T18:28:47.107421Z"Yang, Junjian"https://zbmath.org/authors/?q=ai:yang.junjian"Lu, Linzhang"https://zbmath.org/authors/?q=ai:lu.linzhangSummary: In this short note, we provide alternative proofs for several recent results due to \textit{K. M. R. Audenaert} [Oper. Matrices 9, No. 2, 475--479 (2015; Zbl 1317.15020)], \textit{L. Zou} [Linear Algebra Appl. 562, 154--162 (2019; Zbl 1402.15017)] and \textit{L. Zou} and \textit{Y. Jiang} [J. Math. Inequal. 10, No. 4, 1119--1122 (2016; Zbl 1366.15016)].Norm inequalities for submultiplicative functions involving contraction sector \(2 \times 2\) block matriceshttps://zbmath.org/1503.150282023-03-23T18:28:47.107421Z"Zhou, Xiaoying"https://zbmath.org/authors/?q=ai:zhou.xiaoyingSummary: In this article, we show unitarily invariant norm inequalities for sector \(2\times 2\) block matrices which extend and refine some recent results of \textit{A. Bourahli} et al. [Positivity 25, No. 2, 447--467 (2021; Zbl 1465.15026)].On the joint spectral radius of nonnegative matriceshttps://zbmath.org/1503.150292023-03-23T18:28:47.107421Z"Bui, Vuong"https://zbmath.org/authors/?q=ai:bui.vuongThe author studies the joint spectral radius of nonnegative matrices and gives an effective bound of the joint spectral radius of a finite set of nonnegative matrices. Given a finite set \(\Sigma\) of square matrices in \(\mathbb{C}^{d\times d}\), the joint spectral radius \(\rho(\Sigma)\) of \(\Sigma\) is defined to be the limit
\[
\rho(\Sigma)=\lim_{n\to \infty} \sqrt[n]{\|\Sigma^n\|},
\]
where \(\|\Sigma^n\|=\max_{A_1,\dots,A_n\in \Sigma} \|A_1\dots A_n\|\). The existence of the limit is shown in Proposition 1. Though \(\rho(\Sigma)\) is also defined for infinite bounded sets \(\Sigma\), only finite sets are considered in this paper. For a component \(C\) and every \(m\), denote
\[
\|\Sigma^n\|_C=\max_{A_1,\dots,A_n\in \Sigma} \ \max_{i,j\in C} \ |(A_1\dots A_n)_{i,j}|
\]
and
\[
P_m(\Sigma)=\max_{A_1,\dots,A_m\in \Sigma}\ \rho(A_1\dots A_m).
\]
Given a finite set \(\Sigma\) of nonnegative matrices and \(n\), the author gives the bound of \(\rho(\Sigma)\) in Theorem 2 as follows:
\[
\sqrt[n]{\left(\frac{V}{UD}\right)^D\max_C\|\Sigma^n\|_C}\leq \rho(\Sigma)\leq \sqrt[n]{D\max_C\|\Sigma^n\|_C},
\]
where \(D\times D\) is the dimension of the matrices, \(U\), \(V\) are respectively the largest entry and the smallest entry over all the positive entries of the matrices in \(\Sigma\), and \(C\) is taken over all components in the dependency graph (see Definition 1). The method for estimating the joint spectral radius in Theorem 2 is better than the one for the popular bound \( \sqrt[m]{P_m(\Sigma)}\leq \rho(\Sigma)\leq \sqrt[m]{D\|\Sigma^m\|}\) in [\textit{R. M. Jungers}, The joint spectral radius. Theory and applications. Berlin: Springer (2009; \url{doi:10.1007/978-3-540-95980-9})] by a root of a polynomial of degree \(r\).
The author also gives the following bound on \(\|\Sigma^n\|\):
Theorem. If \(\rho(\Sigma)>0\), then there exist a non-negative integer \(r\) and two positive numbers \(\alpha\), \(\beta\) so that for every \(n\) there holds
\[
\alpha n^r \rho(\Sigma)^n \leq \|\Sigma\|^n \leq \beta n^r\rho(\Sigma)^n.
\]
Reviewer: Tin Yau Tam (Reno)Inverse numerical range and Abel-Jacobi map of Hermitian determinantal representationhttps://zbmath.org/1503.150302023-03-23T18:28:47.107421Z"Chien, Mao-Ting"https://zbmath.org/authors/?q=ai:chien.mao-ting"Nakazato, Hiroshi"https://zbmath.org/authors/?q=ai:nakazato.hiroshiSummary: Let \(A\) be an \(n\times n\) matrix. The Hermitian parts of \(A\) are denoted by \(\Re(A)=(A+A^\ast)/2\) and \(\Im(A)=(A-A^\ast)/(2 i)\). The kernel vectors of the linear pencil \(x\Re(A)+y\Im(A)+zI_n\) play a role for the inverse numerical range of \(A\). This kernel vector technique was applied to perform the inverse numerical range of \(3\times 3\) symmetric matrices. In this paper, we follow the kernel vector method and apply the Abel theorem for \(3\times 3\) Hermitian matrices. We present the elliptic curve group structure of the cubic curve associated to the ternary form of the matrix, and characterize the Abel type additive structure of the divisors of the cubic curve. A numerical example is given to illustrate the characterization related to the Riemann theta representation.Generalized circular projectionshttps://zbmath.org/1503.150312023-03-23T18:28:47.107421Z"Ilišević, Dijana"https://zbmath.org/authors/?q=ai:ilisevic.dijana"Li, Chi-Kwong"https://zbmath.org/authors/?q=ai:li.chi-kwong"Poon, Edward"https://zbmath.org/authors/?q=ai:poon.edwardLet \(X\) be a complex Banach space. The collection \(\{P_1,\ldots, P_r\}\) of non-zero bounded projections on \(X\) is said to be a family of \(r\)-circular projections if:
\begin{itemize}
\item[(i)] \(\sum_{j=1}^r P_j = I\) and \(P_iP_j = \delta_{i,j}P_i\);
\item[(ii)] There are distinct complex numbers \(\mu_1,\ldots, \mu_r\) of modulus \(1\), such that \(\sum_{j=1}^r \mu_j P_j\) is an isometry of \(X\) onto \(X\).
\end{itemize}
The authors carefully analyse the existence of \(r\)-circular projections for classes of matrix norms on \(M_n(\mathbb C)\), including unitarily invariant, unitarily congruence invariant and unitarily similarity invariant norms. Here, the norm \(\Vert\cdot \Vert\) on \(M_n(\mathbb C)\) is unitarily invariant if \(\Vert UAV\Vert = \Vert A\Vert\), unitarily congruence invariant if \(\Vert UAU^{\dagger}\Vert = \Vert A\Vert\), and unitarily similarity invariant if \(\Vert UAU^*\Vert = \Vert A\Vert\) for all unitary matrices \(U, V\) and all \(A \in M_n(\mathbb C)\). The structure of the \(r\)-projections \(\{P_1,\ldots, P_r \}\) is determined, together with the values of \(r\) for which the isometry \(\mu_1 P_1 + \dots + \mu_rP_r\) takes a certain form on \((M_n(\mathbb C), \Vert\cdot\Vert)\). The outcome is quite involved and depends on the isometry group of the given matrix norm.
A sample result is as follows. Let \(\Vert \cdot\Vert\) be a unitarily invariant norm on \(M_n(\mathbb C)\), different from a multiple of the Frobenius norm. Then:
\begin{itemize}
\item[(i)] There is a family of \(r\)-circular projections corresponding to an isometry of the form \(A \mapsto UAV\) if and only if \(2 \le r \le n^2\);
\item[(ii)] There is a family of \(r\)-circular projections corresponding to an isometry of the form \(A \mapsto UA^{\dagger}V\) if and only if \(r \in \{2,\ldots,n^2\} \setminus J_n\), where \(J_4 = \{3,7,11\}\) and \(J_n =\{3,7\}\) for \(n \neq 4\).
\end{itemize}
Reviewer: Hans-Olav Tylli (Helsinki)Erratum to: ``Decomposing a matrix into two submatrices with extremely small operator norm''https://zbmath.org/1503.150322023-03-23T18:28:47.107421Z"Limonova, I. V."https://zbmath.org/authors/?q=ai:limonova.irina-vErratum to the author's paper [Math. Notes 108, No. 1, 137--141 (2020; Zbl 1447.15021); translation from Mat. Zametki 108, No. 1, 153--157 (2020)].Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theoryhttps://zbmath.org/1503.150402023-03-23T18:28:47.107421Z"Bogoya, Manuel"https://zbmath.org/authors/?q=ai:bogoya.manuel"Ekström, Sven-Erik"https://zbmath.org/authors/?q=ai:ekstrom.sven-erik"Serra-Capizzano, Stefano"https://zbmath.org/authors/?q=ai:serra-capizzano.stefanoAuthors' abstract: Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function \(f\). Independently and under the milder hypothesis that \(f\) is even and monotone over \([0, \pi]\), matrix-less algorithms have been developed for the fast eigenvalue computation of large Toeplitz matrices, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions predicted by the simple-loop theory, combined with the extrapolation idea. Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we adapt the matrix-less algorithm to the considered new setting. Numerical experiments show a higher precision (till machine precision) and the same linear computation cost, when compared with the matrix-less procedures already presented in the relevant literature. Among the advantages, we concisely mention the following: (a) when the coefficients of the simple-loop function are analytically known, the algorithm computes them perfectly; (b) while the proposed algorithm is better or at worst comparable to the previous ones for computing the inner eigenvalues, it is vastly better for the computation of the extreme eigenvalues; a mild deterioration in the quality of the numerical experiments is observed when dense Toeplitz matrices are considered, having generating function of low smoothness and not satisfying the simple-loop assumptions.
Reviewer: Anton Iliev (Plovdiv)Generalized Lie triple derivations on generalized matrix algebrashttps://zbmath.org/1503.160552023-03-23T18:28:47.107421Z"Akhtar, Mohd Shuaib"https://zbmath.org/authors/?q=ai:akhtar.mohd-shuaib"Ashraf, Mohammad"https://zbmath.org/authors/?q=ai:ashraf.mohammad"Ansari, Mohammad Afajal"https://zbmath.org/authors/?q=ai:ansari.mohammad-afajalIn this paper, the authors suppose that \(R\) is a commutative ring with identity, \(A, B\) are \(R\)-algebras, \(M\) is an \((A, B)\)-bimodule and \(N\) is a \((B, A)\)-bimodule. The \(R\)-algebra \(g=G(A,M,N, B)\) is a generalized matrix algebra defined by the Morita context \((A, B,M,N, \zeta_{MN}, \chi_{NM})\): More precisely, the authors provide the structure of generalized Lie triple derivations \(G_{L}\) on generalized matrix algebras \(g\) and prove that under certain restrictions \(G_L\) can be written as \(G_{L}=\Delta +\chi\), where \(\Delta\) is a generalized derivation and \(\chi\) is a central valued mapping.
The authors present their results in two main sections: The first one discusses the structure of generalized Lie triple derivations, they find out the structure of generalized Lie triple derivations on generalized matrix algebras; the second section concerning the properness of generalized Lie triple derivations. In this section, they characterize generalized Lie triple derivations on generalized matrix algebras.
Reviewer: Mehsin Atteya (Leicester)Boundedness of fractional integrals on special John-Nirenberg-Campanato and Hardy-type spaces via congruent cubeshttps://zbmath.org/1503.260092023-03-23T18:28:47.107421Z"Jia, Hongchao"https://zbmath.org/authors/?q=ai:jia.hongchao"Tao, Jin"https://zbmath.org/authors/?q=ai:tao.jin"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wen"Zhang, Yangyang"https://zbmath.org/authors/?q=ai:zhang.yangyang(no abstract)On integral inequalities related to the weighted and the extended Chebyshev functionals involving different fractional operatorshttps://zbmath.org/1503.260452023-03-23T18:28:47.107421Z"Çelik, Barış"https://zbmath.org/authors/?q=ai:celik.baris"Gürbüz, Mustafa Ç."https://zbmath.org/authors/?q=ai:gurbuz.mustafa-cagri"Özdemir, M. Emin"https://zbmath.org/authors/?q=ai:ozdemir.muhamet-emin"Set, Erhan"https://zbmath.org/authors/?q=ai:set.erhanSummary: The role of fractional integral operators can be found as one of the best ways to generalize classical inequalities. In this paper, we use different fractional integral operators to produce some inequalities for the weighted and the extended Chebyshev functionals. The results are more general than the available classical results in the literature.Some inequalities for the multilinear singular integrals with Lipschitz functions on weighted Morrey spaceshttps://zbmath.org/1503.260502023-03-23T18:28:47.107421Z"Gürbüz, Ferit"https://zbmath.org/authors/?q=ai:gurbuz.feritSummary: The aim of this paper is to prove the boundedness of the oscillation and variation operators for the multilinear singular integrals with Lipschitz functions on weighted Morrey spaces.A class of Hilbert-type multiple integral inequalities with the kernel of generalized homogeneous function and its applicationshttps://zbmath.org/1503.260542023-03-23T18:28:47.107421Z"Hong, Yong"https://zbmath.org/authors/?q=ai:hong.yong"Liao, Jianquan"https://zbmath.org/authors/?q=ai:liao.jianquan"Yang, Bicheng"https://zbmath.org/authors/?q=ai:yang.bicheng"Chen, Qiang"https://zbmath.org/authors/?q=ai:chen.qiang.1Summary: Let \(x=(x_1,x_2,\dots,x_n)\), and let \(K(u(x),v(y))\) satisfy \(u(rx)=ru(x)\), \(v(ry)=rv(y)\), \(K(ru,v)=r^{\lambda\lambda_1}K(u, r^{-\frac{\lambda_1}{\lambda_2}}v)\), and \(K(u,rv)=r^{\lambda\lambda_2}K(r^{-\frac{\lambda_2}{\lambda_1}}u, v)\). In this paper, we obtain a necessary and sufficient condition and the best constant factor for the Hilbert-type multiple integral inequality with kernel \(K(u(x),v(y))\) and discuss its applications in the theory of operators.On a new Hilbert-type integral inequality involving the upper limit functionshttps://zbmath.org/1503.260682023-03-23T18:28:47.107421Z"Mo, Hongmin"https://zbmath.org/authors/?q=ai:mo.hongmin"Yang, Bicheng"https://zbmath.org/authors/?q=ai:yang.bichengSummary: By applying the weight functions and the idea of introduced parameters we give a new Hilbert-type integral inequality involving the upper limit functions and the beta and gamma functions. We consider equivalent statements of the best possible constant factor related to a few parameters. As applications, we obtain a corollary in the case of a nonhomogeneous kernel and some particular inequalities.Parametrization of the solution set of a matricial truncated hamburger moment problem by a Schur type algorithmhttps://zbmath.org/1503.300872023-03-23T18:28:47.107421Z"Fritzsche, Bernd"https://zbmath.org/authors/?q=ai:fritzsche.bernd"Kirstein, Bernd"https://zbmath.org/authors/?q=ai:kirstein.bernd"Kley, Susanne"https://zbmath.org/authors/?q=ai:kley.susanne"Mädler, Conrad"https://zbmath.org/authors/?q=ai:madler.conradSummary: This paper contains a Schur analytic approach to a truncated matricial moment problem of Hamburger type, which is studied in the most general case. It is shown that a Schur type algorithm constructed by the authors for a related moment problem can be suitably modified to obtain a full description of the solution set with the aid of a linear fractional transformation with polynomial generating matrix-valued function. The main feature of our Schur type algorithm consists of an appropriate synthesis of two different versions of types of algorithms, namely on the one side an algebraic one working for sequences of complex matrices and on the other side a function theoretic one applied to special classes of holomorphic matrix functions.
For the entire collection see [Zbl 1478.30002].Carleson measures on Dirichlet type spaces in the quaternionic unit ballhttps://zbmath.org/1503.301102023-03-23T18:28:47.107421Z"Kumar, Sanjay"https://zbmath.org/authors/?q=ai:kumar.sanjay"Sharma, Vishal"https://zbmath.org/authors/?q=ai:sharma.vishal.1|sharma.vishalSummary: In this paper, we characterize Carleson measures on the Dirichlet-type space \(\mathcal D_{\alpha}^p\) for different values of \(p\), \(q\) and \(\alpha\) and in terms of axially symmetric completion of a pseudohyperbolic disc.Difference of quaternionic weighted composition operators on slice regular Fock spaceshttps://zbmath.org/1503.301122023-03-23T18:28:47.107421Z"Liang, Y."https://zbmath.org/authors/?q=ai:liang.yuxia"Wang, J."https://zbmath.org/authors/?q=ai:wang.j.22Summary: Quaternionic Fock space is a useful generalization of the Fock space in the complex plane, which plays an important role in quantum mechanics. In view of quaternionic operator theory this topic attains more diversity and complexity. In this paper we first explore the connection between the properties and about the difference of quaternionic weighted composition operators acting on slice regular Fock space with the function theoretic properties of the symbols. It can further imply some topological information on the set of quaternionic weighted composition operators.Strongly continuous composition semigroups on analytic Morrey spaceshttps://zbmath.org/1503.301192023-03-23T18:28:47.107421Z"Sun, Fangmei"https://zbmath.org/authors/?q=ai:sun.fangmei"Wulan, Hasi"https://zbmath.org/authors/?q=ai:wulan.hasiSummary: For a semigroup \((\varphi_t)_{t\geq 0}\) consiting of analytic self-maps from the unit disk \(\mathbb{D}\) to itself, a strongly continuous composition semi-group \((C_t)_{t\geq 0}\) onduced by \((\varphi_t)_{t\geq 0}\) on analytic Morrey spaces \(H^{2, \lambda}\), \(0<\lambda<1\), is investigated. By the weak compactness of resolvent operator, we give a complete characteriziation of \(H^{2, \lambda}=[\varphi_t, H^{2, \lambda}]\) for \(0<\lambda < 1\) in terms of the inifinitesimal generator if the Denjoy-Wolff point of \((\varphi_{t})_{t\geq 0}\) is in \(\mathbb{D}\).Toeplitz operators between distinct abstract Hardy spaceshttps://zbmath.org/1503.301212023-03-23T18:28:47.107421Z"Karlovich, Alexei"https://zbmath.org/authors/?q=ai:karlovych.oleksiy|karlovich.alexei-yuFor the entire collection see [Zbl 1478.00019].The generalized Volterra integral operator and Toeplitz operator on weighted Bergman spaceshttps://zbmath.org/1503.301242023-03-23T18:28:47.107421Z"Du, Juntao"https://zbmath.org/authors/?q=ai:du.juntao"Li, Songxiao"https://zbmath.org/authors/?q=ai:li.songxiao"Qu, Dan"https://zbmath.org/authors/?q=ai:qu.danSummary: We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disc. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class membership of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy space \(H^2\).Pseudo-Carleson measures for Fock spaceshttps://zbmath.org/1503.301252023-03-23T18:28:47.107421Z"Liu, Yongqing"https://zbmath.org/authors/?q=ai:liu.yongqing.1"Hou, Shengzhao"https://zbmath.org/authors/?q=ai:hou.shengzhaoSummary: In this paper, we introduce pseudo-Carleson measure, a generalization of Carleson measure, for Fock spaces, and give sufficient and necessary conditions for a complex Borel measure to be pseudo-Carleson. We then give integral representations of functions in a Fock space, and use the measures to characterize boundedness and compactness of small Hankel operators on Fock spaces.Commutative algebras of Toeplitz operators on the Bergman space revisited: spectral theorem approachhttps://zbmath.org/1503.301262023-03-23T18:28:47.107421Z"Rozenblum, Grigori"https://zbmath.org/authors/?q=ai:rozenblum.grigori-v"Vasilevski, Nikolai"https://zbmath.org/authors/?q=ai:vasilevski.nikolai-lLet \(\mathbb{D}\) be the open unit disk, \(\mathbb{T}=\partial\mathbb{D}\), and let \(\Pi\) be the upper half-plane. The paper deals with studying a bijection between commutative \(C^*\)-algebras generated by Toeplitz operators with bounded measurable symbols that act on the standard weighted Bergman space \({\mathcal A}^2_\lambda(\mathbb{D})\) \((\lambda>-1)\) and maximal abelian subgroups of Möbius transforms of \(\mathbb{D}\). Each one-parameter abelian subgroup \(G\) of this type is conjugated to one of the following three model groups:
\begin{itemize}
\item elliptic, \(\mathbb{T}:\mathbb{D}\to\mathbb{D}\), with action \(t\in\mathbb{T}:z\mapsto tz\);
\item parabolic, \(\mathbb{R}:\Pi\to\Pi\), with action \(h\in\mathbb{R}:z\mapsto z+h\);
\item hyperbolic, \(\mathbb{R}_+:\Pi\to\Pi\), with action \(\rho\in\mathbb{R}_+:z\mapsto\rho z\).
\end{itemize}
The orbits of these groups are given by the following parametric equations:
\begin{itemize}
\item group \(\mathbb{T}\) on \(\mathbb{D}\), \(x=R\cos\theta\), \(y=R\sin\theta\), \(\theta\in[0,2\pi)\), with a fixed \(R\in(0,1)\);
\item group \(\mathbb{R}\) on \(\Pi\): \(x=h\), \(y=y_0\), \(h\in\mathbb{R}\), with a fixed \(y_0\in\mathbb{R}_+\);
\item group \(\mathbb{R}_+\) on \(\Pi\): \(x=\rho\cos\theta\), \(y=\rho\sin\theta\), \(\rho\in\mathbb{R}_+\), with a fixed \(\theta\in(0,\pi)\).
\end{itemize}
These orbits are integral curves, respectively, for the following vector fields:
\[
V_{\mathrm{ell}}=-y\frac{\partial}{\partial x}+x\frac{\partial}{\partial y},\quad V_{\mathrm{par}}=\frac{\partial}{\partial x},\quad V_{\mathrm{hyp}}= x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}.
\]
For each \(G\in\{\mathbb{T},\mathbb{R},\mathbb{R}_+\}\), there exists a Borel subset \(X_G\subset\mathbb{R}\) with a measure \(\sigma_G\) and a unitary operator \(R_G:{\mathcal A}^2_\lambda\to L_2(X_G,\sigma_G)\) such that each Toeplitz operator \(T_a\) with \(G\)-invariant symbol \(a\) is unitarily equivalent to the multiplication operator by a certain function \(\gamma_a\) on the space \(L_2(X_G,\sigma_G)\),
\[
R_GT_aR_G^*=\gamma_aI.
\]
The map \(T_a\mapsto\gamma_a\) defined initially for Toeplitz operators with bounded symbols \(a\) can be extended to bounded Toeplitz operators with wider classes of symbols \(a\) by using the following approach.
With each model commutative algebra, the authors associate a certain self-adjoint (unbounded) operator \(N\) which generates this algebra via the functional calculus based on the Spectral Theorem. For example, for \(G=\mathbb{R}\), they construct the spectral measure \(E\) of \(N\) such that the operator \(\varphi(N)=\int_{\mathbb{R}} \varphi(\eta)dE(\eta)\) is well defined and normal on a domain \({\mathcal D}_\varphi \subset{\mathcal A}^2_\lambda(\Pi)\) for every \(E\)-measurable function \(\varphi\). The operator \(\varphi(N)\) is bounded, and thus defined on the whole \({\mathcal A}^2_\lambda(\Pi)\), if and only if the function \(\varphi\) is \(E\)-essentially bounded. The mapping \({\mathcal J}:\varphi\mapsto\varphi(N)\) is an isometric isomorphism of the unital \(C^*\)-algebra \(L_\infty(\mathbb{R},E)\) with involution \(\varphi\mapsto \overline{\varphi}\) onto a commutative unital algebra of bounded linear operators on the space \({\mathcal A}^2_\lambda(\Pi)\) with involution \(H\mapsto H^*\), and \({\mathcal J} (L_\infty(\mathbb{R},E))\) is a von Neumann algebra.
Relations between properties of the spectral functions \(\gamma_a\) of Toeplitz operators \(T_a\), their symbols \(a\) and the spectral representations of the considered commutative algebras of Toeplitz operators are studied.
Reviewer: Yuri I. Karlovich (Cuernavaca)Inner functions, completeness and spectrahttps://zbmath.org/1503.301312023-03-23T18:28:47.107421Z"Poltoratski, Alexei"https://zbmath.org/authors/?q=ai:poltoratski.alexei-gFor the entire collection see [Zbl 1478.00019].Caputo-Fabrizio fractional differential equations with non instantaneous impulseshttps://zbmath.org/1503.340042023-03-23T18:28:47.107421Z"Abbas, Saïd"https://zbmath.org/authors/?q=ai:abbas.said"Benchohra, Mouffak"https://zbmath.org/authors/?q=ai:benchohra.mouffak"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseThe initial value problem of Caputo-Fabrizio fractional differential equations with non-instantaneous impulses is studied. Some existence results based on Schauder's and Mönch's fixed point theorems and the technique of the measure of noncompactness are obtained. Ignoring typos in formulas, the paper provides a good base for further studying of the given problem.
Reviewer: Snezhana Hristova (Plovdiv)Solutions to differential equations via fixed point approaches: new mathematical foundations and applicationshttps://zbmath.org/1503.340052023-03-23T18:28:47.107421Z"Almuthaybiri, Saleh Salhan G."https://zbmath.org/authors/?q=ai:almuthaybiri.saleh-salhan-g(no abstract)Applicability of Mönch's fixed point theorem on existence of a solution to a system of mixed sequential fractional differential equationhttps://zbmath.org/1503.340062023-03-23T18:28:47.107421Z"Awadalla, Muath"https://zbmath.org/authors/?q=ai:awadalla.muath-m(no abstract)On system of mixed fractional hybrid differential equationshttps://zbmath.org/1503.340072023-03-23T18:28:47.107421Z"Awadalla, Muath"https://zbmath.org/authors/?q=ai:awadalla.muath-m"Mahmudov, Nazim I."https://zbmath.org/authors/?q=ai:mahmudov.nazim-idrisoglu(no abstract)On solutions of hybrid-Sturm-Liouville-Langevin equations with generalized versions of Caputo fractional derivativeshttps://zbmath.org/1503.340092023-03-23T18:28:47.107421Z"Boutiara, Abdellatif"https://zbmath.org/authors/?q=ai:boutiara.abdellatif"Wahash, Hanan A."https://zbmath.org/authors/?q=ai:wahash.hanan-abdulrahman"Zahran, Heba Y."https://zbmath.org/authors/?q=ai:zahran.heba-y"Mahmoud, Emad E."https://zbmath.org/authors/?q=ai:mahmoud.emad-e"Abdel-Aty, Abdel-Haleem"https://zbmath.org/authors/?q=ai:abdel-aty.abdel-haleem"Yousef, El Sayed"https://zbmath.org/authors/?q=ai:yousef.el-sayed(no abstract)Finite-approximate controllability of fractional stochastic evolution equations with nonlocal conditionshttps://zbmath.org/1503.340132023-03-23T18:28:47.107421Z"Ding, Yonghong"https://zbmath.org/authors/?q=ai:ding.yonghong"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: This paper deals with the finite-approximate controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. We establish sufficient conditions for the finite-approximate controllability of the control system when the compactness conditions or Lipschitz conditions for the nonlocal term and uniform boundedness conditions for the nonlinear term are not required. The discussion is based on the fixed point theorem, approximation techniques and diagonal argument. In the end, an example is presented to illustrate the abstract theory. Our result improves and extends some relevant results in this area.On a multipoint fractional boundary value problem in a fractional Sobolev spacehttps://zbmath.org/1503.340172023-03-23T18:28:47.107421Z"Guezane-Lakoud, A."https://zbmath.org/authors/?q=ai:guezane-lakoud.assia|lakoud.a-guezane"Khaldi, R."https://zbmath.org/authors/?q=ai:khaldi.rabah"Boucenna, D."https://zbmath.org/authors/?q=ai:boucenna.djalal"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-joseIn this paper, a class of Riemann-Liouville fractional differential equations with multipoint boundary conditions in a Sobolev space are considered. By employing the upper and lower solutions method and Schauder fixed point theorem, the existence of positive solutions for the mentioned problems is established. An example with numerical simulations is given to show the effectiveness of the main results.
Reviewer: Wengui Yang (Sanmenxia)Solvability of infinite systems of fractional differential equations in the double sequence space \(2^c (\triangle)\)https://zbmath.org/1503.340262023-03-23T18:28:47.107421Z"Mehravaran, Hamid"https://zbmath.org/authors/?q=ai:mehravaran.hamid"Kayvanloo, Hojjatollah Amiri"https://zbmath.org/authors/?q=ai:kayvanloo.hojjatollah-amiri"Mursaleen, Mohammad"https://zbmath.org/authors/?q=ai:mursaleen.mohammad|mursaleen.mohammad-ayman(no abstract)A generalized approach of fractional Fourier transform to stability of fractional differential equationhttps://zbmath.org/1503.340272023-03-23T18:28:47.107421Z"Mohanapriya, Arusamy"https://zbmath.org/authors/?q=ai:mohanapriya.arusamy"Sivakumar, Varudaraj"https://zbmath.org/authors/?q=ai:sivakumar.varudaraj"Prakash, Periasamy"https://zbmath.org/authors/?q=ai:prakash.periasamyIn this work, authors consider a fractional order differential equation with impulsive condition. The fractional derivative is taken as Caputo kind. The main result is Mittag-Leffler-Hyers-Ulam stability of the considered equation. Such property implies that a function satisfying the differential equation approximately is close to an exact solution of differential equation. Fixed point theorem, fractional Fourier transform along with other concepts are used to establish the results. At the end, the authors provide three examples for illustration.
Reviewer: Syed Abbas (Mandi)Certain analysis of solution for the nonlinear two-point boundary value problem with Caputo fractional derivativehttps://zbmath.org/1503.340282023-03-23T18:28:47.107421Z"Murad, Shayma Adil"https://zbmath.org/authors/?q=ai:murad.shayma-adil(no abstract)Existence of solutions for a fractional boundary value problem at resonancehttps://zbmath.org/1503.340322023-03-23T18:28:47.107421Z"Silva, Anabela S."https://zbmath.org/authors/?q=ai:silva.anabela-sousaSummary: In this paper, we focus on the existence of solutions to a fractional boundary value problem at resonance. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin.Existence and uniqueness of solutions for a class of higher-order fractional boundary value problems with the nonlinear term satisfying some inequalitieshttps://zbmath.org/1503.340332023-03-23T18:28:47.107421Z"Wang, Fang"https://zbmath.org/authors/?q=ai:wang.fang"Liu, Lishan"https://zbmath.org/authors/?q=ai:liu.lishan"Wu, Yonghong"https://zbmath.org/authors/?q=ai:wu.yonghong.1Summary: This paper focuses on a class of hider-order nonlinear fractional boundary value problems. The boundary conditions contain Riemann-Stieltjes integral and nonlocal multipoint boundary conditions. It is worth mentioning that the nonlinear term and the boundary conditions contain fractional derivatives of different orders. Based on the Schauder fixed point theorem, we obtain the existence of solutions under the hypothesis that the nonlinear term satisfies the Carathéodory conditions. We apply the Banach contraction mapping principle to obtain the uniqueness of solutions. Moreover, by using the theory of spectral radius we prove the uniqueness and nonexistence of positive solutions. Finally, we illustrate our main results by some examples.Sensitivity analysis for a fractional stochastic differential equation with \(S^p\)-weighted pseudo almost periodic coefficients and infinite delayhttps://zbmath.org/1503.340352023-03-23T18:28:47.107421Z"Yan, Zuomao"https://zbmath.org/authors/?q=ai:yan.zuomao(no abstract)Positive solutions of fractional \(p\)-Laplacian equations with integral boundary value and two parametershttps://zbmath.org/1503.340362023-03-23T18:28:47.107421Z"Zhang, Luchao"https://zbmath.org/authors/?q=ai:zhang.luchao"Zhang, Weiguo"https://zbmath.org/authors/?q=ai:zhang.weiguo"Liu, Xiping"https://zbmath.org/authors/?q=ai:liu.xiping"Jia, Mei"https://zbmath.org/authors/?q=ai:jia.meiSummary: We consider a class of Caputo fractional \(p\)-Laplacian differential equations with integral boundary conditions which involve two parameters. By using the Avery-Peterson fixed point theorem, we obtain the existence of positive solutions for the boundary value problem. As an application, we present an example to illustrate our main result.Multiple positive solutions for higher-order fractional integral boundary value problems with singularity on space variablehttps://zbmath.org/1503.340372023-03-23T18:28:47.107421Z"Zhang, Xingqiu"https://zbmath.org/authors/?q=ai:zhang.xingqiu"Shao, Zhuyan"https://zbmath.org/authors/?q=ai:shao.zhuyan"Zhong, Qiuyan"https://zbmath.org/authors/?q=ai:zhong.qiuyan(no abstract)Weakly singular integral inequalities and global solutions for fractional differential equations of Riemann-Liouville typehttps://zbmath.org/1503.340392023-03-23T18:28:47.107421Z"Zhu, Tao"https://zbmath.org/authors/?q=ai:zhu.taoSummary: In this paper, we obtain some new results about weakly singular integral inequalities. These inequalities are used to establish the global existence and uniqueness results for fractional differential equations of Riemann-Liouville type. Some examples are provided to illustrate the applicability of our main results.Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delayshttps://zbmath.org/1503.340462023-03-23T18:28:47.107421Z"Han, Xiaoling"https://zbmath.org/authors/?q=ai:han.xiaoling"Lei, Ceyu"https://zbmath.org/authors/?q=ai:lei.ceyu(no abstract)On the uniform stability of recovering sine-type functions with asymptotically separated zeroshttps://zbmath.org/1503.340532023-03-23T18:28:47.107421Z"Buterin, S. A."https://zbmath.org/authors/?q=ai:buterin.sergey-alexandrovichThe paper deals with the entire function of the form
\[
\theta(z) = S(z) + \int_{-b}^b w(x) exp(izx) \, dx,
\]
where \(S(z) = P_N(z) S_0(z)\), \(P_N(z)\) is a polynomial of degree \(N\), \(S_0(z)\) is a sine-type function of exponential type \(b\) with asymptotically separated zeros, \(w \in L_2(-b,b)\). In particular, characteristic functions of various strongly regular differential operators and pencils of the first and the second orders have this form.
The main result of the paper is the uniform stability of recovering the function \(\theta(z)\) from its zeros. The author proves that the dependence of \(\theta(z)\) on the sequence of its zeros is Lipschitz continuous with respect to a suitable metric on each ball of a finite radius. This result can be applied to investigate the uniform stability of inverse spectral problems for differential operators and pencils. In addition, the author obtains the asymptotics of the zeros and a representation of \(\theta(z)\) as an infinite product by its zeros. The latter results play an auxiliary role for the proof of the main theorem, but also have a separate significance and can be applied to the spectral theory of differential operators.
Reviewer: Natalia Bondarenko (Saratov)Recovering differential pencils with spectral boundary conditions and spectral jump conditionshttps://zbmath.org/1503.340542023-03-23T18:28:47.107421Z"Khalili, Yasser"https://zbmath.org/authors/?q=ai:khalili.yasser"Baleanu, Dumitru"https://zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: In this work, we discuss the inverse problem for second order differential pencils with boundary and jump conditions dependent on the spectral parameter. We establish the following uniqueness theorems: \((i)\) the potentials \(q_k(x)\) and boundary conditions of such a problem can be uniquely established by some information on eigenfunctions at some internal point \(b\in (\frac{\pi }{2},\pi )\) and parts of two spectra; \((ii)\) if one boundary condition and the potentials \(q_k(x)\) are prescribed on the interval \([\pi /2(1-\alpha ),\pi ]\) for some \(\alpha \in (0, 1)\), then parts of spectra \(S\subseteq \sigma (L)\) are enough to determine the potentials \(q_k(x)\) on the whole interval \([0, \pi ]\) and another boundary condition.Analysis of fractional differential inclusion models for COVID-19 via fixed point results in metric spacehttps://zbmath.org/1503.340552023-03-23T18:28:47.107421Z"Alansari, Monairah"https://zbmath.org/authors/?q=ai:alansari.monairah-omar"Shagari, Mohammed Shehu"https://zbmath.org/authors/?q=ai:shagari.mohammed-shehu(no abstract)Boundary value problems for differential inclusions with \(\varphi\)-Laplacianhttps://zbmath.org/1503.340562023-03-23T18:28:47.107421Z"Tebbaa, Ahmed"https://zbmath.org/authors/?q=ai:tebbaa.ahmed"Aitalioubrahim, Myelkebir"https://zbmath.org/authors/?q=ai:aitalioubrahim.myelkebirIn this paper, the authors establish existence results for a second-order nonlinear differential inclusion problem
\[
(\Phi(x'(t)))'\in F(t,x(t)) \text{ for a.e. } t\in [0,T],
\]
satisfying periodic or Dirichlet boundary conditions. The approach is based on the method of upper and lower solutions and on the topological degree technique. The involved multifunction \(F:[0,T]\times \mathbb{R}\to 2^{\mathbb{R}}\) is supposed to be compact, lower semi-continuous and with nonconvex values.
Reviewer: Adrian Petruşel (Cluj-Napoca)On a coupled system of fractional differential equations via the generalized proportional fractional derivativeshttps://zbmath.org/1503.340582023-03-23T18:28:47.107421Z"Abbas, M. I."https://zbmath.org/authors/?q=ai:abbas.mohamed-ibrahim"Ghaderi, M."https://zbmath.org/authors/?q=ai:ghaderi.mehran"Rezapour, Sh."https://zbmath.org/authors/?q=ai:rezapour.shahram"Thabet, S. T. M."https://zbmath.org/authors/?q=ai:thabet.sabri-t-m(no abstract)Tenth order boundary value problem solution existence by fixed point theoremhttps://zbmath.org/1503.340592023-03-23T18:28:47.107421Z"Fabiano, Nicola"https://zbmath.org/authors/?q=ai:fabiano.nicola"Nikolić, Nebojša"https://zbmath.org/authors/?q=ai:nikolic.nebojsa-t"Shanmugam, Thenmozhi"https://zbmath.org/authors/?q=ai:shanmugam.thenmozhi"Radenović, Stojan"https://zbmath.org/authors/?q=ai:radenovic.stojan"Čitaković, Nada"https://zbmath.org/authors/?q=ai:citakovic.nadaSummary: In this paper we consider the Green function for a boundary value problem of generic order. For a specific case, the Leray-Schauder form of the fixed point theorem has been used to prove the existence of a solution for this particular equation. Our theoretical approach generalizes, extends, complements, and enriches several results in the existing literature.Anti-periodic solutions to a class of second-order nonlinear evolution equationshttps://zbmath.org/1503.340642023-03-23T18:28:47.107421Z"Han, Xiangling"https://zbmath.org/authors/?q=ai:han.xiangling"Liu, Zhenhai"https://zbmath.org/authors/?q=ai:liu.zhenhai(no abstract)Inequalities of Green's functions and positive solutions to nonlocal boundary value problemshttps://zbmath.org/1503.340712023-03-23T18:28:47.107421Z"Fan, Shijie"https://zbmath.org/authors/?q=ai:fan.shijie"Wen, Pengxu"https://zbmath.org/authors/?q=ai:wen.pengxu"Zhang, Guowei"https://zbmath.org/authors/?q=ai:zhang.guowei|zhang.guowei.1Summary: In this paper, we discuss the positive solutions of beam equations with the nonlinearities including the slope and bending moment under nonlocal boundary conditions involving Stieltjes integrals. We pose some inequality conditions on nonlinearities and the spectral radius conditions on associated linear operators. These conditions mean that the nonlinearities have superlinear or sublinear growth. The existence of positive solutions is obtained by fixed point index on cones in \(C^2[0,1]\), and some examples are given for beam equations subject to mixed integral and multi-point boundary conditions with sign-changing coefficients.Positive solutions for a system of \(2n\)th-order boundary value problems involving semipositone nonlinearitieshttps://zbmath.org/1503.340722023-03-23T18:28:47.107421Z"Hao, Xinan"https://zbmath.org/authors/?q=ai:hao.xinan"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donal"Xu, Jiafa"https://zbmath.org/authors/?q=ai:xu.jiafaSummary: In this paper we use the fixed point index to study the existence of positive solutions for a system of \(2n\)th-order boundary value problems involving semipositone nonlinearities.Singular fourth-order Sturm-Liouville operators and acoustic black holeshttps://zbmath.org/1503.340742023-03-23T18:28:47.107421Z"Belinskiy, Boris P."https://zbmath.org/authors/?q=ai:belinsky.boris-p"Hinton, Don B."https://zbmath.org/authors/?q=ai:hinton.don-barker"Nichols, Roger A."https://zbmath.org/authors/?q=ai:nichols.roger-aSummary: We derive conditions for a one-term fourth-order Sturm-Liouville operator on a finite interval with one singular endpoint to have essential spectrum equal to \([0, \infty)\) or \(\varnothing\). Of particular usefulness are Kummer-Liouville transformations which have been a valuable tool in the study of second-order equations. Applications to a mechanical beam with a thickness tapering to zero at one of the endpoints are considered. When the thickness \(2h\) satisfies \(c_1x^\nu \leq h(x) \leq c_2x^\nu\), we show that the essential spectrum is empty if and only if \(\nu < 2\). As a final application, we consider a tapered beam on a Winkler foundation and derive sufficient conditions on the beam thickness and the foundational rigidity to guarantee the essential spectrum is equal to \([0, \infty)\).On the investigation of a discontinuous Sturm-Liouville operator of scattering theoryhttps://zbmath.org/1503.340752023-03-23T18:28:47.107421Z"Akcay, Ozge"https://zbmath.org/authors/?q=ai:akcay.ozgeIn the paper, the direct scattering problem for a half-line Sturm-Liouville operator with discontinuity in both the coupling conditions and the coefficient is studied. The equation
\[
-y'' + q(x) y = \lambda^2 \rho(x) y\,,\quad x \in(0,a)\cup (a,\infty)
\]
with the discontinuous coupling conditions at the point \(x=a\), \(a\in (0,\infty)\)
\[
y(a-) = \alpha\, y (a+)\,,\quad y'(a-) = \frac{1}{\alpha} y'(a+)
\]
with \(\alpha> 0\) and the Dirichlet boundary condition \(y(0)= 0\) is investigated. \(\rho(x)\) is a discontinuous function having the positive value \(\beta^2\) for \(x<a\) and being 0 for \(x>a\). The potential \(q\) is assumed to be real and satisfies the condition
\[
\int_0^\infty x|q(x)|\,\mathrm{d}x <\infty.
\]
The author finds the integral representation of the Jost function for the problem and then investigates the properties of the scattering data. The paper is a generalization of previous work by V. A. Marchenko who studied the particular continuous case \(\alpha = 1\), \(\rho \equiv 1\).
Reviewer: Jiři Lipovský (Hradec Králové)Existence of solutions for odd-order multi-point impulsive boundary value problems on time scaleshttps://zbmath.org/1503.340772023-03-23T18:28:47.107421Z"Georgiev, Svetlin G."https://zbmath.org/authors/?q=ai:georgiev.svetlin-georgiev"Doğru Akgöl, Sibel"https://zbmath.org/authors/?q=ai:akgol.sibel-dogru"Eymen Kuş, Murat"https://zbmath.org/authors/?q=ai:eymen-kus.muratThe paper deals with \(2q+1\)st order (\(q\geq1\)) impulsive dynamic equations on time scales with periodic multi-point boundary conditions (PBVP). The authors first obtain the integral representation of the solutions for the impulsive PBVP of the form
\[
\begin{cases}
x^{\Delta^{2q+1}}(t)&= f(t, x(\sigma(t))),\quad t\in J_0,\\
x^{\Delta^p}(0)&= 0,\quad p\in \{2,3, \ldots, 2q\},\\
x^{\Delta^2}(t_k^+)&= x^{\Delta^2}(t_k)+I_k(x(t_k)),\quad k\in \{1, \ldots, m\},\\
x^{\Delta}(0)&= \sum\limits_{j=1}^m \alpha_j x^{\Delta}(\xi_j),\\
x(t_k^+)&= x(t_k)+J_k(x(t_k)),\quad k\in \{1, \ldots, m\},\\
x(0)&= x(\sigma(T)),
\end{cases}\tag{1}
\]
where
\[
z(t_k^\pm)=\lim_{t\to t_k^\pm}z(t),\quad k\in \{1, \ldots, m\}
\]
with \(z(t_k^-)=z(t_k)\), and \(t_k\)'s are right dense for \(k\in \{1, \ldots, m\}\), and \(J_0=[0,T]_{\mathbb{T}}\backslash \{t_k\}_{k=1}^m\). Then, by setting some reasonable conditions on the nonlinear terms \(f(t, x(\sigma(t)))\), \( I_k(x(t_k))\) and \(J_k(x(t_k))\), the authors prove that the impulsive PBVP (1) has at least one solution. The main tool for the proof of existence of solutions is the Schaefer fixed point theorem. An example supporting the results is also given.
Reviewer: Abdullah Özbekler (Ankara)Existence and asymptotic properties of the solution of a nonlinear boundary-value problem on the real axishttps://zbmath.org/1503.340782023-03-23T18:28:47.107421Z"Parasyuk, I. O."https://zbmath.org/authors/?q=ai:parasyuk.igor-o"Protsak, L. V."https://zbmath.org/authors/?q=ai:protsak.l-vIn this paper, there is considered the following nonlinear system of ordinary differential equations defined on the entire real axis with Dirichlet-type boundary conditions at \( \pm\infty\):
\[
\dot{x}=F(t,x),\tag{1}
\]
where \(F(.,.)\in C^{0,1}(\mathbb{R}\times\mathbb{R}^{n}, \mathbb{R}^n)\), \(\lim_{t\rightarrow \pm\infty}F(t,0)=0.\)
It is assumed that the linear part of system (1) has the property of non-uniform strong exponential dichotomy.
The authors analyzes the following questions:
-- is it possible to find at least one solution of system (1) satisfying the Dirichlet boundary conditions
\[
x(\pm\infty)=0.\tag{2}
\]
-- what is the relationship between the asymptotic properties of the solution of the problems (1), (2) and the function \(f_{0}(.):=F(.,0)\)?
To prove the existence theorems, it is applied a Schauder-Tikhonov-type fixed-point principle. In addition, there are also established conditions under which the obtained solution has the same asymptotic properties as the solution of the inhomogeneous linearized system. Indeed, here, two theorems, Theorem 1 and Theorem 2, which provide sufficient conditions, are proved such that the related integral equations and problems have at the least one solution.
Reviewer: Cemil Tunç (Van)A singular periodic Ambrosetti-Prodi problem of Rayleigh equations without coercivity conditionshttps://zbmath.org/1503.340902023-03-23T18:28:47.107421Z"Yu, Xingchen"https://zbmath.org/authors/?q=ai:yu.xingchen"Lu, Shiping"https://zbmath.org/authors/?q=ai:lu.shipingThe authors study a singular periodic problem of the type
\[
x'' + f(x') + g(t,x) = s,
\]
\[
x(0) - x(T) = 0 = x'(0) - x'(T),
\]
where \(f : {\mathbb R} \to {\mathbb R}\) is a continuous function, while \(g : [0,T]\times (0,+\infty)\to{\mathbb R}\) is continuous with an attractive singularity at the origin, and \(s\) is a real constant. Under suitable assumptions, an Ambrosetti-Prodi type result is obtained, i.e., the existence of some \(s^*\) such that
-- if \(s<s^*\) there are no solutions;
-- if \(s=s^*\) there is at least one solution;
-- if \(s>s^*\) there are at least two solutions.
The proof is carried out by the use of Leray-Schauder degree theory.
Reviewer: Alessandro Fonda (Trieste)Existence of rotating-periodic solutions for nonlinear second order vector differential equationshttps://zbmath.org/1503.340912023-03-23T18:28:47.107421Z"Zhang, Jin"https://zbmath.org/authors/?q=ai:zhang.jin"Yang, Xue"https://zbmath.org/authors/?q=ai:yang.xueSummary: In this paper, we establish two existence theorems of rotating-periodic solutions for nonlinear second order vector differential equations via the Leray-Schauder degree theory and the lower and upper solutions method. The concept ``rotating-periodicity'' is a kind of symmetry, which is a general version of periodicity, anti-periodicity, harmonic-periodicity, and it is also a special kind of quasi-periodicity. We also include several examples to illustrate the validity and applicability of our results.An averaging result for periodic solutions of Carathéodory differential equationshttps://zbmath.org/1503.340922023-03-23T18:28:47.107421Z"Novaes, Douglas D."https://zbmath.org/authors/?q=ai:novaes.douglas-duarteThe problem of existence of \(T\)-periodic solutions \(x(t)=x(t+T)\) for Carathéodory differential equations with small a parameter \(\varepsilon\) is analyzed with application of the averaging method. It is assumed that the differential equation is given in the so-called standard form, where its right-hand side has a form \(\varepsilon\cdot f(t,x,\varepsilon)\) and \(f\) is a periodic function in the variable \(t\). Then average of \(f(t,x,0)\) in the variable \(t\) exists and is a continuous function of \(x\). The main result is given in the form of Theorem A which provides sufficient conditions on the averaged equation under which periodic solutions of the Carathéodory differential equations exist. A proof of the main result is mainly based on an abstract continuation result for operator equations and is precisely described step by step. Some details on continuation result for operator equations are given what helps to understand the proof and enables to reproduce it if a reader will be interested in.
Reviewer: Alexander Prokopenya (Warszawa)Fixed points of a mapping generated by a system of ordinary differential equations with relay hysteresishttps://zbmath.org/1503.340932023-03-23T18:28:47.107421Z"Kamachkin, A. M."https://zbmath.org/authors/?q=ai:kamachkin.aleksandr-michailovich"Potapov, D. K."https://zbmath.org/authors/?q=ai:potapov.dmitrii-konstantinovich"Yevstafyeva, V. V."https://zbmath.org/authors/?q=ai:yevstafyeva.victoria-vSummary: We consider an \(n\)-dimensional system of ordinary differential equations with relay hysteresis on the right-hand side. Under certain conditions, the solution of the system defines a self-mapping of bounded sets lying in the discontinuity surfaces. We obtain conditions for the existence of fixed points of the mapping and the uniqueness of the fixed point as well as conditions under which fixed points for various types of mappings exist simultaneously. To each type of mapping there corresponds one type of periodic orbits: either with two switching points (the so-called unimodal orbits) or with an even number of switching points greater than two. In the case of unimodal orbits, examples of the existence of orbits of various configurations are given.Positive periodic solutions for multiparameter nonlinear differential systems with delayshttps://zbmath.org/1503.341202023-03-23T18:28:47.107421Z"Chen, Ruipeng"https://zbmath.org/authors/?q=ai:chen.ruipeng"Li, Xiaoya"https://zbmath.org/authors/?q=ai:li.xiaoyaSummary: We establish several criteria for the existence of positive periodic solutions of the multi-parameter differential systems
\[\begin{cases} u'(t)+a_1(t)g_1(u(t))u(t)=\lambda b_1(t)f(u(t-\tau_1(t)),v(t-\zeta_1(t))), \\ v'(t)+a_2(t)g_2(v(t))v(t)=\mu b_2(t)g(u(t-\tau_2(t)),v(t-\zeta_2(t))), \end{cases}\]
where the functions \(g_1, g_2:[0,\infty)\to[0,\infty)\) are assumed to be unbounded. The analysis in the paper relies on the classical fixed point index theory. Our main findings improve and complement some existing results in the literature.Periodic solution for some class of linear partial differential equation with infinite delay using semi-Fredholm perturbationshttps://zbmath.org/1503.341212023-03-23T18:28:47.107421Z"Elazzouzi, Abdelhai"https://zbmath.org/authors/?q=ai:elazzouzi.abdelhai"Ezzinbi, Khalil"https://zbmath.org/authors/?q=ai:ezzinbi.khalil"Kriche, Mohammed"https://zbmath.org/authors/?q=ai:kriche.mohammedThe authors study the following equation with infinite delay in a Banach space \(\mathbb{E}:\)
\[
\begin{cases}
\frac{d}{dt}\nu (t) = \mathcal{A} \nu(t) + L(\nu_t) + g(t), \quad t \geq 0, \\
\nu_0 = \varphi \in \mathcal{B},
\end{cases}
\]
where \(\nu_t(\theta) = \nu(t + \theta)\), \(\theta\in(-\infty, 0]\) belongs to the Hale-Kato phase space \(\mathcal{B}\) of functions \(\psi \colon (-\infty,0] \to \mathbb{E}\); \(L \colon \mathcal{B} \to \mathbb{E}\) is a bounded linear operator; \(g \colon \mathbb{R}^+ \to \mathbb{E}\) is a continuous \(\omega\)-periodic function and \(\mathcal{A}\) is not necessarily densely defined linear operator on \(\mathbb{E}\) satisfying the Hille-Yosida condition.
By using the theory of perturbations of semi-Fredholm operators, fixed point methods and basing on a priori bounds of the norm \(|L|\), the authors deduce sufficient conditions under which the presence of a bounded solution to the above problem yields the existence of a periodic solution. A particular case when the operator \(\mathcal{A}\) can be decomposed into the sum of a generator of a \(C_0\)-semigroup and a compact operator is considered. Some examples from the sphere of partial differential equations are given.
Reviewer: Valerii V. Obukhovskij (Voronezh)Pseudo compact almost automorphy of neutral type Clifford-valued neural networks with mixed delayshttps://zbmath.org/1503.341242023-03-23T18:28:47.107421Z"Li, Yongkun"https://zbmath.org/authors/?q=ai:li.yongkun.1|li.yongkun"Li, Bing"https://zbmath.org/authors/?q=ai:li.bing.1|li.bingSummary: We consider a class of neutral type Clifford-valued cellular neural networks with discrete delays and infinitely distributed delays. Unlike most previous studies on Clifford-valued neural networks, we assume that the self feedback connection weights of the networks are Clifford numbers rather than real numbers. In order to study the existence of \((\mu, \nu)\)-pseudo compact almost automorphic solutions of the networks, we prove a composition theorem of \((\mu, \nu)\)-pseudo compact almost automorphic functions with varying deviating arguments. Based on this composition theorem and the fixed point theorem, we establish the existence and the uniqueness of \((\mu, \nu)\)-pseudo compact almost automorphic solutions of the networks. Then, we investigate the global exponential stability of the solution by employing differential inequality techniques. Finally, we give an example to illustrate our theoretical finding. Our results obtained in this paper are completely new, even when the considered networks are degenerated into real-valued, complex-valued or quaternion-valued networks.A partial inverse problem for the Sturm-Liouville operator with constant delays on a star graphhttps://zbmath.org/1503.341302023-03-23T18:28:47.107421Z"Wang, Feng"https://zbmath.org/authors/?q=ai:wang.feng.6"Yang, Chuan-Fu"https://zbmath.org/authors/?q=ai:yang.chuanfuSummary: In this work we consider boundary value problems for the Sturm-Liouville operator with constant delays on a star graph. We assume that the potentials and the delay constants are known a priori on all the edges except one, and study the partial inverse problem, which consists in recovering the potential and the delay constant on remaining edge from the part of the spectrum. The uniqueness of the solution of the inverse problem is proved. A constructive method is developed for the solution of the inverse problem.Global solvability for nonlinear nonautonomous evolution inclusions of Volterra-type and its applicationshttps://zbmath.org/1503.341372023-03-23T18:28:47.107421Z"Yu, Yang-Yang"https://zbmath.org/authors/?q=ai:yu.yangyang"Ma, Zhong-Xin"https://zbmath.org/authors/?q=ai:ma.zhong-xinThe authors study the following integro-differential inclusion in a reflexive Banach space \(X:\)
\[
u^\prime (t) - A(t)u(t) \in \int_0^t k(t,s)F(s,u(s),u_s)ds, \,\, t \in \mathbb{R}_0^+ := [0,+\infty).
\]
Here \(A(t),\) \(t \in \mathbb{R}^+_0\) is a family of closed, densely defined linear operators on \(X\) generating an exponentially decreasing evolution family \(U(t,s),\) \(t \geq s \geq 0,\) of linear operators; a multimap \(F \colon \mathbb{R}^+_0 \times X \times C([-\tau,0];X) \multimap X\) is strongly-weakly upper semicontinuous with compact convex values satisfying a regularity condition expressed in terms of the Hausdorff measure of noncompactness; \(x_t(\theta) = x(t + \theta), \theta \in [-\tau,0];\) the kernel \(k(t,s), t \geq s \geq 0\) is a continuous function.
The existence of a bounded mild solutions to the above inclusion on \([-\tau, +\infty)\) satisfying the initial conditions of the form
\[
u(t) = \psi(t), \quad t \in [-\tau, 0]
\]
or
\[
u(t) = N(u)(t), \quad t \in [-\tau, 0],
\]
for a given compact function \(N \colon C_b([-\tau,+\infty];X) \to C([-\tau,0];X)\) is considered.
As example, an initial boundary value problem for a partial integro-differential inclusion is given.
Reviewer: Valerii V. Obukhovskij (Voronezh)On approximate controllability of multi-term time fractional measure differential equations with nonlocal conditionshttps://zbmath.org/1503.341412023-03-23T18:28:47.107421Z"Diop, Amadou"https://zbmath.org/authors/?q=ai:diop.amadou(no abstract)Vallée-Poussin theorem for fractional functional differential equationshttps://zbmath.org/1503.341432023-03-23T18:28:47.107421Z"Domoshnitsky, Alexander"https://zbmath.org/authors/?q=ai:domoshnitsky.alexander-i"Padhi, Seshadev"https://zbmath.org/authors/?q=ai:padhi.seshadev"Srivastava, Satyam Narayan"https://zbmath.org/authors/?q=ai:srivastava.satyam-narayan(no abstract)Analytical approaches on the attractivity of solutions for multiterm fractional functional evolution equationshttps://zbmath.org/1503.341462023-03-23T18:28:47.107421Z"Li, Xiangling"https://zbmath.org/authors/?q=ai:li.xiangling"Niazi, Azmat Ullah Khan"https://zbmath.org/authors/?q=ai:niazi.azmat-ullah-khan"Hafeez, Farva"https://zbmath.org/authors/?q=ai:hafeez.farva"George, Reny"https://zbmath.org/authors/?q=ai:george.reny"Hussain, Azhar"https://zbmath.org/authors/?q=ai:hussain.azhar(no abstract)Asymptotic spreading for general heterogeneous Fisher-KPP type equationshttps://zbmath.org/1503.350022023-03-23T18:28:47.107421Z"Berestycki, Henri"https://zbmath.org/authors/?q=ai:berestycki.henri"Nadin, Grégoire"https://zbmath.org/authors/?q=ai:nadin.gregoireIn this paper, the authors study the heterogeneous reaction-diffusion equations with Fisher-KPP type nonlinearity
\[
\partial_tu -\sum_{i,j=1}^Na_{i,j}(t,x)\partial_{ij}u-\sum_{i=1}^Nq_{i}(t,x)\partial_{i}u=f(t,x,u),
\]
which admit two steady states \(0\) and \(1\), \(0\) being unstable and \(1\) being globally attractive, and the nonlinear term \(f\) is below its tangent at the unstable steady state \(0\). Compactly supported initial data \(u_0\) with \(0 \le u_0\le 1\) are considered. The goal of this paper is to understand the spreading properties for this problem in this general setting.
In order to describe the spreading dynamics, two non-empty star-shaped compact sets \(\overline{S}\) and \(\underline S\) are constructed. Then with help of the generalized principal eigenvalues for linear parabolic operators in unbounded domains, it can be shown that \(\overline{S}=\underline S\), and the exact asymptotic speed of propagation in various frameworks is established. These include: almost periodic, asymptotically almost periodic, uniquely ergodic, slowly varying, radially periodic and random stationary ergodic equations. In dimension \(N\), when the coefficients converge in radial segments, they showed that \(\overline{S}=\underline S\) and this set is characterized using some geometric optics minimization problem. At the end of the article, by analyzing in detail various examples, the optimality of their results is discussed. Moreover, some open problems are given.
Reviewer: Guobao Zhang (Lanzhou)Contracting differential equations in weighted Banach spaceshttps://zbmath.org/1503.350292023-03-23T18:28:47.107421Z"Srinivasan, Anand"https://zbmath.org/authors/?q=ai:srinivasan.anand"Slotine, Jean-Jacques"https://zbmath.org/authors/?q=ai:slotine.jean-jacques-eSummary: Geodesic contraction in vector-valued differential equations is readily verified by linearized operators which are uniformly negative-definite in the Riemannian metric. In the infinite-dimensional setting, however, such analysis is generally restricted to norm-contracting systems. We develop a generalization of geodesic contraction rates to Banach spaces using a smoothly-weighted semi-inner product structure on tangent spaces. We show that negative contraction rates in bijectively weighted spaces imply asymptotic norm-contraction, and apply recent results on asymptotic contractions in Banach spaces to establish the existence of fixed points. We show that contraction in surjectively weighted spaces verifies non-equilibrium asymptotic properties, such as convergence to finite- and infinite-dimensional subspaces, submanifolds, limit cycles, and phase-locking phenomena. We use contraction rates in weighted Sobolev spaces to establish existence and continuous data dependence in nonlinear PDEs, and pose a method for constructing weak solutions using vanishing one-sided Lipschitz approximations. We discuss applications to control and order reduction of PDEs.Estimates for a beam-like partial differential operator and applicationshttps://zbmath.org/1503.350482023-03-23T18:28:47.107421Z"Belahdji, Meriem"https://zbmath.org/authors/?q=ai:belahdji.meriem"Ayad, Setti"https://zbmath.org/authors/?q=ai:ayad.setti"Mortad, Mohammed Hichem"https://zbmath.org/authors/?q=ai:mortad.mohammed-hichemSummary: The aim of this paper is to provide some a priori estimates for a beam-like operator. Some applications and counterexamples are also given.Existence of global weak solutions of \(p\)-Navier-Stokes equationshttps://zbmath.org/1503.351362023-03-23T18:28:47.107421Z"Liu, Jian-Guo"https://zbmath.org/authors/?q=ai:liu.jian-guo.1|liu.jian-guo"Zhang, Zhaoyun"https://zbmath.org/authors/?q=ai:zhang.zhaoyunSummary: This paper investigates the global existence of weak solutions for the incompressible \(p\)-Navier-Stokes equations in \(\mathbb{R}^d \) (\(2\leq d\leq p\)). The \(p\)-Navier-Stokes equations are obtained by adding viscosity term to the \(p\)-Euler equations. The diffusion added is represented by the \(p\)-Laplacian of velocity and the \(p\)-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-\(p\) distances with constraint density to be characteristic functions.Many-body excitations in trapped Bose gas: a non-Hermitian approachhttps://zbmath.org/1503.351952023-03-23T18:28:47.107421Z"Grillakis, Manoussos"https://zbmath.org/authors/?q=ai:grillakis.manoussos-g"Margetis, Dionisios"https://zbmath.org/authors/?q=ai:margetis.dionisios"Sorokanich, Stephen"https://zbmath.org/authors/?q=ai:sorokanich.stephenSummary: We study a physically motivated model for a trapped dilute gas of Bosons with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state. We start with an approximate many-body Hamiltonian, \(\mathcal{H}_{\mathrm{app}}\), in the Bosonic Fock space. This \(\mathcal{H}_{\mathrm{app}}\) conserves the total number of atoms. Inspired by \textit{T. T. Wu} [J. Math. Phys. 2, 105--123 (1961; Zbl 0151.44907)], we apply a non-unitary transformation to \(\mathcal{H}_{\mathrm{app}}\). Key in this procedure is the pair-excitation kernel, which obeys a nonlinear integro-partial differential equation. In the stationary case, we develop an existence theory for solutions to this equation by a variational principle. We connect this theory to a system of partial differential equations for one-particle excitation (``quasiparticle''-) wave functions derived by \textit{A. L. Fetter} [``Nonuniform states of an imperfect Bose gas'', Ann. Phys. 70, No. 1, 67--101 (1972; \url{doi:10.1016/0003-4916(72)90330-2})], and prove existence of solutions for this system. These wave functions solve an eigenvalue problem for a \(J\)-self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter's energy spectrum. We also analytically provide an explicit construction of the excited eigenstates of the reduced Hamiltonian in the \(N\)-particle sector of Fock space.Orbital stability and concentration of standing-wave solutions to a nonlinear Schrödinger system with mass critical exponentshttps://zbmath.org/1503.352092023-03-23T18:28:47.107421Z"Garrisi, Daniele"https://zbmath.org/authors/?q=ai:garrisi.daniele"Gou, Tianxiang"https://zbmath.org/authors/?q=ai:gou.tianxiangSummary: For a nonlinear Schrödinger system with mass critical exponent, we prove the existence and orbital stability of standing-wave solutions obtained as minimizers of the underlying energy functional restricted to a double mass constraint. In addition, we discuss the concentration of a sequence of minimizers as its masses approach to certain critical masses.Decay solutions to abstract impulsive fractional mobile-immobile equations involving superlinear nonlinearitieshttps://zbmath.org/1503.352452023-03-23T18:28:47.107421Z"Anh, Nguyen Thi Van"https://zbmath.org/authors/?q=ai:anh.nguyen-thi-van"Dac, Nguyen Van"https://zbmath.org/authors/?q=ai:van-dac.nguyen"Tuan, Tran Van"https://zbmath.org/authors/?q=ai:tuan.tran-van(no abstract)Wellposedness and stability of fractional stochastic nonlinear heat equation in Hilbert spacehttps://zbmath.org/1503.352462023-03-23T18:28:47.107421Z"Arab, Zineb"https://zbmath.org/authors/?q=ai:arab.zineb"El-Borai, Mahmoud Mohamed"https://zbmath.org/authors/?q=ai:el-borai.mahmoud-mohamed(no abstract)Existence of positive ground state solutions to a nonlinear fractional Schrödinger system with linear couplingshttps://zbmath.org/1503.352572023-03-23T18:28:47.107421Z"Du, Xinsheng"https://zbmath.org/authors/?q=ai:du.xinsheng"Mao, Anmin"https://zbmath.org/authors/?q=ai:mao.anmin"Liu, Ke"https://zbmath.org/authors/?q=ai:liu.keSummary: In this paper, we investigate a nonlinear fractional Schrödinger system with linear couplings as follows:
\[\begin{cases} (-\Delta )^{\alpha }u+(1+a(x))u=F_u(u,v)+\lambda v, \quad & \text{in } \mathbb{R}^3, \\
(-\Delta )^{\alpha }v+(1+b(x))v=F_v(u,v)+\lambda u,& \text{in } \mathbb{R}^3, \\
u,v\in H^{\alpha }(\mathbb{R}^3), \end{cases}\]
where \((-\Delta )^{\alpha }\), \(\alpha \in (0,1)\), denotes the fractional Laplacian and \(\lambda >0\) is the coupling parameter. Under some assumptions, we prove the existence of positive ground state solutions to the above system with the help of the method of Nehari manifold and concentration compactness lemma.Two disjoint and infinite sets of solutions for a concave-convex critical fractional Laplacian equationhttps://zbmath.org/1503.352582023-03-23T18:28:47.107421Z"Echarghaoui, Rachid"https://zbmath.org/authors/?q=ai:echarghaoui.rachid"Masmodi, Mohamed"https://zbmath.org/authors/?q=ai:masmodi.mohamed(no abstract)Hölder regularity for non-autonomous fractional evolution equationshttps://zbmath.org/1503.352632023-03-23T18:28:47.107421Z"He, Jia Wei"https://zbmath.org/authors/?q=ai:he.jiawei"Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yong.1(no abstract)Multiple solutions for a fractional \(p\)\&\(q\)-Laplacian system involving Hardy-Sobolev exponenthttps://zbmath.org/1503.352792023-03-23T18:28:47.107421Z"Zheng, Tiantian"https://zbmath.org/authors/?q=ai:zheng.tiantian"Zhang, Chunyan"https://zbmath.org/authors/?q=ai:zhang.shunyan"Zhang, Jihui"https://zbmath.org/authors/?q=ai:zhang.jihuiSummary: In this paper, we prove the existence of infinitely many solutions for a fractional \(p\)\&\(q\)-Laplacian system involving Hardy-Sobolev exponents and obtain new conclusion under different conditions. The methods used here are based on variational methods and Ljusternik-Schnirelmann theory.Regularization methods for identifying the initial value of time fractional pseudo-parabolic equationhttps://zbmath.org/1503.352842023-03-23T18:28:47.107421Z"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.1"Xu, Jian-Ming"https://zbmath.org/authors/?q=ai:xu.jianming|xu.jian-ming"Li, Xiao-Xiao"https://zbmath.org/authors/?q=ai:li.xiaoxiaoSummary: In this paper, the problem we investigate is that the inverse problem of identifying the initial value for fractional pseudo-parabolic equation. This problem is ill-posed, i.e. the solution (if exists) does not depend on the measurable data. We give the estimate of conditional stability under an a-priori bound assumption for exact solution. Two regularization methods are used to solve this problem, and under an a-priori and an a-posteriori selection rule for the regularization parameter, the error estimates for these methods are obtained. Finally, several numerical examples are given to prove the effectiveness of these regularization methods.Nonstationary iterated Tikhonov regularization: convergence analysis via Hölder stabilityhttps://zbmath.org/1503.352882023-03-23T18:28:47.107421Z"Mittal, Gaurav"https://zbmath.org/authors/?q=ai:mittal.gaurav"Giri, Ankik Kumar"https://zbmath.org/authors/?q=ai:giri.ankik-kumarSummary: In this paper, we study the nonstationary iterated Tikhonov regularization method (NITRM) proposed by \textit{Q. Jin} and \textit{M. Zhong} [Numer. Math. 127, No. 3, 485--513 (2014; Zbl 1297.65062)] to solve the inverse problems, where the inverse mapping fulfills a Hölder stability estimate. The iterates of NITRM are defined through certain minimization problems in the settings of Banach spaces. In order to study the various important characteristics of the sought solution, we consider the non-smooth uniformly convex penalty terms in the minimization problems. In the case of noisy data, we terminate the method via a discrepancy principle and show the strong convergence of the iterates as well as the convergence with respect to the Bregman distance. For noise free data, we show the convergence of the iterates to the sought solution. Additionally, we derive the convergence rates of NITRM method for both the noisy and noise free data that are missing from the literature. In order to derive the convergence rates, we solely utilize the Hölder stability of the inverse mapping that opposes the standard analysis which requires a source condition as well as a nonlinearity estimate to be satisfied by the inverse mapping. Finally, we discuss three numerical examples to show the validity of our results.Subshifts with leading sequences, uniformity of cocycles and spectra of Schreier graphshttps://zbmath.org/1503.370352023-03-23T18:28:47.107421Z"Grigorchuk, Rostislav"https://zbmath.org/authors/?q=ai:grigorchuk.rostislav-i"Lenz, Daniel"https://zbmath.org/authors/?q=ai:lenz.daniel-h"Nagnibeda, Tatiana"https://zbmath.org/authors/?q=ai:nagnibeda.tatiana-v"Sell, Daniel"https://zbmath.org/authors/?q=ai:sell.danielSummary: We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.Center manifolds for ill-posed stochastic evolution equationshttps://zbmath.org/1503.370782023-03-23T18:28:47.107421Z"Li, Zonghao"https://zbmath.org/authors/?q=ai:li.zonghao"Zeng, Caibin"https://zbmath.org/authors/?q=ai:zeng.caibinSummary: The aim of this paper is to develop a center manifold theory for a class of stochastic partial differential equations with a non-dense domain through the Lyapunov-Perron method. We construct a novel variation of constants formula by the resolvent operator to formulate the integrated solutions. Moreover, we impose an additional condition involving a non-decreasing map to deduce the required estimate since Young's convolution inequality is not applicable. As an application, we present a stochastic parabolic equation to illustrate the obtained results.Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noisehttps://zbmath.org/1503.370812023-03-23T18:28:47.107421Z"Bertram, Alexander"https://zbmath.org/authors/?q=ai:bertram.alexander"Grothaus, Martin"https://zbmath.org/authors/?q=ai:grothaus.martinThe authors study the properties of the Kolmogorov-type operator
\[
L = \mathrm{Tr}(\Sigma D^{2}_{v} ) + b(v) \cdot D_{v} + v \cdot D_{x} - D_{x} \Phi(x) \cdot D_{v} ,
\]
called the Langevin operator. This is associated with the Langevin dynamics
\begin{align*}
&dX(t) =V(t) dt ,\\
&dV(t)=b(V(t)) dt -\nabla \Phi(X(t)) dt + \sigma(V(t)) dW(t) ,
\end{align*}
in \({\mathbb R}^{d}\). Here \(W(\cdot)\) is a \(d\)-dimensional Wiener process, \(\Phi\) is a suitable potential, \(\sigma\) is a variable diffusion matrix with weakly differentiable coefficients, \(b\) a vector field with components
\[
b_{i}(v)=\sum_{j=1}^{d} \partial_{j} a_{ij}(v) -a_{ij}(v) v_j,
\]
for \(a_{ij} = \Sigma_{ij}\), \(\Sigma = \sigma \sigma^{T}\) and \(D_{v}, D_{x}\), \(D^{2}_{x}\) denote the gradients and the Hessian matrix. The operator \(L\) is studied in the weighted Hilbert space \(H=L^{2}({\mathbb R}^{2 d}, \mu)\), where \(\mu\) is a Gaussian measure on \(({\mathbb R}^{2 d}, \mathcal{B}({\mathbb R}^{2 d})\) weighted by an exponential factor \(e^{-\Phi(x) }\) corresponding to the action of the potential \(\Phi\).
The authors prove that the Langevin operator \((L, C_{c}({\mathbb R}^{2d}))\) is closable on \(H\) and its closure \((L, D(L))\) generates a strongly continuous contraction semigroup \((T_t )_{t \ge 0}\) on \(H\). Moreover, the following contractivity result is obtained. For all \(\theta_1 \in (1,\infty)\) there exists a \(\theta_2 \in (0,\infty)\) such that
\[
\| T_{t} g - (g,1)_{H} \|_{H} \le \theta_1 e^{-\theta_2 t} \| g -(g,1)_{H}\|_{H}, \quad \forall \, g \in H, \,\, t \ge 0.
\]
Such estimate provides important information on the asymptotic behavior of Langevin dynamics towards equilibrium and extends previous results by \textit{J. Dolbeault} et al. [C. R., Math., Acad. Sci. Paris 347, No. 9--10, 511--516 (2009; Zbl 1177.35054)]
for constant diffusion matrices to the case of variable diffusion matrices with weak derivatives.
Reviewer: Athanasios Yannacopoulos (Athína)Discrete fractional order two-point boundary value problem with some relevant physical applicationshttps://zbmath.org/1503.390062023-03-23T18:28:47.107421Z"Selvam, A. George Maria"https://zbmath.org/authors/?q=ai:selvam.a-george-maria"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Dhineshbabu, R."https://zbmath.org/authors/?q=ai:dhineshbabu.raghupathi"Rashid, S."https://zbmath.org/authors/?q=ai:rashid.sabrina|rashid.sheikh|rashid.sabbir-m|rashid.saima|rashid.suliman|rashid.salim|rashid.shahid"Rehman, M."https://zbmath.org/authors/?q=ai:rehman.mutti-ur|ur-rehman.mujeebSummary: The results reported in this paper are concerned with the existence and uniqueness of solutions of discrete fractional order two-point boundary value problem. The results are developed by employing the properties of Caputo and Riemann-Liouville fractional difference operators, the contraction mapping principle and the Brouwer fixed point theorem. Furthermore, the conditions for Hyers-Ulam stability and Hyers-Ulam-Rassias stability of the proposed discrete fractional boundary value problem are established. The applicability of the theoretical findings has been demonstrated with relevant practical examples. The analysis of the considered mathematical models is illustrated by figures and presented in tabular forms. The results are compared and the occurrence of overlapping/non-overlapping has been discussed.Transportation inequalities for Markov kernels and their applicationshttps://zbmath.org/1503.390172023-03-23T18:28:47.107421Z"Baudoin, Fabrice"https://zbmath.org/authors/?q=ai:baudoin.fabrice"Eldredge, Nathaniel"https://zbmath.org/authors/?q=ai:eldredge.nathanielThe aim of the paper is to study the connection between reverse Poincaré inequalities and Hellinger-Kantorovich contraction inequalities for heat semigroups associated with Markovian kernels. One of the main results of the paper (Theorem 3.7) is that given a heat kernel \(P\) on a metric space \(X\) (assumed to be a length space, in particular, path connected, equipped with a strong upper gradient) then the following inequalities are equivalent: the reverse Poincaré inequality
\[
| \nabla P \, f|^2 \le C (P(f^2)-(P(f))^2), \quad \forall \, f \in \mathrm{Lip}_{b}(X),
\]
the Hellinger-Kantorovich contraction inequalities
\[
He_{2}(\mu_0 P, \mu_1 P) \le HK_{4/C}(\mu_0,\mu_1) \le \sqrt{\frac{C}{4}} W_{2}(\mu_0,\mu_1),
\]
where \(\mu_0,\mu_1\) are probability measures on \(X\) and \(He\), \(HK\), \(W_{2}\) are the Hellinger, the Hellinger-Kantorovich and the Wasserstein distances respectively, and the Harnack type inequality
\[
Pf(x) \le P f(y) + \sqrt{C} \, d(x,y) \sqrt{P(f^2)(x)}, \quad\forall \, x,y \in X, \, f \in B_{b}(X), \quad f \ge 0.
\]
This result builds on the following tools:
\begin{itemize}
\item[(1)] A suitable extension of recent results of \textit{G. Luise} and \textit{G. Savaré} [Discrete Contin. Dyn. Syst., Ser. S 14, No. 1, 273--297 (2021; Zbl 1459.49030)] on the contraction properties of heat semigroups in spaces of measures, to general metric spaces;
\item[(2)] The connection of the resulting Hellinger-Kantorovich contraction inequalities to reverse Poincaré inequalities;
\item[(3)] A dynamic dual formulation of the Hellinger-Kantorovich distance to define a new family of divergences on \(P(X)\) which generalize the Rényi divergence.
This allows the authors to prove that contraction inequalities for these divergences are equivalent to the reverse logarithmic Sobolev and Wang-Harnack inequalities. \end{itemize}
This important result connects the Hellinger-Kantorovich contraction with well known functional inequalities.
The results of the paper are illustrated by various important applications, including subelliptic diffusions arising in sub-Riemannian geometry, non-symmetric Ornstein-Uhlenbeck operators on Carnot groups, Langevin dynamics driven by Lévy processes, and others.
Reviewer: Athanasios Yannacopoulos (Athína)The stability of \(N\)-dimensional quadratic functional inequality in non Archimedean Banach spaceshttps://zbmath.org/1503.390192023-03-23T18:28:47.107421Z"Aribou, Y."https://zbmath.org/authors/?q=ai:aribou.youssef"Kabbaj, S."https://zbmath.org/authors/?q=ai:kabbaj.samirSummary: In this paper, using the direct method, we study the stability of the following inequality:
\[
\begin{multlined}
\left\| f\left( \sum^n_{i=1}x_i\right) +\sum_{1\leq i \prec j\leq n}f(x_i -x_j) -n\sum^n_{i=1} f(x_i)\right\| \\
\qquad \leq \left\| f\left( \frac{\sum^n_{i=1}x_i}{n}\right) +\sum_{1\leq i \prec j\leq n}f\left( \frac{x_i -x_j}{n}\right) -\frac{1}{n}\sum^n_{i=1} f(x_i)\right\|
\end{multlined}
\]
in Banach spaces, and the stability of the following inequality:
\[
\begin{multlined}
\left\| f\left( \frac{\sum^n_{i=1}x_i}{n}\right) +\sum_{1\leq i \prec j\leq n}f\left( \frac{x_i -x_j}{n}\right) -\frac{1}{n}\sum^n_{i=1}f(x_i)\right\|_* \\
\qquad \leq \left\| f\left( \sum^n_{i=1}x_i\right) +\sum_{1\leq i \prec j\leq n}f(x_i -x_j)-n\sum^n_{i=1} f(x_i)\right\|_*
\end{multlined}
\]
in non-Archimedean Banach spaces with \(n\) an integer greater than or equal to \(2\).On the convergence and statistical convergence of difference sequences of fractional orderhttps://zbmath.org/1503.400012023-03-23T18:28:47.107421Z"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad"Baliarsingh, P."https://zbmath.org/authors/?q=ai:baliarsingh.pinakadharSummary: In the present note, we discuss the convergence of the difference sequences defined by
\textit{H.~Kizmaz} [Can. Math. Bull. 24, 169--176 (1981; Zbl 0454.46010)],
\textit{M.~Et} and \textit{R.~Çolak} [Soochow J. Math. 21, No.~4, 377--386 (1995; Zbl 0841.46006)],
\textit{E.~Malkowsky} et al. [Acta Math. Sin., Engl. Ser. 23, No.~3, 521--532 (2007; Zbl 1123.46007)],
\textit{F.~Başar} [Summability theory and its applications. With a foreword by M. Mursaleen. Oak Park, IL: Bentham Science Publishers (2012; Zbl 1342.40001)],
\textit{P.~Baliarsingh}, Appl. Math. Comput. 219, No.~18, 9737--9742 (2013; Zbl 1300.46004)],
\textit{P.~Baliarsingh} and \textit{S.~Dutta}, Appl. Math. Comput. 250, 665--674 (2015; Zbl 1328.46002)]
and many others. In this regard, we establish some results on the convergence and statistical convergence of the fractional difference sequence. We also give some illustrative examples for more clarity and demonstrations of the results obtained.Characterization of boundedness on weighted modulation spaces of \(\tau\)-Wigner distributionshttps://zbmath.org/1503.420072023-03-23T18:28:47.107421Z"Guo, Weichao"https://zbmath.org/authors/?q=ai:guo.weichao"Chen, Jiecheng"https://zbmath.org/authors/?q=ai:chen.jiecheng"Fan, Dashan"https://zbmath.org/authors/?q=ai:fan.dashan"Zhao, Guoping"https://zbmath.org/authors/?q=ai:zhao.guopingSummary: This paper is devoted to give several characterizations on a more general level for the boundedness of \(\tau\)-Wigner distributions acting from weighted modulation spaces to weighted modulation and Wiener amalgam spaces. As applications, sharp exponents are obtained for the boundedness of \(\tau\)-Wigner distributions on modulation spaces with power weights. We also recapture the main theorems of Wigner distribution obtained by \textit{E. Cordero} and \textit{F. Nicola} [ibid. 2018, No. 6, 1779--1807 (2018; Zbl 1440.81057)] and \textit{E. Cordero} [Springer INdAM Ser. 43, 149--166 (2021; Zbl 07332023)]. As consequences, the characterizations of the boundedness on weighted modulation spaces of several types of pseudodifferential operators are established. In particular, we give the sharp exponents for the boundedness of pseudodifferential operators with symbols in Sjöstrand's class and the corresponding Wiener amalgam spaces.Endpoint boundedness for commutators of singular integral operators on weighted generalized Morrey spaceshttps://zbmath.org/1503.420172023-03-23T18:28:47.107421Z"Qi, Jinyun"https://zbmath.org/authors/?q=ai:qi.jinyun"Yan, Xuefang"https://zbmath.org/authors/?q=ai:yan.xuefang"Li, Wenming"https://zbmath.org/authors/?q=ai:li.wenmingSummary: In this paper, we obtain the endpoint boundedness for the commutators of singular integral operators with \(BMO\) functions and the associated maximal operators on weighted generalized Morrey spaces. We also get similar results for the commutators of fractional integral operators with \(BMO\) functions and the associated maximal operators.Hardy operators and the commutators on Hardy spaceshttps://zbmath.org/1503.420192023-03-23T18:28:47.107421Z"Niu, Zhuang"https://zbmath.org/authors/?q=ai:niu.zhuang.1"Guo, Shasha"https://zbmath.org/authors/?q=ai:guo.shasha"Li, Wenming"https://zbmath.org/authors/?q=ai:li.wenmingSummary: In this paper, the boundedness of the classic Hardy operator and its adjoint on Hardy spaces is obtained. We also discuss the boundedness for the commutators generated by the classic Hardy operator and its adjoint with \(BMO\) and \(CMO(\mathbb{R}^+)\) functions on Hardy spaces.Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operatorshttps://zbmath.org/1503.420202023-03-23T18:28:47.107421Z"Ding, Wei"https://zbmath.org/authors/?q=ai:ding.wei"Lu, Guozhen"https://zbmath.org/authors/?q=ai:lu.guozhenAuthors' abstract: To study the boundedness of bi-parameter singular integral operators of non-convolution type in the Journé class, \textit{R. Fefferman} [Ann. Math. (2) 126, 109--130 (1987; Zbl 0644.42017)] discovered a boundedness criterion on bi-parameter Hardy spaces \(H^p(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})\) by considering the action of the operators on rectangle atoms. More recently, the theory of multiparameter local Hardy spaces has been developed by the authors. In this paper, we establish this type of boundedness criterion on weighted bi-parameter local Hardy spaces \(h^p_{\omega}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})\). In comparison with the unweighted case, the uniform boundedness of rectangle atoms on weighted local bi-parameter Hardy spaces, which is crucial to establish the atomic decomposition on bi-parameter weighted local Hardy spaces, is considerably more involved. As an application, we establish the boundedness of bi-parameter pseudodifferential operators, including \(h^p_{\omega}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})\) to \(L_{\omega}^p(\mathbb{R}^{n_1+n_2})\) and \(h^p_{\omega}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})\) to \(h^p_{\omega}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})\) for all \(0<p\leq 1\), which sharpens our earlier result even in the unweighted case requiring
\[
\max\left\{\frac{n_1}{n_1+1},\frac{n_2}{n_2+1}\right\}<p\leq 1.
\]
Reviewer: Mohammed El Aïdi (Bogotá)Commutators of \(\theta \)-type generalized fractional integrals on non-homogeneous spaceshttps://zbmath.org/1503.420212023-03-23T18:28:47.107421Z"Lu, Guanghui"https://zbmath.org/authors/?q=ai:lu.guanghuiSummary: The aim of this paper is to establish the boundednes of the commutator \([b,T_{\alpha }]\) generated by \(\theta \)-type generalized fractional integral \(T_{\alpha }\) and \(b\in \widetilde{\mathrm{RBMO}}(\mu)\) over a non-homogeneous metric measure space. Under the assumption that the dominating function \(\lambda\) satisfies the \(\varepsilon \)-weak reverse doubling condition, the author proves that the commutator \([b,T_{\alpha }]\) is bounded from the Lebesgue space \(L^p(\mu )\) into the space \(L^q(\mu )\) for \(\frac{1}{q}=\frac{1}{p}-\alpha\) and \(\alpha \in (0,1)\), and bounded from the atomic Hardy space \(\widetilde{H}^1_b(\mu )\) with discrete coefficient into the space \(L^{\frac{1}{1-\alpha },\infty }(\mu )\). Furthermore, the boundedness of the commutator \([b,T_{\alpha }]\) on a generalized Morrey space and a Morrey space is also obtained.\(K\)-g-fusion frames in Hilbert spaceshttps://zbmath.org/1503.420262023-03-23T18:28:47.107421Z"Huang, Yongdong"https://zbmath.org/authors/?q=ai:huang.yongdong"Yang, Yuanyuan"https://zbmath.org/authors/?q=ai:yang.yuanyuanSummary: \(K\)-g-fusion frames are generalizations of fusion frames, \(K\)-fusion frames and g-fusion frames. In this paper, we present some new characterizations of \(K\)-g-fusion frames and tight \(K\)-g-fusion frames, and give several properties of \(K\)-g-fusion frames by means of the methods and techniques in frame and operator theories. Then, we discuss the redundancy of \(K\)-g-fusion frames. Finally, we discuss the stability of \(K\)-g-fusion frames.The equivalence of \(F_a\)-frameshttps://zbmath.org/1503.420272023-03-23T18:28:47.107421Z"Hussain, Tufail"https://zbmath.org/authors/?q=ai:hussain.tufail"Li, Yun-Zhang"https://zbmath.org/authors/?q=ai:li.yunzhang.1|li.yunzhangSummary: Structured frames such as wavelet and Gabor frames in \(L^2(\mathbb{R})\) have been extensively studied. But \(L^2(\mathbb{ R}_+)\) cannot admit wavelet and Gabor systems due to \(\mathbb{R}_+\) being not a group under addition. In practice, \(L^2(\mathbb{R}_+)\) models the causal signal space. The function-valued inner product-based \(F_a \)-frame for \(L^2(\mathbb{R}_+)\) was first introduced by Hasankhani Fard and Dehghan, where an \(F_a \)-frame was called a function-valued frame. In this paper, we introduce the notions of \(F_a \)-equivalence and unitary \(F_a\)-equivalence between \(F_a\)-frames, and present a characterization of the \(F_a\)-equivalence and unitary \(F_a\)-equivalence. This characterization looks like that of equivalence and unitary equivalence between frames, but the proof is nontrivial due to the particularity of \(F_a\)-frames.Fractional vector-valued nonuniform MRA and associated wavelet packets on \(L^2\big({\mathbb{R}},{\mathbb{C}}^M\big)\)https://zbmath.org/1503.420292023-03-23T18:28:47.107421Z"Bhat, M. Younus"https://zbmath.org/authors/?q=ai:bhat.mohammad-younus"Dar, Aamir H."https://zbmath.org/authors/?q=ai:dar.aamir-h(no abstract)Strichartz's Radon transforms for mutually orthogonal affine planes and fractional integralshttps://zbmath.org/1503.440012023-03-23T18:28:47.107421Z"Wang, Yingzhan"https://zbmath.org/authors/?q=ai:wang.yingzhan(no abstract)The Weyl matrix balls corresponding to the matricial truncated Hamburger moment problemhttps://zbmath.org/1503.440052023-03-23T18:28:47.107421Z"Fritzsche, Bernd"https://zbmath.org/authors/?q=ai:fritzsche.bernd"Kirstein, Bernd"https://zbmath.org/authors/?q=ai:kirstein.bernd"Kley, Susanne"https://zbmath.org/authors/?q=ai:kley.susanne"Mädler, Conrad"https://zbmath.org/authors/?q=ai:madler.conradSummary: The main goal of the paper is to parametrize the Weyl matrix balls associated with an arbitrary matricial truncated Hamburger moment problem. For the special case of a non-degenerate matricial truncated Hamburger moment problem, the corresponding Weyl matrix balls were computed by
\textit{I.~V. Kovalishina} [Math. USSR, Izv. 22, 419--463 (1984; Zbl 0549.30026); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No.~3, 455--497 (1983)] in the framework of V.~P. Potapov's method of `Fundamental matrix inequalities'.Existence of solutions for implicit functional-integral equations associated with discontinuous functionshttps://zbmath.org/1503.450102023-03-23T18:28:47.107421Z"Cubiotti, Paolo"https://zbmath.org/authors/?q=ai:cubiotti.paolo"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihThe authors study the existence of solutions of the functional-integral equation of the form
\[
h(u(t))=g(t)+f\left(t,\int_Ik(t,z)u(\varphi(z))\mathrm{d}z\right)\text{ for a.e. }t\in I=[0,1],
\]
where \(h:Y\to\mathbb{R}\), \(g:I\to\mathbb{R}\), \(f:I\times\mathbb{R}\to\mathbb{R}\), \(k:I\times I\to[0,\infty)\) and \(\varphi:I\to I\) with \(Y\subseteq\mathbb{R}\).
The function \(f\) is generally taken to be a Carathéodory map in the literature. In this paper, the existence of the solutions \(u\) of the above equation is proved for \(u\in L^p(I)\), \(p\in(1,\infty)\), where the continuity of \(f\) with respect to the second variable is not assumed. The main tool of the proof is a result by \textit{J. Saint Raymond} concerning Riemann-measurable selections
[Set-Valued Anal. 2, No. 3, 481--485 (1994; Zbl 0851.54021), Theorem 3]. In the nonautonomous case, where \(f\) can depend on \(t\) explicitly (keeping \(g\equiv0\) and \(\varphi(s)=s\)), the latter result was partially extended in [\textit{G. Anello} and \textit{P. Cubiotti}, Commentat. Math. Univ. Carol. 45, No. 3, 417--429 (2004; Zbl 1099.45004); J. Integral Equations Appl. 19, No. 4, 391--403 (2007; Zbl 1139.45002)] by imposing a less general and more articulate condition on \(f\). More precisely, it was assumed that \(f\) is almost everywhere equal to a function whose discontinuity points are contained in a closed null-measure set. The authors point out that, for the explicit case \(h(z)=z\) (and keeping \(g\equiv0\) and \(\varphi(s)=s\)), such results were partially extended to the \(n\)-dimensional case in [\textit{F. Cammaroto} and \textit{P. Cubiotti}, Commentat. Math. Univ. Carol. 40, No. 3, 483--490 (1999; Zbl 1065.47505); \textit{P. Cubiotti}, Commentat. Math. Univ. Carol. 42, No. 2, 319--329 (2001; Zbl 1055.45004)].
As a matter of fact, it is only required that the restriction of \(f(t,\cdot)\) to the complement of a null-measure set \(E\) is continuous. Such an assumption is simpler and more general than the ones assumed in [Zbl 1099.45004; Zbl 1139.45002; \textit{F. Cammaroto} and \textit{P. Cubiotti}, Commentat. Math. Univ. Carol. 38, No. 2, 241--246 (1997; Zbl 0886.47031); Zbl 1065.47505; Zbl 1055.45004]. Moreover, it is not difficult to see that such a function \(f\), with respect to the second variable, could be discontinuous even at each point \(x\in\mathbb{R}\). Concerning the function \(h\), it is only required that it is continuous and locally nonconstant. The main tool of the proof is a very recent result in [\textit{P. Cubiotti} and \textit{J.-C. Yao}, J. Nonlinear Convex Anal. 17, No. 5, 853--863 (2016; Zbl 1466.34017), Theorem 2.1], together with a deep result by \textit{O. Naselli Ricceri} and \textit{B. Ricceri} on operator inclusions [Appl. Anal. 38, No. 4, 259--270 (1990; Zbl 0687.47044), Theorem 1].
Reviewer: Abdullah Özbekler (Ankara)The convergence rate of truncated hypersingular integrals generated by the modified Poisson semigrouphttps://zbmath.org/1503.450122023-03-23T18:28:47.107421Z"Eryiğit, Melih"https://zbmath.org/authors/?q=ai:eryigit.melih"Evcan, Sinem Sezer"https://zbmath.org/authors/?q=ai:evcan.sinem-sezer"Çobanoğlu, Selim"https://zbmath.org/authors/?q=ai:cobanoglu.selimSummary: Hypersingular integrals have appeared as effective tools for inversion of multidimensional potential-type operators such as Riesz, Bessel, Flett, parabolic potentials, etc. They represent (at least formally) fractional powers of suitable differential operators. In this paper the family of the so-called ``truncated hypersingular integral operators'' \(\mathbf{D}_{\varepsilon }^{\alpha }f\) is introduced, that is generated by the modified Poisson semigroup and associated with the Flett potentials \(\mathcal{F}^\alpha\varphi = (E + \sqrt{ - \operatorname{\Delta}} )^{- \alpha}\varphi\) (\(0<\alpha <\infty\), \(\varphi \in L_p(\mathbb{R}^n)\)). Then the relationship between the order of ``\(L_p \)-smoothness'' of a function \(f\) and the ``rate of \(L_p\)-convergence'' of the families \(\mathbf{D}_{\varepsilon }^{\alpha } \mathcal{F}^{\alpha }f\) to the function \(f\) as \(\varepsilon \rightarrow 0^+\) is also obtained.Geometric properties of the Bernatsky integral operatorhttps://zbmath.org/1503.450142023-03-23T18:28:47.107421Z"Maĭer, Fedor Fedorovich"https://zbmath.org/authors/?q=ai:maier.fedor-fedorovich"Tastanov, Maĭrambek Gabulievich"https://zbmath.org/authors/?q=ai:tastanov.mairambek-gabulievich"Utemisova, Anar Altaevna"https://zbmath.org/authors/?q=ai:utemisova.anar-altaevnaSummary: In the geometric theory of complex variable functions, the study of mapping of classes of regular functions using various operators has now become an independent trend. The connection \(f(z)\in S^o\Leftrightarrow g(z) = zf'(z) \in S^*\) of the classes \(S^o\) and \(S^*\) of convex and star-shaped functions can be considered as mapping using the differential operator \(G[f](x) = zf'(z)\) of class \(S^o\) to class \(S^*\), that is, \(G: S^o \to S^*\) or \(G(S^o) = S^*\). The impetus for studying this range of issues was M. Bernatsky's assumption that the inverse operator \(G^{-1}[f](x)\), which translates \(S^* \to S^o\) and thereby ``improves'' the properties of functions, maps the entire class \(S\) of single-leaf functions into itself.
At present, a number of articles have been published which study the various integral operators. In particular, they establish sets of values of indicators included in these operators where operators map class \(S\) or its subclasses to themselves or to other subclasses.
This paper determines the values of the Bernatsky parameter included in the generalized integral operator, at which this operator transforms a subclass of star-shaped functions allocated by the condition \(a < \operatorname{Re}z f'(z)/f(z) < b\) (\(0 < a < 1 < b\)), in the class \(K(\gamma)\) of functions, almost convex in order \(\gamma \). The results of the article summarize or reinforce previously known effects.Lectures in nonlinear functional analysis. Synopsis of lectures given at the Faculty of Physics of Lomonosov Moscow State Universityhttps://zbmath.org/1503.460012023-03-23T18:28:47.107421Z"Korpusov, Maxim O."https://zbmath.org/authors/?q=ai:korpusov.maksim-olegovich"Ovchinnikov, Alexey V."https://zbmath.org/authors/?q=ai:ovchinnikov.alexey-vitalevich"Panin, Alexander A."https://zbmath.org/authors/?q=ai:panin.aleksandr-anatolevichThere is a vast literature of textbooks and monographs on Nonlinear Functional Analysis, the most remarkable one probably being \textit{E.~Zeidler}'s legendary encyclopedic work ``Nonlinear functional analysis and its applications'' published by Springer (1985; Zbl 0583.47051, 1986; Zbl 0583.47050, 1990; Zbl 0684.47028 and Zbl 0684.47029, 1993; Zbl 0794.47033). This book is different: it just collects lectures which have been given by the three authors at the Faculty of Physics of the Lomonosov University (MGU) in Moscow. The collection covers linear and nonlinear operators, continuity, compactness, and differentiability, variational methods, monotone operators, and fixed point theory. These topics belong to the standard analysis programme of every serious Math Department, and it is fair and legitimate to suspect that the authors assumed they should not withhold their lectures from the international mathematical community. The reviewer, however, does not share this assumption.
Reviewer: Jürgen Appell (Würzburg)Isometries and approximate local isometries between \(\mathrm{AC}^p (X)\)-spaceshttps://zbmath.org/1503.460062023-03-23T18:28:47.107421Z"Hosseini, Maliheh"https://zbmath.org/authors/?q=ai:hosseini.maliheh"Jiménez-Vargas, A."https://zbmath.org/authors/?q=ai:jimenez-vargas.antonioSummary: Let \(X\) and \(Y\) be compact subsets of \(\mathbb{R}\) with at least two points. For \(p\geq 1\), let \(\mathrm{AC}^p (X)\) be the space of all absolutely continuous complex-valued functions \(f\) on \(X\) such that \(f'\in L^p (X)\), with the norm \(\left\| f\right\|_{\Sigma}= \left\| f\right\|_{\infty} +\Vert f' \Vert_p\). We describe the topological reflexive closure of the set of linear isometries from \(\mathrm{AC}^p (X)\) onto \(\mathrm{AC}^p (Y)\). Using this description, we prove that such a set is algebraically reflexive and 2-algebraically reflexive. Moreover, as another application, it is shown that the sets of isometric reflections and generalized bi-circular projections of \(\mathrm{AC}^p (X)\) are topologically reflexive and 2-topologically reflexive.Geometry of Banach spaces: a new route towards position based cryptographyhttps://zbmath.org/1503.460072023-03-23T18:28:47.107421Z"Junge, Marius"https://zbmath.org/authors/?q=ai:junge.marius"Kubicki, Aleksander M."https://zbmath.org/authors/?q=ai:kubicki.aleksander-m"Palazuelos, Carlos"https://zbmath.org/authors/?q=ai:palazuelos.carlos"Pérez-García, David"https://zbmath.org/authors/?q=ai:perez-garcia.davidSummary: In this work we initiate the study of position based quantum cryptography (PBQC) from the perspective of geometric functional analysis and its connections with quantum games. The main question we are interested in asks for the optimal amount of entanglement that a coalition of attackers have to share in order to compromise the security of any PBQC protocol. Known upper bounds for that quantity are exponential in the size of the quantum systems manipulated in the honest implementation of the protocol. However, known lower bounds are only linear. In order to deepen the understanding of this question, here we propose a position verification (PV) protocol and find lower bounds on the resources needed to break it. The main idea behind the proof of these bounds is the understanding of cheating strategies as vector valued assignments on the Boolean hypercube. Then, the bounds follow from the understanding of some geometric properties of particular Banach spaces, their type constants. Under some regularity assumptions on the former assignment, these bounds lead to exponential lower bounds on the quantum resources employed, clarifying the question in this restricted case. Known attacks indeed satisfy the assumption we make, although we do not know how universal this feature is. Furthermore, we show that the understanding of the type properties of some more involved Banach spaces would allow to drop out the assumptions and lead to unconditional lower bounds on the resources used to attack our protocol. Unfortunately, we were not able to estimate the relevant type constant. Despite that, we conjecture an upper bound for this quantity and show some evidence supporting it. A positive solution of the conjecture would lead to stronger security guarantees for the proposed PV protocol providing a better understanding of the question asked above.Existence of unique fixed point of a mapping defined on an uniquely remotal subset in Hilbert spacehttps://zbmath.org/1503.460122023-03-23T18:28:47.107421Z"Som, Sumit"https://zbmath.org/authors/?q=ai:som.sumit"Savas, Ekrem"https://zbmath.org/authors/?q=ai:savas.ekremSummary: In this paper we introduce the notion of \(f\)-partial statistical continuity of a function (where \(f\) is an unbounded modulus function) which is much weaker than continuity of a function. We give an example to show that \(f\)-partial statistical continuity is weaker than continuity. Then we apply unbounded modulus function to give some answers to farthest point problem in real normed linear space which improves the result in [\textit{A.~Niknam}, Indian J. Pure Appl. Math. 18, 630--632 (1987; Zbl 0622.46014)]. As an application, we provide a sufficient condition for the existence of an unique fixed point of a self mapping defined on a non-empty, closed, bounded uniquely remotal subset in Hilbert space. Lastly, we introduce the notion of \(f\)-statistically maximizing sequence of a non-empty bounded subset \(M\) of a normed linear space \(\mathbb{X}\) and show by an example that this notion is weaker than maximizing sequence.Uncomplemented subspaces in operator and polynomial idealshttps://zbmath.org/1503.460132023-03-23T18:28:47.107421Z"Botelho, Geraldo"https://zbmath.org/authors/?q=ai:botelho.geraldo"Fávaro, Vinícius V."https://zbmath.org/authors/?q=ai:favaro.vinicius-vieira"Pérez, Sergio A."https://zbmath.org/authors/?q=ai:perez.sergio-aThe authors prove various interesting results regarding uncomplemented subspaces in operator and polynomial ideals. Sample results:
Theorem 4.1. Let \(\mathcal{M}\) be a closed symmetric coherent multi-ideal and let \(E\) and \(F\) be Banach spaces such that \(\mathcal{M}\left( E;F\right) \) is not complemented in \(L\left( E;F\right) \). Then \(\mathcal{M}\left( ^{n}E;F\right) \) is not complemented in \(\mathcal{P}\left( ^{n}E;F\right) \) for every \(n\in \mathbb{N}\).
Proposition 4.15. Let \(E\) be an infinite dimensional Banach space, \(n>1\), and let \(\mathcal{M}\) be a closed symmetric coherent multi-ideal whose linear component \(\mathcal{M}_{1}\) is contained in the ideal of compact operators and \(\widehat{ \mathcal{M}}\) is contained in \(P_{wb}\). If \( c_{0}\hookrightarrow \widehat{\mathcal{M}}\left( ^{n}E;F\right)\), then \(\widehat{\mathcal{M}}\left( ^{n}E;F\right) \) is not complemented in \(\mathcal{P}\left( ^{n}E;F\right) \).
Reviewer: Dumitrŭ Popa (Constanţa)New Banach spaces defined by the domain of Riesz-Fibonacci matrixhttps://zbmath.org/1503.460142023-03-23T18:28:47.107421Z"Zengin Alp, Pinar"https://zbmath.org/authors/?q=ai:zengin-alp.pinar"Kara, Emrah Evren"https://zbmath.org/authors/?q=ai:kara.emrah-evrenSummary: The main object of this study is to introduce the spaces \(c_0(\widehat{F}^q)\) and \(c(\widehat{F}^q)\) derived by the matrix \(\widehat{F}^q\) which is the multiplication of Riesz matrix and Fibonacci matrix. Moreover, we find the \(\alpha\)-, \(\beta\)-, \(\gamma\)- duals of these spaces and give the characterization of matrix classes \((\Lambda(\widehat{F}^q),\Omega)\) and \((\Omega,\Lambda(\widehat{F}^q))\) for \(\Lambda\in\{c_0,c\}\) and \(\Omega\in\{\ell_1,c_0,c,\ell_\infty\}\).Weighted estimates for operators of fractional integration of variable order in generalized variable Hölder spaceshttps://zbmath.org/1503.460222023-03-23T18:28:47.107421Z"Karapetyants, Alexey"https://zbmath.org/authors/?q=ai:karapetyants.aleksei-nikolaevich"Morales, Evelyn"https://zbmath.org/authors/?q=ai:morales.evelynSummary: %Summary: The paper is devoted to weighted estimates for operators of fractional integration of variable order of Bergman type in generalized variable Hölder spaces of holomorphic functions on the unit disc \(\mathbb{D}\). Due to the choice of the weight, we can include in consideration the case when the real part of the complex power of the operator degenerates. We prove the estimates of Zygmund type for the modulus of continuity, and then we obtain the corresponding weighted boundedness result.Abstract Hardy spaceshttps://zbmath.org/1503.460252023-03-23T18:28:47.107421Z"Liu, Yin"https://zbmath.org/authors/?q=ai:liu.yin.3Summary: In this paper, we study the Hardy spaces on spaces of homogeneous \(X\). Firstly, we give the definitions of the atomic Hardy spaces \(H_{ato}^p\) and the molecular Hardy spaces \(H_{\epsilon,mol}^p\) (\(0<p<1\)). Secondly, we give out the comparison between our Hardy spaces with some other Hardy spaces. Finally, we prove the continuity theorem of the sublinear operator on the Hardy spaces and give an example.Perturbation ideals and Fredholm theory in Banach algebrashttps://zbmath.org/1503.460392023-03-23T18:28:47.107421Z"Lukoto, Tshikhudo"https://zbmath.org/authors/?q=ai:lukoto.tshikhudo"Raubenheimer, Heinrich"https://zbmath.org/authors/?q=ai:raubenheimer.heinrichLet \(\mathcal{A}\) be a Banach algebra. The sets of all left (resp., right) invertible elements in \(\mathcal{A}\) is denoted by \(\mathcal{A}_l^{-1}\) (resp., by \(\mathcal{A}_r^{-1}\)). For a closed ideal \(I\) of \(\mathcal{A}\), the canonical homomorphism from \(\mathcal{A}\) to \(\mathcal{A}/I\) is denoted by \(\pi\) and \(\Phi_l(I)=\pi^{-1}((\mathcal{A}/I)_l^{-1})\), \(\Phi_r(I)=\pi^{-1}((\mathcal{A}/I)_r^{-1})\). Let \(\mathrm{Rad}(\mathcal{A})\) be the radical of \(\mathcal{A}\) and \(kh(I)=\{a\in\mathcal{A}:a+I\in\mathrm{Rad}(\mathcal{A}/I)\}\). For a set \(\mathcal{S}\subset\mathcal{A}\), the perturbation class \(\mathcal{P}(\mathcal{S})\) is defined by \(\mathcal{P}(\mathcal{S})=\{a\in\mathcal{A}: a+s\in\mathcal{S}\) \(\forall s\in \mathcal{S} \}\). If \(\mathcal{A}\) has minimal left (right) ideals, then the smallest left (right) ideal containing all the minimal left (right) ideals is called the left (right) socle. If \(\mathcal{A}\) has both minimal left and right ideals, and if the left and right socles of \(\mathcal{A}\) are equal, we say that the socle of \(\mathcal{A}\) exists and it is denoted by \(\mathrm{Soc}(\mathcal{A})\).
For a closed trace ideal \(I\), the authors define the so-called index function \(i:\Phi_l(I)\cup\Phi_r(I)\to\mathbb{Z}\), and for a subset \(Z\) of \(\mathbb{Z}\) they consider the set \(\Phi_Z=\{a\in\Phi_l(I)\cup\Phi_r(I):i(a)\in Z\}\). The main result of the paper is the following. Theorem. Let \(\mathcal{A}\) be a semisimple Banach algebra and \(I\) be a closed trace ideal in \(\mathcal{A}\) such that \(\mathrm{Soc}(\mathcal{A})\subset I\subset kh(\mathrm{Soc}(\mathcal{A}))\). If \(\Phi_Z\ne\emptyset\), then \(\mathcal{P}(\Phi_Z)=\pi^{-1}(\mathrm{Rad}(\mathcal{A}/I))\).
Reviewer: Oleksiy Karlovych (Lisboa)Non-unital operator systems that are dual spaceshttps://zbmath.org/1503.460432023-03-23T18:28:47.107421Z"Jia, Yu-Shu"https://zbmath.org/authors/?q=ai:jia.yu-shu"Ng, Chi-Keung"https://zbmath.org/authors/?q=ai:ng.chi-keung\textit{D. P. Blecher} and \textit{B. Magajna} [Bull. Lond. Math. Soc. 43, No. 2, 311--320 (2011; Zbl 1234.46046)]
proved that a unital operator system (operator space) which is the dual of an operator system (operator space) can be represented completely isometrically and weak* homeomorphically as a weak* closed operator subsystem of $B(H)$. The proof used the fact that involution in unital operator system is weak*-continuous and the Krein-Smulian theorem to show that in unital operator systems the matrix cones are weak* closed. Non-unital operator systems lose these properties. In the present paper, the authors generalize the Blecher-Magajana result to non-unital operator systems by adding the assumptions that the involution is weak*-continuous and matrix cones are weak*-closed.
Reviewer: Preeti Luthra (Delhi)Radial Schur multipliers on some generalisations of treeshttps://zbmath.org/1503.460452023-03-23T18:28:47.107421Z"Vergara, Ignacio"https://zbmath.org/authors/?q=ai:vergara.ignacioA function \(\phi: X\times X\rightarrow\mathbb{C}\) is said to be a Schur multiplier on an nonempty set \(X\) if the map
\[
M_\phi:(T_{x,y})_{x,y\in X}\mapsto(\phi(x,y)T_{x,y})_{x,y\in X}
\]
defines a bounded operator on \(B(\ell_2(X))\). An interesting characterization of Schur multipliers was given by [\textit{A. Grothendieck}, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8, 1--79 (1956; Zbl 0074.32303)]. Equipped with the completely bounded norm \(\|\phi\|_{cb}=\|M_\phi\|\), the set of Schur multipliers on \(X\) becomes a Banach space.
In the case when \(X\) is (the set of vertices of) an infinite graph, a function \(\phi: X\times X\rightarrow\mathbb{C}\) is said to be \textit{radial} if there exists \(\dot{\phi}:\mathbb{N}\rightarrow\mathbb{C}\) such that if \(d:X \times X \rightarrow \mathbb{N}\) is the combinatorial distance associated to \(X\),
\[
\phi(x,y)=\dot{\phi}(d(x,y)),\quad\forall x,y\in X.
\]
Conversely, \(\dot{\phi}\) defines a radial function \(\phi:X\times X\rightarrow\mathbb{C}\) if the previous relation holds for \(\phi\). The author's main interest is to give a characterisation of radial Schur multipliers on \(X\) (infinite graph), and to extend it to finite products of trees. Equivalent conditions of this characterisation in terms of Hankel matrices and also in terms of Besov spaces are proved. The characterisation in terms of Hankel matrices gives an extension of Haagerup, Steenstrup and Szwarc's result for trees [\textit{U. Haagerup} et al., Int. J. Math. 21, No. 10, 1337--1382 (2010; Zbl 1272.43003)] to finite products of trees:
Let \(N \geq1 \) and let \(\dot{\phi}:\mathbb{N}\rightarrow\mathbb{C}\) be a bounded function. Then \(\dot{\phi}\) defines a radial Schur multiplier on any product of \(N\) infinites trees \(T_1,\dots,T_N\) of minimum degrees \(d_1,\dots,d_N\geq 3\) if and only if the generalised Hankel matrix
\[
H=\left(\begin{pmatrix}N+i-1\\ N-1\end{pmatrix}^{\frac{1}{2}} \begin{pmatrix}N+i-1\\ N-1\end{pmatrix}^{\frac{1}{2}}\mathfrak{d}_2^N\dot{\phi}(i+j)\right)_{i,j\in\mathbb{N}}
\]
is an element of \(S_1(\ell_2(\mathbb{N}))\). In that case, the following limits exist
\[
\lim_{n\to\infty}\dot{\phi}(2n),\quad \lim_{n\to\infty}\dot{\phi}(2n+1),
\]
and the corresponding Schur multiplier \(\phi\) satisfies
\[
\left[\prod_{i=1}^N\frac{d_i-2}{d_i}\right] \|H\|_{S_1} + |c_+|+|c_-|\leq\|\phi\|_{cb} \leq \|H\|_{S_1} + |c_+|+|c_-|,
\]
where
\[
c_\pm=\frac{1}{2}\lim_{n\rightarrow\infty}\dot{\phi}(2n)\pm \frac{1}{2}\lim_{n\rightarrow\infty}\dot{\phi}(2n+1).
\]
A similar result is obtained for products of hyperbolic graphs and a sufficient condition for a function to define a radial Schur multiplier on a finite-dimensional CAT\((0)\) cube complex is provided.
Reviewer: Essé Julien Atto (Lomé)Orthogonality preserving pairs of operators on Hilbert \(C_0(Z)\)-moduleshttps://zbmath.org/1503.460462023-03-23T18:28:47.107421Z"Asadi, Mohammad B."https://zbmath.org/authors/?q=ai:asadi.mohammad-b"Olyaninezhad, Fatemeh"https://zbmath.org/authors/?q=ai:olyaninezhad.fatemeh\textit{M. Frank} et al. [Aequationes Math. 95, No. 5, 867--887 (2021; Zbl 1477.46065)] showed that if \(\mathscr{A}\) is a \(C^*\)-algebra and \(T, S:\mathscr{E}\to \mathscr{F}\) are two bounded \({\mathscr A}\)-linear operators between full Hilbert \(\mathscr{A}\)-modules, then \(\langle x, y\rangle = 0\) implies \(\langle T(x), S(y)\rangle = 0\) for all \(x, y\in \mathscr{E}\) if and only if there exists an element \(\gamma\) of the center \(Z(M({\mathscr A}))\) of the multiplier algebra \(M({\mathscr A})\) of \({\mathscr A}\) such that \(\langle T(x), S(y)\rangle = \gamma \langle x, y\rangle\) for all \(x, y\in \mathscr{E}\). Moreover, they showed that the assumptions of fullness and boundedness can be removed if \(\mathscr{A}\) is a standard \(C^*\)-algebra. In the paper under review, the authors provide a proof of the above statement for Hilbert \(C^*\)-modules over a commutative \(C^*\)-algebra without the assumptions of fullness and boundedness. In fact, they show that if \(T\) and \(S\) have the same range, then they are automatically continuous. To achieve their result, the authors follow the lines of the proofs of the main theorems in [\textit{H.-L. Gau} et al., J. Aust. Math. Soc. 74, No.~1, 101--109 (2003; Zbl 1052.47017); \textit{C.-W. Leung} et al., J. Aust. Math. Soc. 89, No.~2, 245--254 (2010; Zbl 1242.46068)].
Reviewer: Mohammad Sal Moslehian (Mashhad)Additivity violation of the regularized minimum output entropyhttps://zbmath.org/1503.460502023-03-23T18:28:47.107421Z"Collins, Benoît"https://zbmath.org/authors/?q=ai:collins.benoit"Youn, Sang-Gyun"https://zbmath.org/authors/?q=ai:youn.sang-gyunSummary: The problem of additivity of the Minimum Output Entropy is of fundamental importance in Quantum Information Theory (QIT). It was solved by \textit{Matthew B. Hastings} [``Superadditivity of communication capacity using entangled inputs'', Nature Physics 5, 255--257 (2009; \url{doi:10.1038/nphys1224})] in the one-shot case by exhibiting a pair of random quantum channels. However, the initial motivation was arguably to understand regularized quantities, and there was so far no way to solve additivity questions in the regularized case. The purpose of this paper is to give a solution to this problem. Specifically, we exhibit a pair of quantum channels that unearths additivity violation of the regularized minimum output entropy. Unlike previously known results in the one-shot case, our construction is non-random, infinite-dimensional, and in the commuting-operator setup. The commuting-operator setup is equivalent to the tensor-product setup in the finite-dimensional case for this problem, but their difference in the infinite-dimensional setting has attracted substantial attention and legitimacy recently in QIT with the celebrated resolutions of Tsirelson's and Connes embedding problem [\textit{Z.-F. Ji} et al., ``\(\mathsf{MIP}^*= \mathsf{RE}\)'', Preprint (2020), \url{arXiv:2001.04383}], Likewise, it is not clear that our approach works in the finite-dimensional setup. Our strategy of proof relies on developing a variant of the Haagerup inequality optimized for a product of free groups.A new decomposition for multivalued \(3\times 3\) matriceshttps://zbmath.org/1503.470012023-03-23T18:28:47.107421Z"Ammar, Aymen"https://zbmath.org/authors/?q=ai:ammar.aymen"Jeribi, Aref"https://zbmath.org/authors/?q=ai:jeribi.aref"Saadaoui, Bilel"https://zbmath.org/authors/?q=ai:saadaoui.bilelSummary: In this paper, a new concept for a \(3\times 3\) block relation matrix is studied in a Banach space. It is shown that, under certain condition, we can investigate the Frobenius-Schur decomposition of relation matrices. Furthermore, we present some conditions which should allow the multivalued \(3\times 3\) matrices linear operator to be closable.Spectra of \(3 \times 3\) upper triangular operator matriceshttps://zbmath.org/1503.470022023-03-23T18:28:47.107421Z"Wu, Xiufeng"https://zbmath.org/authors/?q=ai:wu.xiufeng"Huang, Junjie"https://zbmath.org/authors/?q=ai:huang.junjie"Chen, Alatancang"https://zbmath.org/authors/?q=ai:chen.alatancangSummary: Let \(\mathcal H_1\), \(\mathcal H_2\), and \(\mathcal H_3\) be complex separable Hilbert spaces. Given \(A\in \mathcal B(\mathcal H_1)\), \(B\in \mathcal B(\mathcal H_2)\), and \(C\in \mathcal B(\mathcal H_3)\), write \(M_{D,E,F} = \begin{pmatrix} A&D&E \\ 0&B&F \\ 0&0&C \end{pmatrix}\), where \(D\in \mathcal B(\mathcal H_2,\mathcal H_1)\), \(E\in \mathcal B(\mathcal H_3,\mathcal H_1)\), and \(F\in \mathcal B(\mathcal H_3,\mathcal H_2)\) are unknown operators. This paper gives a complete description of the intersection \(\bigcap_{D,E,F}\sigma(M_{D,E,F})\), where \(D\), \(E\), and \(F\) range over the respective sets of bounded linear operators. Further, we show that \(\sigma(A)\cup \sigma(B)\cup \sigma(C) = \sigma(M_{D,E,F})\cup W\), where \(W\) is the union of certain gaps in \(\sigma(M_{D,E,F})\), which are subsets of \((\sigma(A)\cap \sigma(B))\cup (\sigma(B)\cap \sigma(C))\cup (\sigma(A)\cap \sigma(C))\). Finally, we obtain a necessary and sufficient condition for the relation \(\sigma(M_{D,E,F}) = \sigma(A)\cup \sigma(B)\cup \sigma(C)\) to hold for any \(D\), \(E\), and \(F\).Local spectral theory and quasinilpotent operatorshttps://zbmath.org/1503.470032023-03-23T18:28:47.107421Z"Yoo, Jong-Kwang"https://zbmath.org/authors/?q=ai:yoo.jong-kwangSummary: In this paper we show that if \(A\in L(X)\) and \(R\in L(X)\) is a quasinilpotent operator commuting with \(A\) then \(X_A(F) = X_{A+R}(F)\) for all subset \(F\subseteq \mathbb{C}\) and \(\sigma_{loc}(A) = \sigma_{loc}(A + R)\). Moreover, we show that \(A\) and \(A + R\) share many common local spectral properties such as SVEP, property \((C)\), property \((\delta)\), property \((\beta)\) and decomposability. Finally, we show that quasisimility preserves local spectrum.Refinements of numerical radius inequalities via Specht's ratiohttps://zbmath.org/1503.470042023-03-23T18:28:47.107421Z"Khatib, Yaser"https://zbmath.org/authors/?q=ai:khatib.yaser"Hassani, Mahmoud"https://zbmath.org/authors/?q=ai:hassani.mahmoud"Amyari, Maryam"https://zbmath.org/authors/?q=ai:amyari.maryamSummary: We present some new numerical radius inequalities of Hilbert space operators. We improve and generalize some inequalities with respect to Specht's ratio. Let \(A\) and \(B\) be two positive invertible operators on a Hilbert space \(H\) and let \(X\) be a bounded operator on \(H\). Then
\[
\omega((A\natural B)X)\leq \frac{1}{2S(\sqrt{h})}\|X^\ast BX+A\| \quad (h>0,\ h\neq 1)
\] where \(\|\cdot\|\), \(\omega(\cdot)\), \(S(\cdot)\), and \(\natural\) denote the usual operator norm, numerical radius, the Specht's ratio, and the operator geometric mean, respectively.Joint spectra of spherical Aluthge transforms of commuting \(n\)-tuples of Hilbert space operatorshttps://zbmath.org/1503.470052023-03-23T18:28:47.107421Z"Benhida, Chafiq"https://zbmath.org/authors/?q=ai:benhida.chafiq"Curto, Raúl E."https://zbmath.org/authors/?q=ai:curto.raul-enrique"Lee, Sang Hoon"https://zbmath.org/authors/?q=ai:lee.sanghoon"Yoon, Jasang"https://zbmath.org/authors/?q=ai:yoon.jasangSummary: Let \(\mathbf{T} \equiv(T_1, \dots, T_n)\) be a commuting \(n\)-tuple of operators on a Hilbert space \(\mathcal{H}\), and let \(T_i \equiv V_i P\) (\(1 \leq i \leq n\)) be its canonical joint polar decomposition (i.e., \(P : = \sqrt{T_1^\ast T_1 + \dots + T_n^\ast T_n}\), (\(V_1, \dots, V_n\)) a joint partial isometry, and \(\bigcap_{i = 1}^n \ker T_i = \bigcap_{i = 1}^n \ker V_i = \ker P\)). The spherical Aluthge transform of \(\mathbf{T}\) is the (necessarily commuting) \(n\)-tuple \(\hat{\mathbf{T}} : = (\sqrt{P} V_1 \sqrt{P}, \dots, \sqrt{P} V_n \sqrt{P})\). We prove that \(\sigma_{\mathrm T}(\hat{\mathbf{T}}) = \sigma_{\mathrm{T}}(\mathbf{T})\), where \(\sigma_{\mathrm{T}}\) denotes the Taylor spectrum. We do this in two stages: away from the origin, we use tools and techniques from criss-cross commutativity; at the origin, we show that the left invertibility of \(\mathbf{T}\) or \(\hat{\mathbf{T}}\) implies the invertibility of \(P\). As a consequence, we can readily extend our main result to other spectral systems that rely on the Koszul complex for their definitions.\(A\)-isometries and Hilbert-\(A\)-modules over product domainshttps://zbmath.org/1503.470062023-03-23T18:28:47.107421Z"Didas, Michael"https://zbmath.org/authors/?q=ai:didas.michaelAuthor's abstract: For a compact set \(K\subset \mathbb{C}^n\), let \(A\subset C(K)\) be a function algebra containing the polynomials \(\mathbb{C}[z_1,\dots,z_n]\). Assuming that a certain regularity condition holds for \(A\), we prove a commutant-lifting theorem for \(A\)-isometries that contains the known results for isometric subnormal tuples in its different variants as special cases, e.g., [\textit{W. Mlak}, Stud. Math. 43, 219--233 (1972; Zbl 0257.46081)] and [\textit{A. Athavale}, J. Oper. Theory 23, No. 2, 339--350 (1990; Zbl 0738.47005); Rocky Mt. J. Math. 48, No. 1, 19--46 (2018; Zbl 06866698); Complex Anal. Oper. Theory 2, No. 3, 417--428 (2008; Zbl 1182.47024); New York J. Math. 25, 934--948 (2019; Zbl 07118595)]. In the context od Hilbert-\(A\)-modules, our result implies the existence of an extension map \(\varepsilon: \mathrm{Hom}_A(\mathscr{S}_1,\mathscr{S}_2)\rightarrow \mathrm{Hom}_{C(\partial_A)}(\mathscr{K}_1,\mathscr{K}_2)\) for hypo-Shilov-modules \(\mathscr{S}_i\subset\mathscr{K}_i\) (\(i=1,2\)). By standard arguments, we obtain an identification \(\mathrm{Hom}_A(\mathscr{S}_1,\mathscr{S}_2)\cong\mathrm{Hom}_A(\mathscr{K}_1\ominus\mathscr{S}_1,\mathscr{K}_2\ominus\mathscr{S}_2)\) where \(\mathscr{K}_i\) is the minimal \(C(\partial_A)\)-extension of \(\mathscr{S}_i\) (\(i=1,2\)), provided that \(\mathscr{K}_1\) is projective and \(\mathscr{S}_2\) is pure. Using embedding techniques, we show that these results apply in particular to the domain algebra \(A=A(D)=C(\overline{D})\cap \mathscr{O}(D)\) over a product domain \(D=D_1\times\dots\times D_k\subset\mathbb{C}^n\) where each factor \(D_i\) is either a smoothly bounded, strictly pseudoconvex domain or a bounded symmetric and circled domain in some \(\mathbb{C}^{d_i}\) (\(1\leq i\leq k\)). This extends known results from the ball and poydisc-case [\textit{K. Guo}, Stud. Math. 135, No. 1, 1--12 (1999; Zbl 0944.46050)] and [\textit{X. Chen} and \textit{K. Guo}, J. Oper. Theory 43, No. 1, 69--81 (2000; Zbl 0992.46038)].
Reviewer: Andrzej Sołtysiak (Poznań)Holomorphic spectral theory: a geometric approachhttps://zbmath.org/1503.470072023-03-23T18:28:47.107421Z"Martin, Mircea"https://zbmath.org/authors/?q=ai:martin.mirceaOverall, the purpose of the article is to analyze holomorphic mappings with values in the following spaces: Grassmann manifolds of Hilbert spaces, Hermitian holomorphic vector bundles, and Cowen-Douglas classes of operators.
Reviewer: Mohammed El Aïdi (Bogotá)Some results on b-AM-compact operatorshttps://zbmath.org/1503.470082023-03-23T18:28:47.107421Z"Cheng, Na"https://zbmath.org/authors/?q=ai:cheng.naSummary: We show that for positive operator \(B: E\to E\) on Banach lattices, if there exists a positive operator \(S: E\to E\) such that: (1) \(SB\leq BS\), (2) \(S\) is quasinilpotent at some \(x_0>0\), (3) \(S\) dominates a non-zero b-AM-compact operator, then \(B\) has a non-trivial closed invariant subspace. Also, we prove that for two commuting non-zero positive operators on Banach lattices, if one of them is quasinilpotent at a non-zero positive vector and the other dominates a non-zero b-AM-compact operator, then both of them have a common non-trivial closed invariant ideal. Then we introduce the class of b-AM-compact-friendly operators and show that a non-zero positive b-AM-compact-friendly operator which is quasinilpotent at some \(x_0>0\) has a non-trivial closed invariant ideal.Local spectral properties of typical contractions on \(\ell_p\)-spaceshttps://zbmath.org/1503.470092023-03-23T18:28:47.107421Z"Grivaux, S."https://zbmath.org/authors/?q=ai:grivaux.sophie"Matheron, É."https://zbmath.org/authors/?q=ai:matheron.etienneThis nice article is a follow-up of the two important papers [\textit{S. Grivaux} et al., Linear dynamical systems on Hilbert spaces: typical properties and explicit examples. Providence, RI: American Mathematical Society (AMS) (2021; Zbl 1479.47002)] and [\textit{S. Grivaux} et al., Trans. Am. Math. Soc. 374, No. 10, 7359--7410 (2021; Zbl 07398006)]. Let \(X\) be one of the \(l_p\) spaces, where \(1<p<\infty\). The closed unit ball of the space \(B(X)\) of bounded linear operators on \(X\) is a Polish space when equipped with the \(SOT^*\) topology, defined as convergence of \(T\) and \(T^*\) in the weak operator topology. Hence it is natural to investigate the typical properties, in the Baire category sense, of a contraction on \(X\).
The gist of the present work is that the methods from local spectral theory are not helpful for showing that a typical contraction on \(l_p\), where \(1 < p < \infty\) and \(p\not= 2\), has a nontrivial invariant subspace (a~basic open problem). Note that in the case \(p=2\), it follows from the Brown-Chevreau-Pearcy theorem that this typical existence holds true. Precisely, it is shown here that a typical contraction on \(l_p\) fails the properties \((\delta)\) and \((\beta)\), is completely indecomposable and has empty localizable spectrum. More results are shown on the typical behaviour of orbits, which happens to be chaotic as could be expected, and on contractions which are mixing in the Gaussian sense.
Several open questions conclude the article: for instance, it is not known if a typical contraction on \(l_p\) fails to satisfy the assumptions of the Lomonosov theorem.
Reviewer: Gilles Godefroy (Paris)On operators with orbits dense relative to nontrivial subspaceshttps://zbmath.org/1503.470102023-03-23T18:28:47.107421Z"Rezaei, H."https://zbmath.org/authors/?q=ai:rezaei.hassan|rezaei.hossein|rezaei.hamed|rezaei.hamid|rezaei.humanSummary: In the present paper we consider bounded linear operators which have orbits dense relative to nontrivial subspaces. We give nontrivial examples of such operators and establish many of their basic properties. An example of an operator which has an orbit dense relative to a certain subspace but is not subspace-hypercyclic for this subspace is given. This, in turn, provides a new answer to a question posed by the author in [J. Math. Anal. Appl. 397, No. 1, 428--433 (2013; Zbl 1272.47010)]. Other hypercyclic-like properties of such operators are also considered.Isometric dilations of commuting contractions and Brehmer positivityhttps://zbmath.org/1503.470112023-03-23T18:28:47.107421Z"Barik, Sibaprasad"https://zbmath.org/authors/?q=ai:barik.sibaprasad"Das, B. Krishna"https://zbmath.org/authors/?q=ai:krishna-das.bLet \(T = (T_1, \dots, T_n)\) be an \(n\)-tuple of commuting contractions on a (complex and separable) Hilbert space \(\mathcal{H}\). A commuting tuple of isometries \((V_1, \dots, V_n)\) on some Hilbert space \(\mathcal{K} ( \supseteq \mathcal{H})\) is said to be an isometric dilation (or co-isometric extension, to be precise) of \(T\) if
\[
T_{1}^{*k_1} \dots T_n^{*k_n} = (V_{1}^{*k_1} \cdots V_n^{*k_n})|_{\mathcal{H}},
\]
for all \((k_1, \dots, k_n) \in \mathbb{Z}_+^n\). The existence of isometric dilations of single contractions is the classical result of \textit{B. Sz.-Nagy} and \textit{C. Foiaş} [Harmonic analysis of operators on Hilbert space. Amsterdam-London: North-Holland Publishing Company (1970; Zbl 0201.45003)], and the seminal paper of \textit{T. Andô} [Acta Sci. Math. 24, 88--90 (1963; Zbl 0116.32403)] assures the same for pairs of commuting contractions. However, the existence of isometric dilations of \(n\)-tuples of commuting contractions, for \(n \geq 3\), is not true in general. Counterexamples appear even at the level of triples of commuting matrices. Subsequently, identifying classes of \(n\)-tuples of commuting contractions, \(n \geq 3\), admitting isometric dilations is a challenging problem. There is a handful of papers that address this delicate issue. For instance, see [\textit{A. Grinshpan} et al., J. Funct. Anal. 256, No. 9, 3035--3054 (2009; Zbl 1167.47009)] and [\textit{S. Barik} et al., Trans. Am. Math. Soc. 372, No. 2, 1429--1450 (2019; Zbl 1475.47007)].
This paper identifies a class of \(n\)-tuples of commuting contractions that admit isometric dilations. The tuples satisfy certain positivity conditions related to the classical Brehmer positivity for commuting contractions. Part of the motivation comes from [\textit{J. R. Archer}, Oper. Theory: Adv. Appl. 171, 17--35 (2006; Zbl 1122.47005)] and [Barik et al., loc. cit.].
Reviewer: Jaydeb Sarkar (Bangalore)Uniform ergodicity of Lotz-Räbiger nets of Markov operators on abstract state spaceshttps://zbmath.org/1503.470122023-03-23T18:28:47.107421Z"Erkurşun Özcan, Nazife"https://zbmath.org/authors/?q=ai:erkursun-ozcan.nazife"Mukhamedov, Farrukh"https://zbmath.org/authors/?q=ai:mukhamedov.farruh-mSummary: It is known that Dobrushin's ergodicity coefficient is one of the powerful tools in the investigation of limiting behavior of Markov chains. Several interesting properties of the ergodicity coefficient of a positive mapping defined on an abstract state space have been studied. In this paper, we consider uniform ergodicity of Lotz-Räbiger nets of Markov operators on abstract state spaces. We prove a uniform mean ergodicity criterion in terms of the ergodicity coefficient. This result allows us to investigate perturbations of uniformly mean ergodic operators. Moreover, our main results open new perspectives in quantum Markov processes defined on von Neumann algebras.Uniform ergodicities and perturbation bounds of Markov chains on base norm spaceshttps://zbmath.org/1503.470132023-03-23T18:28:47.107421Z"Erkurşun-Özcan, Nazife"https://zbmath.org/authors/?q=ai:erkursun-ozcan.nazife"Mukhamedov, Farrukh"https://zbmath.org/authors/?q=ai:mukhamedov.farruh-mSummary: It is known that Dobrushin's ergodicity coefficient is one of the effective tools in the investigations of limiting behavior of Markov processes. Several interesting properties of the ergodicity coefficient of a positive mapping defined on base norm spaces have been studied. In this paper, we consider uniformly mean ergodic and asymptotically stable Markov operators on such spaces. In terms of the ergodicity coefficient, we establish uniform mean ergodicity criterion. Moreover, we develop the perturbation theory for uniformly asymptotically stable Markov chains on base norm spaces. In particularly, main results open new perspectives in the perturbation theory for quantum Markov processes defined on von Neumann algebras.On nonergodic uniform Lotka-Volterra operatorshttps://zbmath.org/1503.470142023-03-23T18:28:47.107421Z"Mukhamedov, F. M."https://zbmath.org/authors/?q=ai:mukhamedov.farruh-m"Jamilov, U. U."https://zbmath.org/authors/?q=ai:jamilov.uygun-u"Pirnapasov, A. T."https://zbmath.org/authors/?q=ai:pirnapasov.abror-tSummary: In this paper, we introduce uniform Lotka-Volterra operators and construct Lyapunov functions for them. We establish that the ergodic averages associated with operators of such kind diverge.A fast multilevel iteration method for solving linear ill-posed integral equationshttps://zbmath.org/1503.470152023-03-23T18:28:47.107421Z"Yang, Hongqi"https://zbmath.org/authors/?q=ai:yang.hongqi"Zhang, Rong"https://zbmath.org/authors/?q=ai:zhang.rongSummary: We propose a new concept of noise level: \(\mathcal{R}(K^\ast)\)-noise level for ill-posed linear integral equations in Tikhonov regularization, which extends the range of regularization parameter. This noise level allows us to choose a more suitable regularization parameter. Moreover, we also analyze error estimates of the approximate solution with respect to this noise level. For ill-posed integral equations, finding fast and effective numerical methods is a challenging problem. For this, we formulate a matrix truncated strategy based on multiscale Galerkin method to generate the linear system of Tikhonov regularization for ill-posed linear integral equations, which greatly reduce the computational complexity. To further reduce the computational cost, a fast multilevel iteration method for solving the linear system is established. At the same time, we also prove convergence rates of the approximate solution obtained by this fast method with respect to the \(\mathcal{R}(K^\ast)\)-noise level under the balance principle. By numerical results, we show that \(\mathcal{R}(K^\ast)\)-noise level is very useful and the proposed method is a fast and effective method, respectively.Spectral theory for structured perturbations of linear operatorshttps://zbmath.org/1503.470162023-03-23T18:28:47.107421Z"Adler, Martin"https://zbmath.org/authors/?q=ai:adler.martin"Engel, Klaus-Jochen"https://zbmath.org/authors/?q=ai:engel.klaus-jochenSummary: We characterize the spectrum (and its parts) of operators which can be represented as \(G=A+BC\) for a ``simpler'' operator \(A\) and a structured perturbation \(BC\). The interest in this kind of perturbations is motivated, e.g., by perturbations of the domain of an operator \(A\) but also arises in the theory of closed-loop systems in control theory. In many cases our results yield the spectral values of \(G\) as zeros of a ``\textit{characteristic equation}''.Perturbations of \(C\)-normal operatorshttps://zbmath.org/1503.470172023-03-23T18:28:47.107421Z"Amara, Zouheir"https://zbmath.org/authors/?q=ai:amara.zouheir"Oudghiri, Mourad"https://zbmath.org/authors/?q=ai:oudghiri.mouradThe authors are interested in perturbations of normal operators that remain \(C\)-normal for a certain conjugation \(C\). They also show that the only operator that remains \(C\)-normal under all complex symmetric perturbations is the scalar multiple of the identity.
Reviewer: Mohammed Hichem Mortad (Oran)On the stability of the approximate point spectrum under commuting perturbationshttps://zbmath.org/1503.470182023-03-23T18:28:47.107421Z"Aponte, E."https://zbmath.org/authors/?q=ai:aponte.elvis"Biondi, M. T."https://zbmath.org/authors/?q=ai:biondi.maria-teresaSummary: The spectrum, the approximate point spectrum and the surjectivity spectrum of an operator \(T\) are studied under commuting perturbations \(K\) for which \(K^n\) is a finite rank operator for some natural \(n\).Operator-valued Mazur-Orlicz and moment problems in spaces of analytic functionshttps://zbmath.org/1503.470192023-03-23T18:28:47.107421Z"Olteanu, Cristian-Octav"https://zbmath.org/authors/?q=ai:olteanu.cristian-octav"Mihaila, Janina Mihaela"https://zbmath.org/authors/?q=ai:mihaila.janina-mihaelaSummary: The aim of the present work is to prove new applications of some earlier general abstract results on the subject to spaces of analytic functions. The Cauchy inequalities are used systematically, as well as constrained extension of linear operators. One gives necessary and sufficient conditions and only sufficient conditions for the existence of solutions of an operator valued moment problem and of Mazur-Orlicz problems. The upper constraint appears naturally from the corresponding computations, while the lower constraint is sometimes the positivity of the solution. Considering Markov moment problem, one solves a concrete interpolation problem with two constraints. In the case of Mazur-Orlicz problems, the interpolation conditions are replaced by the corresponding inequalities mentioned in Section~2. Operator valued solutions are obtained.Mazur-Orlicz theorem in concrete spaces and inverse problems related to the moment problemhttps://zbmath.org/1503.470202023-03-23T18:28:47.107421Z"Olteanu, Octav"https://zbmath.org/authors/?q=ai:olteanu.octavSummary: In the first part of this work, we derive some new applications of a version of Mazur-Orlicz theorem, in concrete spaces of absolutely integrable functions and respectively continuous functions of several real variables. The second part is devoted to inverse problems related to the Markov moment problem. A~geometric approach of approximating the solutions of a system with infinitely many equations involving transcendent functions, with infinitely many unknowns, is briefly discussed.Further generalizations of Bebiano-Lemos-Providência inequalityhttps://zbmath.org/1503.470212023-03-23T18:28:47.107421Z"Fujii, Masatoshi"https://zbmath.org/authors/?q=ai:fujii.masatoshi"Matsumoto, Akemi"https://zbmath.org/authors/?q=ai:matsumoto.akemi"Nakamoto, Ritsuo"https://zbmath.org/authors/?q=ai:nakamoto.ritsuoThe authors discuss Furuta-type inequalities with negative parameters. In fact, they prove the following result. Let \(A,B\) be positive invertible bounded linear operators on a Hilbert space. Then
\[
A\geq B>0\ \Longrightarrow\ A^{-r}\natural_{\frac{1+r}{p+r}} B^{p}\leq A
\]
for all \(p\leq -1\) and \(r\in [-1,0]\), or for \(p\in [0,1]\) and \(r\leq -1\), where \(\natural_{\alpha}\) is defined by the same formula of the \(\alpha\)-weighted operator geometric mean for \(\alpha\in \mathbb{R}\). If \(p\geq 1\) and \(r\geq 0\), then the above inequality is called the Furuta inequality.
Next, the authors consider reverse implications of the Furuta-type inequalities. The authors obtain the following. For \(p\geq -1\) and \(r\in [-1,0]\), or for \(p\in [0,1]\) and \(r\leq -1\),
\[
A^{r}\natural_{1/p}B^{p+r}\leq A^{1+r} \ \Longrightarrow\ B^{1+s}\leq A^{1+s}
\]
for all \(-1\leq s \leq r\), or for \(r\leq s\leq -1\), respectively. By using these results, the authors obtain some norm inequalities which have a similar form as in [\textit{N. Bebiano} et al., Linear Algebra Appl. 401, 159--172 (2005; Zbl 1076.15019)].
Moreover, the authors obtain several inequalities by using the generalized Kantorovich constant or the generalized Furuta-type inequalities. Some of them are inequalities of relative operator entropy, and some of them are log-majorization relations.
Reviewer: Takeaki Yamazaki (Kawagoe)Operator maps of Jensen-typehttps://zbmath.org/1503.470222023-03-23T18:28:47.107421Z"Hansen, Frank"https://zbmath.org/authors/?q=ai:hansen.frank-peter"Moslehian, Mohammad Sal"https://zbmath.org/authors/?q=ai:moslehian.mohammad-sal"Najafi, Hamed"https://zbmath.org/authors/?q=ai:najafi.hamedSummary: Let \(\mathbb {B}_J({\mathcal {H}})\) denote the set of self-adjoint operators acting on a Hilbert space \(\mathcal {H}\) with spectra contained in an open interval \(J\). A map \(\Phi :\mathbb {B}_J({\mathcal {H}})\rightarrow \mathbb {B}({\mathcal {H}})_{\mathrm{sa}}\) is said to be of Jensen-type if
\[
\Phi (C^*AC+D^*BD)\leq C^*\Phi (A)C+D^*\Phi (B)D
\]
for all \(A, B \in \mathbb {B}_J({\mathcal {H}})\) and bounded linear operators \(C\), \(D\) acting on \(\mathcal {H}\) with \(C^*C+D^*D=I\), where \(I\) denotes the identity operator. We show that a Jensen-type map on an infinite dimensional Hilbert space is of the form \(\Phi (A)=f(A)\) for some operator convex function \(f\) defined in \(J\).Some lower and upper bounds for relative operator entropyhttps://zbmath.org/1503.470232023-03-23T18:28:47.107421Z"Moradi, H. R."https://zbmath.org/authors/?q=ai:moradi.hamid-reza"Hosseini, M. Shah"https://zbmath.org/authors/?q=ai:hosseini.mohsen-shah"Omidvar, M. E."https://zbmath.org/authors/?q=ai:omidvar.mohsen-erfanian"Dragomir, S. S."https://zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: The relation between the related operator entropy \(S(A\vert B)\) and Tsallis relative operator entropy \(T_p(A\vert B)\) was firstly considered by \textit{J. I. Fujii} and \textit{E. Kamei} [Math. Japon. 34, No. 3, 341--348 (1989; Zbl 0699.46048)], as follows: \[ T_p(A\vert B)\le S(A\vert B)\le T_p(A\vert B), \quad p\in(0,1], \] where \(A, B\) are positive invertible operators. The aim of this paper is to establish some refinements for the above inequality.Operator inequalities related to \(p\)-angular distanceshttps://zbmath.org/1503.470242023-03-23T18:28:47.107421Z"Tab, Davood Afkhami"https://zbmath.org/authors/?q=ai:tab.davood-afkhami"Dehghan, Hossein"https://zbmath.org/authors/?q=ai:dehghan.hosseinSummary: For any nonzero elements \(x,y\) in a normed space \(X\), the angular and skew-angular distance is respectively defined by \(\alpha[x,y]= \| \frac{x}{\|x\|} - \frac{y}{\|y\|} \|\) and \(\beta[x,y]= \| \frac{x}{\|y\|} - \frac{y}{\|x\|} \|\). Also, the inequality \(\alpha \leq \beta\) characterizes inner product spaces. An operator version of \(\alpha_p\) has been studied by \textit{J. Pečarić} and \textit{R. Rajić} [J. Math. Inequal. 4, No. 1, 1--10 (2010; Zbl 1186.26020)], \textit{K.-S. Saito} and \textit{M. Tominaga} [Linear Algebra Appl. 432, No. 12, 3258--3264 (2010; Zbl 1195.26044)], and \textit{L.-M. Zou} et al. [Linear Algebra Appl. 438, No. 1, 436--442 (2013; Zbl 1267.47028)].
In this paper, we study the operator version of \(p\)-angular distance \(\beta_p\) by using Douglas' lemma. We also prove that the operator version of inequality \(\alpha_ p \leq \beta_p\) holds for normal and double commute operators. Some examples are presented to show essentiality of these conditions.An extension of the Maurey factorization theoremhttps://zbmath.org/1503.470252023-03-23T18:28:47.107421Z"Popa, Dumitru"https://zbmath.org/authors/?q=ai:popa.dumitruSummary: Let \(1\leq p, q, r<\infty\) be such that \(1/p =(1/q) + (1/r)\), \(X, Y\) Banach spaces, \((\Omega, \Sigma,\mu)\) a~measure space, and \(U:X\to L_p (\mu,Y)\) a bounded linear operator. The Maurey factorization theorem gives the necessary and sufficient condition that \(U\) can be factored under the form \(U=M_g \circ V\), where \(V:X\to L_q (\mu,Y)\) is bounded linear and \(g\in L_r(\mu)\). In this paper, we define two new classes of bounded linear operators \(U:X\to L_p (\mu,Y)\): the class of all operators which satisfies the \((q,p,S)\)-Maurey factorization, i.e., there exists \(g\in L_r (\mu)\) and a \((q,S)\)-summing operator \(V:X\to L_q(\mu,Y)\) such that \(U=M_g\circ V\) and the class of \((q,p,S)\)-Maurey operators. Our main result, which extends the Maurey factorization theorem, asserts that these two classes coincide. Under the assumptions that \(Y\) has cotype 2 we prove the equality between the class of operators with values in \(L_p (\mu,Y)\) which satisfies the \((2,p,S)\)-Maurey factorization, the class of \((2,p,S)\)-Maurey operators, the class of \(S\)-almost summing operators, and the class of \((2,S)\)-summing operators. Applications are given.Representation and approximation of the polar factor of an operator on a Hilbert spacehttps://zbmath.org/1503.470262023-03-23T18:28:47.107421Z"Mbekhta, Mostafa"https://zbmath.org/authors/?q=ai:mbekhta.mostafaSummary: Let \(H\) be a complex Hilbert space and let \(\mathcal{B}(H)\) be the algebra of all bounded linear operators on \(H\). The polar decomposition theorem asserts that every operator \(T \in \mathcal{B}(H)\) can be written as the product \(T = V P\) of a partial isometry \(V\in \mathcal{B}(H) \) and a positive operator \(P \in \mathcal{B}(H)\) such that the kernels of \(V\) and \(P\) coincide. Then this decomposition is unique. \(V\) is called the polar factor of \(T\). Moreover, we have automatically \(P = \vert T\vert = (T^*T)^{\frac{1}{2}} \). Unlike \(P\), we have no representation formula that is required for \(V\).
In this paper, we introduce, for \(T\in \mathcal{B}(H)\), a family of functions called a ``polar function'' for \(T\), such that the polar factor of \(T\) is obtained as a limit of a net built via continuous functional calculus from this family of functions. We derive several explicit formulas representing different polar factors. These formulas allow new for methods of approximations of the polar factor of \(T\).Conditional positive definiteness in operator theoryhttps://zbmath.org/1503.470272023-03-23T18:28:47.107421Z"Jabłoński, Zenon Jan"https://zbmath.org/authors/?q=ai:jablonski.zenon-jan"Jung, Il Bong"https://zbmath.org/authors/?q=ai:jung.il-bong"Stochel, Jan"https://zbmath.org/authors/?q=ai:stochel.janSummary: In this paper, we extensively investigate the class of conditionally positive definite operators, namely, operators generating conditionally positive definite sequences. This class itself contains subnormal operators, \(2\)-and \(3\)-isometries, complete hypercontractions of order \(2\) and much more beyond them. Quite a large part of the paper is devoted to the study of conditionally positive definite sequences of exponential growth with emphasis put on finding criteria for their positive definiteness, where both notions are understood in the semigroup sense. As a consequence, we obtain semispectral and dilation type representations for conditionally positive definite operators. We also show that the class of conditionally positive definite operators is closed under the operation of taking powers. On the basis of Agler's hereditary functional calculus, we build an \(L^{\infty}(M)\)-functional calculus for operators of this class, where \(M\) is an associated semispectral measure. We provide a variety of applications of this calculus to inequalities involving polynomials and analytic functions. In addition, we derive new necessary and sufficient conditions for a conditionally positive definite operator to be a subnormal contraction (including a telescopic condition).Polar decomposition and characterization of binormal operatorshttps://zbmath.org/1503.470282023-03-23T18:28:47.107421Z"Karizaki, Mehdi Mohammadzadeh"https://zbmath.org/authors/?q=ai:karizaki.mehdi-mohammadzadehSummary: We illustrate the matrix representation of the closed range operator that enables us to determine the polar decomposition with respect to the orthogonal complemented submodules. This result proves that the reverse order law for the Moore-Penrose inverse of operators holds. Also, it is given some new characterizations of the binormal operators via the generalized Aluthge transformation. New characterizations of the binormal operators enable us to obtain equivalent conditions when the inner product of the binormal operator with its generalized Aluthge transformation is positive in the general setting of adjointable operators on Hilbert \(C^\ast\)-modules.\((m, n)\)-paranormal composition operatorshttps://zbmath.org/1503.470292023-03-23T18:28:47.107421Z"Kour, Baljinder"https://zbmath.org/authors/?q=ai:kour.baljinder"Ram, Sonu"https://zbmath.org/authors/?q=ai:ram.sonuSummary: In this paper, we prove some characterizations for the class of \((m, n)\)-paranormal operators acting on the complex Hilbert space \(\mathcal{H}\). The class of \((m, n)\)-paranormal operators is characterized in terms of the Radon-Nikodym derivative of the measure \(\lambda T^{-1}\) with respect to \(\lambda\). Moreover, we discuss the conditions under which the classes of composition operators, weighted composition operators, multiplication composition operators are \((m, n)\)-paranormal.
For the entire collection see [Zbl 1492.26003].On commutator of generalized Aluthge transformations and Fuglede-Putnam theoremhttps://zbmath.org/1503.470302023-03-23T18:28:47.107421Z"Rashid, M. H. M."https://zbmath.org/authors/?q=ai:rashid.mohammad-hussein-mohammad"Tanahashi, Kotaro"https://zbmath.org/authors/?q=ai:tanahashi.kotaroSummary: Let \(A=U|A|\) be the polar decomposition of \(A\) on a complex Hilbert space \(\mathscr {H}\) and \(0<s,t\). Then \(\widetilde{A}_{s, t}=|A|^sU|A|^t\) and \(\widetilde{A}_{s, t}^{(*)}=|A^*|^sU|A^*|^t\) are called the generalized Aluthge transformation and generalized \(\ast\)-Aluthge transformation of \(A\), respectively. A pair \((A,B)\) of operators is said to have the Fuglede-Putnam property (briefly, the FP-property) if \(AX=XB\) implies \(A^*X=XB^*\) for every operator \(X\). We prove that if \((A,B)\) has the FP-property, then \((\widetilde{A}_{s, t},\widetilde{B}_{s, t})\) and \(((\widetilde{A}_{s, t})^{*},(\widetilde{B}_{s, t})^{*})\) has the FP-property for every \(s,t>0\) with \(s+t=1\). Also, we prove that \((\widetilde{A}_{s, t},\widetilde{B}_{s, t})\) has the FP-property if and only if \(((\widetilde{A}_{s, t})^{*},(\widetilde{B}_{s, t})^{*})\) has the FP-property, where \(A, B\) are invertible and \(0 < s, t\) with \(s + t =1\). Moreover, we prove that if \(0 < s, t\) and \(\widetilde{A}_{s, t}\) is positive and invertible, then \(\| \widetilde{A}_{s, t}X-X\widetilde{A}_{s, t}\| \leq \| A\|^{2t}\| (\widetilde{A}_{s, t})^{-1}\| \| X\|\) for every operator \(X\). Also, if \( 0 <s, t\) and \(X\) is positive, then \(\| |\widetilde{A}_{s, t}|^{2r} X-X|\widetilde{A}_{s, t}|^{2r}\| \leq \frac{1}{2}\| |A|\| ^{2r}\| X\|\) for every \(r>0\).Weighted composition operators on analytic Lipschitz spaceshttps://zbmath.org/1503.470312023-03-23T18:28:47.107421Z"Amiri, S."https://zbmath.org/authors/?q=ai:amiri.sasan"Golbaharan, A."https://zbmath.org/authors/?q=ai:golbaharan.azin"Mahyar, H."https://zbmath.org/authors/?q=ai:mahyar.hakimehSummary: We study boundedness and compactness of weighted composition operators on spaces of analytic Lipschitz functions \(\mathrm{Lip}_A(X, \alpha )\) where \(X\) is a compact plane set and \(0<\alpha \leq 1\). We give necessary conditions for these operators to be compact, we also provide some sufficient conditions for the compactness of such operators. In the case of \(0<\alpha <1\), to obtain the necessary condition we consider the relationship between these spaces and Bloch type spaces \(\mathcal {B}^\alpha \). We then conclude some results about boundedness and compactness of weighted composition operators on \(\mathcal {B}^\alpha \). Finally, we determine the spectra of compact (Riesz) weighted composition operators acting on analytic Lipschitz spaces or on Bloch type spaces. Also as a consequence, we characterize power compact composition operators on these spaces.Weakly compact weighted composition operators on spaces of Lipschitz functionshttps://zbmath.org/1503.470322023-03-23T18:28:47.107421Z"Golbaharan, A."https://zbmath.org/authors/?q=ai:golbaharan.azinSummary: We prove that if \(X\) is a compact metric space and \(\operatorname{lip}(X,d)\) has the uniform separation property, then weakly compact weighted composition operators on spaces of Lipschitz functions \(\operatorname{Lip}(X,d)\) and \(\operatorname{lip}(X,d)\) are compact.Operators of composition between Orlicz spaceshttps://zbmath.org/1503.470332023-03-23T18:28:47.107421Z"Komal, B. S."https://zbmath.org/authors/?q=ai:komal.b-s"Gupta, Shally"https://zbmath.org/authors/?q=ai:gupta.shally"Kour, Tejinder"https://zbmath.org/authors/?q=ai:kour.tejinderSummary: A study of composition operators between Orlicz spaces is made in this paper. It is shown that if \(\varphi_1\) is not stronger than \(\varphi_2\), then no composition operator exists from \(L^{\varphi_1} (\mu)\) into \(L^{\varphi_2}(\mu)\).\(q\)-quasi-2-isometric composition operatorshttps://zbmath.org/1503.470342023-03-23T18:28:47.107421Z"Lal, E. Shine"https://zbmath.org/authors/?q=ai:lal.e-shine"Prasad, T."https://zbmath.org/authors/?q=ai:prasad.tribhuan|prasad.t-jayachandra|prasad.t-v-s-r-k|prasad.t-ram|prasad.t-b-aruna"Devadas, V."https://zbmath.org/authors/?q=ai:devadas.vinaySummary: In this paper we characterize \(q\)-quasi-2-isometric and \((2, q)\)-partial-isometric composition operators on \(L^2\) space.Commutators of weighted composition operators on Hardy space of the unit ballhttps://zbmath.org/1503.470352023-03-23T18:28:47.107421Z"Xu, Ning"https://zbmath.org/authors/?q=ai:xu.ning"Zhou, Zehua"https://zbmath.org/authors/?q=ai:zhou.zehua"Ding, Ying"https://zbmath.org/authors/?q=ai:ding.yingSummary: In this paper, we study commutators of weighted composition operators with linear fractional non-automorphisms on Hardy space of the unit ball. First, we obtain the formula of commutators of weighted composition operators. Then, we characterize compactness of commutators according to two special situations of linear fractional maps. Finally, we obtain that commutators are compact when linear fractional maps are parabolic and commutators are not compact when linear fractional maps are hyperbolic.Unbounded complex symmetric Toeplitz operatorshttps://zbmath.org/1503.470362023-03-23T18:28:47.107421Z"Han, Kaikai"https://zbmath.org/authors/?q=ai:han.kaikai"Wang, Maofa"https://zbmath.org/authors/?q=ai:wang.maofa"Wu, Qi"https://zbmath.org/authors/?q=ai:wu.qiSummary: In this paper, we study unbounded complex symmetric Toeplitz operators on the Hardy space \(H^2(\mathbb{D})\) and the Fock space \(\mathscr{F}^2\). The technique used to investigate the complex symmetry of unbounded Toeplitz operators is different from that used to investigate the complex symmetry of bounded Toeplitz operators.Schatten class Hankel operators on the Segal-Bargmann space and the Berger-Coburn phenomenonhttps://zbmath.org/1503.470372023-03-23T18:28:47.107421Z"Hu, Zhangjian"https://zbmath.org/authors/?q=ai:hu.zhangjian"Virtanen, Jani A."https://zbmath.org/authors/?q=ai:virtanen.jani-aLet \(F^{2}\) be the Segal-Bargmann space of Gaussian square-integrable entire functions on \(\mathbb{C}^{n}\). The authors consider \(F^{2}(\varphi ),\) the weighted Segal-Bargmann spaces (also referred to as Fock spaces and Bargmann-Fock spaces). By using the notion of integral distance to analytic functions in \(\mathbb{C}^{n}\) and Hörmander's \(\bar{\partial}\)-theory, a complete characterization of Schatten class Hankel operators \(H_{f}\) acting on \(F^{2}(\varphi )\) is given. Among other results, the authors prove that, for all \(f\in L^{p}\) (\(1<p\leq \infty \)), \(H_{f}\) is in the Schatten class \( \mathcal{S}_{p}\) if and only if \(H_{\bar{f}}\in \mathcal{S}_{p}\).
Reviewer: Elhadj Dahia (Bou Saâda)Matrix valued truncated Toeplitz operators: basic propertieshttps://zbmath.org/1503.470382023-03-23T18:28:47.107421Z"Khan, Rewayat"https://zbmath.org/authors/?q=ai:khan.rewayat"Timotin, Dan"https://zbmath.org/authors/?q=ai:timotin.danSummary: Matrix valued truncated Toeplitz operators act on vector-valued model spaces. They represent a generalization of block Toeplitz matrices. A characterization of these operators analogue to the scalar case is obtained, as well as the determination of the symbols that produce the zero operator.Compact Hankel operators on compact abelian groupshttps://zbmath.org/1503.470392023-03-23T18:28:47.107421Z"Mirotin, A. R."https://zbmath.org/authors/?q=ai:mirotin.adolf-ruvimovich|mirotin.adolf-rThe author analyses Hankel operators on compact abelian groups (with linearly ordered group of characters) and some of their classical related properties, avoiding successfully to use some known conditions of previous works. Thus, within this framework, results such as Beurling's theorem on invariant subspaces, the existence of finite Blaschke products, and the Kronecker, Hartman, Peller, and Adamyan-Arov-Krein theorems are exhibited.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)On Toeplitz operators with biharmonic symbolshttps://zbmath.org/1503.470402023-03-23T18:28:47.107421Z"Yousef, A."https://zbmath.org/authors/?q=ai:yousef.abdulaziz|yousef.ahmed-ali|yousef.abdeljabbar-talal|yousef.abdel|yousef.abdelrahman|yousef.ahmed-m|yousef.ali-a|yousef.a-r|yousef.ali-s"Al-Naimi, R."https://zbmath.org/authors/?q=ai:al-naimi.rSummary: In this article, we study the commuting problem of Toeplitz operators acting on the Bergman space of the unit disk when one of the symbols is biharmonic. Particularly, we prove that if a Toeplitz operator with right-terminating bounded symbol commutes with a biharmonic symbol, of special form, then it must be a linear combination of the latter one and the constant function.Sharp phase transitions for the almost Mathieu operatorhttps://zbmath.org/1503.470412023-03-23T18:28:47.107421Z"Avila, Artur"https://zbmath.org/authors/?q=ai:avila.artur"You, Jiangong"https://zbmath.org/authors/?q=ai:you.jiangong"Zhou, Qi"https://zbmath.org/authors/?q=ai:zhou.qiSummary: It is known that the spectral type of the almost Mathieu operator (AMO) depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya's conjecture [\textit{S. Ya. Jitomirskaya}, in: Proceedings of the XIth international congress on mathematical physics, Paris, France, July 18--23, 1994. Cambridge, MA: International Press. 373--382 (1995; Zbl 1052.82539); Proc. Symp. Pure Math. 76, 613--647 (2007; Zbl 1129.82018)]. Together with [\textit{A. Avila}, ``The absolutely continuous spectrum of the almost Mathieu operator'', Preprint (2008), \url{arXiv:0810.2965}], this gives the sharp description of phase transitions for the AMO for the a.e.\ phase.Subordinacy theory for extended CMV matriceshttps://zbmath.org/1503.470422023-03-23T18:28:47.107421Z"Guo, Shuzheng"https://zbmath.org/authors/?q=ai:guo.shuzheng"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Ong, Darren C."https://zbmath.org/authors/?q=ai:ong.darren-cSummary: We develop subordinacy theory for extended Cantero-Moral-Velázquez (CMV) matrices, i.e., we provide explicit supports for the singular and absolutely continuous parts of the canonical spectral measure associated with a given extended CMV matrix in terms of the presence or absence of subordinate solutions to the generalized eigenvalue equation. Some corollaries and applications of this result are described as well.Norm of the Hilbert matrix operator on the Korenblum spacehttps://zbmath.org/1503.470432023-03-23T18:28:47.107421Z"Dai, Jineng"https://zbmath.org/authors/?q=ai:dai.jinengSummary: In this paper, the norm representation of the Hilbert matrix operator \(\mathcal{H}\) on the Korenblum space \(H_\alpha^\infty\) is given for \(0 < \alpha < 1\). In particular, we find an exact value \(\alpha_0\) in \((0, 1)\) such that the norm of \(\mathcal{H}\) is equal to \(\frac{\pi}{\sin (\pi \alpha)}\) for \(0 < \alpha \leq \alpha_0\), and the norm is greater than \(\frac{\pi}{\sin (\pi \alpha)}\) for \(\alpha_0 < \alpha < 1\).The quadratic hyponormality of one-step extension of the Bergman-type shifthttps://zbmath.org/1503.470442023-03-23T18:28:47.107421Z"Li, Chunji"https://zbmath.org/authors/?q=ai:li.chunji"Qi, Wentao"https://zbmath.org/authors/?q=ai:qi.wentaoSummary: Let \(p > 1\) and \(\alpha^{[p]}(x) : \sqrt{x}\), \(\sqrt{\frac{p}{2p-1}}\), \(\sqrt{\frac{2p-1}{3p-2}}, \cdots\), with \(0 < x \leq \frac{p}{2p-1}\). In [\textit{C.-J. Li} et al., Bull. Korean Math. Soc. 57, No. 1, 81--93 (2020; Zbl 07232972)], the authors considered the subnormality, \(n\)-hyponormality and positive quadratic hyponormality of \(W_{\alpha [p]_{(x)}}\). By continuing to study, in this paper, we give a sufficient condition of quadratic hyponormality of \(W_{\alpha [p]_{(x)}}\). Finally, we give an example to characterize the gaps of \(W_{\alpha [p]_{(x)}}\) distinctively.Property of reflexivity for multiplication operators on Banach function spaceshttps://zbmath.org/1503.470452023-03-23T18:28:47.107421Z"Yousefi, Bahmann"https://zbmath.org/authors/?q=ai:yousefi.bahmann"Zangeneh, Fatemeh"https://zbmath.org/authors/?q=ai:zangeneh.fatemehSummary: In this paper, we give conditions under which the powers of the multiplication operator \(M_{z}\) are reflexive on a Banach space of functions analytic on a plane domain.Some results for the fractional integral operator defined on the Sobolev spaceshttps://zbmath.org/1503.470462023-03-23T18:28:47.107421Z"Gurdal, Mehmet"https://zbmath.org/authors/?q=ai:gurdal.mehmet"Nabiev, Anar Adiloglu"https://zbmath.org/authors/?q=ai:nabiev.anar-adiloglu"Ayyildiz, Meral"https://zbmath.org/authors/?q=ai:ayyildiz.meralSummary: We investigate the invariant subspaces of the fractional integral operator in the Sobolev space \(W^k_p [0,1]\) and prove unicellularity of the operator \(J^\alpha\) by using the Duhamel product.Generalized Cesàro operators on Dirichlet-type spaceshttps://zbmath.org/1503.470472023-03-23T18:28:47.107421Z"Jin, Jianjun"https://zbmath.org/authors/?q=ai:jin.jianjun"Tang, Shuan"https://zbmath.org/authors/?q=ai:tang.shuanSummary: In this note, we introduce and study a new kind of generalized Cesàro operator, \(\mathcal{C}_\mu\), induced by a positive Borel measure \(\mu\) on \([0, 1)\) between Dirichlet-type spaces. We characterize the measures \(\mu\) for which \(\mathcal{C}_\mu\) is bounded (compact) from one Dirichlet-type space, \(\mathcal{D}_\alpha\), into another one, \(\mathcal{D}_\beta\).Extrapolation of compactness on weighted Morrey spaceshttps://zbmath.org/1503.470482023-03-23T18:28:47.107421Z"Lappas, Stefanos"https://zbmath.org/authors/?q=ai:lappas.stefanosAuthors' summary: In a previous work, ``compact versions'' of Rubio de Francia's weighted extrapolation theorem were proved, which allow one to extrapolate the compactness of a linear operator from just one space to the full range of weighted Lebesgue spaces where this operator is bounded [\textit{T. Hytönen} and \textit{S. Lappas}, ``Extrapolation of compactness on weighted spaces'', Preprint (2020), \url{arXiv:2003.01606} and Rev. Mat. Iberoam. (2021; \url{doi:10.4171/rmi/1325})]. In this paper, we extend these results to the setting of weighted Morrey spaces. As applications, we easily obtain new results on the weighted compactness of commutators of Calderón-Zygmund singular integrals, rough singular integrals and Bochner-Riesz multipliers.
Reviewer: Ferit Gürbüz (Hakkari)Bochner representable operators on Banach function spaceshttps://zbmath.org/1503.470492023-03-23T18:28:47.107421Z"Nowak, Marian"https://zbmath.org/authors/?q=ai:nowak.marianSummary: Let \((E,\| \cdot \|_E)\) be a Banach function space, \(E'\) the Köthe dual of \(E\) and \((X,\| \cdot \|_X)\) be a Banach space. It is shown that every Bochner representable operator \(T:E\rightarrow X\) maps relatively \(\sigma (E,E')\)-compact sets in \(E\) onto relatively norm compact sets in \(X\). If, in particular, the associated norm \(\| \cdot \|_{E'}\) on \(E'\) is order continuous, then every Bochner representable operator \(T:E\rightarrow X\) is \((\gamma_E,\| \cdot \|_X)\)-compact, where \(\gamma_E\) stands for the natural mixed topology on \(E\). Applications to Bochner representable operators on Orlicz spaces are given.Direct sums of \(S\)-decomposable and \(S\)-spectral operator systemshttps://zbmath.org/1503.470502023-03-23T18:28:47.107421Z"Şerbănescu, Cristina"https://zbmath.org/authors/?q=ai:serbanescu.cristina"Zamfir, Mariana"https://zbmath.org/authors/?q=ai:zamfir.marianaSummary: This paper is trying to extend and generalize several results of the spectral theory for a single $S$-decomposable ($S$-spectral) operator to $S$-decomposable ($S$-spectral) operator systems. The goal of the work is to establish the behaviour of $S$-decomposable ($S$-spectral) systems related to direct sums, by showing that the direct sum of two systems is an $S$-decomposable ($S$-spectral) system if and only if both systems are $S$-decomposable ($S$-spectral). These spectral decompositions are related to differential equations and to systems of differential equations [\textit{B. Nagy}, in: Functions, series, operators, Proc. int. Conf., Budapest 1980, Vol. II, Colloq. Math. Soc. János Bolyai 35, 891--917 (1983; Zbl 0543.47024)] and can have applications in quantum mechanics and fractal theory.Spectral resolutions for non-self-adjoint block convolution operatorshttps://zbmath.org/1503.470512023-03-23T18:28:47.107421Z"Zalot, Ewelina"https://zbmath.org/authors/?q=ai:zalot.ewelinaThe author gives a spectral decomposition of block non-self-adjoint convolution operators with respect to a chain in a Banach space. Moreover, an alternative method of constructing an invariant chain under the operator \(A\) acting in a Hilbert space is presented. Also, certain applications of the main results to Jacobi-type matrices are given.
Reviewer: Ömer Gök (İstanbul)Strong accretive property of fractional differentiation operator in the Kipriyanov sensehttps://zbmath.org/1503.470522023-03-23T18:28:47.107421Z"Kukushkin, Maksim"https://zbmath.org/authors/?q=ai:kukushkin.maksim-vladimirovichSummary: In this paper we will prove theorem establishes the strong accretive property for the operator of fractional differentiation in the Kipriyanov sense.A new proof of Singer-Wermer theorem with some results on \(\{g, h\}\)-derivationshttps://zbmath.org/1503.470532023-03-23T18:28:47.107421Z"Hosseini, Amin"https://zbmath.org/authors/?q=ai:hosseini.aminSummary: \textit{I. M. Singer} and \textit{J. Wermer} [Math. Ann. 129, 260--264 (1955; Zbl 0067.35101)] proved that if \(\mathcal{A}\) is a commutative Banach algebra and \(d: \mathcal{A} \rightarrow \mathcal{A}\) is a continuous derivation, then \(d(\mathcal{A}) \subseteq rad(\mathcal{A})\), where \(rad(\mathcal{A})\) denotes the Jacobson radical of \(\mathcal{A}\). In this paper, we establish a new proof of that theorem. Moreover, we prove that every continuous Jordan derivation on a finite dimensional Banach algebra, under certain conditions, is identically zero. As another objective of this article, we study \(\{g, h\}\)-derivations on algebras. In this regard, we prove that if \(f\) is a \(\{g, h\}\)-derivation on a unital algebra, then \(f, g\) and \(h\) are generalized derivations. Additionally, we achieve some results concerning the automatic continuity of \(\{g, h\}\)-derivations on Banach algebras. In the last section of the article, we introduce the concept of a \(\{g, h\}\)-homomorphism and then we present a characterization of it under certain conditions.Bijections on the unit ball of \(B(H)\) preserving \({}^*\)-Jordan triple producthttps://zbmath.org/1503.470542023-03-23T18:28:47.107421Z"Hejazian, Shirin"https://zbmath.org/authors/?q=ai:hejazian.shirin"Safarizadeh, Mozhdeh"https://zbmath.org/authors/?q=ai:safarizadeh.mozhdehSummary: Let \(\mathcal{B}_1\) denote the closed unit ball of \(\mathcal{B}(H)\), the von Neumann algebra of all bounded linear operators on a complex Hilbert space \(H\) with \(\dim H\geq 2\). Suppose that \(\phi\) is a bijection on \(\mathcal{B}_1\) (with no linearity assumption) satisfying
\[
\phi (AB^*A)=\phi (A)\phi (B)^*\phi (A), \quad A, B\in \mathcal{B}_1.
\]
If \(I\) and \(\mathbb{T}\) denote the identity operator on \(H\) and the unit circle in \(\mathbb{C}\), respectively, and if \(\phi\) is continuous on \(\{\lambda I: \lambda\in \mathbb{T}\}\), then we show that \(\phi(I)\) is a unitary operator and \(\phi (I)\phi\) extends to a linear or conjugate linear Jordan \({}^*\)-automorphism on \(\mathcal{B}(H)\). As a consequence, there is either a unitary or an antiunitary operator \(U\) on \(H\) such that \(\phi (A)=\phi (I) UAU^*\), \(A\in\mathcal{B}_1\), or \(\phi (A)=\phi (I) UA^*U^*\), \(A\in\mathcal{B}_1\).Ascent, descent and additive preserving problemshttps://zbmath.org/1503.470552023-03-23T18:28:47.107421Z"Oudghiri, Mourad"https://zbmath.org/authors/?q=ai:oudghiri.mourad"Souilah, Khalid"https://zbmath.org/authors/?q=ai:souilah.khalidSummary: Given an integer \(n\ge 1\), we provide a complete description of all additive surjective maps, on the algebra of all bounded linear operators acting on a complex separable infinite-dimensional Hilbert space, preserving in both directions the set of all bounded linear operators with ascent (resp.\ descent) nongreater than \(n\). In the context of Banach spaces, we consider the additive preserving problem for semi-Fredholm operators with ascent or descent non-greater than \(n\).Nonlinear maps preserving the second mixed Lie triple products on factor von Neumann algebrashttps://zbmath.org/1503.470562023-03-23T18:28:47.107421Z"Yang, Zhujun"https://zbmath.org/authors/?q=ai:yang.zhujun"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhuaSummary: Let \(\mathcal{M}\) and \(\mathcal{N}\) be factor von Neumann algebras with \(\dim\mathcal{M}>4\) and \(\dim\mathcal{N} >1\), and let \(\Phi :\mathcal{M}\to\mathcal{N}\) be a bijective map preserving the second mixed Lie triple products \([[A,B], C]_*\). Then there exists \(\varepsilon \in \{1,-1\}\) such that \(\Phi =\varepsilon\Psi\), where \(\Psi :\mathcal{M}\to\mathcal{N}\) is a linear \(*\)-isomorphism or a conjugate linear \(*\)-isomorphism. Also, we give the structure of this map when \(\dim\mathcal{M}=4\).A generalization of order convergence in the vector latticeshttps://zbmath.org/1503.470572023-03-23T18:28:47.107421Z"Azar, Kazem Haghnejad"https://zbmath.org/authors/?q=ai:azar.kazem-haghnejadSummary: Let \(E\) be a sublattice of a vector lattice \(F\). \(\left( x_\alpha \right)\subseteq E\) is said to be order convergent to a vector \(x \) (in symbols, \(x_\alpha \xrightarrow{Fo} x)\), whenever there exists another net \(\left(y_\alpha\right)\) in \(F\) with the some index set satisfying \(y_\alpha\downarrow 0\) in \(F\) and \(\vert x_\alpha - x \vert \leq y_\alpha\) for all indexes \(\alpha \). If \(F=E^{\sim\sim} \), this convergence is called \(b\)-order convergence and we write \(x_\alpha \xrightarrow{bo} x\). In this paper, first we study some properties of order convergence nets and we extend the same results to the general case. In the second part, we introduce \(b\)-order continuous operators and we investigate some properties of this new concept. An operator \(T\) between two vector lattices \(E\) and \(F\) is said to be \(b\)-order continuous if \(x_\alpha \xrightarrow{bo} 0\) in \(E\) implies \(Tx_\alpha \xrightarrow{bo} 0\) in \(F\).On two-weight inequalities for Hausdorff operators of special kind in Lebesgue spaceshttps://zbmath.org/1503.470582023-03-23T18:28:47.107421Z"Bandaliyev, Rovshan"https://zbmath.org/authors/?q=ai:bandaliev.rovshan-alifaga-ogly"Safarova, Kamala"https://zbmath.org/authors/?q=ai:safarova.kamala-hSummary: In this paper, we establish necessary and sufficient conditions on monotone weight functions for the boundedness for Hausdorff operators of special kind in weighted Lebesgue spaces. In particular, we get similar results for important operators of harmonic analysis which are special cases of the Hausdorff operators. The weights are illustrated by examples at the end of the paper.Spectral transitions for the square Fibonacci Hamiltonianhttps://zbmath.org/1503.470592023-03-23T18:28:47.107421Z"Damanik, David"https://zbmath.org/authors/?q=ai:damanik.david"Gorodetski, Anton"https://zbmath.org/authors/?q=ai:gorodetskii.anton-semenovichSummary: We study the spectrum and the density of states measure of the square Fibonacci Hamiltonian. We describe where the transitions from positive-measure to zero-measure spectrum and from absolutely continuous to singular density of states measure occur. This shows in particular that for almost every parameter from some open set, a positive-measure spectrum and a singular density of states measure coexist. This provides the first physically relevant example exhibiting this phenomenon.Contraction quasi semigroups and their applications in decomposing Hilbert spaceshttps://zbmath.org/1503.470602023-03-23T18:28:47.107421Z"Sutrima, S."https://zbmath.org/authors/?q=ai:sutrima.sutrima"Indrati, C. R."https://zbmath.org/authors/?q=ai:indrati.christiana-rini"Aryati, L."https://zbmath.org/authors/?q=ai:aryati.linaSummary: This paper addresses the problem of implementations of a strongly continuous quasi semigroup in analyzing non-autonomous Cauchy problems induced by dissipative operators. The implementations are closely related to contraction quasi semigroups. Lumer-Phillips theorem for the contraction quasi semigroups is proved. Relationships between the contraction quasi semigroups and their cogenerator are also explored. Furthermore, we show that the contraction quasi semigroups are applicable in decomposing Hilbert spaces.On extensions of singular fourth order dynamic operators on time scaleshttps://zbmath.org/1503.470612023-03-23T18:28:47.107421Z"Tuna, Hüseyin"https://zbmath.org/authors/?q=ai:tuna.huseyin"Bayrak, Songül"https://zbmath.org/authors/?q=ai:bayrak.songulSummary: In this work, we consider a singular fourth order dynamic operator on time scales. We construct a space of boundary values. Later, we give a description of all maximal dissipative, self-adjoint and other extensions of singular fourth order differential operators on unbounded time scales.Study of new class of \(q\)-fractional integral operatorhttps://zbmath.org/1503.470622023-03-23T18:28:47.107421Z"Momenzadeh, M."https://zbmath.org/authors/?q=ai:momenzadeh.mariam|momenzadeh.mohammad"Mahmudov, N. I."https://zbmath.org/authors/?q=ai:mahmudov.nazim-idrisogluSummary: In this paper, we study on the new class of \(q\)-fractional integral operators. In the aid of iterated Cauchy integral approach to fractional integral operator, we applied \(t^p f(t)\) in these integrals and a new class of \(q\)-fractional integral operator with parameter \(p\) is introduced. Recently, the \(q\)-analogue of fractional differential integral operator has been studied and all of the operators defined in these studies are \(q\)-analogues of the Riemann fractional differential operator. We show that our new class of operators generalizes all the operators in use, and additionally, it can cover the \(q\)-analogue of Hadamard fractional differential operators as well. Some properties of this operator are investigated.Hausdorff operators associated with the Opdam-Cherednik transform in Lebesgue spaceshttps://zbmath.org/1503.470632023-03-23T18:28:47.107421Z"Mondal, Shyam Swarup"https://zbmath.org/authors/?q=ai:mondal.shyam-swarup"Poria, Anirudha"https://zbmath.org/authors/?q=ai:poria.anirudhaSummary: In this paper, we introduce the Hausdorff operator associated with the Opdam-Cherednik transform and study the boundedness of this operator in various Lebesgue spaces. In particular, we prove the boundedness of the Hausdorff operator in Lebesgue spaces, in grand Lebesgue spaces, and in quasi-Banach spaces that are associated with the Opdam-Cherednik transform. Also, we give necessary and sufficient conditions for the boundedness of the Hausdorff operator in these spaces.On some applications of Duhamel operatorshttps://zbmath.org/1503.470642023-03-23T18:28:47.107421Z"Tapdigoglu, Ramiz"https://zbmath.org/authors/?q=ai:tapdigoglu.ramiz-g"Altwaijry, Najla"https://zbmath.org/authors/?q=ai:altwaijry.najla-aSummary: Let \(\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}\) be the unit disk and \(\operatorname{Hol}(\mathbb{D}\times \mathbb{D})\) be the space of all holomorphic functions on the bi-disc \(\mathbb{D}\times \mathbb{D}\). We consider the double convolution operator \(\mathcal{K}_f\) on the subspace Hol\(_{zw}(\mathbb{D}\times\mathbb{D}):=\{f\in\operatorname{Hol}(\mathbb{D}\times\mathbb{D}):f(z,w)=g(zw)\text{ for some }g\in\operatorname{Hol}(\mathbb{D})\}\) defined by
\[
(\mathcal{K}_fh)(zw)=(f*h)(zw):=\int\limits_0^z\int\limits_0^wf((z-u)(w-v))h(uv)\,\mathrm{d}v\,\mathrm{d}u.
\]
We study extended eigenvalues of \(\mathcal{K}_f\). We characterize extended eigenvectors of \(\mathcal{K}_f\) in terms of Duhamel operators. Moreover, we describe cyclic vectors of operator \(\mathcal{K}_f\) by applying the Duhamel product method.Dixmier traces for discrete pseudo-differential operatorshttps://zbmath.org/1503.470652023-03-23T18:28:47.107421Z"Cardona, Duván"https://zbmath.org/authors/?q=ai:cardona.duvan"del Corral, César"https://zbmath.org/authors/?q=ai:del-corral.cesar"Kumar, Vishvesh"https://zbmath.org/authors/?q=ai:kumar.vishveshThe authors look for sharp conditions for the Dixmier traceability of discrete pseudo-differential operators on \(\ell^2(\mathbb{Z}^n)\). In this setting, the pseudo-differential operators on \(\mathbb{Z}^n\) are defined by \[ t_m f(n'):=\int\limits_{\mathbb{T}^n} e^{i2\pi n'\cdot\xi}m(n',\xi)(\mathscr{F}f)(\xi)\,d\xi \] (with \(f\in \mathscr{S}(\mathbb{Z}^n)\) and \(n'\in\mathbb{Z}^n\)). Under suitable conditions on the symbol \(m\), \(t_m\) is a continuous linear operator from the discrete Schwartz space \(\mathscr{S}(\mathbb{Z}^n)\) into itself. Assuming \(t_m\) in the class of periodic pseudo-differential operators \(\Psi^{-n}_{cl}(\mathbb{Z}^n)\), and that \(t_m\) is a positive discrete pseudo-differential operator, the main conclusion of this work is that its Dixmier trace is given by \[ \operatorname{Tr}_\omega(t_m)= \frac{1}{n(2\pi)^n} \int\limits_{\mathbb{T}^n}\int\limits_{\mathbb{S}^{n-1}}[i^*(\tau)(x,\xi)]_{(-n)}\,d\Sigma(\xi) \,d_{\mathrm{Vol}}x, \] where \(\tau(x,k):=\overline{m(-k,x)}\), \(d\Sigma(\xi)\) is the surface measure associated to the sphere \(\mathbb{S}^{n-1}\subset \mathbb{R}^n,\) and \(d_{\mathrm{Vol}}x\) is the volume form on \(M=[0,1)^n\).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Characterizations, adjoints and products nuclear pseudo-differential operators on compact and Hausdorff groupshttps://zbmath.org/1503.470662023-03-23T18:28:47.107421Z"Ghaemi, M. B."https://zbmath.org/authors/?q=ai:ghaemi.mohammad-bagher"Jamalpourbirgani, M."https://zbmath.org/authors/?q=ai:jamalpourbirgani.majid"Wong, M. W."https://zbmath.org/authors/?q=ai:wong.man-wahSummary: Characterizations of nuclear pseudo-differential operators from \(L^{p_1}\) into \(L^{p_2}\) on compact and Hausdorff groups are given for \(1\le p_1,p_2<\infty\). Explicit formulas for the adjoints from \(L^{p'_2}\) into \(L^{p'_1}\) and products of nuclear pseudo-differential operators from \(L^p\) into \(L^p\), \(1\le p<\infty\), on compact and Hausdorff groups are given.On the boundedness of pseudodifferential operators defined by amplitudes in generalized weighted grand Lebesgue spaceshttps://zbmath.org/1503.470672023-03-23T18:28:47.107421Z"Gordadze, Eteri"https://zbmath.org/authors/?q=ai:gordadze.eteri"Kokilashvili, Vakhtang"https://zbmath.org/authors/?q=ai:kokilashvili.vakhtang-mIn this survey article, the authors present several results concerning pseudo-differential operators whose symbols are of limited regularity in the spatial variable. In particular, they give several sufficient conditions on the symbols such that the operators are continuous on the grand Lebesgue spaces.
Reviewer: Bojan Prangoski (Skopje)Stochastic relaxed inertial forward-backward-forward splitting for monotone inclusions in Hilbert spaceshttps://zbmath.org/1503.470682023-03-23T18:28:47.107421Z"Cui, Shisheng"https://zbmath.org/authors/?q=ai:cui.shisheng"Shanbhag, Uday"https://zbmath.org/authors/?q=ai:shanbhag.uday-v"Staudigl, Mathias"https://zbmath.org/authors/?q=ai:staudigl.mathias"Vuong, Phan"https://zbmath.org/authors/?q=ai:vuong.phan-tuThe problem of monotone inclusions of an operator, given by the sum of a maximal monotone operator and a single-valued monotone, Lipschitz continuous, and expectation-valued operator, defined on a Hilbert space. A stochastic extension of the relaxed inertial forward-backward-forward method is considered. Employing an online variance reduction strategy via a mini-batch approach, it is shown that the proposed method produces a sequence that weakly converges to the solution set. The rate at which the discrete velocity of the stochastic process vanishes, is estimated. Under strong monotonicity, strong convergence, a comprehensive assessment of the iteration and oracle complexity of the scheme, are demonstrated. It is shown, for instance, that when the mini-batch is raised at a geometric rate, the rate statement can be strengthened to a linear rate while the oracle complexity of computing an \(\epsilon\)-solution improves to \(\mathcal{O}(1/\epsilon)\). This allows for possibly biased oracles, which in turn, permits a far broader applicability. Defining a restricted gap function based on the Fitzpatrick function, it is shown that the expected gap of an averaged sequence diminishes at a sublinear rate of \(\mathcal{O}(1/k)\) while the oracle complexity of computing a suitably defined \(\epsilon\)-solution is \(\mathcal{O}(1/{\epsilon}^{1+a})\) for \(a > 1.\) Numerical results on two-stage games and an overlapping group Lasso problem are presented as illustrations.
Reviewer: K. C. Sivakumar (Chennai)On the Rockafellar function associated to a non-cyclically monotone mappinghttps://zbmath.org/1503.470692023-03-23T18:28:47.107421Z"Precupanu, Teodor"https://zbmath.org/authors/?q=ai:precupanu.teodorSummary: In an earlier paper [\textit{T. Precupanu}, J. Convex Anal. 24, No. 1, 319--331 (2017; Zbl 06704311)], we have given a definition of the Rockafellar integration function associated to a cyclically monotone mapping considering only systems of distinct elements in its domain. Thus, this function can be proper for certain non-cyclically monotone mappings. In this paper, we establish general properties of Rockafellar function if the graph of mapping does not contain finite set of accumulation elements where the mapping is not cyclically monotone. Also, some dual properties are given.Characterization of approximate monotone operatorshttps://zbmath.org/1503.470702023-03-23T18:28:47.107421Z"Rezaie, Mahboubeh"https://zbmath.org/authors/?q=ai:rezaie.mahboubeh"Mirsaney, Zahra Sadat"https://zbmath.org/authors/?q=ai:mirsaney.zahra-sadatThe authors study approximate monotone operators. They show that a well-known property of monotone operators, namely, representing by convex functions, remains valid for this larger class of operators. In this general framework, results of \textit{S. Fitzpatrick} [in: Functional analysis and optimization, Workshop/Miniconf., Canberra/Australia Proc. Cent. Math. Anal. Aust. Natl. Univ. 20, 59--65 (1988; Zbl 0669.47029)] and by \textit{J.-E. Martinez-Legaz} and \textit{M. Théra} [J. Nonlinear Convex Anal. 2, No. 2, 243--247 (2001; Zbl 0999.47037)] are proved. In particular, it is shown that the set of maximal \(\epsilon\)-monotone operators between a normed linear space \(X\) and its continuous dual \(X^*\) can be identified as some subset of convex functions on \(X \times X^*\).
Reviewer: K. C. Sivakumar (Chennai)Coincidence points for non-self mapshttps://zbmath.org/1503.470712023-03-23T18:28:47.107421Z"Latif, Abdul"https://zbmath.org/authors/?q=ai:latif.abdul"Al-Mezel, Saleh A."https://zbmath.org/authors/?q=ai:al-mezel.saleh-abdullah-rasheedSummary: In this paper we obtain some results on the existence of coincidence points for non-self \(f\)-contraction and \(f\)-nonexpensive multivalued maps satisfying weaker form of the weakly inward condition. These results unify and extend the corresponding results of a number of authors.On the fixed point property for nonexpansive mappings in hyperbolic geodesic spaceshttps://zbmath.org/1503.470722023-03-23T18:28:47.107421Z"Piatek, Bozena"https://zbmath.org/authors/?q=ai:piatek.bozenaSummary: Motivated by the well-known results concerning the complex Hilbert ball with the hyperbolic metric and metric trees, we give a characterization of the convex and closed subsets of Busemann and hyperbolic geodesic spaces in terms of the fixed point property for nonexpansive and firmly nonexpansive mappings. Furthermore, motivated by Goebel and Reich's results concerning the complex Hilbert ball with the hyperbolic metric, we describe how the fixed point free mappings behave in a much more general class of spaces.DeMarr's fixed point theorem in modular vector spaceshttps://zbmath.org/1503.470732023-03-23T18:28:47.107421Z"Abdou, Afrah A. N."https://zbmath.org/authors/?q=ai:abdou.afrah-ahmad-noan"Khamsi, Mohamed A."https://zbmath.org/authors/?q=ai:khamsi.mohamed-amineSummary: In this work, we discuss the existence of common fixed points of a family of mappings defined in modular vector spaces. This is the extension of the original DeMarr's common fixed point theorem [\textit{R. DeMarr}, Pac. J. Math. 13, 1139--1141 (1963; Zbl 0191.14901)] to modular vector spaces. In order to do this, we prove that the fixed point set of one map is a one-local retract. This property is crucial to prove the main common fixed point theorem of this work.Fixed point theory for sums of operatorshttps://zbmath.org/1503.470742023-03-23T18:28:47.107421Z"Djebali, Smaïl"https://zbmath.org/authors/?q=ai:djebali.smail"Mebarki, Karima"https://zbmath.org/authors/?q=ai:mebarki.karimaSummary: After a brief survey of the fixed point theory for sums of operators and related operator theory, we present recent results for sums of expansive mappings and \(k\)-set contractions developed from a fixed point index. Some new Kranosel'skii fixed point theorems are then derived.Fixed point theorems in Fréchet algebras and Fréchet spaces and applications to nonlinear integral equationshttps://zbmath.org/1503.470752023-03-23T18:28:47.107421Z"Dudek, Szymon"https://zbmath.org/authors/?q=ai:dudek.szymonSummary: In this paper, we present two new fixed point theorems in Fréchet algebras and Fréchet spaces. Our fixed point results are expressed with the help of family of measures of noncompactness and generalizes Darbo theorem. As an application, we establish some existence results for various types of nonlinear integral equations.Existence results for nonlinear boundary value problemshttps://zbmath.org/1503.470762023-03-23T18:28:47.107421Z"Ghanmi, Abdeljabbar"https://zbmath.org/authors/?q=ai:ghanmii.abdeljabbar"Horrigue, Samah"https://zbmath.org/authors/?q=ai:horrigue.samahSummary: In the present paper, we are concerned to prove under some hypothesis the existence of fixed points of the operator \(L\) defined on \(C(I)\) by
\[
Lu(t)=\int^w_0 G(t,s)h(s)f(u(s))\,ds,\ t \in I,\ w\in \{1,\infty\},
\]
where the functions \(f \in C([0,\infty);[0,\infty))\), \(h\in C(I;[0,\infty))\), \(G\in C(I \times I)\), and
\[
\begin{cases} &I=[0,1], \text{ if } w=1, \\
&I=[0,\infty), \text{ if } w=\infty. \end{cases}
\]
By using Guo-Krasnoselskii fixed point theorem [\textit{D. Guo} and \textit{V. Lakshmikantham}, Nonlinear problems in abstract cones. Boston, MA: Academic Press, Inc. (1988; Zbl 0661.47045); \textit{M. A. Krasnosel'skiĭ}, Positive solutions of operator equations. Translated from the Russian by Richard E. Flaherty. Edited by Leo F. Boron. Groningen: P. Noordhoff Ltd. (1964; Zbl 0121.10604)], we establish the existence of at least one fixed point of the
operator \(L\).Fixed-point theorems for multivalued operator matrix under weak topology with an applicationhttps://zbmath.org/1503.470772023-03-23T18:28:47.107421Z"Jeribi, Aref"https://zbmath.org/authors/?q=ai:jeribi.aref"Kaddachi, Najib"https://zbmath.org/authors/?q=ai:kaddachi.najib"Krichen, Bilel"https://zbmath.org/authors/?q=ai:krichen.bilelSummary: In the present paper, we establish some fixed-point theorems for a \(2\times 2\) block operator matrix involving multivalued maps acting on Banach spaces. These results are formulated in terms of weak sequential continuity and the technique of measures of weak noncompactness. The results obtained are then applied to a coupled system of nonlinear equations.Fixed point theorems for single valued mappings satisfying the ordered non-expansive conditions on ultrametric and non-Archimedean normed spaceshttps://zbmath.org/1503.470782023-03-23T18:28:47.107421Z"Mamghaderi, Hamid"https://zbmath.org/authors/?q=ai:mamghaderi.hamid"Masiha, Seyed Hashem Parvaneh"https://zbmath.org/authors/?q=ai:masiha.seyed-hashem-parvanehSummary: In this paper, some fixed point theorems for nonexpansive mappings in partially ordered spherically complete ultrametric spaces are proved. In addition, we investigate the existence of fixed points for nonexpansive mappings in partially ordered non-Archimedean normed spaces. Finally, we give some examples to discuss the assumptions and support our results.Relatively nonexpansive mappings in \(k\)-uniformly convex Banach spaceshttps://zbmath.org/1503.470792023-03-23T18:28:47.107421Z"Normohamadi, Z."https://zbmath.org/authors/?q=ai:normohamadi.z"Moosaei, M."https://zbmath.org/authors/?q=ai:moosaei.mohammad"Gabeleh, M."https://zbmath.org/authors/?q=ai:gabeleh.moosaSummary: Let \(A\) and \(B\) be two nonempty subsets of a normed linear space \(X\). A~mapping \(T:A \cup B \to A \cup B\) is said to be noncyclic if \(T(A) \subseteq A\) and \(T(B) \subseteq B\). In the present paper, we consider the problem of finding the best proximity pair for the noncyclic mapping \(T\), that is, two fixed points of \(T\) which achieve the minimum distance between the sets \(A\) and \(B\). We do it from some different approaches. The common condition on these results is relatively nonexpansivity of the mapping \(T\). At the first conclusion, we obtain the existence of best proximity pairs in the setting of uniformly convex in every direction Banach spaces where the pair \((A, B)\) is nonconvex. Then we conclude a similar result by replacing the geometric property of Opial's property of the Banach space and adding another assumption on the mapping \(T\), called condition \((C)\). We also show that the same result is true when \(X\) is a 2-uniformly convex Banach space. In the setting of \(k\)-uniformly convex Banach spaces, we prove that every nonempty, and convex pair of subsets has a geometric notion of proximal normal structure and then, we deduce the existence of best proximity pairs for relatively nonexpansive mappings in such spaces.A fixed point theorem in Hilbert \(C^\ast\)-moduleshttps://zbmath.org/1503.470802023-03-23T18:28:47.107421Z"Ranjbar, Hasan"https://zbmath.org/authors/?q=ai:ranjbar.hasan"Niknam, Asadollah"https://zbmath.org/authors/?q=ai:niknam.assadollahSummary: Fixed point theory has many useful applications in applied sciences. The object of this paper is to obtain fixed point for continuous self mappings in Hilbert \(C^*\)-module with rational conditions.New nonlinear operators and split common fixed point problems in Banach spaces and applicationshttps://zbmath.org/1503.470812023-03-23T18:28:47.107421Z"Takahashi, Wataru"https://zbmath.org/authors/?q=ai:takahashi.wataruSummary: In this article, motivated by the split feasibility problem, the split common null point problem and the split common fixed point problem in Hilbert spaces, we consider such problems in Banach spaces. We first introduce two new classes of nonlinear operators in Banach spaces. Then, using the geometry of Banach spaces, the hybrid method and the shrinking projection method, we establish strong convergence theorems for finding a solution of the split common fixed point problem in Banach spaces. It seems that these results are the first of their kind in Banach spaces. Using these results, we solve the split feasibility problem and the split common null point problem in Banach spaces.Ergodic behaviour of a Douglas-Rachford operator away from the originhttps://zbmath.org/1503.470822023-03-23T18:28:47.107421Z"Borwein, Jonathan M."https://zbmath.org/authors/?q=ai:borwein.jonathan-michael"Giladi, Ohad"https://zbmath.org/authors/?q=ai:giladi.ohadSummary: It is shown that away from the origin, the Douglas-Rachford operator with respect to a sphere and a convex set in a Hilbert space can be approximated by a another operator which satisfies a weak ergodic theorem. Similar results for other projection and reflection operators are also discussed.General history-dependent operators with applications to differential equationshttps://zbmath.org/1503.470832023-03-23T18:28:47.107421Z"Li, Xiuwen"https://zbmath.org/authors/?q=ai:li.xiuwen"Zeng, Biao"https://zbmath.org/authors/?q=ai:zeng.biaoSummary: In this paper, we introduce a class of nonlinear operators -- the class of general history-dependent operators. These are the operators defined on spaces of functions endowed with a structure of Banach space (the case of bounded interval of time) or Fréchet space (the case of unbounded interval of time). We state and prove various properties of such operators, including fixed point properties. Moreover, we also study several classes of differential equations in Banach spaces, for which we our previous results can be applied.Linear Lyapunov functions of infinite dimensional Volterra operatorshttps://zbmath.org/1503.470842023-03-23T18:28:47.107421Z"Mukhamedov, F."https://zbmath.org/authors/?q=ai:mukhamedov.farruh-m"Embong, A. F."https://zbmath.org/authors/?q=ai:embong.ahmad-fadillahSummary: In the present paper, we investigate infinite dimensional Volterra quadratic stochastic operators. We construct linear Lyapunov functions for such kind of operators, which allow to explore limiting set of associated with these kinds of operators.On surjective second order non-linear Markov operators and associated nonlinear integral equationshttps://zbmath.org/1503.470852023-03-23T18:28:47.107421Z"Mukhamedov, Farrukh"https://zbmath.org/authors/?q=ai:mukhamedov.farruh-m"Khakimov, Otabek"https://zbmath.org/authors/?q=ai:khakimov.otabek-n"Embong, Ahmad Fadillah"https://zbmath.org/authors/?q=ai:embong.ahmad-fadillahSummary: It is known that orthogonality preserving property and surjectivity of nonlinear Markov operators, acting on finite dimensional simplices, are equivalent. It turns out that these notions are no longer equivalent when such kind of operators are considered over on infinite dimensional spaces. In the present paper, we find necessary and sufficient condition to be equivalent of these notions, for the second order nonlinear Markov operators. To do this, we fully describe all surjective second order nonlinear Markov operators acting on infinite dimensional simplex. As an application of this result, we provided some sufficient conditions for the existence of positive solutions of nonlinear integral equations whose domain are not compact.Nontrivial solutions for semilinear equations with a discontinuous nonlinear termhttps://zbmath.org/1503.470862023-03-23T18:28:47.107421Z"Kim, In-Sook"https://zbmath.org/authors/?q=ai:kim.in-sook|kim.insookSummary: We study the existence of at least one nontrivial solution for semilinear equations of the form \(Lu\in Nu\) in the case where \(N\) interacts with a finite number of eigenvalues of finite multiplicity of \(L\). It is assumed that \(L\) is a self-adjoint closed densely defined linear operator in Hilbert space \(L_2\) having infinite dimensional kernel and \(N\) is a bounded set-valued operator generated by certain nonlinearity \(g\). The method of approach is to use a topological degree theory for a class of operators associated with the given problem. As an application, we consider the periodic Dirichlet problem for semilinear wave equations with a discontinuous nonlinear term.Multiple solutions for semilinear equations with discontinuous nonlinearities crossing one simple eigenvalueshttps://zbmath.org/1503.470872023-03-23T18:28:47.107421Z"Kim, In-Sook"https://zbmath.org/authors/?q=ai:kim.in-sook|kim.insookSummary: Let \(L\) be a closed self-adjoint densely defined linear operator in Hilbert space \(L_2\) with infinite dimensional kernel, and let \(N\) be a bounded monotone multi-valued operator generated by a possibly discontinuous function. We study the existence of two nontrivial solutions for a semilinear equation of the form \(Lu\in Nu\) when \(N\) interacts with one simple eigenvalue of \(L\). To do this, we establish a continuation theorem on pseudomonotone operators which connects the given problem to reference maps having nonzero degree. Our result is applied to the problem of finding multiple periodic solutions for semilinear wave equations with discontinuous nonlinearities under Dirichlet boundary conditions.Co-proximal operators for solving generalized co-variational inclusion problems in \(q\)-uniformly smooth Banach spaceshttps://zbmath.org/1503.470882023-03-23T18:28:47.107421Z"Ahmad, Rais"https://zbmath.org/authors/?q=ai:ahmad.rais"Irfan, Syed Shakaib"https://zbmath.org/authors/?q=ai:irfan.syed-shakaib"Ahmad, Iqbal"https://zbmath.org/authors/?q=ai:ahmad.iqbal"Rahaman, Mijanur"https://zbmath.org/authors/?q=ai:rahaman.mijanurSummary: In this paper, we introduce a new proximal operator which is called co-proximal operator and by using it, we establish an existence result for a unique solution of the generalized co-variational inclusion problem. Based on this co-proximal operator, we propose an iterative algorithm for solving generalized co-variational inclusion problems in \(q\)-uniformly smooth Banach spaces. The convergence analysis of the proposed iterative scheme is also discussed.Generalized split equilibrium problems for countable family of relatively quasi-nonexpansive mappingshttps://zbmath.org/1503.470892023-03-23T18:28:47.107421Z"Al-Homidan, Suliman"https://zbmath.org/authors/?q=ai:al-homidan.suliman-s"Ali, Bashir"https://zbmath.org/authors/?q=ai:ali.bashir"Suleiman, Yusuf I."https://zbmath.org/authors/?q=ai:suleiman.yusuf-iSummary: In this paper, we consider a generalized split equilibrium problem in the setting of uniformly smooth and uniformly convex Banach spaces. We construct an iterative scheme and prove its convergence to a solution of the generalized split equilibrium problem. Our scheme contains several well-known schemes as special cases.A new preconditioning algorithm for finding a zero of the sum of two monotone operators and its application to image restoration problemshttps://zbmath.org/1503.470902023-03-23T18:28:47.107421Z"Altiparmak, Ebru"https://zbmath.org/authors/?q=ai:altiparmak.ebru"Karahan, Ibrahim"https://zbmath.org/authors/?q=ai:karahan.ibrahimSummary: Finding a zero of the sum of two monotone operators is one of the most important problems in monotone operator theory, and the forward-backward algorithm is the most prominent approach for solving this type of problem. The aim of this paper is to present a new preconditioning forward-backward algorithm to obtain the zero of the sum of two operators in which one is maximal monotone and the other one is \(M\)-cocoercive, where \(M\) is a linear bounded operator. Furthermore, the strong convergence of the proposed algorithm, which is a broader variant of previously known algorithms, has been proven in Hilbert spaces. We also use our algorithm to tackle the convex minimization problem and show that it outperforms existing algorithms. Finally, we discuss several image restoration applications.Convergence theorems by using a projection method without the monotonicity in Hilbert spaceshttps://zbmath.org/1503.470912023-03-23T18:28:47.107421Z"Arunchai, Areerat"https://zbmath.org/authors/?q=ai:arunchai.areerat"Plubtieng, Somyot"https://zbmath.org/authors/?q=ai:plubtieng.somyot"Seangwattana, Thidaporn"https://zbmath.org/authors/?q=ai:seangwattana.thidapornSummary: In this paper, we present a projection iterative algorithm for finding the common solution of variational inequality problem without monotonicity, fixed point problem of a nonexpansive mapping, and zero point problem of the sum of two monotone mappings in Hilbert spaces. When setting the solution set of the dual variational inequality is nonempty, strong convergence is established under some suitable control conditions. Finally, we reduce some mappings in our main result to study several problems.Coupling Popov's algorithm with subgradient extragradient method for solving equilibrium problemshttps://zbmath.org/1503.470922023-03-23T18:28:47.107421Z"Kassay, Gábor"https://zbmath.org/authors/?q=ai:kassay.gabor"Trinh Ngoc Hai"https://zbmath.org/authors/?q=ai:trinh-ngoc-hai."Nguyen The Vinh"https://zbmath.org/authors/?q=ai:nguyen-the-vinh.Summary: Based on the recent works by \textit{Y. Censor} et al. [J. Optim. Theory Appl. 148, No. 2, 318--335 (2011; Zbl 1229.58018)], \textit{Yu. V. Malitsky} and \textit{V. V. Semenov} [Cybern. Syst. Anal. 50, No. 2, 271--277 (2014; Zbl 1311.49024); translation from Kibern. Sist. Anal. No. 2, 125--131 (2014)], and \textit{S. I. Lyashko} and \textit{V. V. Semenov} [Springer Optim. Appl. 115, 315--325 (2016; Zbl 1354.90172)], we propose a new scheme for solving pseudomonotone equilibrium problems in real Hilbert spaces. Weak and strong convergence results are suitably established. Our algorithm improves the recent one announced by Lyashko and Semenov not only from computational point of view, but also in some assumptions imposed on their main result. A comparative numerical study is carried out between the algorithms of \textit{D. Q. Tran} et al. [Optimization 57, No. 6, 749 - 776 (2008; Zbl 1152.90564)], Lyashko-Semenov [loc.\,cit.], and the new one. Some numerical examples are given to illustrate the efficiency and performance of the proposed method.A weak convergence theorem by Mann type iteration for a finite family of new demimetric mappings in a Hilbert spacehttps://zbmath.org/1503.470932023-03-23T18:28:47.107421Z"Lin, Chien-Nan"https://zbmath.org/authors/?q=ai:lin.chien-nan"Takahashi, Wataru"https://zbmath.org/authors/?q=ai:takahashi.wataru"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfengSummary: In this paper, using Mann type iteration, we prove a weak convergence theorem for finding a common element of the set of common fixed points for a finite family of new demimetric mappings and the set of common solutions of generalized variational inequality problems for a finite family of inverse strongly monotone mappings in a Hilbert space. Using this result, we obtain well-known and new weak convergence theorems in a Hilbert space.Strong convergence theorem for finding a common solution of convex minimization and fixed point problems in \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1503.470942023-03-23T18:28:47.107421Z"Ndlovu, P. V."https://zbmath.org/authors/?q=ai:ndlovu.p-v"Jolaoso, L. O."https://zbmath.org/authors/?q=ai:jolaoso.lateef-olakunle"Aphane, Maggie"https://zbmath.org/authors/?q=ai:aphane.maggieSummary: In this paper, we introduce a proximal point algorithm for approximating a common solution of finite family of convex minimization problems and fixed point problems for \(k\)-demicontractive mappings in complete \(\mathrm{CAT}(0)\) spaces. We prove a strong convergence result and obtain other consequence results which generalize and extend some recent results in the literature. We further provide a numerical example to illustrate the convergence behaviour of the sequence generated by our algorithm.Application of the combination of variational inequalities for fixed point problems and optimization problemshttps://zbmath.org/1503.470952023-03-23T18:28:47.107421Z"Saechou, Kanyanee"https://zbmath.org/authors/?q=ai:saechou.kanyanee"Kangtunyakarn, Atid"https://zbmath.org/authors/?q=ai:kangtunyakarn.atidSummary: In this paper, we obtain a strong convergence theorem for finding a common element of the set of solutions of variational inequality problems and equilibrium problems. Moreover, we apply our main result to obtain a strong convergence theorem for finding a common element of the set of fixed point problems of strictly pseudo-contractive mappings and a convergence theorem involving minimization problems. Furthermore, we utilize our main theorem for numerical examples.A new generalized forward-backward splitting method in reflexive Banach spaceshttps://zbmath.org/1503.470962023-03-23T18:28:47.107421Z"Sunthrayuth, Pongsakorn"https://zbmath.org/authors/?q=ai:sunthrayuth.pongsakorn"Yang, Jun"https://zbmath.org/authors/?q=ai:yang.jun.7"Cholamjiak, Prasit"https://zbmath.org/authors/?q=ai:cholamjiak.prasitSummary: We propose a new generalized forward-backward splitting method for finding a common zero of a finite family of the sum of maximal monotone and Bregman inverse strongly monotone operators in the framework of a reflexive Banach space. We then prove the strong convergence result of the sequence generated by our proposed method under suitable conditions. Some numerical experiments are presented to illustrate the efficiency of the proposed algorithm. The results presented in this paper improve and generalize many known results in this research field.The split common fixed point problem and the shrinking projection method in Banach spaceshttps://zbmath.org/1503.470972023-03-23T18:28:47.107421Z"Takahashi, Wataru"https://zbmath.org/authors/?q=ai:takahashi.wataruSummary: We consider the split common fixed point problem in Banach spaces. Using the shrinking projection method, we prove a strong convergence theorem for finding a solution of the common fixed point problem in Banach spaces. Using this result, we get well-known and new results which are connected with the split feasibility problem and the split common null point problem in Banach spaces.Two projection-based methods for bilevel pseudomonotone variational inequalities involving non-Lipschitz operatorshttps://zbmath.org/1503.470982023-03-23T18:28:47.107421Z"Tan, Bing"https://zbmath.org/authors/?q=ai:tan.bing.1"Cho, Sun Young"https://zbmath.org/authors/?q=ai:cho.sun-youngSummary: In this paper, we propose two new iterative algorithms to discover solutions of bilevel pseudomonotone variational inequalities with non-Lipschitz continuous operators in real Hilbert spaces. Our proposed algorithms need to compute the projection on the feasible set only once in each iteration although they employ Armijo line search methods. Strong convergence theorems of the suggested algorithms are established under suitable and weaker conditions. Some numerical experiments and applications are given to demonstrate the performance of the offered algorithms compared to some known ones.Strong convergence of a general iterative process in Hilbert spaceshttps://zbmath.org/1503.470992023-03-23T18:28:47.107421Z"Yang, Yantao"https://zbmath.org/authors/?q=ai:yang.yantaoSummary: We investigate a general iterative process with a strongly positive linear bounded self-adjoint operator for solving a inclusion problem of two maximal monotone operators, a common fixed point problem of an infinite family of nonexpansive mappings, and a generalized equilibrium problem in a Hilbert space.Projection algorithms for a general system of variational inequality and fixed point problemshttps://zbmath.org/1503.471002023-03-23T18:28:47.107421Z"Yu, Youli"https://zbmath.org/authors/?q=ai:yu.youli"Liou, Yeong-Cheng"https://zbmath.org/authors/?q=ai:liou.yeongcheng"Yao, Zhangsong"https://zbmath.org/authors/?q=ai:yao.zhangsong"Zhu, Li-Jun"https://zbmath.org/authors/?q=ai:zhu.lijunSummary: A projection algorithm with Meir-Keeler contraction is presented for solving a general system of variational inequality and fixed point problem of the pseudocontractive operator. Strong convergence analysis of the suggested algorithm is given.An extragradient method for equilibrium problems and fixed point problems of an asymptotically nonexpansive mapping in \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1503.471012023-03-23T18:28:47.107421Z"Zamani Eskandani, Gholamreza"https://zbmath.org/authors/?q=ai:zamani-eskandani.gholamreza"Kim, Jong Kyu"https://zbmath.org/authors/?q=ai:kim.jong-kyu"Moharami, Ruholah"https://zbmath.org/authors/?q=ai:moharami.ruholah"Lim, Won Hee"https://zbmath.org/authors/?q=ai:lim.won-heeSummary: Using a hybrid extragradient method, we introduce a new iterative process for approximating a common element of the set of solutions of an equilibrium problem and a fixed point problem of an asymptotically nonexpansive mapping in \(\mathrm{CAT}(0)\) spaces. We also give a numerical example to solve an optimization problem in a \(\mathrm{CAT}(0)\) space to support our main result.Approximation of common solutions of nonlinear problems in Banach spaceshttps://zbmath.org/1503.471022023-03-23T18:28:47.107421Z"Zegeye, Solomon B."https://zbmath.org/authors/?q=ai:zegeye.solomon-bekele"Sangago, Mengistu G."https://zbmath.org/authors/?q=ai:sangago.mengistu-g"Zegeye, Habtu"https://zbmath.org/authors/?q=ai:zegeye.habtuSummary: In this article, we propose a hybrid projection iterative algorithm for finding a common element of solution set of a finite family of generalized mixed equilibrium problem, semi-fixed point set of a finite family of continuous semi-pseudocontractive mappings and solution set of a finite family of variational inequality for a finite family of monotone and \(L\)-Lipschitz mappings in Banach spaces and prove a strong convergence theorem. The main result of the article generalizes some known results in the literature. Furthermore, we give a numerical example to demonstrate the convergence behaviour of the algorithm.Fixed point approximation of Suzuki generalized nonexpansive mappings via new faster iteration processhttps://zbmath.org/1503.471032023-03-23T18:28:47.107421Z"Hussain, Nawab"https://zbmath.org/authors/?q=ai:hussain.nawab"Ullah, Kifayat"https://zbmath.org/authors/?q=ai:ullah.kifayat"Arshad, Muhammad"https://zbmath.org/authors/?q=ai:arshad.muhammadSummary: In this paper we propose a new iteration process, called the \(K\) iteration process, for approximation of fixed points. We show that our iteration process is faster than the existing well-known iteration processes using numerical examples. Finally, we prove some weak and strong convergence theorems for Suzuki generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. Our results are the extension, improvement and generalization of many well-known results in the literature of iterations in fixed point theory.A strong convergence theorem for countable families of nonlinear nonself mappings in Hilbert spaces and applicationshttps://zbmath.org/1503.471042023-03-23T18:28:47.107421Z"Kawasaki, Toshiharu"https://zbmath.org/authors/?q=ai:kawasaki.toshiharu"Takahashi, Wataru"https://zbmath.org/authors/?q=ai:takahashi.wataruSummary: In [J. Convex Anal. 24, No. 3, 1015--1028 (2017; Zbl 1503.47097)], the second author introduced the concept of demimetric mappings in Banach spaces and \textit{S. M. Alsulami} and the second author [J. Nonlinear Convex Anal. 17, No. 12, 2511--2527 (2016; Zbl 1382.47012)] showed strong convergence theorems for demimetric mappings in Hilbert spaces. On the other hand, in [J. Nonlinear Convex Anal. 14, No. 1, 71--87 (2013; Zbl 1269.47041)] the authors introduced the concept of widely more generalize hybrid mappings in Hilbert spaces. A widely more generalize hybrid mapping is not demimetric generally even if the set of fixed points of the mapping is nonempty. In this paper, we extend the class of demimetric mappings to a more broad class of mappings in Banach spaces and prove a strong convergence theorem applicable to the class of widely more generalized hybrid mappings in Hilbert spaces. Using this result we obtain strong convergence theorems which are connected to the class of widely more generalized hybrid mappings in Hilbert spaces.Rate of convergence and data dependency of almost Prešić contractive operatorshttps://zbmath.org/1503.471052023-03-23T18:28:47.107421Z"Khan, Abdul Rahmi"https://zbmath.org/authors/?q=ai:khan.abdul-rahmi"Fukhar-Ud-Din, Hafiz"https://zbmath.org/authors/?q=ai:fukhar-ud-din.hafiz"Gürsoyy, Faik"https://zbmath.org/authors/?q=ai:gursoy.faikSummary: We introduce the notion of almost Prešić contractive operator and approximate its unique fixed point through some well-known iterative algorithms. We show that Picard-Picard hybrid iterative algorithm is faster than the others. Moreover, Mann and Ishikawa iterative algorithms have the same rate of convergence. The corresponding data dependence results and examples to validate our findings are also given. Our results hold in normed spaces and \(\mathrm{CAT}(0)\) spaces, simultaneously.Comparison rate of convergence and data dependence for a new iteration methodhttps://zbmath.org/1503.471062023-03-23T18:28:47.107421Z"Maldar, Samet"https://zbmath.org/authors/?q=ai:maldar.samet"Atalan, Yunus"https://zbmath.org/authors/?q=ai:atalan.yunus"Dogan, Kadri"https://zbmath.org/authors/?q=ai:dogan.kadriSummary: In this paper, we have defined hyperbolic type of some iteration methods. The new iteration has been investigated convergence for mappings satisfying certain condition in hyperbolic spaces. It has been proved that this iteration is equivalent in terms of convergence with another iteration method in the same spaces. The rate of convergence of these two iteration methods have been compared. We have investigated data dependence result using hyperbolic type iteration. Finally, we have given numerical examples about rate of convergence and data dependence.Strong convergence results for fixed points of nearly weak uniformly \(L\)-Lipschitzian mappings of \(I\)-dominated mappingshttps://zbmath.org/1503.471072023-03-23T18:28:47.107421Z"Mogbademu, A. A."https://zbmath.org/authors/?q=ai:mogbademu.adesanmi-alao"Kim, J. K."https://zbmath.org/authors/?q=ai:kim.jang-kyo|kim.jeom-keun|kim.jae-kyeong|kim.joung-kook|kim.jeong-kyun|kim.jong-kyoung|kim.jung-kuk|kim.jong-kwan|kim.jae-kwang|kim.jong-kyou|kim.jae-kyoon|kim.jae-kyung|kim.jwa-k|kim.jong-kyu|kim.jae-kyoung|kim.jin-kon|kim.jung-kon|kim.jae-kyu|kim.ju-kyong|kim.jong-kook|kim.jung-kyungSummary: In this paper, we prove strong convergence results for a modified Mann iterative process for a new class of \(I\)-nearly weak uniformly \(L\)-Lipschitzian mappings in a real Banach space. The class of \(I\)-nearly weak uniformly \(L\)-Lipschitzian mappings is an interesting generalization of the class of nearly weak uniformly \(L\)-Lipschitzian mappings which in turn is a generalization of the class of nearly uniformly \(L\)-Lipschitzian mappings which in turn generalises uniformly \(L\)-Lipschitzian mappings. Our theorems include some very recent results in fixed point theory and applications, in the context of nearly uniformly \(L\)-Lipschitzian mappings.On convergence theorems of an implicit iterative process with errors for a finite family of asymptotically quasi \(I\)-nonexpansive mappingshttps://zbmath.org/1503.471082023-03-23T18:28:47.107421Z"Mukhamedov, Farrukh"https://zbmath.org/authors/?q=ai:mukhamedov.farruh-m"Saburov, Mansoor"https://zbmath.org/authors/?q=ai:saburov.mansoorSummary: In this paper we prove the weak and strong convergence of the implicit iterative process with errors to a common fixed point of a finite family \(\{T_{j}\}^{N}_{i=1}\) of asymptotically quasi \(I_{j}\)-nonexpansive mappings as well as a family of \(\{I_j\}^{N}_{j=1}\) of asymptotically quasi nonexpansive mappings in the framework of Banach spaces. The obtained results improve and generalize the corresponding results in the existing literature.Rates of convergence for asymptotically weakly contractive mappings in normed spaceshttps://zbmath.org/1503.471092023-03-23T18:28:47.107421Z"Powell, Thomas"https://zbmath.org/authors/?q=ai:powell.thomas-d|powell.thomas-m|powell.thomas-g"Wiesnet, Franziskus"https://zbmath.org/authors/?q=ai:wiesnet.franziskusSummary: We study Krasnoselskii-Mann style iterative algorithms for approximating fixpoints of asymptotically weakly contractive mappings, with a focus on providing generalized convergence proofs along with explicit rates of convergence. More specifically, we define a new notion of being asymptotically \(\psi \)-weakly contractive with modulus, and present a series of abstract convergence theorems which both generalize and unify known results from the literature. Rates of convergence are formulated in terms of our modulus of contractivity, in conjunction with other moduli and functions which form quantitative analogues of additional assumptions that are required in each case. Our approach makes use of ideas from proof theory, in particular, our emphasis on abstraction and on formulating our main results in a quantitative manner. As such, the paper can be seen as a contribution to the \textit{proof mining} program [\textit{U. Kohlenbach}, Applied proof theory. Proof interpretations and their use in mathematics. Berlin: Springer (2008; Zbl 1158.03002)].Convergence of Halpern's iteration method with applications in optimizationhttps://zbmath.org/1503.471102023-03-23T18:28:47.107421Z"Qi, Huiqiang"https://zbmath.org/authors/?q=ai:qi.huiqiang"Xu, Hong-Kun"https://zbmath.org/authors/?q=ai:xu.hong-kunSummary: \textit{B. Halpern}'s iteration method [Bull. Am. Math. Soc. 73, 957--961 (1967; Zbl 0177.19101)] is an iterative algorithm for finding fixed points of a nonexpansive mapping in Hilbert and Banach spaces. Since many optimization problems can be cast into fixed point problems of nonexpansive mappings, Halpern's method plays an important role in optimization methods. This paper discusses recent advances in convergence and rate of convergence results of Halpern's method, and applications in optimization problems, including variational inequalities, monotone inclusions, Douglas-Rachford splitting method, and minimax problems.Convergence of inexact iterates of uniformly locally nonexpansive mappingshttps://zbmath.org/1503.471112023-03-23T18:28:47.107421Z"Reich, Simeon"https://zbmath.org/authors/?q=ai:reich.simeon"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: In our joint paper [in: Proceedings of the 7th international conference on fixed-point theory and its applications, Guanajuato, Mexico, July 17--23, 2005. Yokohama: Yokohama Publishers. 11--32 (2006; Zbl 1116.47052)] with \textit{D. Butnariu}, it was shown that, if all exact orbits of a nonexpansive mapping converge to a fixed point, then this convergence property also holds for all its inexact orbits with summable errors. In the present paper, we establish an analogous result for inexact orbits of uniformly locally nonexpansive mappings.Convergence theorems for new iteration scheme and comparison resultshttps://zbmath.org/1503.471122023-03-23T18:28:47.107421Z"Sahu, Vinod Kumar"https://zbmath.org/authors/?q=ai:sahu.vinod-kumar"Pathak, H. K."https://zbmath.org/authors/?q=ai:pathak.hemant-kumar"Tiwari, Rakesh"https://zbmath.org/authors/?q=ai:tiwari.rakesh-prabhatSummary: In this article, we propose a new three step iteration process for approximation of fixed points of the nonexpansive mappings. We show that our iteration scheme is faster than some known iterative algorithms for the contractive mapping. We support analytic proof by a numerical example in which we approximate the fixed point by a computer using Matlab program. We also prove convergence results for the nonexpansive mappings.Strong convergence of three-step iteration processes for multivalued mappings in some \(\mathrm{CAT}(k)\) spaceshttps://zbmath.org/1503.471132023-03-23T18:28:47.107421Z"Shabani, Saeed"https://zbmath.org/authors/?q=ai:shabani.saeedSummary: In this paper, we prove the strong convergence of the three-step iteration processes for some generalized nonexpansive multivalued mappings in the framework of
\(\mathrm{CAT}(1)\) spaces. The obtained results extend some recent known results.A strong convergence of a modified Krasnoselskii-Mann algorithm for a finite family of demicontractive mappings in Banach spaceshttps://zbmath.org/1503.471142023-03-23T18:28:47.107421Z"Sow, T. M. M."https://zbmath.org/authors/?q=ai:sow.thierno-mohamadane-mansour|sow.thierno-mohadamane-mansour|sow.thierno-m-mSummary: In this paper, we propose an iterative algorithm, which is based on the Krasnoselskii-Mann iterative algorithm for fixed point problems of a finite family of demicontractive mappings in the setting of real Banach spaces. We prove that the sequence generated by the proposed method converges strongly to a common fixed point of a finite family of demicontractive mappings which is also the solution of a variational inequality. The iterative algorithm and results presented in this paper generalize, unify and improve some previously known results of this area.Weak and strong convergence theorems for noncommutative two generalized hybrid mappings in Hilbert spaceshttps://zbmath.org/1503.471152023-03-23T18:28:47.107421Z"Takahashi, Wataru"https://zbmath.org/authors/?q=ai:takahashi.wataruSummary: In this paper, we first obtain a weak convergence theorem of Mann's type iteration for noncommutative two generalized hybrid mappings in a Hilbert space. Next, we obtain a strong convergence theorem of Halpern's type iteration for the mappings in a Hilbert space. Using these results, we get new weak and strong convergence theorems in a Hilbert space.New iteration process and numerical reckoning fixed points in Banach spaceshttps://zbmath.org/1503.471162023-03-23T18:28:47.107421Z"Ullah, Kifayat"https://zbmath.org/authors/?q=ai:ullah.kifayat"Arshad, Muhammad"https://zbmath.org/authors/?q=ai:arshad.muhammadSummary: In this paper, we propose a new iteration process, called \(M^*\) iteration process, for approximation of fixed points. Some weak and strong convergence theorems for fixed point of Suzuki generalized nonexpansive mappings are proved in the setting of uniformly convex Banach spaces. A~numerical example is given to show the efficiency of \(M^*\) iteration process. Our results are the extension, improvement and generalization of many known results in the literature of iterations in fixed point theory.Mixed iterative algorithms for the multiple-set split equality common fixed-point problem of demicontractive mappingshttps://zbmath.org/1503.471172023-03-23T18:28:47.107421Z"Wang, Yaqin"https://zbmath.org/authors/?q=ai:wang.yaqin"Liu, Wei"https://zbmath.org/authors/?q=ai:liu.wei.19"Song, Yanlai"https://zbmath.org/authors/?q=ai:song.yanlai"Fang, Xiaoli"https://zbmath.org/authors/?q=ai:fang.xiaoliSummary: In this paper, we consider the multiple-set split equality common fixed-point problem governed by the general class of demicontractive mappings. For establishing strong convergence result we introduce a new mixed cyclic and simultaneous iterative algorithm combining viscosity approximation methods with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. Our results improve and extend previously discussed related problems and algorithms.Self-adaptive step-sizes choice for split common fixed point problemshttps://zbmath.org/1503.471182023-03-23T18:28:47.107421Z"Yao, Yonghong"https://zbmath.org/authors/?q=ai:yao.yonghong"Qin, Xiaolong"https://zbmath.org/authors/?q=ai:qin.xiaolong"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: The split common fixed point problem attempts to find a fixed point of an operator in one space whose image under a linear transformation is a fixed point of another operator in the image space. We formulate and analyze a self-adaptive algorithm for solving this split common fixed problem for the class of demicontractive operators. Strong convergence result is given.Nonlinear evolutionary systems driven by set-valued mixed equilibrium problemshttps://zbmath.org/1503.471192023-03-23T18:28:47.107421Z"Naraghirad, Eskandar"https://zbmath.org/authors/?q=ai:naraghirad.eskandar"Shi, Luoyi"https://zbmath.org/authors/?q=ai:shi.luoyi"Wong, Ngai-Ching"https://zbmath.org/authors/?q=ai:wong.ngai-chingSummary: This paper is concerned with the system (NEESSVMEP) obtained by mixing a nonlinear evolutionary equation and a strong set-valued mixed equilibrium problem (SSVMEP). In our setting, the set of constraints is not necessarily compact and the problem is driven by a not necessarily monotone set-valued mapping. We show that the solution set for (SSVMEP) is nonempty, closed, convex and bounded. We then establish the upper semicontinuity and the measurability properties of the involved functions in the nonlinear evolutionary equation. Utilizing these results, we prove that the solution set for (NEESSVMEP) is nonempty and compact.\(L^p\)-operator algebras with approximate identities. Ihttps://zbmath.org/1503.471202023-03-23T18:28:47.107421Z"Blecher, David P."https://zbmath.org/authors/?q=ai:blecher.david-p"Phillips, N. Christopher"https://zbmath.org/authors/?q=ai:phillips.n-christopherThe authors initiate a systematic study of \(L^p\)-operator algebras for \(p \neq 2\), that is, Banach algebras \(A\) which are isometrically isomorphic to a norm closed subalgebra of the bounded operators on an \(L^p(\mu)\)-space for some measure space \((X,\mu)\). The general aim is to investigate how much of the existing theory of (non-selfadjoint) operator algebras on Hilbert spaces extends to this setting. Moreover, the applicability of the recent theory of real positivity, developed by Blecher, Read, Neal, Ozawa and others for general Banach algebras, is studied for \(L^p\)-operator algebras having a contractive approximate identity (called approximately unital). Let \(A\) be a unital Banach algebra. Recall from [\textit{D. P. Blecher} and \textit{N. Ozawa}, Pac. J. Math. 277, No. 1, 1--59 (2015; Zbl 1368.46038)] that the element \(a \in A\) is real positive (or accretive) if \(\Re(\phi(a)) \ge 0\) for all states \(\phi\) on \(A\). If \(A\) is only approximately unital, then \(a \in A\) is real positive if \(a\) is real positive in the multiplier unitization \(A^1\) of \(A\). Moreover, the functional \(\phi \in A^*\) is real positive if \(\Re(\phi(a)) \ge 0\) for all real positive elements \(a \in A\).
In view of the length and wide scope of the results of this paper, it is only possible to indicate its contents, which is suggested by the following list of (sub)chapters: Miscellaneous results on \(L^p\)-operator algebras (quotients and bi-approximately unital algebras -- unitization of nonunital \(L^p\)-operator algebras -- the Cayley and \(\mathcal F\) transforms -- support idempotents -- some consequences of strict convexity of \(L^p\)-spaces -- Hahn-Banach smoothness of \(L^p\)-operator algebras). \(M\)-ideals. Scaled \(L^p\)-operator algebras. Kaplansky density.
There is a list of \(L^p\)-operator algebras which is used as basic examples or counterexamples. For instance, it follows from the results that the Calkin algebra \(\mathcal B(\ell^p)/\mathcal K(\ell^p)\) is an \(L^p\)-operator algebra for \(1 < p < \infty\). This can also be seen from earlier results of \textit{M. T. Boedihardjo} and \textit{W. B. Johnson} [Proc. Am. Math. Soc. 143, No. 6, 2451--2457 (2015; Zbl 1339.47014)].
Reviewer: Hans-Olav Tylli (Helsinki)Maximal \(C^\ast\)-covers and residual finite-dimensionalityhttps://zbmath.org/1503.471212023-03-23T18:28:47.107421Z"Thompson, Ian"https://zbmath.org/authors/?q=ai:thompson.ian-j|thompson.ian-mSummary: We study residually finite-dimensional (or RFD) operator algebras which may not be self-adjoint. An operator algebra may be RFD while simultaneously possessing completely isometric representations whose generating \({C}^\ast \)-algebra is not RFD. This has provided many hurdles in characterizing residual finite-dimensionality for operator algebras. To better understand the elusive behavior, we explore the \({C}^\ast \)-covers of an operator algebra. First, we equate the collection of \({C}^\ast \)-covers with a complete lattice arising from the spectrum of the maximal \({C}^\ast \)-cover. This allows us to identify a largest RFD \({C}^\ast \)-cover whenever the underlying operator algebra is RFD. The largest RFD \({C}^\ast \)-cover is shown to be similar to the maximal \({C}^\ast \)-cover in several different facets and this provides supporting evidence to a previous query of whether an RFD operator algebra always possesses an RFD maximal \({C}^\ast \)-cover. In closing, we present a non self-adjoint version of Hadwin's characterization of separable RFD \({C}^\ast \)-algebras.A note on partial \(*\)-algebras and spaces of distributionshttps://zbmath.org/1503.471222023-03-23T18:28:47.107421Z"Tschinke, F."https://zbmath.org/authors/?q=ai:tschinke.francescoSummary: Given a rigged Hilbert space \(\mathcal{D}\hookrightarrow\mathcal{H}\hookrightarrow\mathcal{D}'\), the spaces \(\mathcal{D}_{\mathrm{loc}}\) are considered. It is shown that, if \(\mathcal{D}\) is a Hilbert \(*\)-algebra, \(\mathcal{D}_{\mathrm{loc}}\) carry out a natural structure of partial \(*\)-algebra. Furthermore, on \(\mathcal{D}_{\mathrm{loc}}\) it is defined a topology, so that \(\mathcal{D}_{\mathrm{loc}}\) is an interspace. Examples from distributions theory are considered.Applications of measure of non-compactness and modified simulation function for solvability of nonlinear functional integral equationshttps://zbmath.org/1503.471232023-03-23T18:28:47.107421Z"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipan"Arab, Reza"https://zbmath.org/authors/?q=ai:arab.reza"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumarSummary: In this work we introduce a modified version of simulation function, define simulation type contraction mappings involving measure of noncompactness in the framework of Banach spaces, and derive some basic Darbo type fixed point results. Also, our theorem generalizes Theorem~4 of [\textit{R. Arab}, Miskolc Math. Notes 18, No. 2, 595--610 (2017; Zbl 1399.54082)] and extends some recent results. Further, we show the applicability of obtained results to the theory of integral equations, followed by two concrete examples.Optimal exponential decay for the linear inhomogeneous Boltzmann equation with hard potentialshttps://zbmath.org/1503.471242023-03-23T18:28:47.107421Z"Sun, Baoyan"https://zbmath.org/authors/?q=ai:sun.baoyanSummary: In this paper, we consider the asymptotic behavior of solutions to the linear spatially inhomogeneous Boltzmann equation for hard potentials in the torus. We obtain an optimal rate of exponential convergence towards equilibrium in a Lebesgue space with polynomial weight \(L_v^1 L_x^2(\langle v\rangle^k)\). This model is analyzed from a spectral point of view and from the point of view of semigroups. Our strategy is taking advantage of the spectral gap estimate in the Hilbert space with inverse Gaussian weight, the factorization argument and the enlargement method.Non-self-adjointness of bent optical waveguide eigenvalue problemhttps://zbmath.org/1503.471252023-03-23T18:28:47.107421Z"Kumar, Rakesh"https://zbmath.org/authors/?q=ai:kumar.rakesh.6|kumar.rakesh.4|kumar.rakesh.5"Hiremath, Kirankumar R."https://zbmath.org/authors/?q=ai:hiremath.kirankumar-rSummary: In the literature, the mathematical problem of optical wave propagation in dielectric straight waveguides has been systematically studied as a self-adjoint eigenvalue problem with real eigenvalues. In terms of the underlying physics, such real eigenvalues meant no losses during the wave propagation. However, when the waveguides were bent, experiments showed that the wave propagation became lossy. In this paper, optical wave propagation in dielectric bent waveguides is mathematically analyzed. It is shown that the corresponding eigenvalue problem is a non-self-adjoint eigenvalue problem and has complex-valued eigenvalues. The imaginary part of the eigenvalues is a measure of loss. For large-bend radii, the eigenvalue problem for bent waveguides behaves as an eigenvalue problem for straight waveguides, and the complex-valued eigenvalues approach the real-valued eigenvalues of the straight waveguide problem. By expressing the bent waveguide eigenvalue operator as a sum of the self-adjoint operator and the non-self-adjoint operator, asymptotic behaviour of guided modes and their lossy nature are investigated.Spectral analysis for infinite rank perturbations of unbounded diagonal operatorshttps://zbmath.org/1503.471262023-03-23T18:28:47.107421Z"Diagana, Toka"https://zbmath.org/authors/?q=ai:diagana.tokaSummary: In this paper we study the spectral theory for the class of linear operators \(A_\infty\) defined on the so-called non-archimedean Hilbert space \(\mathbb E_\omega\) by, \(A_\infty:= D + F_\infty\) where \(D\) is an unbounded diagonal linear operator and \(F_\infty:=\sum^\infty_{k=1} u_k \otimes v_k\) is an operator of infinite rank on \(\mathbb E_\omega\).Nonlinear ordered variational inclusion problem involving XOR operation with fuzzy mappingshttps://zbmath.org/1503.490082023-03-23T18:28:47.107421Z"Ahmad, Iqbal"https://zbmath.org/authors/?q=ai:ahmad.iqbal"Irfan, Syed Shakaib"https://zbmath.org/authors/?q=ai:irfan.syed-shakaib"Farid, Mohammad"https://zbmath.org/authors/?q=ai:farid.mohammad"Shukla, Preeti"https://zbmath.org/authors/?q=ai:shukla.preetiSummary: In the setting of real ordered positive Hilbert spaces, a nonlinear fuzzy ordered variational inclusion problem with its corresponding nonlinear fuzzy ordered resolvent equation problem involving XOR operation has been recommended and solved by employing an iterative algorithm. We establish the equivalence between nonlinear fuzzy ordered variational inclusion problem and nonlinear fuzzy ordered resolvent equation problem. The existence and convergence analysis of the solution of nonlinear fuzzy ordered variational inclusion problem involving XOR operation has been substantiated by applying a new resolvent operator method with XOR operation technique. The iterative algorithm and results demonstrated in this article have witnessed a significant improvement in many previously known results of this domain.Error bounds for mixed set-valued vector inverse quasi-variational inequalitieshttps://zbmath.org/1503.490092023-03-23T18:28:47.107421Z"Chang, Shih-sen"https://zbmath.org/authors/?q=ai:chang.shih-sen"Salahuddin"https://zbmath.org/authors/?q=ai:salahuddin."Wang, L."https://zbmath.org/authors/?q=ai:wang.lin.1"Wang, G."https://zbmath.org/authors/?q=ai:wang.gang.9"Ma, Z. L."https://zbmath.org/authors/?q=ai:ma.zao-li|ma.zhaoliSummary: The purpose of this paper is to introduce and study the mixed set-valued vector inverse quasi-variational inequality problems (MSVIQVIPs) and to obtain error bounds for this kind of MSVIQVIP in terms of the residual gap function, the regularized gap function, and the D-gap function. These bounds provide effective estimated distances between an arbitrary feasible point and the solution set of mixed set-valued vector inverse quasi-variational inequality problem. The results presented in the paper improve and generalize some recent results.Extragradient-like method for pseudomonotone equilibrium problems on Hadamard manifoldshttps://zbmath.org/1503.490102023-03-23T18:28:47.107421Z"Chen, Junfeng"https://zbmath.org/authors/?q=ai:chen.junfeng"Liu, Sanyang"https://zbmath.org/authors/?q=ai:liu.sanyangSummary: This paper presents an extragradient-like method for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition on Hadamard manifolds. The algorithm only needs to know the existence of the Lipschitz-type constants of the bifunction, and the stepsize of each iteration is determined by the adjacent iterations. Convergence of the algorithm is analyzed, and its application to variational inequalities is also provided. Finally, several experiments are made to verify the effectiveness of the algorithms.Common solutions to variational inequality problem via parallel and cyclic hybrid inertial CQ-subgradient extragradient algorithms in (HSs)https://zbmath.org/1503.490112023-03-23T18:28:47.107421Z"Hammad, Hasanen A."https://zbmath.org/authors/?q=ai:hammad.hasanen-abuelmagd"Diallo, Mamadou Alouma"https://zbmath.org/authors/?q=ai:diallo.mamadou-aloumaIn this paper, the authors investigate two new algorithms, so-called strongly convergent parallel and cyclic hybrid inertial CQ-subgradient extragradient algorithms. The choice of the intersection of the sets \(C_{n+1}^i\) instead of only one single set based on the furthest intermediate approximation from the current iterate in the parallel algorithm allowed to increase the precision of the computations and, therefore, to reduce the number of computations to get the required precision. Proposed algorithms are applied to find common solutions to the variational inequality problem (CSVIP) in the Hilbert spaces. The generated sequences converge strongly to the nearest point projection of the starting point onto the solution set of the CSVIP. Ultimately, numerical experiments are presented here to examine the efficiency of the proposed algorithms.
For the entire collection see [Zbl 1478.54001].
Reviewer: Liya Liu (Chengdu)A new class of hyperbolic variational-hemivariational inequalities driven by non-linear evolution equationshttps://zbmath.org/1503.490142023-03-23T18:28:47.107421Z"Migórski, Stanisław"https://zbmath.org/authors/?q=ai:migorski.stanislaw"Han, Weimin"https://zbmath.org/authors/?q=ai:han.weimin"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengdaSummary: The aim of the paper is to introduce and investigate a dynamical system which consists of a variational-hemivariational inequality of hyperbolic type combined with a non-linear evolution equation. Such a dynamical system arises in studies of complicated contact problems in mechanics. Existence, uniqueness and regularity of a global solution to the system are established. The approach is based on a new semi-discrete approximation with an application of a surjectivity result for a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelastic frictional contact model with adhesion is studied as an application, in which the contact boundary condition is described by a generalised normal damped response condition with unilateral constraint and a multivalued frictional contact law.The extragradient method for quasi-monotone variational inequalitieshttps://zbmath.org/1503.490152023-03-23T18:28:47.107421Z"Salahuddin"https://zbmath.org/authors/?q=ai:salahuddin.anjum-r|salahuddin.taimoor|salahuddin.salahuddin|salahuddin.2|salahuddin.m|salahuddin.1|salahuddin.|salahuddin.k-mThe author studies the quasi-monotone variational inequalities in infinite dimensional Hilbert space. The main result of the paper shows that the iterative sequence suggested by the extragradient algorithm for solving quasi-monotone variational inequalities converges weakly to a solution.
Reviewer: Leszek Gasiński (Kraków)Tykhonov well-posedness of split problemshttps://zbmath.org/1503.490162023-03-23T18:28:47.107421Z"Shu, Qiao-yuan"https://zbmath.org/authors/?q=ai:shu.qiao-yuan"Sofonea, Mircea"https://zbmath.org/authors/?q=ai:sofonea.mircea"Xiao, Yi-bin"https://zbmath.org/authors/?q=ai:xiao.yibinSummary: In [J. Optim. Theory Appl. 183, No. 1, 139--157 (2019; Zbl 1425.49005)], the last two authors introduced and studied the concept of well-posedness in the sense of Tykhonov for abstract problems formulated on metric spaces. Our aim of this current paper is to extend the results in [loc. cit.] to a system which consists of two independent problems denoted by \(P\) and \(Q\), coupled by a nonlinear equation. Following the terminology used in literature we refer to such a system as a split problem. We introduce the concept of well-posedness for the abstract split problem and provide its characterization in terms of metric properties for a family of approximating sets and in terms of the well-posedness for the problems \(P\) and \(Q\), as well. Then we illustrate the applicability of our results in the study of three relevant particular cases: a split variational-hemivariational inequality, an elliptic variational inequality and a history-dependent variational inequality. We describe each split problem and clearly indicate the family of approximating sets. We provide necessary and sufficient conditions which guarantee the well-posedness of the split variational-hemivariational inequality. Moreover, under appropriate assumptions on the data, we prove the well-posedness of the split elliptic variational inequality as well as the well-posedness of the split history-dependent variational inequality. We illustrate our abstract results with various examples, part of them arising in contact mechanics.On vector quasi-equilibrium problems via a Browder-type fixed-point theoremhttps://zbmath.org/1503.490192023-03-23T18:28:47.107421Z"Capătă, Adela"https://zbmath.org/authors/?q=ai:capata.adela-elisabetaIn this paper, new sufficient conditions for the existence of solutions to a vector quasi-equilibrium problem with set-valued mappings are provided. Using a Browder-type fixed-point theorem, which allows the author to relax the common lower semicontinuity assumptions, the results improve some theorems from the literature.
Reviewer: Radu Ioan Boţ (Wien)A vanishing theorem for the Mathai-Zhang indexhttps://zbmath.org/1503.531072023-03-23T18:28:47.107421Z"Zhang, Xin"https://zbmath.org/authors/?q=ai:zhang.xin.12Consider an even dimensional spin manifold admitting a proper cocompact Lie group action. The author shows that the Mathai-Zhang index vanishes under the condition that the connected component of the identity element in the action group is non-unimodular (instead of the condition of positive scalar curvature). The idea involves the stability of the index of Fredholm operators.
Reviewer: Wen Lu (Wuhan)Fixed point theory in generalized metric spaceshttps://zbmath.org/1503.540012023-03-23T18:28:47.107421Z"Karapınar, Erdal"https://zbmath.org/authors/?q=ai:karapinar.erdal"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pPublisher's description: This book presents fixed point theory, one of the crucial tools in applied mathematics, functional analysis, and topology, which has been used to solve distinct real-world problems in computer science, engineering, and physics. The authors begin with an overview of the extension of metric spaces. Readers are introduced to general fixed-point theorems while comparing and contrasting important and insignificant metric spaces. The book is intended to be self-contained and serves as a unique resource for researchers in various disciplines.Generalized integral type mappings on orthogonal metric spaceshttps://zbmath.org/1503.540242023-03-23T18:28:47.107421Z"Acar, Ö."https://zbmath.org/authors/?q=ai:acar.ozlem|acar.ozgur"Erdoğan, E."https://zbmath.org/authors/?q=ai:erdogan.emin|erdogan.eren|erdogan.ezgi"Özkapu, A. S."https://zbmath.org/authors/?q=ai:ozkapu.a-sSummary: This study is devoted to investigate the problem whether the existence and uniqueness of integral type contraction mappings on orthogonal metric spaces. At the end, we give an example to illustrative our main result.On some fixed point theorem in generalized complex valued metric spaces for BKC-contractionhttps://zbmath.org/1503.540252023-03-23T18:28:47.107421Z"Bhaumik, Shantanu"https://zbmath.org/authors/?q=ai:bhaumik.shantanu"Yadav, Devnarayan"https://zbmath.org/authors/?q=ai:yadav.devnarayan"Tiwari, Surendra Kumar"https://zbmath.org/authors/?q=ai:tiwari.surendra-kumarSummary: In this paper, we introduce a new concept of BKC-contraction in generalized complex valued metric spaces for some partial order and establish a fixed point theorem. In addition an example ispresent which illustrates our main result.Some notes on the paper ``On best proximity points of interpolative proximal contractions''https://zbmath.org/1503.540292023-03-23T18:28:47.107421Z"Gabeleh, M."https://zbmath.org/authors/?q=ai:gabeleh.moosa"Markin, J."https://zbmath.org/authors/?q=ai:markin.jack-tSummary: In a recent paper [\textit{I. Altun} and \textit{A. Taşdemir}, Quaest. Math. 44, No. 9, 1233--1241 (2021; Zbl 1477.54046)] the authors proved two best proximity point theorems for a new class of non-self mappings, called interpolative Reich-Rus-Ćirić type proximal contractions of the first and second kinds. Our purpose is to show that their results follow from a fixed point theorem for interpolative Reich-Rus-Ćirić type contraction mappings [\textit{E. Karapinar} et al., Mathematics 6, No. 11, Paper No. 256, 7 p. (2018; Zbl 1469.54127)].On the algebra of operators corresponding to the union of smooth submanifoldshttps://zbmath.org/1503.580112023-03-23T18:28:47.107421Z"Poluektova, D. A."https://zbmath.org/authors/?q=ai:poluektova.d-a"Savin, A. Yu."https://zbmath.org/authors/?q=ai:savin.anton-yu"Sternin, B. Yu."https://zbmath.org/authors/?q=ai:sternin.boris-yuConsider two transversally intersecting submanifolds in a smooth closed manifold. In this article, the authors give an algebraic description of all types of operators generated by pseudodifferential operators, boundary operators and coboundary operators corresponding to submanifolds. Following [\textit{B. Yu. Sternin}, Sov. Math., Dokl. 8, 41--45 (1967; Zbl 0177.37103); translation from Dokl. Akad. Nauk SSSR 172, 44--47 (1967)] a general element of this operator algebra is called a morphism. Generators of these morphisms are completely classified into 18 types. The notion of symbol of a morphism is introduced and the formula for the composition of two morphisms is derived as well. However, the study of ellipticity of morphisms is postponed to a sequel of this article.
Reviewer: Gihyun Lee (Ghent)Generating M-indeterminate probability densities by way of quantum mechanicshttps://zbmath.org/1503.600182023-03-23T18:28:47.107421Z"Mayato, Rafael Sala"https://zbmath.org/authors/?q=ai:sala-mayato.rafael"Loughlin, Patrick"https://zbmath.org/authors/?q=ai:loughlin.patrick-j"Cohen, Leon"https://zbmath.org/authors/?q=ai:cohen.leon-wSummary: Probability densities that are not uniquely determined by their moments are said to be ``moment-indeterminate,'' or ``M-indeterminate.'' Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison with standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given.Convex ordering of Pólya random variables and approximation monotonicity of Bernstein-Stancu operatorshttps://zbmath.org/1503.600222023-03-23T18:28:47.107421Z"Meleşteu, Alexandra D."https://zbmath.org/authors/?q=ai:melesteu.alexandra-diana"Pascu, Mihai N."https://zbmath.org/authors/?q=ai:pascu.mihai-nicolae"Pascu, Nicolae R."https://zbmath.org/authors/?q=ai:pascu.nicolae-raduSummary: In the present paper we show that in Pólya's urn model, for an arbitrarily fixed initial distribution of the urn, the corresponding random variables satisfy a natural convex ordering with respect to the replacement parameter. As an application, we show that in the class of convex functions, the error of approximation for Bernstein-Stancu operators is a non-decreasing (strictly increasing under an additional hypothesis) function of the corresponding parameter. The proofs rely on two results of independent interest: an interlacing lemma of three sets and the monotonicity of the (partial) first moment of Pólya random variables with respect to the replacement parameter.On weak solution of SDE driven by inhomogeneous singular Lévy noisehttps://zbmath.org/1503.600682023-03-23T18:28:47.107421Z"Kulczycki, Tadeusz"https://zbmath.org/authors/?q=ai:kulczycki.tadeusz"Kulik, Alexei"https://zbmath.org/authors/?q=ai:kulik.alexey-m"Ryznar, Michał"https://zbmath.org/authors/?q=ai:ryznar.michalSummary: We study a time-inhomogeneous SDE in \(\mathbb{R}^d\) driven by a cylindrical Lévy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and complicates the analysis of the model significantly. We prove that the weak solution to the SDE is uniquely defined, is Markov, and has the strong Feller property. The heat kernel of the process is presented as a combination of an explicit `principal part' and a `residual part', subject to certain \(L^\infty (dx)\otimes L^1(dy)\) and \(L^\infty (dx)\otimes L^\infty (dy)\)-estimates showing that this part is negligible in a short time, in a sense. The main tool of the construction is the analytic parametrix method, specially adapted to Lévy-type generators with strong spatial inhomogeneities.The Trotter product formula for nonlinear Fokker-Planck flowshttps://zbmath.org/1503.600782023-03-23T18:28:47.107421Z"Barbu, Viorel"https://zbmath.org/authors/?q=ai:barbu.viorelSummary: One proves herein that the flow \(S(t)\), generated by the nonlinear Fokker-Planck equation \(\rho_t - \Delta\beta (\rho) + \operatorname{div}(a(\rho) \rho) = 0\) in \((0, \infty) \times \mathbb{R}^d\), is expressed by the Trotter product formula
\[
S(t) \rho_0 = \lim_{n \to \infty} \left( S_{A_1} \left( \frac{t}{n}\right) S_{A_2} \left( \frac{t}{n}\right)\right)^n \rho_0 \text{ in } L^1 (\mathbb{R}^d),
\]
where \(S_{A_1}(t)\) is the flow (continuous semigroup) generated in \(L^1 (\mathbb{R}^d)\) by the nonlinear diffusion operator \(A_1 (\rho) = - \Delta\beta (\rho)\), while \(S_{A_2}(t)\) is that generated in \(L^1 (\mathbb{R}^d)\) by the conservation law operator \(A_2 (\rho) = \operatorname{div}(a(\rho) \rho)\) defined in the entropy sense. As an application, one obtains a split-product formula for the McKean-Vlasov stochastic differential equation associated with the Fokker-Planck equation.Comparison theorem for path dependent SDEs driven by \(G\)-Brownian motionhttps://zbmath.org/1503.601152023-03-23T18:28:47.107421Z"Huang, Xing"https://zbmath.org/authors/?q=ai:huang.xing"Yang, Fen-Fen"https://zbmath.org/authors/?q=ai:yang.fenfenThe authors present necessary and sufficient conditions for the comparison of solutions of path-dependent \(G\)-SDEs. They apply a probabilistic method to prove these results which is different from the one used for the appropriate study of path-independent \(G\)-SDEs. The derived results extend the ones obtained in the linear expectation case.
Reviewer: Pavel Gapeev (London)Numerical integration as a finite matrix approximation to multiplication operatorhttps://zbmath.org/1503.650502023-03-23T18:28:47.107421Z"Sarmavuori, Juha"https://zbmath.org/authors/?q=ai:sarmavuori.juha"Särkkä, Simo"https://zbmath.org/authors/?q=ai:sarkka.simoGaussian quadrature for the integral \(\int_\Omega f(x)w(x)dx\) is captured by the Golub-Meurant formula as the top left element in the matrix \(f(J_n)\) where \(J_n\) is the Jacobi matrix for the underlying orthonormal polynomial sequence. This matrix is the truncated form of the matrix representation of the multiplication operator \(M(x):f(x)\mapsto xf(x)\), with respect to the orthogonal polynomial basis. In this paper the idea is extended considerably in three essential ways. (1) One may replace \(M(x)\) by \(M(g(x)): f(x)\mapsto g(x)f(x)\) for a bounded \(g\) in a separable Hilbert space. (2) The polynomial basis can be replaced by any, not necessary orthogonal, basis. (3) \(\Omega\) can be a subset of \(\mathbb{R}^d\). The result will always be a scalar integration problem for \(f(x)\). The interpretation that can be given to the entries of the matrix \(f(M(g))\) is explored. Convergence and other properties are discussed. The paper is very readable with one-liner proofs referring to known literature. The possibilities are illustrated with some numerical examples that need symbolic computation when the condition is bad.
Reviewer: Adhemar Bultheel (Leuven)Interpolating self consistent field for eigenvector nonlinearitieshttps://zbmath.org/1503.651132023-03-23T18:28:47.107421Z"Claes, Rob"https://zbmath.org/authors/?q=ai:claes.rob"Meerbergen, Karl"https://zbmath.org/authors/?q=ai:meerbergen.karlSummary: One of the most common approaches for solving eigenvalue problems with eigenvector nonlinearities (NEPv) is the Self Consistent Field (SCF) method that uses a vector from the previous iteration to build a zeroth order approximation of the nonlinearity. This approach is often slow and unreliable, which is why most applications use mixing scheme extensions of SCF, that use a linear combination of multiple previous iterates to build the zeroth order approximation. In this paper, we present a method that uses multiple previous iterates to build an interpolating first order approximation of the nonlinearity. It can be shown that this method converges superlinearly to the desired eigenpair.Conditioning and error analysis of nonlocal operators with local boundary conditionshttps://zbmath.org/1503.651142023-03-23T18:28:47.107421Z"Aksoylu, Burak"https://zbmath.org/authors/?q=ai:aksoylu.burak"Kaya, Adem"https://zbmath.org/authors/?q=ai:kaya.ademSummary: We study the conditioning and error analysis of novel nonlocal operators in 1D with local boundary conditions. These operators are used, for instance, in peridynamics (PD) and nonlocal diffusion. The original PD operator uses nonlocal boundary conditions (BC). The novel operators agree with the original PD operator in the bulk of the domain and simultaneously enforce local periodic, antiperiodic, Neumann, or Dirichlet BC. We prove sharp bounds for their condition numbers in the parameter \(\delta\) only, the size of nonlocality. We accomplish sharpness both rigorously and numerically. We also present an error analysis in which we use the Nyström method with the trapezoidal rule for discretization. Using the sharp bounds, we prove that the error bound scales like \(\mathcal{O}(h^2 \delta^{-2})\) and verify the bound numerically.
The conditioning analysis of the original PD operator was studied by \textit{B. Aksoylu} and \textit{Z. Unlu} [SIAM J. Numer. Anal. 52, No. 2, 653--677 (2014; Zbl 1297.65201)]. For that operator, we had to resort to a discretized form because we did not have access to the eigenvalues of the analytic operator. Due to analytical construction, we now have direct access to the explicit expression of the eigenvalues of the novel operators in terms of \(\delta\). This gives us a big advantage in finding sharp bounds for the condition number without going to a discretized form and makes our analysis easily accessible. We prove that the novel operators have ill-conditioning indicated by \(\delta^{-2}\) sharp bounds. For the original PD operator, we had proved the similar \(\delta^{-2}\) ill-conditioning when the mesh size approaches 0. From the conditioning perspective, we conclude that the modification made to the original PD operator to obtain the novel operators that accommodate local BC is minor. Furthermore, the sharp \(\delta^{-2}\) bounds shed light on the role of \(\delta\) in nonlocal problems.On the local convergence study for an efficient \(k\)-step iterative methodhttps://zbmath.org/1503.651152023-03-23T18:28:47.107421Z"Amat, S."https://zbmath.org/authors/?q=ai:amat.sergio-p"Argyros, I. K."https://zbmath.org/authors/?q=ai:argyros.ioannis-konstantinos"Busquier, S."https://zbmath.org/authors/?q=ai:busquier.sonia"Hernández-Verón, M. A."https://zbmath.org/authors/?q=ai:hernandez-veron.miguel-angel"Martínez, E."https://zbmath.org/authors/?q=ai:martinez.eulaliaSummary: This paper is devoted to a family of Newton-like methods with frozen derivatives used to approximate a locally unique solution of an equation. The methods have high order of convergence but only using first order derivatives. Moreover only one LU decomposition is required in each iteration. In particular, the methods are real alternatives to the classical Newton method. We present a local convergence analysis based on hypotheses only on the first derivative. These types of local results were usually proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these types of methods [\textit{S. Amat} et al., Appl. Math. Lett. 25, No. 12, 2209--2217 (2012; Zbl 1252.65090); \textit{M. S. Petković} et al., Multipoint methods for solving nonlinear equations. Amsterdam: Elsevier/Academic Press (2013; Zbl 1286.65060); \textit{J. F. Traub}, Iterative methods for the solution of equations. Englewood Cliffs: Prentice-Hall, Inc. (1964; Zbl 0121.11204)]. We apply these methods to an equation related to the nonlinear complementarity problem. Finally, we find the most efficient method in the family for this problem and we perform a theoretical and a numerical study for it.Local convergence comparison between frozen Kurchatov and Schmidt-Schwetlick-Kurchatov solvers with applicationshttps://zbmath.org/1503.651182023-03-23T18:28:47.107421Z"Moysi, A."https://zbmath.org/authors/?q=ai:moysi.a"Argyros, M."https://zbmath.org/authors/?q=ai:argyros.michael-i"Argyros, I. K."https://zbmath.org/authors/?q=ai:argyros.ioannis-konstantinos"Magreñán, Á. A."https://zbmath.org/authors/?q=ai:magrenan.angel-alberto"Sarría, Í."https://zbmath.org/authors/?q=ai:sarria.inigo"González, D."https://zbmath.org/authors/?q=ai:gonzalez.danielSummary: In this work we are going to use the Kurchatov-Schmidt-Schwetlick-like solver (KSSLS) and the Kurchatov-like solver (KLS) to locate a zero, denoted by \(x^\ast\) of operator \(F\). We define \(F\) as \(F : D \subseteq B_1 \longrightarrow B_2\) where \(B_1\) and \(B_2\) stand for Banach spaces, \( D \subseteq B_1\) be a convex set and \(F\) be a differentiable mapping according to Fréchet. Under these conditions, for all \(n = 0 , 1 , 2 , \ldots\) and \(0 \leq i \leq m - 1\) using Taylor expansion, KSSLS and KLS, when \(B_1 = B_2\) and high order derivatives and divided differences not appearing in these solvers, the results obtained are the restart of the utilization of these iterative solvers. Moreover, we show under the same set of conditions that the local convergence radii are the same, the uniqueness balls coincide but the error estimates on \(\| x_n - x_\ast \|\) differ. It is worth noticing our results improve the corresponding ones [\textit{M. Grau-Sánchez} et al., J. Comput. Appl. Math. 235, No. 6, 1739--1743 (2011; Zbl 1204.65051); \textit{V. A. Kurcatov}, Sov. Math., Dokl. 835--838 (1971; Zbl 0252.65044); translation from Dokl. Akad. Nauk SSSR 198, 524--526 (1971); \textit{S. M. Shakhno}, J. Comput. Appl. Math. 231, No. 1, 222--235 (2009; Zbl 1225.65062)]. Finally, we apply our theoretical results to some numerical examples in order to prove the improvement.A new interpretation of (Tikhonov) regularizationhttps://zbmath.org/1503.651192023-03-23T18:28:47.107421Z"Gerth, Daniel"https://zbmath.org/authors/?q=ai:gerth.danielSummary: Tikhonov regularization with square-norm penalty for linear forward operators has been studied extensively in the literature. However, the results on convergence theory are based on technical proofs and sometimes difficult to interpret. It is also often not clear how those results translate into the discrete, numerical setting. In this paper we present a new strategy to study the properties of a regularization method on the example of Tikhonov regularization. The technique is based on the well-known observation that Tikhonov regularization approximates the unknown exact solution in the range of the adjoint of the forward operator. This is closely related to the concept of approximate source conditions, which we generalize to describe not only the approximation of the unknown solution, but also noise-free and noisy data; all from the same source space. Combining these three approximation results we derive the well-known convergence results in a concise way and improve the understanding by tightening the relation between concepts such as convergence rates, parameter choice, and saturation. The new technique is not limited to Tikhonov regularization, it can be applied also to iterative regularization, which we demonstrate by relating Tikhonov regularization and Landweber iteration. All results are accompanied by numerical examples.Optimal-order convergence of Nesterov acceleration for linear ill-posed problemshttps://zbmath.org/1503.651202023-03-23T18:28:47.107421Z"Kindermann, Stefan"https://zbmath.org/authors/?q=ai:kindermann.stefanSummary: We show that Nesterov acceleration is an optimal-order iterative regularization method for linear ill-posed problems provided that a parameter is chosen accordingly to the smoothness of the solution. This result is proven both for an \textit{a priori} stopping rule and for the discrepancy principle under Hölder source conditions. Furthermore, some converse results and logarithmic rates are verified. The essential tool to obtain these results is a representation of the residual polynomials via Gegenbauer polynomials.Convergence rates for iteratively regularized Gauss-Newton method subject to stability constraintshttps://zbmath.org/1503.651212023-03-23T18:28:47.107421Z"Mittal, Gaurav"https://zbmath.org/authors/?q=ai:mittal.gaurav"Giri, Ankik Kumar"https://zbmath.org/authors/?q=ai:giri.ankik-kumarSummary: In this paper we formulate the convergence rates of the iteratively regularized Gauss-Newton method by defining the iterates via convex optimization problems in a Banach space setting. We employ the concept of conditional stability to deduce the convergence rates in place of the well known concept of variational inequalities. To validate our abstract theory, we also discuss an ill-posed inverse problem that satisfies our assumptions. We also compare our results with the existing results in the literature.Variational regularization theory based on image space approximation rateshttps://zbmath.org/1503.651222023-03-23T18:28:47.107421Z"Miller, Philip"https://zbmath.org/authors/?q=ai:miller.philip-hSummary: We present a new approach to convergence rate results for variational regularization. Avoiding Bregman distances and using image space approximation rates as source conditions we prove a nearly minimax theorem showing that the modulus of continuity is an upper bound on the reconstruction error up to a constant. Applied to Besov space regularization we obtain convergence rate results for \(0, 2, q\)- and \(0, p, p\)-penalties without restrictions on \(p,q\in (1,\infty)\). Finally we prove equivalence of Hölder-type variational source conditions, bounds on the defect of the Tikhonov functional, and image space approximation rates.Weak and strong convergence Bregman extragradient schemes for solving pseudo-monotone and non-Lipschitz variational inequalitieshttps://zbmath.org/1503.651342023-03-23T18:28:47.107421Z"Jolaoso, Lateef Olakunle"https://zbmath.org/authors/?q=ai:jolaoso.lateef-olakunle"Aphane, Maggie"https://zbmath.org/authors/?q=ai:aphane.maggieSummary: In this paper, we introduce Bregman subgradient extragradient methods for solving variational inequalities with a pseudo-monotone operator which are not necessarily Lipschitz continuous. Our algorithms are constructed such that the stepsizes are determined by an Armijo line search technique, which improves the convergence of the algorithms without prior knowledge of any Lipschitz constant. We prove weak and strong convergence results for approximating solutions of the variational inequalities in real reflexive Banach spaces. Finally, we provide some numerical examples to illustrate the performance of our algorithms to related algorithms in the literature.Inertial hybrid algorithm for variational inequality problems in Hilbert spaceshttps://zbmath.org/1503.651362023-03-23T18:28:47.107421Z"Tian, Ming"https://zbmath.org/authors/?q=ai:tian.ming"Jiang, Bing-Nan"https://zbmath.org/authors/?q=ai:jiang.bingnanSummary: For a variational inequality problem, the inertial projection and contraction method have been studied. It has a weak convergence result. In this paper, we propose a strong convergence iterative method for finding a solution of a variational inequality problem with a monotone mapping by projection and contraction method and inertial hybrid algorithm. Our result can be used to solve other related problems in Hilbert spaces.Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination conditionhttps://zbmath.org/1503.652112023-03-23T18:28:47.107421Z"Huntul, Mousa J."https://zbmath.org/authors/?q=ai:huntul.mousa-j"Oussaeif, Taki-Eddine"https://zbmath.org/authors/?q=ai:taki-eddine.oussaeif"Tamsir, Mohammad"https://zbmath.org/authors/?q=ai:tamsir.mohammad"Aiyashi, Mohammed A."https://zbmath.org/authors/?q=ai:aiyashi.mohammed-aSummary: In this work, the solvability for an inverse problem of a nonlinear parabolic equation with nonlocal integral overdetermination supplementary condition is examined. The proof of the existence and uniqueness of the solution of the inverse nonlinear parabolic problem upon the data is established by using the fixed-point technique. In addition, the inverse problem is investigated by using the cubic B-spline collocation technique together with the Tikhonov regularization. The resulting nonlinear system of parabolic equation is approximated using the MATLAB subroutine \textit{lsqnonlin}. The obtained results demonstrate the accuracy and efficiency of the technique, and the stability of the approximate solutions even in the existence of noisy data. The stability analysis is also conducted for the discretized system of the direct problem.Improved convergence of the Arrow-Hurwicz iteration for the Navier-Stokes equation via grad-div stabilization and Anderson accelerationhttps://zbmath.org/1503.652952023-03-23T18:28:47.107421Z"Geredeli, Pelin G."https://zbmath.org/authors/?q=ai:geredeli.pelin-guven"Rebholz, Leo G."https://zbmath.org/authors/?q=ai:rebholz.leo-g"Vargun, Duygu"https://zbmath.org/authors/?q=ai:vargun.duygu"Zytoon, Ahmed"https://zbmath.org/authors/?q=ai:zytoon.ahmedSummary: We consider two modifications of the Arrow-Hurwicz (AH) iteration for solving the incompressible steady Navier-Stokes equations for the purpose of accelerating the algorithm: grad-div stabilization, and Anderson acceleration. AH is a classical iteration for general saddle point linear systems and it was later extended to Navier-Stokes iterations in the 1970's which has recently come under study again. We apply recently developed ideas for grad-div stabilization and divergence-free finite element methods along with Anderson acceleration of fixed point iterations to AH in order to improve its convergence. Analytical and numerical results show that each of these methods improves AH convergence, but the combination of them yields an efficient and effective method that is competitive with more commonly used solvers.A fixed-point iteration method for high frequency Helmholtz equationshttps://zbmath.org/1503.652982023-03-23T18:28:47.107421Z"Luo, Songting"https://zbmath.org/authors/?q=ai:luo.songting"Liu, Qing Huo"https://zbmath.org/authors/?q=ai:liu.qinghuoSummary: For numerically solving the high frequency Helmholtz equation, the conventional finite difference and finite element methods based on discretizing the equation on meshes usually suffer from the numerical dispersion errors (`pollution effect') that require very refined meshes [\textit{I. M. Babuska} and \textit{S. A. Sauter}, SIAM Rev. 42, No. 3, 451--484 (2000; Zbl 0956.65095)]. Asymptotic methods like geometrical optics provide an alternative way to compute the solutions without `pollution effect', but they generally can only compute locally valid approximations for the solutions and fail to capture the caustics faithfully. In order to obtain globally valid solutions efficiently without `pollution effect', we transfer the problem into a fixed-point problem related to an exponential operator, and the associated functional evaluations are achieved by unconditionally stable operator-splitting based pseudospectral schemes such that large step sizes are allowed to reach the approximated fixed point efficiently for certain prescribed accuracy requirement. And the Anderson acceleration is incorporated to accelerate the convergence. Both two-dimensional and three-dimensional numerical experiments are presented to demonstrate the method.New approach to solve two-dimensional Fredholm integral equationshttps://zbmath.org/1503.653192023-03-23T18:28:47.107421Z"Kazemi, Manochehr"https://zbmath.org/authors/?q=ai:kazemi.manochehr"Golshan, Hamid Mottaghi"https://zbmath.org/authors/?q=ai:golshan.hamid-mottaghi"Ezzati, Reza"https://zbmath.org/authors/?q=ai:ezzati.reza"Sadatrasoul, Mohsen"https://zbmath.org/authors/?q=ai:sadatrasoul.mohsenSummary: In this paper, we present an efficient iterative method based on quadrature formula to solve two-dimensional nonlinear Fredholm integral equations. We investigate the convergence analysis of the proposed method. The error estimation is given in terms of uniform and partial modulus of continuity. In addition, we show the numerical stability analysis of the method with respect to the choice of the first iteration. Finally, some numerical experiments confirm the theoretical results and illustrate the accuracy of the method.A royal road to quantum theory (or thereabouts), extended abstracthttps://zbmath.org/1503.810092023-03-23T18:28:47.107421Z"Wilce, Alexander"https://zbmath.org/authors/?q=ai:wilce.alexanderSummary: A representation of finite-dimensional probabilistic models in terms of formally real Jordan algebras is obtained, in a strikingly easy way, from simple assumptions. This provides a framework in which real, complex and quaternionic quantum mechanics can be treated on an equal footing, and allows some (but not too much) room for other alternatives. This is based on earlier work [the author, ``Conjugates, filters and quantum mechanics'', Preprint, \url{arXiv:1206:2897}], but the development here is further simplified, and also extended in several ways. I also discuss the possibilities for organizing probabilistic models, subject to the assumptions discussed here, into symmetric monoidal categories, showing that such a category will automatically have a dagger-compact structure. (Recent joint work with \textit{H. Barnum} and \textit{M. Graydon} [``Some nearly quantum theories'', Preprint, \url{arXiv:1507.06278}] exhibits several categories of this kind.)
For the entire collection see [Zbl 1434.03013].An interaction-free quantum measurement-driven enginehttps://zbmath.org/1503.810122023-03-23T18:28:47.107421Z"Elouard, Cyril"https://zbmath.org/authors/?q=ai:elouard.cyril"Waegell, Mordecai"https://zbmath.org/authors/?q=ai:waegell.mordecai"Huard, Benjamin"https://zbmath.org/authors/?q=ai:huard.benjamin"Jordan, Andrew N."https://zbmath.org/authors/?q=ai:jordan.andrew-nSummary: Recently highly-efficient quantum engines were devised by exploiting the stochastic energy changes induced by quantum measurement. Here we show that such an engine can be based on an interaction-free measurement, in which the meter seemingly does not interact with the measured object. We use a modified version of the Elitzur-Vaidman bomb tester, an interferometric setup able to detect the presence of a bomb triggered by a single photon without exploding it. In our case, a quantum bomb subject to a gravitational force is initially in a superposition of being inside and outside one of the 60G10
erometer arms. We show that the bomb can be lifted without blowing up. This occurs when a photon traversing the interferometer is detected at a port that is always dark when the bomb is located outside the arm. The required potential energy is provided by the photon (which plays the role of the meter) even though it was not absorbed by the bomb. A natural interpretation is that the photon traveled through the arm which does not contain the bomb -- otherwise the bomb would have exploded -- but it implies the surprising conclusion that the energy exchange occurred at a distance despite a local interaction Hamiltonian. We use the weak value formalism to support this interpretation and find evidence of contextuality. Regardless of interpretation, this interaction-free quantum measurement engine is able to lift the most sensitive bomb without setting it off.On \(L^2\) convergence of the Hamiltonian Monte Carlohttps://zbmath.org/1503.810262023-03-23T18:28:47.107421Z"Ghosh, Soumyadip"https://zbmath.org/authors/?q=ai:ghosh.soumyadip"Lu, Yingdong"https://zbmath.org/authors/?q=ai:lu.yingdong"Nowicki, Tomasz"https://zbmath.org/authors/?q=ai:nowicki.tomasz-j|nowicki.tomaszSummary: We represent the abstract Hamiltonian (Hybrid) Monte Carlo (HMC) algorithm as iterations of an operator on densities in a Hilbert space, and recognize two invariant properties of Hamiltonian motion sufficient for convergence. Under a mild coverage assumption, we present a proof of strong convergence of the algorithm to the target density. The proof relies on the self-adjointness of the operator, and we extend the result to the general case of the motions beyond Hamiltonian ones acting on a finite dimensional space, to the motions acting an abstract space equipped with a reference measure, as long as they satisfy the two sufficient properties. For standard Hamiltonian motion, the convergence is also geometric in the case when the target density satisfies a log-convexity condition.A Glazman-Povzner-Wienholtz theorem on graphshttps://zbmath.org/1503.810352023-03-23T18:28:47.107421Z"Kostenko, Aleksey"https://zbmath.org/authors/?q=ai:kostenko.aleksey-s"Malamud, Mark"https://zbmath.org/authors/?q=ai:malamud.mark-m"Nicolussi, Noema"https://zbmath.org/authors/?q=ai:nicolussi.noemaSummary: The Glazman-Povzner-Wienholtz theorem states that the semiboundedness of a Schrödinger operator, when combined with suitable local regularity assumptions on its potential and the completeness of the underlying manifold, guarantees its essential self-adjointness. Our aim is to extend this result to Schrödinger operators on graphs. We first obtain the corresponding theorem for Schrödinger operators on metric graphs, allowing in particular distributional potentials which are locally \(H^{- 1}\). Moreover, we exploit recently discovered connections between Schrödinger operators on metric graphs and weighted graphs in order to prove a discrete version of the Glazman-Povzner-Wienholtz theorem.Scattering in quantum dots via noncommutative rational functionshttps://zbmath.org/1503.810362023-03-23T18:28:47.107421Z"Erdős, László"https://zbmath.org/authors/?q=ai:erdos.laszlo"Krüger, Torben"https://zbmath.org/authors/?q=ai:kruger.torben"Nemish, Yuriy"https://zbmath.org/authors/?q=ai:nemish.yurii-nikolaevichSummary: In the customary random matrix model for transport in quantum dots with \(M\) internal degrees of freedom coupled to a chaotic environment via \(N\ll M\) channels, the density \(\rho\) of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large \(N\) regime allowing for (i) arbitrary ratio \(\phi:=N/M\le 1\); and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit \(\phi \rightarrow 0\), we recover the formula for the density \(\rho\) that \textit{C. W. J. Beenakker} [``Random-matrix theory of quantum transport'', Rev. Mod. Phys. 69, No. 3, 731--808 (1997; \url{doi:10.1103/RevModPhys.69.731})] has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker's formula persists for any \(\phi <1\) but in the borderline case \(\phi=1\) an anomalous \(\lambda^{-2/3}\) singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries.Quantifying dip-ramp-plateau for the Laguerre unitary ensemble structure functionhttps://zbmath.org/1503.810372023-03-23T18:28:47.107421Z"Forrester, Peter J."https://zbmath.org/authors/?q=ai:forrester.peter-jSummary: The ensemble average of \(| \sum_{j=1}^N e^{i k \lambda_j} |^2\) is of interest as a probe of quantum chaos, as is its connected part, the structure function. Plotting this average for model systems of chaotic spectra reveals what has been termed a dip-ramp-plateau shape. Generalising earlier work of Brézin and Hikami for the Gaussian unitary ensemble, it is shown how the average in the case of the Laguerre unitary ensemble can be reduced to an expression involving the spectral density of the Jacobi unitary ensemble. This facilitates studying the large \(N\) limit, and so quantifying the dip-ramp-plateau effect. When the parameter \(a\) in the Laguerre weight \(x^a e^{-x}\) scales with \(N\), quantitative agreement is found with the characteristic features of this effect known for the Gaussian unitary ensemble. However, for the parameter \(a\) fixed, the bulk scaled structure function is shown to have the simple functional form \(\frac{2}{\pi} \text{Arctan} \, k\), and so there is no ramp-plateau transition.The foundations of quantum field theory.https://zbmath.org/1503.810512023-03-23T18:28:47.107421Z"Glimm, James"https://zbmath.org/authors/?q=ai:glimm.james-gFrom the text: In this article we will deal with the mathematical consistency of Quantum Field Theory, or in other words with the problem of finding rigorously defined mathematical objects which correspond to the operators discussed in Quantum Field Theory.The \(\lambda(\varphi^4)_2\) quantum field theory without cutoffs. III: The physical vacuum.https://zbmath.org/1503.810522023-03-23T18:28:47.107421Z"Glimm, James"https://zbmath.org/authors/?q=ai:glimm.james-g"Jaffe, Arthur"https://zbmath.org/authors/?q=ai:jaffe.arthur-mFrom the text: In this series of papers [I: Zbl 0177.28203; II: Zbl 0191.27005] we construct a quantum field theory model. This model describes a spin-zero boson field \(\varphi\) with a nonlinear \(\varphi^4\) selfinteraction in two dimensional space time.
The section headings areas follows:
\begin{itemize}
\item[1.] Introduction.
\item[2.] The physical representation and renormalization.
\item[3.] Local number operators.
\begin{itemize}
\item[3.1.] One particle operators.
\item[3.2.] Quantized operators in Fock space.
\item[3.3.] Vacuum expectation values of local number operators.
\end{itemize}
\item[4.] Norm compactness of the approximate vacuums.
\item[5.] The vacuum self energy per unit volume is finite.
\end{itemize}
The introduction gives a perfect overview of the results of this paper. For the last Part IV in this series see the authors [J. Math. Phys. 13, 1568--1584 (1972)].The \(\lambda\varphi^4_2\) quantum field theory without cutoffs. IV: Perturbations of the Hamiltonian.https://zbmath.org/1503.810532023-03-23T18:28:47.107421Z"Glimm, James"https://zbmath.org/authors/?q=ai:glimm.james-g"Jaffe, Arthur"https://zbmath.org/authors/?q=ai:jaffe.arthur-mSummary: We introduce an inductive method to estimate the shift \(\delta E\) in the vacuum energy, caused by a perturbation \(\delta H\) of the \(\mathcal P(\varphi)_2\) Hamiltonian \(H\). We prove that if \(\delta H\) equals the field bilinear form \(\varphi(x,t)\), then \(\delta E\) is finite. We show that the vacuum expectation values of products of fields (Wightman functions) exist and are tempered distributions. They determine, via the reconstruction theorem, essentially self-adjoint field operators \(\varphi(f)\), for real test functions \(f\in\mathcal S(\mathbb R^2)\). We also bound the perturbation of the \(\mathcal P(\varphi)_2\) Hamiltonian by a polynomial \((\mathcal P_1(\varphi))(h) = \delta H\), so long as \(\mathcal P + \mathcal P_1\) is formally positive. In that case, and with \(\Vert h\Vert_\infty \le 1\), \(\delta E\) is bounded by \(\text{const}(1 + \text{diam supp}\, h)\).Lorentzian Toda field theorieshttps://zbmath.org/1503.810552023-03-23T18:28:47.107421Z"Fring, Andreas"https://zbmath.org/authors/?q=ai:fring.andreas"Whittington, Samuel"https://zbmath.org/authors/?q=ai:whittington.samuelSummary: We propose several different types of construction principles for new classes of Toda field theories based on root systems defined on Lorentzian lattices. In analogy to conformal and affine Toda theories based on root systems of semi-simple Lie algebras, also their Lorentzian extensions come about in conformal and massive variants. We carry out the Painlevé integrability test for the proposed theories, finding in general only one integer valued resonance corresponding to the energy-momentum tensor. Thus most of the Lorentzian Toda field theories are not integrable, as the remaining resonances, that grade the spins of the W-algebras in the semi-simple cases, are either non-integer or complex valued. We analyze in detail the classical mass spectra of several massive variants. Lorentzian Toda field theories may be viewed as perturbed versions of integrable theories equipped with an algebraic framework.Gravitational model of compact spherical Reissner-Nordström-type star under \(f(R, T)\) gravityhttps://zbmath.org/1503.830042023-03-23T18:28:47.107421Z"Islam, Safiqul"https://zbmath.org/authors/?q=ai:islam.safiqul"Kumar, Praveen"https://zbmath.org/authors/?q=ai:kumar.praveen(no abstract)Sharp decay estimates for massless Dirac fields on a Schwarzschild backgroundhttps://zbmath.org/1503.830092023-03-23T18:28:47.107421Z"Ma, Siyuan"https://zbmath.org/authors/?q=ai:ma.siyuan"Zhang, Lin"https://zbmath.org/authors/?q=ai:zhang.lin.4Summary: We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin \(s = \pm \frac{ 1}{ 2}\) components of the Dirac field a uniform bound of a positive definite energy and an integrated local energy decay estimate from a symmetric hyperbolic wave system. Based on these estimates, we further show that these components have globally pointwise decay \(f v^{- 3 / 2 - s} \tau^{- 5 / 2 + s}\) as both an upper and a lower bound outside the black hole, with function \(f\) finite and explicitly expressed in terms of the initial data and the coordinates. This establishes the validity of the conjectured Price's law for massless Dirac fields outside a Schwarzschild black hole.Existence and uniqueness of solutions of the semiclassical Einstein equation in cosmological modelshttps://zbmath.org/1503.830192023-03-23T18:28:47.107421Z"Meda, Paolo"https://zbmath.org/authors/?q=ai:meda.paolo"Pinamonti, Nicola"https://zbmath.org/authors/?q=ai:pinamonti.nicola"Siemssen, Daniel"https://zbmath.org/authors/?q=ai:siemssen.danielSummary: We prove existence and uniqueness of solutions of the semiclassical Einstein equation in flat cosmological spacetimes driven by a quantum massive scalar field with arbitrary coupling to the scalar curvature. In the semiclassical approximation, the backreaction of matter to curvature is taken into account by equating the Einstein tensor to the expectation values of the stress-energy tensor in a suitable state. We impose initial conditions for the scale factor at finite time, and we show that a regular state for the quantum matter compatible with these initial conditions can be chosen. Contributions with derivative of the coefficient of the metric higher than the second are present in the expectation values of the stress-energy tensor and the term with the highest derivative appears in a non-local form. This fact forbids a direct analysis of the semiclassical equation, and in particular, standard recursive approaches to approximate the solution fail to converge. In this paper, we show that, after partial integration of the semiclassical Einstein equation in cosmology, the non-local highest derivative appears in the expectation values of the stress-energy tensor through the application of a linear unbounded operator which does not depend on the details of the chosen state. We prove that an inversion formula for this operator can be found, furthermore, the inverse happens to be more regular than the direct operator and it has the form of a retarded product, hence, causality is respected. The found inversion formula applied to the traced Einstein equation has thus the form of a fixed point equation. The proof of local existence and uniqueness of the solution of the semiclassical Einstein equation is then obtained applying the Banach fixed point theorem.Inertial proximal incremental aggregated gradient method with linear convergence guaranteeshttps://zbmath.org/1503.901372023-03-23T18:28:47.107421Z"Zhang, Xiaoya"https://zbmath.org/authors/?q=ai:zhang.xiaoya"Peng, Wei"https://zbmath.org/authors/?q=ai:peng.wei"Zhang, Hui"https://zbmath.org/authors/?q=ai:zhang.hui.5Summary: In this paper, we propose an inertial version of the Proximal Incremental Aggregated Gradient (abbreviated by iPIAG) method for minimizing the sum of smooth convex component functions and a possibly nonsmooth convex regularization function. First, we prove that iPIAG converges linearly under the gradient Lipschitz continuity and the strong convexity, along with an upper bound estimation of the inertial parameter. Then, by employing the recent Lyapunov-function-based method, we derive a weaker linear convergence guarantee, which replaces the strong convexity by the quadratic growth condition. At last, we present two numerical tests to illustrate that iPIAG outperforms the original PIAG.Functional-differential games with nonatomic difference operatorhttps://zbmath.org/1503.910332023-03-23T18:28:47.107421Z"Vlasenko, L. A."https://zbmath.org/authors/?q=ai:vlasenko.l-a"Rutkas, A. G."https://zbmath.org/authors/?q=ai:rutkas.a-g"Chikrii, A. O."https://zbmath.org/authors/?q=ai:chikrii.arkadii-aAuthors' abstract: We study a differential pursuit game in a system dynamically described by a linear functional differential equation. The coefficients of the equation are closed linear operators acting in Hilbert spaces. The operator at the derivative of state depends on the current time and is, generally speaking, not invertible. Our main assumption is a restriction imposed on the characteristic operator pencil of the equation on a ray of the real positive semiaxis. The solutions of the equation are represented with the help of the formula of variation of constants in which the effect of delay is taken into account as a result of summation of shift-type operators. To establish conditions under which the dynamic vector of the system approaches a cylindrical terminal set, we use constraints imposed on the support functionals of two sets determined by the behaviors of the pursuer and evader. We also present an example of differential game in a pseudoparabolic system described by a partial functional-differential equation.
Reviewer: Catherine Rainer (Brest)Internally Hankel \(k\)-positive systemshttps://zbmath.org/1503.930272023-03-23T18:28:47.107421Z"Grussler, Christian"https://zbmath.org/authors/?q=ai:grussler.christian"Burghi, Thiago"https://zbmath.org/authors/?q=ai:burghi.thiago-b"Sojoudi, Somayeh"https://zbmath.org/authors/?q=ai:sojoudi.somayehSummary: There has been an increased interest in the variation diminishing properties of controlled linear time-invariant (LTI) systems and time-varying linear systems without inputs. In controlled LTI systems, these properties have recently been studied from the external perspective of \(k\)-positive Hankel operators. Such systems have Hankel operators that diminish the number of sign changes (the variation) from past input to future output if the input variation is at most \(k-1\). For \(k=1\), this coincides with the classical class of externally positive systems. For linear systems without inputs, the focus has been on the internal perspective of \(k\)-positive state-transition matrices, which diminish the variation of the initial system state. In the LTI case and for \(k=1\), this corresponds to the classical class of (unforced) positive systems. This paper bridges the gap between the internal and external perspectives of \(k\)-positivity by analyzing \textit{internally} Hankel \(k\)-positive systems, which we define as state-space LTI systems where controllability and observability operators as well as the state-transition matrix are \(k\)-positive. We show that the existing notions of external Hankel and internal \(k\)-positivity are subsumed under internal Hankel \(k\)-positivity, and we derive tractable conditions for verifying this property in the form of internal positivity of the first \(k\) compound systems. As such, this class provides new means to verify external Hankel \(k\)-positivity, and lays the foundation for future investigations of variation diminishing controlled linear systems. As an application, we use our framework to derive new bounds for the number of over- and undershoots in the step responses of LTI systems. Since our characterization defines a new positive realization problem, we also discuss geometric conditions for the existence of minimal internally Hankel \(k\)-positive realizations.