Recent zbMATH articles in MSC 47https://zbmath.org/atom/cc/472022-11-17T18:59:28.764376ZWerkzeugStationary measures on infinite graphshttps://zbmath.org/1496.051162022-11-17T18:59:28.764376Z"Baraviera, Alexandre"https://zbmath.org/authors/?q=ai:baraviera.alexandre-tavares"Duarte, Pedro"https://zbmath.org/authors/?q=ai:duarte.pedro"Torres, Maria Joana"https://zbmath.org/authors/?q=ai:torres.maria-joanaTwo arithmetic applications of perturbations of composition operatorshttps://zbmath.org/1496.110382022-11-17T18:59:28.764376Z"Bettin, Sandro"https://zbmath.org/authors/?q=ai:bettin.sandro"Drappeau, Sary"https://zbmath.org/authors/?q=ai:drappeau.saryAuthors' abstract: We estimate the spectral radius of perturbations of a particular family of composition operators, in a setting where the usual choices of norms do not account for the typical size of the perturbation. We apply this to estimate the growth rate of large moments of a Thue-Morse generating function and of the Stern sequence. This answers in particular a question of \textit{C. Mauduit} et al. [J. Anal. Math. 135, No. 2, 713--724 (2018; Zbl 1448.11059)].
Reviewer: Michel Rigo (Liège)Towards a fractal cohomology: spectra of Polya-Hilbert operators, regularized determinants and Riemann zeroshttps://zbmath.org/1496.111202022-11-17T18:59:28.764376Z"Cobler, Tim"https://zbmath.org/authors/?q=ai:cobler.tim"Lapidus, Michel L."https://zbmath.org/authors/?q=ai:lapidus.michel-lSummary: Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil in the case of curves and eventually, by Deligne in the case of varieties over finite fields. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field. The zeta function is then expressed as a determinant, allowing the properties of the function to relate to the properties of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis has continued to this day. In this paper we study the properties of the derivative operator \(D = \frac{d} {dz}\) on a particular family of weighted Bergman spaces of entire functions on \(\mathbb{C}\). The operator \(D\) can be naturally viewed as the ``infinitesimal shift of the complex plane'' since it generates the group of translations of \(\mathbb{C}\). Furthermore, this operator is meant to be the replacement for the Frobenius operator in the general case and is used to construct an operator associated with any given meromorphic function. With this construction, we show that for a wide class of meromorphic functions, the function can be recovered by using a regularized determinant involving the operator constructed from the meromorphic function. This is illustrated in some important special cases: rational functions, zeta functions of algebraic curves (or, more generally, varieties) over finite fields, the Riemann zeta function, and culminating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. This shows that all of the information about the given meromorphic function is encoded into the special operator we constructed. Our construction is motivated in part by work of Herichi and the second author on the infinitesimal shift of the real line (instead of the complex plane) and the associated spectral operator, as well as by earlier work and conjectures of Deninger on the role of cohomology in analytic number theory, and a conjectural ``fractal cohomology theory'' envisioned in work of the second author and of Lapidus and van Frankenhuijsen on complex fractal dimensions.
For the entire collection see [Zbl 1381.11005].Approximation in the mean on rational curveshttps://zbmath.org/1496.140592022-11-17T18:59:28.764376Z"Biswas, Shibananda"https://zbmath.org/authors/?q=ai:biswas.shibananda"Putinar, Mihai"https://zbmath.org/authors/?q=ai:putinar.mihaiFor a positive Borel measure \(\mu\), compactly supported on the complex plane and without point masses, a theorem of Thomson, subsequently generalized to rational functions by Brennan, states that the closure \(P^2(\mu)\) in \(L^2(\mu)\) of the polynomials in one complex variable is different from \(L^2(\mu)\) if and only if there exist analytic bounded point evaluations. Now, given a complex affine curve \(\mathcal{V} \in \mathbb{C}^n\) and a positive Borel measure \(\mu\) supported by a compact subset \(K\) of \(\mathcal{V}\), the goal of the paper is to establish conditions that ensure the validity of Thomson's Theorem on algebraic curves \(\mathcal{V}\), thus relating the density of polynomials in Lebesgue \(L^2\)-space to the existence of analytic bounded point evaluations. Analogues to the complex plane results of Thomson and Brennan on rational curves are also obtained.
Reviewer: Carlos Hermoso Ortíz (Madrid)Logarithmic mean of multiple accretive matriceshttps://zbmath.org/1496.150152022-11-17T18:59:28.764376Z"Luo, Wenhui"https://zbmath.org/authors/?q=ai:luo.wenhuiSummary: Using the integral representation of logarithmic mean, we define the logarithmic mean of multiple accretive matrices. When the number of matrices is two, it coincides with the recent studies carried out by \textit{F. Tan} and \textit{A. Xie} [Filomat 33, No. 15, 4747--4752 (2019; Zbl 07537437)]. Several inequalites are presented along with the studies.Nonlinear Jordan derivations of incidence algebrashttps://zbmath.org/1496.160432022-11-17T18:59:28.764376Z"Yang, Yuping"https://zbmath.org/authors/?q=ai:yang.yupingSummary: Let \((X, \leq)\) be a locally finite preordered set, \(\mathcal{R}\) a two-torsion-free commutative ring with unity and \(I(X,\mathcal{R})\) the incidence algebra of \(X\) over \(\mathcal{R}\) this paper, all the nonlinear Jordan derivations of \(I(X,\mathcal{R})\) are determined. In particular, if all the connected components of \(X\) are nontrivial, we prove that every nonlinear Jordan derivation of \(I(X,\mathcal{R})\) is proper and can be presented as a sum of an inner derivation, a transitive induced derivation, and an additive induced derivation.Fractional integral operators in linear spaceshttps://zbmath.org/1496.260062022-11-17T18:59:28.764376Z"Kuang, Jichang"https://zbmath.org/authors/?q=ai:kuang.jichangSummary: In this chapter, we introduce some new fractional integral operators and fractional area balance operators in \(n\)-dimensional linear spaces. The corresponding integral operator inequalities are established. They are significant improvement and generalizations of many known and new classes of fractional integral operators.
For the entire collection see [Zbl 1483.00042].Clarke Jacobians, Bouligand Jacobians, and compact connected sets of matriceshttps://zbmath.org/1496.260122022-11-17T18:59:28.764376Z"Bartl, David"https://zbmath.org/authors/?q=ai:bartl.david"Fabian, Marián"https://zbmath.org/authors/?q=ai:fabian.marian-j"Kolář, Jan"https://zbmath.org/authors/?q=ai:kolar.janThis note is dedicated to extending from Clarke Jacobians to Bouligand Jacobians various recent results of the first two named authors. The main statement reveals that every nonempty compact connected set of matrices can be expressed as the Bouligand Jacobian at the origin of a suitable Lipschitzian mapping which is moreover either countably piecewise affine or \(C^\infty\)-smooth outside the neighbourhoods of the origin.
Reviewer: Sorin-Mihai Grad (Paris)Some new classes of higher order strongly generalized preinvex functionshttps://zbmath.org/1496.260142022-11-17T18:59:28.764376Z"Noor, Muhammad Aslam"https://zbmath.org/authors/?q=ai:noor.muhammad-aslam"Noor, Khalida Inayat"https://zbmath.org/authors/?q=ai:noor.khalida-inayatSummary: In this chapter, we define and introduce some new concepts of the higher order strongly generalized preinvex functions and higher order strongly monotone operators with respect to two auxiliary bifunctions. Some new relationships among various concepts of higher order strongly generalized preinvex functions have been established. As special cases, one can obtain various new and known results from our results. Results obtained in this chapter can be viewed as refinement and improvement of previously known results.
For the entire collection see [Zbl 1483.00042].Some new Gronwall-Bihari type inequalities associated with generalized fractional operators and applicationshttps://zbmath.org/1496.260272022-11-17T18:59:28.764376Z"Ayari, Amira"https://zbmath.org/authors/?q=ai:ayari.amira"Boukerrioua, Khaled"https://zbmath.org/authors/?q=ai:boukerrioua.khaledSummary: In this paper, we derive some generalizations of certain Gronwall-Bihari type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators for functions in one variable, which provide explicit bounds on unknown functions. To show the feasibility of the obtained inequalities, two illustrative examples are also introduced.On quasi-affinity and reducing subspaces of multiplication operator on a certain closed subspacehttps://zbmath.org/1496.320052022-11-17T18:59:28.764376Z"Li, Yucheng"https://zbmath.org/authors/?q=ai:li.yucheng"Song, Lina"https://zbmath.org/authors/?q=ai:song.lina"Lan, Wenhua"https://zbmath.org/authors/?q=ai:lan.wenhuaSummary: Let \(H\) denote a certain closed subspace of the Bergman space \(A_\alpha^2(\mathbb{B}_n)\) (\(\alpha>-1\)) of the unit ball in \(\mathbb{C}^n \). In this paper, we prove that the operator \(\bigoplus\limits_1^m M_{z^{(s_1, \cdots, s_n)}}\) is quasi-affine to the multiplication operator \(M_{z^{(ms_1, \cdots, ms_n)}}\) on \(H\). Furthermore, the reducing subspaces of \(M_{z^{(ms_1, \cdots, ms_n)}}\) are characterized on \(H\).Existence of positive solutions for weighted fractional order differential equationshttps://zbmath.org/1496.340062022-11-17T18:59:28.764376Z"Abdo, Mohammed S."https://zbmath.org/authors/?q=ai:abdo.mohammed-salem"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Ali, Saeed M."https://zbmath.org/authors/?q=ai:ali.saeed-m"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Jarad, Fahd"https://zbmath.org/authors/?q=ai:jarad.fahdSummary: In this paper, we deliberate two classes of initial value problems for nonlinear fractional differential equations under a version weighted generalized of Caputo fractional derivative given by \textit{F. Jarad} et al. [Fractals 28, No. 8, Article ID 2040011, 12 p. (2020; Zbl 1489.26006)]. We get a formula for the solution through the equivalent fractional integral equations to the proposed problems. The existence and uniqueness of positive solutions have been obtained by using lower and upper solutions. The acquired results are demonstrated by building the upper and lower control functions of the nonlinear term with the aid of Banach and Schauder fixed point theorems. The acquired results are demonstrated by pertinent numerical examples along with the Bashforth Moulton prediction correction scheme and Matlab.A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers-Ulam stabilityhttps://zbmath.org/1496.340072022-11-17T18:59:28.764376Z"Alam, Mehboob"https://zbmath.org/authors/?q=ai:alam.mehboob"Zada, Akbar"https://zbmath.org/authors/?q=ai:zada.akbar"Popa, Ioan-Lucian"https://zbmath.org/authors/?q=ai:popa.ioan-lucian"Kheiryan, Alireza"https://zbmath.org/authors/?q=ai:kheiryan.alireza"Rezapour, Shahram"https://zbmath.org/authors/?q=ai:rezapour.shahram"Kaabar, Mohammed K. A."https://zbmath.org/authors/?q=ai:kaabar.mohammedSummary: In this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.Some properties of implicit impulsive coupled system via \(\varphi \)-Hilfer fractional operatorhttps://zbmath.org/1496.340082022-11-17T18:59:28.764376Z"Almalahi, Mohammed A."https://zbmath.org/authors/?q=ai:almalahi.mohammed-a"Panchal, Satish K."https://zbmath.org/authors/?q=ai:panchal.satish-kushabaSummary: The major goal of this work is investigating sufficient conditions for the existence and uniqueness of solutions for implicit impulsive coupled system of \(\varphi \)-Hilfer fractional differential equations (FDEs) with instantaneous impulses and terminal conditions. First, we derive equivalent fractional integral equations of the proposed system. Next, by employing some standard fixed point theorems such as Leray-Schauder alternative and Banach, we obtain the existence and uniqueness of solutions. Further, by mathematical analysis technique we investigate the Ulam-Hyers (UH) and generalized UH (GUH) stability of solutions. Finally, we provide a pertinent example to corroborate the results obtained.On a nonlocal implicit problem under Atangana-Baleanu-Caputo fractional derivativehttps://zbmath.org/1496.340092022-11-17T18:59:28.764376Z"Alnahdi, Abeer S."https://zbmath.org/authors/?q=ai:alnahdi.abeer-s"Jeelani, Mdi Begum"https://zbmath.org/authors/?q=ai:jeelani.mdi-begum"Abdo, Mohammed S."https://zbmath.org/authors/?q=ai:abdo.mohammed-salem"Ali, Saeed M."https://zbmath.org/authors/?q=ai:ali.saeed-m"Saleh, S."https://zbmath.org/authors/?q=ai:saleh.sagvan|saleh.s-q|saleh.s-a|saleh.sami|saleh.saleh-a|saleh.saad-j|saleh.sahar-mohamad|saleh.siti-hidayah-muhad|saleh.sina|saleh.samera-m|saleh.s-v|saleh.shanti-faridah|saleh.shokryaSummary: In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana-Baleanu-Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder's and Banach's fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.Multiplicity solutions for a class of \(p\)-Laplacian fractional differential equations via variational methodshttps://zbmath.org/1496.340122022-11-17T18:59:28.764376Z"Chen, Yiru"https://zbmath.org/authors/?q=ai:chen.yiru"Gu, Haibo"https://zbmath.org/authors/?q=ai:gu.haiboSummary: While it is known that one can consider the existence of solutions to boundary-value problems for fractional differential equations with derivative terms, the situations for the multiplicity of weak solutions for the \(p\)-Laplacian fractional differential equations with derivative terms are less considered. In this article, we propose a new class of \(p\)-Laplacian fractional differential equations with the Caputo derivatives. The multiplicity of weak solutions is proved by the variational method and critical point theorem. At the end of the article, two examples are given to illustrate the validity and practicality of our main results.Existence results for higher order fractional differential equations with integral boundary conditions via Kuratowski measure of noncompactneshttps://zbmath.org/1496.340152022-11-17T18:59:28.764376Z"Lachouri, Adel"https://zbmath.org/authors/?q=ai:lachouri.adel"Ardjouni, Abdelouaheb"https://zbmath.org/authors/?q=ai:ardjouni.abdelouaheb"Gouri, Nesrine"https://zbmath.org/authors/?q=ai:gouri.nesrine"Khelil, Kamel Ali"https://zbmath.org/authors/?q=ai:khelil.kamel-aliThe authors prove the existence of solutions for a higher order fractional differential equation involving a Caputo-Hadamard fractional derivative operator, supplemented with integral boundary conditions by applying the technique of Kuratowski measure of non-compactness combined with Mönch's fixed point theorem. An example illustrating the main result is presented.
Reviewer: Bashir Ahmad (Jeddah)Existence of solutions for tripled system of fractional differential equations involving cyclic permutation boundary conditionshttps://zbmath.org/1496.340182022-11-17T18:59:28.764376Z"Matar, Mohammed M."https://zbmath.org/authors/?q=ai:matar.mohammed-m"Amra, Iman Abo"https://zbmath.org/authors/?q=ai:amra.iman-abo"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-oSummary: In this paper, we introduce and study a tripled system of three associated fractional differential equations. Prior to proceeding to the main results, the proposed system is converted into an equivalent integral form by the help of fractional calculus. Our approach is based on using the addressed tripled system with cyclic permutation boundary conditions. The existence and uniqueness of solutions are investigated. We employ the Banach and Krasnoselskii fixed point theorems to prove our main results. Illustrative examples are presented to explain the theoretical results.On a coupled nonlinear fractional integro-differential equations with coupled non-local generalised fractional integral boundary conditionshttps://zbmath.org/1496.340192022-11-17T18:59:28.764376Z"Muthaiah, Subramanian"https://zbmath.org/authors/?q=ai:muthaiah.subramanianSummary: We investigate a coupled Liouville-Caputo fractional integro-differential equations (CLCFIDEs) with nonlinearities that depend on the lower order fractional derivatives of the unknown functions, and also fractional integrals of the unknown functions supplemented with the coupled non-local generalised Riemann-Liouville fractional integral (GRLFI) boundary conditions. The existence and uniqueness results have endorsed by Leray-Schauder nonlinear alternative, and Banach fixed point theorem respectively. Sufficient examples have also been supplemented to substantiate the proof, and we have discussed some variants of the given problem.Existence and uniqueness results for sequential \(\psi\)-Hilfer fractional differential equations with multi-point boundary conditionshttps://zbmath.org/1496.340202022-11-17T18:59:28.764376Z"Ntouyas, Sotiris K."https://zbmath.org/authors/?q=ai:ntouyas.sotiris-k"Vivek, Devaraj"https://zbmath.org/authors/?q=ai:vivek.devarajIn this paper, the authors establish existence and uniqueness results of solution for multi-point boundary value problems for sequential fractional differential equations involving $\psi$-Hilfer fractional derivative. The existence and uniqueness of a solution are obtained by the Banach contraction mapping principle. The nonlinear alternative of Leray-Schauder is applied to obtain the existence of at least one solution. Finally, they give some examples to illustrate their main results.
Reviewer: Thanin Sitthiwirattham (Bangkok)On the novel existence results of solutions for fractional Langevin equation associating with nonlinear fractional ordershttps://zbmath.org/1496.340232022-11-17T18:59:28.764376Z"Sintunavarat, Wutiphol"https://zbmath.org/authors/?q=ai:sintunavarat.wutiphol"Turab, Ali"https://zbmath.org/authors/?q=ai:turab.aliSummary: The Langevin equation is a core premise of the Brownian motion, which describes the development of essential processes in continuously changing situations. As a generalization of the classical one, the fractional Langevin equation offers a fractional Gaussian mechanism with two indices as parametrization, which is more flexible to model fractal systems. This paper aims to deals with a nonlinear fractional Langevin equation that involves two fractional orders with nonlocal integral boundary conditions. Our goal is to find the results related to the existence of solutions for the proposed Langevin equation by using the appropriate fixed point methods. An example is also presented to illustrate the importance of our result.Implicit nonlinear fractional differential equations of variable orderhttps://zbmath.org/1496.340262022-11-17T18:59:28.764376Z"Benkerrouche, Amar"https://zbmath.org/authors/?q=ai:benkerrouche.amar"Souid, Mohammed Said"https://zbmath.org/authors/?q=ai:souid.mohammed-said"Sitthithakerngkiet, Kanokwan"https://zbmath.org/authors/?q=ai:sitthithakerngkiet.kanokwan"Hakem, Ali"https://zbmath.org/authors/?q=ai:hakem.aliSummary: In this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.The abstract Cauchy problem in locally convex spaceshttps://zbmath.org/1496.340282022-11-17T18:59:28.764376Z"Kruse, Karsten"https://zbmath.org/authors/?q=ai:kruse.karstenSummary: We derive necessary and sufficient criteria for the uniqueness and existence of solutions of the abstract Cauchy problem in locally convex Hausdorff spaces. Our approach is based on a suitable notion of an asymptotic Laplace transform and extends results of Langenbruch beyond the class of Fréchet spaces.Uniqueness theorems for the impulsive Dirac operator with discontinuityhttps://zbmath.org/1496.340332022-11-17T18:59:28.764376Z"Zhang, Ran"https://zbmath.org/authors/?q=ai:zhang.ran.1|zhang.ran.2"Yang, Chuan-Fu"https://zbmath.org/authors/?q=ai:yang.chuanfuThe paper deals with an impulsive Dirac operator with discontinuity
\[ly:=-By^{\prime}+Q(x)y=\lambda\rho y,\qquad x\in(0,\pi),\] with the boundary conditions \[U(y):=y_{1}(0)\cos\alpha+y_{2}(0)\sin\alpha=0,\,\,V(y):=y_{1}(\pi)\cos\beta+y_{2}(\pi)\sin\beta=0\] and the jump conditions \[y_{1}(b+0)=a_{1}y_{1}(b-0),y_{2}(b+0)=a_{1}^{-1}y_{2}(b-0)+a_{2}y_{1}(b-0).\] Here \(\alpha\in(-\pi/2,\pi/2]\) and \(\beta\in(-\pi/2,\pi/2).\)
For studying the inverse problem for \(l\), the author introduced the new supplementary data to prove the uniqueness theorems. It is shown that the potential on the whole interval can be uniquely determined by these given data, which are the analogues of Borg, Marchenko and McLaughlin-Rundell theorems. The results in this paper can be viewed as the generalizing in [\textit{T. N. Harutyunyan}, Lobachevskii J. Math. 40, No. 10, 1489--1497 (2019; Zbl 1483.34028)].
Reviewer: Zhaoying Wei (Xi'an)Existence of solutions for functional boundary value problems at resonance on the half-linehttps://zbmath.org/1496.340412022-11-17T18:59:28.764376Z"Sun, Bingzhi"https://zbmath.org/authors/?q=ai:sun.bingzhi"Jiang, Weihua"https://zbmath.org/authors/?q=ai:jiang.weihuaSummary: By defining the Banach spaces endowed with the appropriate norm, constructing a suitable projection scheme, and using the coincidence degree theory due to Mawhin, we study the existence of solutions for functional boundary value problems at resonance on the half-line with \(\operatorname{dim}\operatorname{Ker}L = 1\). And an example is given to show that our result here is valid.On resonant mixed Caputo fractional differential equationshttps://zbmath.org/1496.340432022-11-17T18:59:28.764376Z"Guezane-Lakoud, Assia"https://zbmath.org/authors/?q=ai:guezane-lakoud.assia"Kılıçman, Adem"https://zbmath.org/authors/?q=ai:kilicman.ademSummary: The purpose of this study is to discuss the existence of solutions for a boundary value problem at resonance generated by a nonlinear differential equation involving both right and left Caputo fractional derivatives. The proofs of the existence of solutions are mainly based on Mawhin's coincidence degree theory. We provide an example to illustrate the main result.Existence results for first derivative dependent \(\varphi \)-Laplacian boundary value problemshttps://zbmath.org/1496.340442022-11-17T18:59:28.764376Z"Talib, Imran"https://zbmath.org/authors/?q=ai:talib.imran"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabetSummary: Our main concern in this article is to investigate the existence of solution for the boundary-value problem
\[
\begin{aligned}
& (\phi \bigl(x^\prime(t)\bigr)^\prime=g_1 \bigl(t,x(t),x^\prime(t)\bigr),\quad \forall t\in [0,1], \\
& \Upsilon_1\bigl(x(0),x(1),x^\prime(0)\bigr)=0, \\
& \Upsilon_2\bigl(x(0),x(1),x^\prime(1)\bigr)=0,
\end{aligned}
\] where \(g_1:[0,1]\times \mathbb{R}^2\rightarrow \mathbb{R}\) is an \(L^1\)-Carathéodory function, \( \Upsilon_i:\mathbb{R}^3\rightarrow \mathbb{R}\) are continuous functions, \(i=1,2\), and \(\phi :(-a,a)\rightarrow \mathbb{R}\) is an increasing homeomorphism such that \(\phi (0)=0\), for \(0< a< \infty \). We obtain the solvability results by imposing some new conditions on the boundary functions. The new conditions allow us to ensure the existence of at least one solution in the sector defined by well ordered functions. These ordered functions do not require one to check the definitions of lower and upper solutions. Moreover, the monotonicity assumptions on the arguments of boundary functions are not required in our case. An application is considered to ensure the applicability of our results.Weak and strong singularities problems to Liénard equationhttps://zbmath.org/1496.340472022-11-17T18:59:28.764376Z"Xin, Yun"https://zbmath.org/authors/?q=ai:xin.yun"Hu, Guixin"https://zbmath.org/authors/?q=ai:hu.guixinSummary: This paper is devoted to an investigation of the existence of a positive periodic solution for the following singular Liénard equation:
\[
x^{\prime\prime}+f\bigl(x(t)\bigr)x^\prime(t)+a(t)x= \frac{b(t)}{x^{\alpha}}+e(t),
\] where the external force \(e(t)\) may change sign, \( \alpha\) is a constant and \(\alpha >0\). The novelty of the present article is that for the first time we show that weak and strong singularities enables the achievement of a new existence criterion of positive periodic solution through an application of the Manásevich-Mawhin continuation theorem. Recent results in the literature are generalized and significantly improved, and we give the existence interval of periodic solution of this equation. At last, two examples and numerical solution (phase portraits and time portraits of periodic solutions of the example) are given to show applications of the theorem.Existence and Ulam-Hyers stability of positive solutions for a nonlinear model for the antarctic circumpolar currenthttps://zbmath.org/1496.340482022-11-17T18:59:28.764376Z"Fečkan, Michal"https://zbmath.org/authors/?q=ai:feckan.michal"Li, Qixiang"https://zbmath.org/authors/?q=ai:li.qixiang"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrongAuthors' abstract: In this paper, we study the existence of positive solutions for the nonlinear model of the antarctic circumpolar current and analyze their Ulam-Hyers stability. By introducing some conditions on the ocean nonlinear vorticity function depending on other functions and initial values, we establish sufficient conditions to guarantee the existence, multiplicity, location and construction of positive solutions via the approach of fixed point theorem in cones, upper and lower solutions, and monotone method, respectively. Finally, we show the existence and uniqueness and Ulam-Hyers stability of positive solutions when the ocean nonlinear vorticity function has uniformly Lipschitz continuity.
Reviewer: Hanying Feng (Shijiazhuang)Existence-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal controlhttps://zbmath.org/1496.340502022-11-17T18:59:28.764376Z"Song, Shu"https://zbmath.org/authors/?q=ai:song.shu"Zhang, Lingling"https://zbmath.org/authors/?q=ai:zhang.lingling"Zhou, Bibo"https://zbmath.org/authors/?q=ai:zhou.bibo"Zhang, Nan"https://zbmath.org/authors/?q=ai:zhang.nan.1Summary: In this thesis, we investigate a kind of impulsive fractional order differential systems involving control terms. By using a class of \(\varphi \)-concave-convex mixed monotone operator fixed point theorem, we obtain a theorem on the existence and uniqueness of positive solutions for the impulsive fractional differential equation, and the optimal control problem of positive solutions is also studied. As applications, an example is offered to illustrate our main results.Multiple positive solutions for one dimensional third order \(p\)-Laplacian equations with integral boundary conditionshttps://zbmath.org/1496.340512022-11-17T18:59:28.764376Z"Yang, You-yuan"https://zbmath.org/authors/?q=ai:yang.youyuan"Wang, Qi-ru"https://zbmath.org/authors/?q=ai:wang.qiruThis paper is devoted to the following third order equation coupled to integral boundary conditions: \[ \left\{\begin{array}{l} \left(\Phi_{p}\left(u^{\prime \prime}\right)\right)^{\prime}+h(t) f(t, u(t))=0, \quad t \in(0,1), \\
u(0)-\alpha \,u^{\prime}(0)=\int_{0}^{1} g_{1}(s) u(s) d s, \\
u(1)+\beta \,u^{\prime}(1)=\int_{0}^{1} g_{2}(s) u(s) d s, \quad u^{\prime \prime}(0)=0. \end{array}\right. \] Here \(\alpha, \beta \geq 0\), \(p>1\), \(\Phi_{p}(u)=|u|^{p-2} u\) is called the one-dimensional \(p\)-Laplacian operator and \( \Phi_{p}^{-1}(u)=\Phi_{q}(u)=|u|^{q-2} u\) with \(\frac{1}{p}+\frac{1}{q}=1.\)
Under suitable assumptions on the function \(f\), the authors deduce the existence of at least three positive solutions of the considered problem. The results are obtained from Avery-Peterson fixed point theorem. The used arguments consist on the construction of the Green's function related to the linear problem
\[
u''(t)+y(t)=0, \; t \in(0,1), \quad u(0)-\alpha\, u^{\prime}(0)=0, \; u(1)+\beta\, u^{\prime}(1)=0,
\]
and recursive methods for kernel functions.
Reviewer: Alberto Cabada (Santiago de Compostela)Existence-uniqueness and monotone iteration of positive solutions to nonlinear tempered fractional differential equation with \(p\)-Laplacian operatorhttps://zbmath.org/1496.340522022-11-17T18:59:28.764376Z"Zhou, Bibo"https://zbmath.org/authors/?q=ai:zhou.bibo"Zhang, Lingling"https://zbmath.org/authors/?q=ai:zhang.lingling"Xing, Gaofeng"https://zbmath.org/authors/?q=ai:xing.gaofeng"Zhang, Nan"https://zbmath.org/authors/?q=ai:zhang.nan.1Summary: In this paper, without requiring the complete continuity of integral operators and the existence of upper-lower solutions, by means of the sum-type mixed monotone operator fixed point theorem based on the cone \(P_h\), we investigate a kind of \(p\)-Laplacian differential equation Riemann-Stieltjes integral boundary value problem involving a tempered fractional derivative. Not only the existence and uniqueness of positive solutions are obtained, but also we can construct successively sequences for approximating the unique positive solution. As an application of our fundamental aims, we offer a realistic example to illustrate the effectiveness and practicability of the main results.The existence of solutions for Sturm-Liouville differential equation with random impulses and boundary value problemshttps://zbmath.org/1496.340532022-11-17T18:59:28.764376Z"Li, Zihan"https://zbmath.org/authors/?q=ai:li.zihan"Shu, Xiao-Bao"https://zbmath.org/authors/?q=ai:shu.xiaobao"Miao, Tengyuan"https://zbmath.org/authors/?q=ai:miao.tengyuanSummary: In this article, we consider the existence of solutions to the Sturm-Liouville differential equation with random impulses and boundary value problems. We first study the Green function of the Sturm-Liouville differential equation with random impulses. Then, we get the equivalent integral equation of the random impulsive differential equation. Based on this integral equation, we use Dhage's fixed point theorem to prove the existence of solutions to the equation, and the theorem is extended to the general second order nonlinear random impulsive differential equations. Then we use the upper and lower solution method to give a monotonic iterative sequence of the generalized random impulsive Sturm-Liouville differential equations and prove that it is convergent. Finally, we give two concrete examples to verify the correctness of the results.Existence of positive solutions for singular Dirichlet boundary value problems with impulse and derivative dependencehttps://zbmath.org/1496.340592022-11-17T18:59:28.764376Z"Jin, Fengfei"https://zbmath.org/authors/?q=ai:jin.fengfei"Yan, Baoqiang"https://zbmath.org/authors/?q=ai:yan.baoqiangSummary: In this paper, we present a theorem for some impulsive boundary problems with derivative dependence by the upper and lower solutions method. Using the theorem obtained, we consider the existence of positive solutions of some class of singular impulsive boundary problems.Correctness conditions for high-order differential equations with unbounded coefficientshttps://zbmath.org/1496.340612022-11-17T18:59:28.764376Z"Ospanov, Kordan N."https://zbmath.org/authors/?q=ai:ospanov.kordan-nauryzkhanovichSummary: We give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.On Landesman-Lazer conditions and the fundamental theorem of algebrahttps://zbmath.org/1496.340752022-11-17T18:59:28.764376Z"Amster, Pablo"https://zbmath.org/authors/?q=ai:amster.pabloIn this paper, the author deals with the differential system
\[
u'(t)+g(u(t))=p(t), \tag{1}
\]
where \(g:\mathbb{R}^2 \rightarrow \mathbb{R}^2\) is bounded and \(p\) is continuous and \(T\)-periodic. Two results for the existence of at least one \(T\)-periodic solution for system (1) are obtained when \(g\) satisfies Landesman-Lazer type conditions. The connection of the second result with the fundamental theorem of algebra is stated.
Furthermore, the author treats the following delay systems
\[
u'(t)=g(u(t))+p(t,u(t),u(t-\tau)), \tag{2}
\]
where \(\tau>0\) and \(p\) is bounded, continuous and \(T\)-periodic in the first coordinate. Under similar conditions, two theorems for the existence of at least one \(T\)-periodic solution for system (2) are proved.
Reviewer: Chun Li (Chongqing)Subharmonic solutions in reversible non-autonomous differential equationshttps://zbmath.org/1496.340772022-11-17T18:59:28.764376Z"Eze, Izuchukwu"https://zbmath.org/authors/?q=ai:eze.izuchukwu"García-Azpeitia, Carlos"https://zbmath.org/authors/?q=ai:garcia-azpeitia.carlos"Krawcewicz, Wieslaw"https://zbmath.org/authors/?q=ai:krawcewicz.wieslaw-z"Lv, Yanli"https://zbmath.org/authors/?q=ai:lv.yanliLet \(p>0\) be a fixed number. The authors are interested in subharmonic solutions of the system
\[
\ddot{u}(t) = f(t,u(t)),\ u(t)\in V
\]
where \(f(t,u)\) is a continuous map, \(p\)-periodic with respect to the temporal variable. More precisely, let \(V := \mathbb{R}^k\) and let \(p = 2 \pi\) without loss of generality. Assume that \(f: \mathbb{R}\times V \rightarrow V\) is a continuous function satisfying the following symmetry conditions:
\begin{enumerate}
\item[(\(S_1\))] For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(t+2\pi,x) = f(t,x)\) (\textit{dihedral symmetry});
\item[(\(S_2\))] For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(-t,x) = f(t,x)\) (\textit{time-reversibility});
\item[(\(S_3\))] For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(t,-x) = -f(t,x)\) (\textit{antipodal \(\mathbb{Z}_2\)-symmetry}).
\end{enumerate}
The symmetric properties of the system of study allow reformulation of the problem of existence of the subharmonic \(2\pi m\)-periodic solutions as a question about the operator equation \(\mathcal{F}(u)=0\) with \(D_m\times \mathbb{Z}_2\)-symmetries in the functional space \(\mathcal{E} := C_{2\pi m}^2(\mathbb{R};V)\). The authors introduce an additional symmetry to the system of study before proving several results on the existence and multiplicity of subharmonic solutions. Namely, let \(\Gamma\) be a finite group then
\begin{enumerate}
\item[(\(S_4\))] For all \(t \in \mathbb{R}\), \(x \in V\), and \(\sigma \in \Gamma\), we have \(f(t,\sigma x) = \sigma f(t,x)\) (\textit{\(\Gamma\)-equivariant}).
\end{enumerate}
The last condition allows for the restatement of the original problem as the \(G\)-equivariant operator equation with respect to the full group
\[
G := \Gamma \times D_m \times \mathbb{Z}_2.
\]
If the isotropy group \(G_u\) of a solution \(u\) satisfies \(\{ e \} \times \mathbb{Z}_m \times\{ 1 \} \nleq G_u\), then \(u\) is a subharmonic solution.
The authors prove several novel results in the paper. Most notably Theorems 2.6 and 2.10. The main technical tool is Brower \(\textbf{G}\)-equivariant degree theory. Given a group \(G\) corresponding \(\textbf{G}\)-equivariant Brower degree is computed using the computer algebra system GAP. In addition, the authors discuss the bifurcation problem of subharmonic solutions in the case of a system depending on an extra parameter \(\alpha\). The paper is clear and easy to follow.
Reviewer: Predrag Punosevac (Pittsburgh)Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problemshttps://zbmath.org/1496.340962022-11-17T18:59:28.764376Z"Magal, Pierre"https://zbmath.org/authors/?q=ai:magal.pierre"Seydi, Ousmane"https://zbmath.org/authors/?q=ai:seydi.ousmaneLet \(X\) be a Banach space and \(A:D(A)\to X\) be a linear operator with possibly non-dense domain. Denote \(\overline{D(A)}=X_0\). Let \(\{B(t)\}_{t\in\mathbb{R}}\subset \mathcal L(X_0,X)\) be a locally bounded and strongly continuous family of linear operators.
Assume that \(\exists \omega\in\mathbb{R}\) and \(M\geq1\) s.t. \((\omega,+\infty)\subset\rho(A)\),
\[
\|(\lambda I-A)^{-k}\|_{\mathcal{L}(X_0,X)}\leq M(\lambda-\omega)^{-k}\quad\forall \lambda>\omega,k\geq1
\]
and \(\lim_{\lambda\to\infty}(\lambda-A)^{-1}x=0\;\forall x\in X\). Suppose that for each \(\tau>0\) and \(f\in C([0,\tau],X)\) the equation \(u'_f=Au_f+f\) has a unique mild solution \(u_f\in C([0,\tau],X_0)\) with \(u_f(0)=0\). Suppose also that \(\sup_{t\in[-n,n]}\|B(t)\|_{\mathcal{L}(X_0,X)}<+\infty\) for all integer \(n\geq1\).
In \(X\) consider the differential non-homogeneous equation
\[
\frac{du(t)}{dt}=(A+B(t))u(t)+f(t),\quad t\geq t_0,\quad u(t_0)=x_0\in X_0.
\]
Then for each \(t_0\), \(x_0\in X_0\) and \(f\in C([t_0,+\infty],X)\) the equation has a unique mild solution
\[
u(t)=U_B(t,t_0)x_0+\lim_{\lambda\to+\infty} \int_{t_0}^tU_{B}(t,s)\lambda(\lambda I-A)^{-1}f(s)ds.
\]
Here \(U_B(t,s)\) is an evolution family for the related homogeneous equation.
If in addition \(\sup_{\mathbb{R}}\|B(t)\|_{\mathcal{L}(X_0,X)}<+\infty\), then the evolution \(U_B\) has an exponantial dichotomy. A related representation for the solution \(u\) is obtained.
Applications to PDEs with non-local conditions are given.
Reviewer: Nikita V. Artamonov (Moskva)Controllability of fractional evolution systems of Sobolev type via resolvent operatorshttps://zbmath.org/1496.340972022-11-17T18:59:28.764376Z"Yang, He"https://zbmath.org/authors/?q=ai:yang.he"Zhao, Yanjie"https://zbmath.org/authors/?q=ai:zhao.yanjieSummary: In this paper, we consider the nonlocal controllability of \(\alpha\in (1,2)\)-order fractional evolution systems of Sobolev type in abstract spaces. By utilizing fixed point theorems and the theory of resolvent operators we establish some sufficient conditions for the nonlocal controllability of Sobolev-type fractional evolution systems.Existence of positive periodic solutions for super-linear neutral Liénard equation with a singularity of attractive typehttps://zbmath.org/1496.341072022-11-17T18:59:28.764376Z"Zhu, Yu"https://zbmath.org/authors/?q=ai:zhu.yuSummary: In this paper, the existence of positive periodic solutions is studied for super-linear neutral Liénard equation with a singularity of attractive type
\[
\bigl(x(t)-cx(t-\sigma)\bigr)''+f\bigl(x(t) \bigr)x'(t)-\varphi(t)x^{\mu}(t)+ \frac{\alpha(t)}{x^{\gamma}(t)}=e(t),
\] where \(f:(0,+\infty)\rightarrow \mathbb{R}\), \(\varphi(t)>0\) and \(\alpha(t)>0\) are continuous functions with \(T\)-periodicity in the \(t\) variable, \(c, \gamma\) are constants with \(|c|<1, \gamma\geq1\). Many authors obtained the existence of periodic solutions under the condition \(0<\mu\leq1\), and we extend the result to \(\mu>1\) by using Mawhin's continuation theorem as well as the techniques of a priori estimates. At last, an example is given to show applications of the theorem.Globally exponential stability of piecewise pseudo almost periodic solutions for neutral differential equations with impulses and delayshttps://zbmath.org/1496.341082022-11-17T18:59:28.764376Z"He, Jianxin"https://zbmath.org/authors/?q=ai:he.jianxin"Kong, Fanchao"https://zbmath.org/authors/?q=ai:kong.fanchao"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-jose"Qiu, Hongjun"https://zbmath.org/authors/?q=ai:qiu.hongjunImpulsive differential equations are very important class of differential equations whose dynamics is very rich. In this work, authors consider a delayed impulsive neutral differential equations. The coefficients are assumed to be bounded. The main objective is to establish the existence of piecewise pseudo almost periodic solution. The techniques used are contraction mapping principle and generalized Gronwall-Bellmain inequality. Moreover, the stability of such solution is also shown. The stability is globally exponential. At the end, an example with numerical illustration is provided by the authors.
Reviewer: Syed Abbas (Mandi)A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systemshttps://zbmath.org/1496.341112022-11-17T18:59:28.764376Z"Dineshkumar, C."https://zbmath.org/authors/?q=ai:dineshkumar.c"Udhayakumar, R."https://zbmath.org/authors/?q=ai:udhayakumar.r"Vijayakumar, V."https://zbmath.org/authors/?q=ai:vijayakumar.velusamy"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: This manuscript is mainly focusing on the approximate controllability of Hilfer fractional neutral stochastic integro-differential equations. The principal results of this article are proved based on the theoretical concepts related to the fractional calculus and Schauder's fixed-point theorem. Initially, we discuss the approximate controllability of the fractional evolution system. Then, we extend our results to the concept of nonlocal conditions. Finally, we provide theoretical and practical applications to assist in the effectiveness of the discussion.Existence and uniqueness of solutions for abstract integro-differential equations with state-dependent delay and applicationshttps://zbmath.org/1496.341132022-11-17T18:59:28.764376Z"Hernandez, Eduardo"https://zbmath.org/authors/?q=ai:hernandez.eduardo-m"Rolnik, Vanessa"https://zbmath.org/authors/?q=ai:rolnik.vanessa"Ferrari, Thauana M."https://zbmath.org/authors/?q=ai:ferrari.thauana-mIn this paper, the authors study the existence and uniqueness of solutions for a general class of abstract ordinary integro-differential equation with state dependent delay. The results are obtained by using a fixed point theorem. Some examples arising in the population dynamics and in the Solow's theory of economic growth are presented.
Reviewer: Krishnan Balachandran (Coimbatore)Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operatorhttps://zbmath.org/1496.341152022-11-17T18:59:28.764376Z"Zhang, Xuping"https://zbmath.org/authors/?q=ai:zhang.xuping"Sun, Pan"https://zbmath.org/authors/?q=ai:sun.pan(no abstract)A study on controllability of impulsive fractional evolution equations via resolvent operatorshttps://zbmath.org/1496.341162022-11-17T18:59:28.764376Z"Gou, Haide"https://zbmath.org/authors/?q=ai:gou.haide"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the \((\alpha ,\beta)\)-resolvent operator, we concern with the term \(u^\prime(\cdot)\) and finding a control \(v\) such that the mild solution satisfies \(u(b)=u_b\) and \(u^\prime b)=u^\prime_b\). Finally, we present an application to support the validity study.Mathematical analysis of nonlinear integral boundary value problem of proportional delay implicit fractional differential equations with impulsive conditionshttps://zbmath.org/1496.341172022-11-17T18:59:28.764376Z"Ali, Arshad"https://zbmath.org/authors/?q=ai:ali.arshad"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Abdeljawad, Thabet"https://zbmath.org/authors/?q=ai:abdeljawad.thabet"Mahariq, Ibrahim"https://zbmath.org/authors/?q=ai:mahariq.ibrahim"Rashdan, Mostafa"https://zbmath.org/authors/?q=ai:rashdan.mostafaSummary: The current study is devoted to deriving some results about existence and stability analysis for a nonlinear problem of implicit fractional differential equations (FODEs) with impulsive and integral boundary conditions. The concerned problem involves proportional type delay term. By using Schaefer's fixed point theorem and Banach's contraction principle, the required conditions are developed. Also, different kinds of Ulam stability results are derived by using nonlinear analysis. Providing a pertinent example, we demonstrate our main results.Qualitative analysis of nonlinear coupled pantograph differential equations of fractional order with integral boundary conditionshttps://zbmath.org/1496.341182022-11-17T18:59:28.764376Z"Alrabaiah, Hussam"https://zbmath.org/authors/?q=ai:alrabaiah.hussam"Ahmad, Israr"https://zbmath.org/authors/?q=ai:ahmad.israr"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamal"Rahman, Ghaus Ur"https://zbmath.org/authors/?q=ai:rahman.ghaus-urSummary: In this research article, we develop a qualitative analysis to a class of nonlinear coupled system of fractional delay differential equations (FDDEs). Under the integral boundary conditions, existence and uniqueness for the solution of this system are carried out. With the help of Leray-Schauder and Banach fixed point theorem, we establish indispensable results. Also, some results affiliated to Ulam-Hyers (UH) stability for the system under investigation are presented. To validate the results, illustrative examples are given at the end of the manuscript.Complete asymptotic expansions of the spectral function for symbolic perturbations of almost periodic Schrödinger operators in dimension onehttps://zbmath.org/1496.341272022-11-17T18:59:28.764376Z"Galkowski, Jeffrey"https://zbmath.org/authors/?q=ai:galkowski.jeffreyThe abstract of the paper itself includes the most accurate and complete explanation about the content of the article under review. Here we quote it in full:
``In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let \(P\colon L^2(\mathbb{R})\to L^2(\mathbb{R})\) have the form \(P:=-\frac{d^2}{dx^2}+W\), where \(W\) is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, \(\mathbf{1}_{(-\infty,\lambda^2]}(P)\) has a full asymptotic expansion in powers of \(\lambda\). In particular, our class of potentials \(W\) is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supported coefficients. Moreover, the class of potentials includes certain potentials with dense pure point spectrum. The proof combines the gauge transform methods of Parnovski-Shterenberg and Sobolev with Melrose's scattering calculus.''
Reviewer: Erdogan Sen (Tekirdağ)Existence and uniqueness of solutions for a nonlinear coupled system of fractional differential equations on time scaleshttps://zbmath.org/1496.341312022-11-17T18:59:28.764376Z"Rao, Sabbavarapu Nageswara"https://zbmath.org/authors/?q=ai:rao.sabbavarapu-nageswaraSummary: In this paper, we establish the criteria for the existence and uniqueness of solutions of a two-point BVP for a system of nonlinear fractional differential equations on time scales.
\[
\begin{aligned}
\Delta_{a^{\star}}^{\alpha_1-1}x(t)&=f_1(t, x(t), y(t)),\quad t\in J:=[a,b]\cap \mathbb{T},\\
\Delta_{a^{\star}}^{\alpha_2-1}y(t)&=f_2(t, x(t), y(t)),\quad t\in J:=[a,b]\cap \mathbb{T},
\end{aligned}
\]
subject to the boundary conditions
\[
\begin{aligned}
x(a)=0,&\quad x^{\Delta}(b)=0,\quad x^{\Delta \Delta }(b)=0,\\
y(a)=0,&\quad y^{\Delta}(b)=0,\quad y^{\Delta \Delta }(b)=0.
\end{aligned}
\]
where \(\mathbb{T}\) is any time scale (nonempty closed subsets of the reals), \(2<\alpha_i<3\) and \(f_i\in C_{rd}([a,b]\times \mathbb{R}\times \mathbb{R}, \mathbb{R})\) and \(\Delta_{a^{\star }}^{\alpha_i-1}\) denotes the delta fractional derivative on time scales \(\mathbb{T}\) of order \(\alpha_i-1\) for \(i=1, 2\). By using the Banach contraction principle. Finally, an example is given to illustrate the main result.Scattering theory for transport phenomena. With a foreword by Peter Laxhttps://zbmath.org/1496.350022022-11-17T18:59:28.764376Z"Emamirad, Hassan"https://zbmath.org/authors/?q=ai:emamirad.hassan-aliIn this book, the author develops a part of the progress of the scattering theory for transport phenomena. Scattering theory is a powerful technical tool in mathematical physics, and transport theory falls within the province of statistical physics.
This book is divided into seven chapters. In Chapter 1, as a preliminaries, the author gives the theory of semigroups and C*-algebra, different types of semigroups, Schatten-von Neumann classes of operators, and some facts about ultraweak operator topology.
In Chapter 2, the author goes into the abstract scattering theory in a general Banach space, and defines the wave and scattering operators and their basic properties. Some abstract methods such as smooth perturbations and the limiting absorption principle are presented.
Chapter 3 is devoted to the transport or linearized Boltzmann equation, which is the advection equation perturbed by the sum of absorption and production operators.
In Chapter 4, the author introduces the Lax and Phillips formalism in scattering theory for the transport equation.
Chapter 5 is devoted to introduce the scattering theory for a charged particle transport problem.
Chapter 6 is the highlight of the book in which the author explains how the scattering operator for the transport problem can lead us to formulate the computerized tomography in medical science.
In the last chapter, the author introduces the Wigner function and shows how this function connects the Schrödinger equation to statistical physics and the Husimi distribution function.
Reviewer: Jiqiang Zheng (Beijing)Hardy-Littlewood-Sobolev inequalities for a class of non-symmetric and non-doubling hypoelliptic semigroupshttps://zbmath.org/1496.350192022-11-17T18:59:28.764376Z"Garofalo, Nicola"https://zbmath.org/authors/?q=ai:garofalo.nicola"Tralli, Giulio"https://zbmath.org/authors/?q=ai:tralli.giulioAim of the authors is to combine semigroup theory and nonlocal calculus for these hypoelliptic operators to prove new inequalities of Hardy-Littlewood-Sobolev-type in the case that the drift matrix has nonnegative trace.
The study is inspired by the previous study made by Elias Stein and Nicolas Th. Varopoulos in the framework of symmetric semigroups.
The authors of the article under review point out that this kind of study can be pushed to successfully handle the present degenerate non-symmetric setting.
Reviewer: Maria Alessandra Ragusa (Catania)Evolution equations with eventually positive solutionshttps://zbmath.org/1496.350302022-11-17T18:59:28.764376Z"Glück, Jochen"https://zbmath.org/authors/?q=ai:gluck.jochenSummary: We discuss linear autonomous evolution equations on function spaces which have the property that a positive initial value leads to a solution which initially changes sign, but then becomes -- and stays -- positive again for sufficiently large times. This eventual positivity phenomenon has recently been discovered for various classes of differential equations, but so far a general theory to explain this type of behaviour exists only under additional spectral assumptions.Singular perturbation and initial layer for the abstract Moore-Gibson-Thompson equationhttps://zbmath.org/1496.350332022-11-17T18:59:28.764376Z"Alvarez, Edgardo"https://zbmath.org/authors/?q=ai:alvarez.edgardo"Lizama, Carlos"https://zbmath.org/authors/?q=ai:lizama.carlosSummary: We investigate the singular limit of a third-order abstract equation in time, in relation to the complete second-order Cauchy problem on Banach spaces, where the principal operator is the generator of a strongly continuous cosine family. Assuming that an initial datum is ill prepared, the initial layer problem is studied. We show convergence, which is uniform on compact sets that stay away from zero, as long as initial data are sufficiently smooth. Our method employs suitable results from the theory of general resolvent families of operators. The abstract formulation of the third-order in time equation is inspired by the Moore-Gibson-Thompson equation, which is the linearization of a model that currently finds applications in the propagation of ultrasound waves, displacement of certain viscoelastic materials, flexible structural systems that possess internal damping and the theory of thermoelasticity.A strange non-local monotone operator arising in the homogenization of a diffusion equation with dynamic nonlinear boundary conditions on particles of critical size and arbitrary shapehttps://zbmath.org/1496.350402022-11-17T18:59:28.764376Z"Diaz, Jesús Ildefonso"https://zbmath.org/authors/?q=ai:diaz-diaz.jesus-ildefonso"Shaposhnikova, Tatiana A."https://zbmath.org/authors/?q=ai:shaposhnikova.tatiana-a"Zubova, Maria N."https://zbmath.org/authors/?q=ai:zubova.maria-nSummary: We characterize the homogenization limit of the solution of a Poisson equation in a bounded domain, either periodically perforated or containing a set of asymmetric periodical small particles and on the boundaries of these particles a nonlinear dynamic boundary condition holds involving a Hölder nonlinear \(\sigma(u)\). We consider the case in which the diameter of the perforations (or the diameter of particles) is critical in terms of the period of the structure. As in many other cases concerning critical size, a ``strange'' nonlinear term arises in the homogenized equation. For this case of asymmetric critical particles we prove that the effective equation is a semilinear elliptic equation in which the time arises as a parameter and the nonlinear expression is given in terms of a nonlocal operator H which is monotone and Lipschitz continuous on \(L^2(0, T)\), independently of the regularity of \(\sigma\).Hopf bifurcation theorem for second-order semi-linear Gurtin-MacCamy equationhttps://zbmath.org/1496.350522022-11-17T18:59:28.764376Z"Ducrot, Arnaud"https://zbmath.org/authors/?q=ai:ducrot.arnaud"Kang, Hao"https://zbmath.org/authors/?q=ai:kang.hao"Magal, Pierre"https://zbmath.org/authors/?q=ai:magal.pierreSummary: In this paper, we prove a Hopf bifurcation theorem for second-order semi-linear equations involving non-densely defined operators. Here, we use the Crandall and Rabinowitz's approach based on a suitable application of the implicit function theorem. As a special case, we obtain the existence of periodic wave trains for the so-called Gurtin-MacCamy problem arising in population dynamics and that couples both spatial diffusion and age structure.Gradient flows and nonlinear power methods for the computation of nonlinear eigenfunctionshttps://zbmath.org/1496.350672022-11-17T18:59:28.764376Z"Bungert, Leon"https://zbmath.org/authors/?q=ai:bungert.leon"Burger, Martin"https://zbmath.org/authors/?q=ai:burger.martinSummary: This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how these can be approximated using \(\Gamma\)-convergence. We review several flows from literature, which were proposed to compute nonlinear eigenfunctions, and show that they all relate to normalized gradient flows. Furthermore, we show that the implicit Euler discretization of gradient flows gives rise to a nonlinear power method of the proximal operator and we demonstrate their convergence to nonlinear eigenfunctions. Finally, we prove that \(\Gamma\)-convergence of functionals implies convergence of their ground states.
For the entire collection see [Zbl 1492.49003].Dynamics for a two-phase free boundary system in an epidemiological model with couple nonlocal dispersalshttps://zbmath.org/1496.350882022-11-17T18:59:28.764376Z"Nguyen, Thanh-Hieu"https://zbmath.org/authors/?q=ai:nguyen.thanh-hieu"Vo, Hoang-Hung"https://zbmath.org/authors/?q=ai:vo.hoang-hungSummary: The present paper is devoted to the investigation of the long time dynamics for a double free boundary system with nonlocal diffusions, which models the infectious diseases transmitted via digestive system such as fecal-oral diseases, cholera, hand-foot and mouth, etc \ldots We start by proving the existence and uniqueness of the Cauchy problem, which is not a trivial step due to presence of couple nonlocal dispersals and new types of nonlinear reaction terms. Next, we provide simple conditions on comparing the basic reproduction numbers \(\mathcal{R}_0\) and \(\mathcal{R}^\ast\) with 1 to characterize the global dynamics, as \(t \to \infty \). We further obtain the sharp criteria for the spreading and vanishing in term of the initial data. This is also called the vanishing-spreading phenomena. The couple dispersals yield significant obstacle that we cannot employ the approach of \textit{M. Zhao} et al. [J. Differ. Equations 269, No. 4, 3347--3386 (2020; Zbl 1442.35486)] and \textit{Y. Du} and \textit{W. Ni} [Nonlinearity 33, No. 9, 4407--4448 (2020; Zbl 1439.35220)]. To overcome this, we must prove the existence and the variational formula for the principal eigenvalue of a linear system with nonlocal dispersals, then use it to obtain the right limits as the dispersal rates and domain tend to zero or infinity. The maximum principle and sliding method for the nonlocal operator are ingeniously employed to achieve the desired results.Convergence of the weighted Yamabe flowhttps://zbmath.org/1496.350942022-11-17T18:59:28.764376Z"Yan, Zetian"https://zbmath.org/authors/?q=ai:yan.zetianSummary: We introduce the weighted Yamabe flow
\[
\begin{cases}
\frac{\partial g}{\partial t} = (r_{\phi}^m - R_{\phi}^m) g \\
\frac{\partial \phi}{\partial t} = \frac{m}{2} (R_{\phi}^m - r_{\phi}^m)
\end{cases}
\]
on a smooth metric measure space \((M^n, g, e^{-\phi} \operatorname{dvol}_g, m)\), where \(R_{\phi}^m\) denotes the associated weighted scalar curvature, and \(r_{\phi}^m\) denotes the mean value of the weighted scalar curvature. We prove long-time existence and convergence of the weighted Yamabe flow if the dimension \(n\) satisfies \(n \geqslant 3\).Sufficient conditions for the continuity of inertial manifolds for singularly perturbed problemshttps://zbmath.org/1496.350972022-11-17T18:59:28.764376Z"Bonfoh, Ahmed"https://zbmath.org/authors/?q=ai:bonfoh.ahmed-sSummary: We consider a nonlinear evolution equation in the form
\[
\mathrm{U_t + A_\varepsilon U + N_\varepsilon G_\varepsilon (U)} = 0,
\tag{\(\mathrm{E}_{\varepsilon}\)}
\]
together with its singular limit problem as \(\varepsilon\to 0\)
\[
U_t+ A U+ \mathrm{N} G(U) = 0,
\tag{E}
\]
where \(\varepsilon\in (0,1]\) (possibly \(\varepsilon = 0\)), \(\mathrm{A}_\varepsilon\) and \(\mathrm{A}\) are linear closed (possibly) unbounded operators, \(\mathrm{N}_\varepsilon\) and \(\mathrm{N}\) are linear (possibly) unbounded operators, \(\mathrm{G}_\varepsilon\) and \(\mathrm{G}\) are nonlinear functions. We give sufficient conditions on \(\mathrm{A}_\varepsilon\), \(\mathrm{N}_\varepsilon\) and \(\mathrm{G}_\varepsilon\) (and also \(\mathrm{A}, \mathrm{N}\) and \(\mathrm{G})\) that guarantee not only the existence of an inertial manifold of dimension independent of \(\varepsilon\) for \((E_\varepsilon)\) on a Banach space \(\mathcal{H}\), but also the Hölder continuity, lower and upper semicontinuity at \(\varepsilon = 0\) of the intersection of the inertial manifold with a bounded absorbing set. Applications to higher order viscous Cahn-Hilliard-Oono equations, the hyperbolic type equations and the phase-field systems, subject to either Neumann or Dirichlet boundary conditions (BC) (in which case \(\Omega\subset \mathbb{R}^d\) is a bounded domain with smooth boundary) or periodic BC (in which case \(\Omega = \Pi_{i = 1}^d (0,L_i), \, L_i>0)\), \(d = 1\), 2 or 3, are considered. These three classes of dissipative equations read
\[
\phi_t+N(\varepsilon \phi_t+N^{\alpha+1} \phi +N\phi + g(\phi))+\sigma\phi = 0,\quad\alpha\in\mathbb{N},\\
\tag{\(\mathrm{P}_\varepsilon\)}
\]
\[
\varepsilon \phi_{tt}+\phi_t+N^\alpha(N \phi + g(\phi))+ \sigma\phi = 0,\quad\alpha = 0, 1,\\
\tag{\(\mathrm{H}_\varepsilon\)}
\]
and
\[
\begin{cases}
\phi_t+N^\alpha (N \phi + g(\phi)-u)+\sigma\phi = 0,\\
\varepsilon u_t+\phi_t+N u = 0,
\end{cases}
\alpha = 0, 1
\tag{\(\mathrm{S}_\varepsilon\)}
\]
respectively, where \(\sigma\ge 0\) and the Laplace operator is defined as
\[
N = -\Delta:\mathscr{D}(N) = \{\psi\in H^2(\Omega),\,\psi \text{ subject to the BC}\}\to \dot L^2(\Omega) \text{ or }L^2(\Omega).
\]
We assume that, for a given real number \(\mathfrak{c}_1>0,\) there exists a positive integer \(n = n(\mathfrak{c}_1)\) such that \(\lambda_{n+1}-\lambda_n>\mathfrak{c}_1\), where \(\{\lambda_k\}_{k\in\mathbb{N}^*}\) are the eigenvalues of \(N\). There exists a real number \(\mathscr{M}>0\) such that the nonlinear function \(g: V_j\to V_j\) satisfies the conditions \(\|g(\psi)\|_j\le\mathscr{M}\) and \(\|g(\psi)-g(\varphi)\|_{V_j}\le\mathscr{M}\|\psi-\varphi\|_{V_j}\), \(\forall\psi\), \(\varphi\in V_j\), where \(V_j = \mathscr{D}(N^{j/2})\), \(j = 1\) for Problems \((P_\epsilon)\) and \((S_\epsilon)\) and \(j = 0\), \(2\alpha\) for Problem \((H_\epsilon)\). We further require \(g\in{\mathcal C}^1(V_j, V_j)\), \(\|g'(\psi)\varphi\|_j\le\mathscr{M}\|\varphi\|_j\) for Problems \((H_\epsilon)\) and \((S_\epsilon)\).Bounded invertibility and separability of a parabolic type singular operator in the space \(L_2(R^2)\)https://zbmath.org/1496.351212022-11-17T18:59:28.764376Z"Muratbekov, Mussakan"https://zbmath.org/authors/?q=ai:muratbekov.mussakan-baipakbaevich"Muratbekov, Madi"https://zbmath.org/authors/?q=ai:muratbekov.madi-m"Suleimbekova, Ainash"https://zbmath.org/authors/?q=ai:suleimbekova.ainashSummary: In this paper, we consider the operator of parabolic type
\[
Lu = \frac{\partial u}{\partial t} - \frac{\partial^2u}{\partial x^2} + q(x)u,
\]
in the space \(L_2(\mathbb{R}^2)\) with a greatly growing coefficient at infinity. The operator is originally defined on \(C_0^\infty(\mathbb{R}^2)\), where \(C_0^\infty(\mathbb{R}^2)\) is the set of infinitely differentiable and compactly supported functions.
Assume that the coefficient \(q(x)\) is a continuous function in \(\mathbb{R}=(-\infty, \infty)\), and it can be a strongly increasing function at infinity.
The operator \(L\) admits closure in space \(L_2(\mathbb{R}^2)\), and the closure is also denoted by \(L\).
In the paper, we proved the bounded invertibility of the operator \(L\) in the space \(L_2(\mathbb{R}^2)\) and the existence of the estimate
\[
\left\|\frac{\partial u}{\partial t}\right\|_{L_2(\mathbb{R}^2)} + \left\|\frac{\partial^2 u}{\partial x^2}\right\|_{L_2(\mathbb{R}^2)} + \|q(x)u\|_{L_2(\mathbb{R}^2)} \leq C(\|Lu\|_{L_2(\mathbb{R}^2)}+\|u\|_{L_2(\mathbb{R}^2)}),
\]
under certain restrictions on \(q(x)\) in addition to the conditions indicated above. Example. \(q(x)=e^{100 x}\), \(-\infty < x <\infty\).Generation of analytic semigroups for some generalized diffusion operators in \(L^p\)-spaceshttps://zbmath.org/1496.351332022-11-17T18:59:28.764376Z"Labbas, Rabah"https://zbmath.org/authors/?q=ai:labbas.rabah"Maingot, Stéphane"https://zbmath.org/authors/?q=ai:maingot.stephane"Thorel, Alexandre"https://zbmath.org/authors/?q=ai:thorel.alexandreSummary: We consider some generalized diffusion operators of fourth order and their corresponding abstract Cauchy problem. Then, using semigroups techniques and functional calculus, we study the invertibility and the spectral properties of each operator. Therefore, we prove that we have generation of \(C_0\)-semigroup in each case. We also point out when these semigroups become analytic.On the relationship between the solutions of an abstract Euler-Poisson-Darboux equation and fractional powers of the operator coefficient in the equationhttps://zbmath.org/1496.351562022-11-17T18:59:28.764376Z"Glushak, A. V."https://zbmath.org/authors/?q=ai:glushak.a-vSummary: We consider an incomplete initial value problem for an abstract singular Euler-Poisson-Darboux equation. It is established that, under weakened requirements on the operator coefficient in the equation, fractional powers of this operator coefficient should be used to construct solutions. It is also shown that a fractional power of the operator coefficient relates the Dirichlet condition and the weighted Neumann condition in the case of a boundary value problem for the Euler-Poisson-Darboux equation.An application of semigroup theory to the coagulation-fragmentation modelshttps://zbmath.org/1496.351652022-11-17T18:59:28.764376Z"Das, Arijit"https://zbmath.org/authors/?q=ai:das.arijit"Das, Nilima"https://zbmath.org/authors/?q=ai:das.nilima"Saha, Jitraj"https://zbmath.org/authors/?q=ai:saha.jitrajSummary: We present the existence and uniqueness of strong solutions for the continuous coagulation-fragmentation equation with singular fragmentation and essentially bounded coagulation kernel using semigroup theory of operators. Initially, we reformulate the coupled coagulation-fragmentation problem into the semilinear abstract Cauchy problem (ACP) and consider it as the nonlinear perturbation of the linear fragmentation operator. The existence of the substochastic semigroup is proved for the pure fragmentation equation. Using the substochastic semigroup and some related results for the pure fragmentation equation, we prove the existence of global nonnegative, strong solution for the coagulation-fragmentation equation.Uniform \(L^p\) resolvent estimates on the torushttps://zbmath.org/1496.351732022-11-17T18:59:28.764376Z"Hickman, Jonathan"https://zbmath.org/authors/?q=ai:hickman.jonathanThe author presents a new range of uniform \(L^p\) resolvent estimates, in the setting of the flat torus \({\mathbb T}^N:={\mathbb R}^N/{\mathbb Z}^N\) for \(N\ge 3\), improving previous results. Specifically, if \(\Delta_{{\mathbb T}^N}\) represents the Laplacian on the flat torus, then for all \(\varepsilon>0\) there exists a \(C_\varepsilon>0\) such that for all
\[
z\in\{z=(\lambda + i\mu)^2\in\mathbb{C}:\, \lambda, \mu\in\mathbb{R},\ \lambda\ge 1,\, |\mu|\ge \lambda^{-1/3+\varepsilon}\},
\]
the following holds
\[
\|u\|_{L^{2^*}({\mathbb T}^N)}\le C_\varepsilon\, \big\|(\Delta_{{\mathbb T}^N}+z)u\big\|_{L^{(2^*)'}({\mathbb T}^N)},
\]
where \(2^*:=\frac{2N}{N-2}\) is the critical Sobolev exponent, and \((2^*)'=\frac{2N}{N+2}\) is its conjugate.
Their arguments rely on the \(l^2\)-decoupling theorem and multidimensional Weyl sum estimates.
Reviewer: Rosa Maria Pardo San Gil (Madrid)On the limiting absorption principle for a new class of Schrödinger Hamiltonianshttps://zbmath.org/1496.351802022-11-17T18:59:28.764376Z"Martin, Alexandre"https://zbmath.org/authors/?q=ai:martin.alexandre.1|martin.alexandreSummary: We prove the limiting absorption principle and discuss the continuity properties of the boundary values of the resolvent for a class of form bounded perturbations of the Euclidean Laplacian \(\Delta\) that covers both short and long range potentials with an essentially optimal behaviour at infinity. For this, we give an extension of \textit{S. Nakamura}'s results [J. Spectr. Theory 4, No. 3, 613--619 (2014; Zbl 1308.81165)].Continuum covariance propagation for understanding variance loss in advective systemshttps://zbmath.org/1496.352502022-11-17T18:59:28.764376Z"Gilpin, Shay"https://zbmath.org/authors/?q=ai:gilpin.shay"Matsuo, Tomoko"https://zbmath.org/authors/?q=ai:matsuo.tomoko"Cohn, Stephen E."https://zbmath.org/authors/?q=ai:cohn.stephen-eVacuum isolating and blow-up analysis for edge hyperbolic system on edge Sobolev spaceshttps://zbmath.org/1496.352532022-11-17T18:59:28.764376Z"Kalleji, Morteza Koozehgar"https://zbmath.org/authors/?q=ai:kalleji.morteza-koozehgar"Kadkhoda, Nematollah"https://zbmath.org/authors/?q=ai:kadkhoda.nematollahSummary: This paper deals with the study of the initial-boundary value problem of edge-hyperbolic system with damping term on the manifold with edge singularity. More precisely, it is analyzed the invariance and vacuum isolating of the solution sets to the edge-hyperbolic systems on edge Sobolev spaces. Then, by using a family of modified potential wells and concavity methods, it is obtained existence and nonexistence results of global solutions with exponential decay and is shown the blow-up in finite time of solutions on the manifold with edge singularities.Principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann typehttps://zbmath.org/1496.352592022-11-17T18:59:28.764376Z"Kang, Hao"https://zbmath.org/authors/?q=ai:kang.hao"Ruan, Shigui"https://zbmath.org/authors/?q=ai:ruan.shiguiSummary: In this paper we study the principal spectral theory and asynchronous exponential growth for age-structured models with nonlocal diffusion of Neumann type. First, we provide two general sufficient conditions to guarantee existence of the principal eigenvalue of the age-structured operator with nonlocal diffusion. Then we show that such conditions are also enough to ensure that the semigroup generated by solutions of the age-structured model with nonlocal diffusion exhibits asynchronous exponential growth. Compared with previous studies, we prove that the semigroup is essentially compact instead of eventually compact, where the latter is usually obtained by showing the compactness of solution trajectories. Next, following the technique developed in Vo (Principal spectral theory of time-periodic nonlocal dispersal operators of Neumann type. arXiv:1911.06119, 2019), we overcome the difficulty that the principal eigenvalue of a nonlocal Neumann operator is not monotone with respect to the domain and obtain some limit properties of the principal eigenvalue with respect to the diffusion rate and diffusion range. Finally, we establish the strong maximum principle for the age-structured operator with nonlocal diffusion.Cutoff Boltzmann equation with polynomial perturbation near Maxwellianhttps://zbmath.org/1496.352732022-11-17T18:59:28.764376Z"Cao, Chuqi"https://zbmath.org/authors/?q=ai:cao.chuqiSummary: In this paper, we consider the cutoff Boltzmann equation near Maxwellian, we proved the global existence and uniqueness for the cutoff Boltzmann equation in polynomial weighted space for all \(\gamma \in(- 3, 1]\). We also proved initially polynomial decay for the large velocity in \(L^2\) space will induce polynomial decay rate, while initially exponential decay will induce exponential rate for the convergence. Our proof is based on newly established inequalities for the cutoff Boltzmann equation and semigroup techniques. Moreover, by generalizing the \(L_x^\infty L_v^1 \cap L_{x, v}^\infty\) approach, we prove the global existence and uniqueness of a mild solution to the Boltzmann equation with bounded polynomial weighted \(L_{x, v}^\infty\) norm under some small condition on the initial \(L_x^1 L_v^\infty\) norm and entropy so that this initial data allows large amplitude oscillations.Bogoliubov theory for many-body quantum systemshttps://zbmath.org/1496.353352022-11-17T18:59:28.764376Z"Schlein, Benjamin"https://zbmath.org/authors/?q=ai:schlein.benjaminSummary: We review some recent applications of rigorous Bogoliubov theory. We show how Bogoliubov theory can be used to approximate quantum fluctuations, both in the analysis of the energy spectrum and in the study of the dynamics of many-body quantum systems.
For the entire collection see [Zbl 1465.35005].Optimal semigroup regularity for velocity coupled elastic systems: a degenerate fractional damping casehttps://zbmath.org/1496.353802022-11-17T18:59:28.764376Z"Kuang, Zhaobin"https://zbmath.org/authors/?q=ai:kuang.zhaobin"Liu, Zhuangyi"https://zbmath.org/authors/?q=ai:liu.zhuangyi"Tebou, Louis"https://zbmath.org/authors/?q=ai:tcheugoue-tebou.louis-roderSummary: In this note, we consider an abstract system of two damped elastic systems. The damping involves the average velocity and a fractional power of the principal operator, with power \(\theta\) in [0, 1], The damping matrix is degenerate, which makes the regularity analysis more delicate. First, using a combination of the frequency domain method and multipliers technique, we prove the following regularity for the underlying semigroup:
\begin{itemize}
\item[--] The semigroup is of Gevrey class \(\delta\) for every \(\delta > 1/2\theta\), for each \(\theta\) in \((0, 1/2)\).
\item[--] The semigroup is analytic for \(\theta = 1/2\).
\item[--] The semigroup is of Gevrey class \(\delta\) for every \(\delta > 1/2(1 - \theta)\), for each \(\theta\) in \((1/2, 1)\).
\end{itemize}
Next, we analyze the point spectrum, and derive the optimality of our regularity results. We also prove
that the semigroup is not differentiable for \(\theta = 0\) or \(\theta = 1\). Those results strongly improve upon some recent results presented in [\textit{K. Ammari} et al., J. Evol. Equ. 21, No. 4, 4973--5002 (2021; Zbl 07452693)].Global subelliptic estimates for Kramers-Fokker-Planck operators with some class of polynomialshttps://zbmath.org/1496.353872022-11-17T18:59:28.764376Z"Ben Said, Mona"https://zbmath.org/authors/?q=ai:said.mona-benSummary: In this article, we study some Kramers-Fokker-Planck operators with a polynomial potential \(V(q)\) of degree greater than two having quadratic limiting behaviour. This work provides an accurate global subelliptic estimate for Kramers-Fokker-Planck operators under some conditions imposed on the potential.Telegraph systems on networks and port-Hamiltonians. I. Boundary conditions and well-posednesshttps://zbmath.org/1496.354042022-11-17T18:59:28.764376Z"Banasiak, Jacek"https://zbmath.org/authors/?q=ai:banasiak.jacek"Błoch, Adam"https://zbmath.org/authors/?q=ai:bloch.adamSummary: The paper is concerned with a system of linear hyperbolic differential equations on a network coupled through general transmission conditions of Kirchhoff's-type at the nodes. We discuss the reduction of such a problem to a system of 1-dimensional hyperbolic problems for the associated Riemann invariants and provide a semigroup-theoretic proof of its well-posedness. A number of examples showing the relation of our results with recent research is also provided.Global classical and weak solutions of the prion proliferation model in the presence of chaperone in a Banach spacehttps://zbmath.org/1496.354082022-11-17T18:59:28.764376Z"Choudhary, Kapil Kumar"https://zbmath.org/authors/?q=ai:choudhary.kapil-kumar"Kumar, Rajiv"https://zbmath.org/authors/?q=ai:kumar.rajiv"Kumar, Rajesh"https://zbmath.org/authors/?q=ai:kumar.rajesh-sSummary: The present work is based on the coupling of prion proliferation system together with chaperone which consists of two ODEs and a partial integro-differential equation. The existence and uniqueness of a positive global classical solution of the system is proved for the bounded degradation rates by the idea of evolution system theory in the state space \(\mathbb{R} \times \mathbb{R} \times L_1(Z, \, zdz)\). Moreover, the global weak solutions for unbounded degradation rates are discussed by weak compactness technique.Fractional truncated Laplacians: representation formula, fundamental solutions and applicationshttps://zbmath.org/1496.354202022-11-17T18:59:28.764376Z"Birindelli, Isabeau"https://zbmath.org/authors/?q=ai:birindelli.isabeau"Galise, Giulio"https://zbmath.org/authors/?q=ai:galise.giulio"Topp, Erwin"https://zbmath.org/authors/?q=ai:topp.erwinSummary: We introduce some nonlinear extremal nonlocal operators that approximate the, so called, truncated Laplacians. For these operators we construct representation formulas that lead to the construction of what, with an abuse of notation, could be called ``fundamental solutions''. This, in turn, leads to Liouville type results. The interest is double: on one hand we wish to ``understand'' what is the right way to define the nonlocal version of the truncated Laplacians, on the other, we introduce nonlocal operators whose nonlocality is on one dimensional lines, and this dramatically changes the prospective, as is quite clear from the results obtained that often differ significantly with the local case or with the case where the nonlocality is diffused. Surprisingly this is true also for operators that approximate the Laplacian.Divergence \& curl with fractional orderhttps://zbmath.org/1496.354342022-11-17T18:59:28.764376Z"Liu, Liguang"https://zbmath.org/authors/?q=ai:liu.liguang"Xiao, Jie"https://zbmath.org/authors/?q=ai:xiao.jie.1Summary: This paper presents a novel analysis for Function Space Norms (F.S.N.) \& Partial Differential Equations (P.D.E.) within the fractional-nonlocal pair \(\{\operatorname{div}^*\mathbf{v},\operatorname{curl}^*\mathbf{v}\}\) that extends the classical-local pair \(\{\operatorname{div}\mathbf{v},\operatorname{curl}\mathbf{v}\}\) which has an inherent physical
content because of causing the conservation of mass \& the rotation produced by fluid elements in motion.Lower and upper solutions for delay evolution equations with nonlocal and impulsive conditionshttps://zbmath.org/1496.354432022-11-17T18:59:28.764376Z"Zhang, Xuping"https://zbmath.org/authors/?q=ai:zhang.xupingSummary: In this article, we apply the method of lower and upper solutions for studying delay evolution equations with nonlocal and impulsive conditions in infinite dimensional Banach spaces. Under wide monotone conditions and noncompactness measure condition of nonlinear term, we obtain the existence of extremal solutions and a unique solution between lower and upper solutions. A concrete application to partial differential equations is considered.Numerical study for time fractional stochastic semi linear advection diffusion equationshttps://zbmath.org/1496.354682022-11-17T18:59:28.764376Z"Sweilam, N. H."https://zbmath.org/authors/?q=ai:sweilam.nasser-hassan"El-Sakout, D. M."https://zbmath.org/authors/?q=ai:elsakout.d-m"Muttardi, M. M."https://zbmath.org/authors/?q=ai:muttardi.m-mSummary: In this work, a stochastic fractional advection diffusion model with multiplicative noise is studied numerically. The Galerkin finite element method in space and finite difference in time are used, where the fractional derivative is in Caputo sense. The error analysis is investigated via Galerkin finite element method. In terms of the Mittag Leffler function, the mild solution is obtained. For the error estimates, the strong convergence for the semi and fully discrete schemes are proved in a semigroup structure. Finally, two numerical examples are given to confirm the theoretical results.The theorem of composition for a class of degenerate pseudodifferential operators with symbol-dependent complex parameterhttps://zbmath.org/1496.354702022-11-17T18:59:28.764376Z"Baev, Alexander D."https://zbmath.org/authors/?q=ai:baev.aleksander-d"Babichev, Andrey A."https://zbmath.org/authors/?q=ai:babichev.andrey-a"Kharchenko, Victoria Dmitrievna"https://zbmath.org/authors/?q=ai:kharchenko.victoria-dmitrievna"Naydyuk, Philip Olegovich"https://zbmath.org/authors/?q=ai:naydyuk.philip-olegovichSummary: The article is devoted to the proof of the composition theorem for a class of pseudo-differential equations with degeneration. A new class of symbol variables that also depend on a complex parameter is considered. Pseudodifferential operators are constructed by a special integral transformation. A formula for representing the superposition of degenerate pseudo-differential operators is obtained. Obtained the asymptotic formula of the symbol of a superposition of degenerate pseudodifferential operators.An operator theoretical approach to the sequence entropy of dynamical systemshttps://zbmath.org/1496.370132022-11-17T18:59:28.764376Z"Rahimi, M."https://zbmath.org/authors/?q=ai:rahimi.mahboobeh|rahimi.mohammad-reza-ostad|rahimi.mohammad-a|rahimi.mohammad-naqib|rahimi.mohamadtaghi|rahimi.m-y|rahimi.mostafa|rahimi.maryam|rahimi.morteza|rahimi.m-ostad|rahimi.mona|rahimi.mehdi|rahimi.mansour|rahimi.mehran"Mohammadi Anjedani, M."https://zbmath.org/authors/?q=ai:mohammadi-anjedani.mSummary: In this paper, given a sequence of positive integers, we assign a linear operator on a Hilbert space, to any compact topological dynamical system of finite entropy. Then we represent the sequence entropy of the systems in terms of the eigenvalues of the linear operator. In this way, we present a spectral approach to the sequence entropy of the dynamical systems. This spectral representation to the sequence entropy of a system is given for systems with some additional condition called admissibility condition. We also prove that, there exist a large family of dynamical systems, satisfying the admissibility condition.On essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpointhttps://zbmath.org/1496.370562022-11-17T18:59:28.764376Z"Zhu, Li"https://zbmath.org/authors/?q=ai:zhu.li"Sun, Huaqing"https://zbmath.org/authors/?q=ai:sun.huaqingSummary: This paper is concerned with essential numerical ranges and essential spectra of Hamiltonian systems with one singular endpoint. For semi-bounded systems, the characterization of each element of the essential numerical range in terms of certain singular sequences is given, the concept of form perturbation small at the singular endpoint is introduced, and the stability of the essential numerical range is obtained under this perturbation, which shows the stability of the infimum or supremum of the essential spectrum. Some sufficient conditions for the invariance of the essential numerical range are given in terms of coefficients of Hamiltonian systems.\(q\)-Hamiltonian systemshttps://zbmath.org/1496.390042022-11-17T18:59:28.764376Z"Paşaoğlu, Bilender"https://zbmath.org/authors/?q=ai:pasaoglu.bilender-p"Tuna, Hüseyin"https://zbmath.org/authors/?q=ai:tuna.huseyinSummary: In this paper, we develop the basic theory of linear \(q\)-Hamiltonian systems. In this context, we establish an existence and uniqueness result. Regular spectral problems are studied. Later, we introduce the corresponding maximal and minimal operators for this system. Finally, we give a spectral resolution.On the spectral and scattering properties of eigenparameter dependent discrete impulsive Sturm-Liouville equationshttps://zbmath.org/1496.390112022-11-17T18:59:28.764376Z"Aygar Küçükevcilioğlu, Yelda"https://zbmath.org/authors/?q=ai:aygar-kucukevcilioglu.yelda"Bayram, Elgiz"https://zbmath.org/authors/?q=ai:bayram.elgiz"Özbey, Güher Gülçehre"https://zbmath.org/authors/?q=ai:ozbey.guher-gulcehreSummary: This work develops scattering and spectral analysis of a discrete impulsive Sturm-Liouville equation with spectral parameter in boundary condition. Giving the Jost solution and scattering solutions of this problem, we find scattering function of the problem. Discussing the properties of scattering function, scattering solutions, and asymptotic behavior of the Jost solution, we find the Green function, resolvent operator, continuous and point spectrum of the problem. Finally, we give an example in which the main results are made explicit.Ternary biderivations and ternary bihomorphisms in \(C^\ast\)-ternary algebrashttps://zbmath.org/1496.390142022-11-17T18:59:28.764376Z"Lee, Jung Rye"https://zbmath.org/authors/?q=ai:lee.jung-rye"Park, Choonkil"https://zbmath.org/authors/?q=ai:park.choonkil"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: In [Rocky Mt. J. Math. 49, No. 2, 593--607 (2019; Zbl 1417.39078)], the second author et al. introduced the following bi-additive \(s\)-functional inequality
\[
\begin{aligned}
\| f(x&+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\| \\
\quad \le \, &\bigg\|s \left(2f \left(\frac{x+y}{2}, z-w \right) + 2f \left(\frac{x-y}{2}, z+w \right) - 2f(x,z)+ 2 f(y, w) \right) \bigg\|,
\end{aligned}
\tag{1}
\]
where \(s\) is a fixed nonzero complex number with \(|s| < 1\). Using the fixed point method, we prove the Hyers-Ulam stability of ternary biderivations and ternary bihomomorphism in \(C^\ast\)-ternary algebras, associated with the bi-additive \(s\)-functional inequality (1).
For the entire collection see [Zbl 1485.65002].Solvability, stability, smoothness and compactness of the set of solutions for a nonlinear functional integral equationhttps://zbmath.org/1496.390162022-11-17T18:59:28.764376Z"Thuc, Nguyen Dat"https://zbmath.org/authors/?q=ai:thuc.nguyen-dat"Ngoc, Le Thi Phuong"https://zbmath.org/authors/?q=ai:le-thi-phuong-ngoc."Long, Nguyen Thanh"https://zbmath.org/authors/?q=ai:nguyen-thanh-long.Summary: This paper is devoted to the study of the following nonlinear functional integral equation
\[
f(x)=\sum\limits_{i=1}^q\alpha_i(x)f(\tau_i(x)) + \int_0^{\sigma_1(x)}\Psi\left(x, t, f(\sigma_2(t)), \int_0^{\sigma_3(t)}f(s)ds\right) dt + g(x),\;\forall x\in [0,1], \tag{E}
\]
where \(\tau_i, \sigma_1, \sigma_2, \sigma_3 :[0,1]\rightarrow [0,1]\); \(\alpha_i, g: [0,1]\rightarrow \mathbb{R}\); \(\Psi: [0,1]\times [0,1]\times\mathbb{R}^2\rightarrow \mathbb{R}\) are the given continuous functions and \(f:[0,1]\,\rightarrow\mathbb{R}\) is an unknown function. First, two sufficient conditions for the existence and some properties of solutions of Eq. (E) are proved. By using Banach's fixed point theorem, we have the first sufficient condition yielding existence, uniqueness and stability of the solution. By applying Schauder's fixed point theorem, we have the second sufficient condition for the existence and compactness of the solution set. An example is also given in order to illustrate the results obtained here. Next, in the case of \(\Psi\in C^2([0, 1]\times [0,1]\times \mathbb{R}^2; \mathbb{R})\), we investigate the quadratic convergence for the solution of Eq. (E). Finally, the smoothness of the solution depending on data is established.Approximate generalized Jensen mappings in 2-Banach spaceshttps://zbmath.org/1496.390172022-11-17T18:59:28.764376Z"Almahalebi, Muaadh"https://zbmath.org/authors/?q=ai:almahalebi.muaadh"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-m"Al-Ali, Sadeq"https://zbmath.org/authors/?q=ai:al-ali.sadeq-a-a"Hryrou, Mustapha E."https://zbmath.org/authors/?q=ai:hryrou.mustapha-esseghyrSummary: Our aim is to investigate the generalized Hyers-Ulam-Rassias stability for the following general Jensen functional equation:
\[
\sum_{k=0}^{n-1} f(x+ b_ky)=nf(x),
\]
where \(n \in \mathbb{N}_2\), \(b_k=\exp (\frac{2i\pi k}{n})\) for \(0 \leq\) k \(\leq\) n \(- 1\), in 2-Banach spaces by using a new version of Brzdȩk's fixed point theorem. In addition, we prove some hyperstability results for the considered equation and the general inhomogeneous Jensen equation
\[
\sum_{k=0}^{n-1} f(x+ b_ky)=nf(x)+G(x,y).
\]
For the entire collection see [Zbl 1485.65002].Some hyperstability results in non-Archimedean 2-Banach space for a \(\sigma\)-Jensen functional equationhttps://zbmath.org/1496.390182022-11-17T18:59:28.764376Z"El Ghali, Rachid"https://zbmath.org/authors/?q=ai:el-ghali.rachid"Kabbaj, Samir"https://zbmath.org/authors/?q=ai:kabbaj.samirSummary: By combining the two versions of Brzdȩk's fixed point theorem in non-Archimedean Banach spaces [\textit{J. Brzdȩk} and \textit{K. Ciepliński}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 6861--6867 (2011; Zbl 1237.39022)] and that in 2-Banach spaces [\textit{J. Brzdȩk} and \textit{K. Ciepliński}, Acta Math. Sci., Ser. B, Engl. Ed. 38, No. 2, 377--390 (2018; Zbl 1399.39063)], we will investigate the hyperstability of the following \(\sigma\)-Jensen functional equation:
\[
f(x+y)+f(x+\sigma (y))=2f(x),
\]
where \(f : X \to Y\) such that \(X\) is a normed space, \(Y\) is a non-Archimedean 2-Banach space, and \(\sigma\) is a homomorphism of \(X\). In addition, we prove some interesting corollaries corresponding to some inhomogeneous outcomes and particular cases of our main results in \(C^\ast\)-algebras.
For the entire collection see [Zbl 1485.65002].Hyers-Ulam stability of an additive-quadratic functional equationhttps://zbmath.org/1496.390192022-11-17T18:59:28.764376Z"Lee, Jung Rye"https://zbmath.org/authors/?q=ai:lee.jung-rye"Park, Choonkil"https://zbmath.org/authors/?q=ai:park.choonkil"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of Lie biderivations and Lie bihomomorphisms in Lie Banach algebras, associated with the bi-additive functional inequality
\[
\begin{aligned}
\| f(x+&y, z+w) + f(x+y, z-w) + f(x-y, z+w)\\
+ &f(x-y, z-w) -4f(x,z)\|\\
\le \| &s (2f(x+ y, z-w) + 2f(x-y, z+ w) - 4f(x,z)+ 4 f(y, w)) \|,
\end{aligned}
\tag{1}
\]
where \(s\) is a fixed nonzero complex number with \(|s| < 1\).
For the entire collection see [Zbl 1485.65002].Hyers-Ulam stability of symmetric biderivations on Banach algebrashttps://zbmath.org/1496.390202022-11-17T18:59:28.764376Z"Lee, Jung Rye"https://zbmath.org/authors/?q=ai:lee.jung-rye"Park, Choonkil"https://zbmath.org/authors/?q=ai:park.choonkil"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: In [Indian J. Pure Appl. Math. 50, No. 2, 413--426 (2019; Zbl 1428.39031)], the second author introduced the following bi-additive \(s\)-functional inequality:
\[
\begin{aligned}
\| f(x&+y, z-w) + f(x-y, z+w) -2f(x, z)+2 f(y, w)\|\\
\le &\bigg\|s \left(2f \left(\frac{x+y}{2}, z-w\right) + 2f \left(\frac{x-y}{2}, z+w\right) - 2f(x, z)+ 2 f(y, w)\right) \bigg\|,
\end{aligned}
\tag{1}
\]
where \(s\) is a fixed nonzero complex number with \(|s| < 1\). Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of symmetric biderivations and a skew-symmetric biderivation on Banach algebras and unital \(C^\ast\)-algebras, associated with the bi-additive \(s\)-functional inequality (1).
For the entire collection see [Zbl 1483.00042].Multi-dimensional \(c\)-almost periodic type functions and applicationshttps://zbmath.org/1496.420092022-11-17T18:59:28.764376Z"Kostic, Marko"https://zbmath.org/authors/?q=ai:kostic.markoSummary: In this article, we analyze multi-dimensional Bohr \((\mathcal{B}, c)\)-almost periodic type functions. The main structural characterizations for the introduced classes of Bohr \((\mathcal{B}, c)\)-almost periodic type functions are established. Several applications of our abstract theoretical results to the abstract Volterra integro-differential equations in Banach spaces are provided, as well.Positive definiteness and infinite divisibility of certain functions of hyperbolic cosine functionhttps://zbmath.org/1496.420102022-11-17T18:59:28.764376Z"Kosaki, Hideki"https://zbmath.org/authors/?q=ai:kosaki.hidekiLet \(\alpha \geq 0\), \(t>-1\) and \(f_{\alpha },\) \(g_{\alpha }\) two real functions defined by
\[
f_{\alpha }(x)=\frac{\cosh (\alpha x)}{\cosh (3x)+t\cosh x}
\]
and
\[
g_{\alpha }(x)=\frac{\cosh (\alpha x)}{\cosh (2x)+t\cosh x}.
\]
The author investigates infinite divisibility and positive definiteness of the functions \(f_{\alpha }\) and of \(g_{\alpha }\). Furthermore, he uses the positive definiteness criterion to study certain norm comparison results for operator means.
Reviewer: Elhadj Dahia (Bou Saâda)Erratum to: ``The boundedness of commutators of sublinear operators on Herz Triebel-Lizorkin spaces''https://zbmath.org/1496.420172022-11-17T18:59:28.764376Z"Fang, Chenglong"https://zbmath.org/authors/?q=ai:fang.chenglong"Zhou, Jiang"https://zbmath.org/authors/?q=ai:zhou.jiangErratum to the authors' paper [ibid. 52, No. 2, 375--383 (2021; Zbl 1480.42019)].Weighted endpoint estimates for the composition of Calderón-Zygmund operators on spaces of homogeneous typehttps://zbmath.org/1496.420192022-11-17T18:59:28.764376Z"Liu, Dongli"https://zbmath.org/authors/?q=ai:liu.dongli"Zhao, Jiman"https://zbmath.org/authors/?q=ai:zhao.jimanSummary: Let \(T_1\), \(T_2\) be two Calderón-Zygmund operators, by establishing bilinear sparse domination of \(T_1 T_2\), we obtain the weighted endpoint estimate for the composite operator \(T_1 T_2\).Hardy spaces meet harmonic weightshttps://zbmath.org/1496.420302022-11-17T18:59:28.764376Z"Preisner, Marcin"https://zbmath.org/authors/?q=ai:preisner.marcin"Sikora, Adam"https://zbmath.org/authors/?q=ai:sikora.adam-s"Yan, Lixin"https://zbmath.org/authors/?q=ai:yan.lixinSummary: We investigate the Hardy space \(H^1_L\) associated with a self-adjoint operator \(L\) defined in a general setting by \textit{S. Hofmann} et al. [Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1232.42018)]. We assume that there exists an \(L\)-harmonic non-negative function \(h\) such that the semigroup \(\exp (-tL)\), after applying the Doob transform related to \(h\), satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space \(H^1_L\) in terms of a simple atomic decomposition associated with the \(L\)-harmonic function \(h\). Our approach also yields a natural characterisation of the \(BMO\)-type space corresponding to the operator \(L\) and dual to \(H^1_L\) in the same circumstances.
The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in \({\mathbb{R}^n} \), Schrödinger operators with certain potentials, and Bessel operators.Order structures of \((\mathcal{D,E})\)-quasi-bases and constructing operators for generalized Riesz systemshttps://zbmath.org/1496.420432022-11-17T18:59:28.764376Z"Inoue, Hiroshi"https://zbmath.org/authors/?q=ai:inoue.hiroshiSummary: The main purpose of this paper is to investigate the relationship between the two order structures of constructing operators for a generalized Riesz system and \((\mathcal{D,E})\)-quasi bases for two fixed biorthogonal sequences \(\{\varphi_n\}\) and \(\{\Psi_n\}\). In a previous paper, we have studied the order structure of the set \(C_\varphi\) of all constructing operators for a generalized Riesz system \(\{\varphi_n\}\), and furthermore we have shown that the notion of generalized Riesz systems has a close relation with that of \((\mathcal{D,E})\)-quasi bases. For this reason, in this paper we define an order structure in the set \(\mathfrak{M}_{\varphi,\psi}\) of all pairs of dense subspaces \(\mathcal{D}\) and \(\mathcal{E}\) in \(\mathcal{H}\) such that \(\{\varphi_n\}\) and \(\{\psi_n\}\) are \((\mathcal{D,E})\)-quasi bases, and shall investigate the relationships between the order sets \(C_\varphi\), \(C_\psi\) and \(M_{\varphi,\psi}\). These results seem to be useful to find suitable constructing operators for each physical model.Integral resolvent for Volterra equations and Favard spaceshttps://zbmath.org/1496.450012022-11-17T18:59:28.764376Z"Fadili, A."https://zbmath.org/authors/?q=ai:fadili.ahmed"Maragh, F."https://zbmath.org/authors/?q=ai:maragh.fouadThe authors study the Volterra integral equation
\[
x(t)=x_0 + \int_0^t a(t-s) Ax(s)\, ds,\quad t\geq 0,
\]
in a Banach space \(X\), where \(a\in L^1_{\text{loc}}(\mathbb R^+)\) and \(A\) is a densely define closed operator in \(X\). The integral resolvent associated with this equation is a strongly continuous family \(R(t)\) of bounded operators such that \(R(t)\) commutes with \(A\) and
\[
R(t)x=a(t)x+ \int_0^t a(t-s)AR(s)x\, ds,
\]
for all \(x\in D(A)\).
The crucial assumption used by the authors is that there exists \(\varepsilon_a>0\) and \(t_a>0\) such that for all \(0< t\leq t_a\) one has
\[
\left | \int_0^t a(t-s)a(s)\, ds\right | \geq \varepsilon_a \int_0^t |a(s)|\, ds. \]
They show that
\[
D(A)= \left \{x\in X\,:\, \lim_{t\to 0+} \frac {R(t)x-a(t)x}{(a*a)(t)}\text{ exists}\right \},
\]
and the limit is \(Ax\).
Furthermore, they show that if in addition the integral resolvent is bounded and \(\int_0^\infty \text{e}^{-\omega t} |a(t)|\, dt <\infty\) for some \(\omega >0\) then the following result on the (frequency and temporal) Favard spaces associated with \((A,a)\) holds:
\[
\left \{x\in X\,:\,\sup_{\lambda > \omega}\left \| \frac 1{\hat a(\lambda)} A\left (\frac 1{\hat a(\lambda)}I-A\right)x\right\|<\infty\right \} \]
\[
= \left \{x\in X\,:\,\sup_{0<t\leq 1}\frac{\|R(t)x-a(t)x\|}{ |(a*a)(t)|}<\infty\right \}.
\]
Reviewer: Gustaf Gripenberg (Aalto)A fixed point theorem using condensing operators and its applications to Erdélyi-Kober bivariate fractional integral equationshttps://zbmath.org/1496.450032022-11-17T18:59:28.764376Z"Das, Anupam"https://zbmath.org/authors/?q=ai:das.anupam"Rabbani, Mohsen"https://zbmath.org/authors/?q=ai:rabbani.mohsen"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipan"Panda, Sumati Kumari"https://zbmath.org/authors/?q=ai:panda.sumati-kumariSummary: The primary aim of this article is to discuss and prove fixed point results using the operator type condensing map, and to obtain the existence of solution of Erdélyi-Kober bivariate fractional integral equation in a Banach space. An instance is given to explain the results obtained, and we construct an iterative algorithm by sinc interpolation to find an approximate solution of the problem with acceptable accuracy.On Hyers-Ulam-Rassias stability of a Volterra-Hammerstein functional integral equationhttps://zbmath.org/1496.450042022-11-17T18:59:28.764376Z"Ciplea, Sorina Anamaria"https://zbmath.org/authors/?q=ai:ciplea.sorina-anamaria"Lungu, Nicolaie"https://zbmath.org/authors/?q=ai:lungu.nicolaie"Marian, Daniela"https://zbmath.org/authors/?q=ai:marian.daniela"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: The aim of this paper is to study the Hyers-Ulam-Rassias stability for a Volterra-Hammerstein functional integral equation in three variables via Picard operators.
For the entire collection see [Zbl 1485.65002].Iterative algorithm and theoretical treatment of existence of solution for \((k, z)\)-Riemann-Liouville fractional integral equationshttps://zbmath.org/1496.450052022-11-17T18:59:28.764376Z"Das, Anupam"https://zbmath.org/authors/?q=ai:das.anupam"Rabbani, Mohsen"https://zbmath.org/authors/?q=ai:rabbani.mohsen"Mohiuddine, S. A."https://zbmath.org/authors/?q=ai:mohiuddine.syed-abdul"Deuri, Bhuban Chandra"https://zbmath.org/authors/?q=ai:deuri.bhuban-chandraAfter an introduction to fractional integral equations involving Riemann-Liouville fractional integrals the authors establish a new Darbo-type fixed point theorem. This allows them to discuss the existence of solutions for certain fractional integral equations. The last part of this work is devoted to the construction of a convergent iterative algorithm based on the modified homotopy perturbation method to find the solutions of given fractional integral equations.
Reviewer: Yogesh Sharma (Sardarpura)Global attractivity, asymptotic stability and blow-up points for nonlinear functional-integral equations' solutions and applications in Banach space \(BC( R_+)\) with computational complexityhttps://zbmath.org/1496.450062022-11-17T18:59:28.764376Z"Karaca, Yeliz"https://zbmath.org/authors/?q=ai:karaca.yelizExistence results of fractional neutral integrodifferential equations with deviating argumentshttps://zbmath.org/1496.450082022-11-17T18:59:28.764376Z"Kamalapriya, B."https://zbmath.org/authors/?q=ai:kamalapriya.b"Balachandran, K."https://zbmath.org/authors/?q=ai:balachandran.krishnan"Annapoorani, N."https://zbmath.org/authors/?q=ai:annapoorani.natarajanSummary: In this paper we prove the existence of solutions of fractional neutral integrodifferential equations with deviating arguments by using the resolvent operators and fixed point theorem. Examples are discussed to illustrate the theory.A new aspect of generalized integral operator and an estimation in a generalized function theoryhttps://zbmath.org/1496.450132022-11-17T18:59:28.764376Z"Al-Omari, Shrideh"https://zbmath.org/authors/?q=ai:al-omari.shrideh-khalaf-qasem"Almusawa, Hassan"https://zbmath.org/authors/?q=ai:almusawa.hassan"Nisar, Kottakkaran Sooppy"https://zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: In this paper we investigate certain integral operator involving Jacobi-Dunkl functions in a class of generalized functions. We utilize convolution products, approximating identities, and several axioms to allocate the desired spaces of generalized functions. The existing theory of the Jacobi-Dunkl integral operator [\textit{N. B. Salem} and \textit{A. O. A. Salem}, Ramanujan J. 12, No. 3, 359--378 (2006; Zbl 1122.44002)] is extended and applied to a new addressed set of Boehmians. Various embeddings and characteristics of the extended Jacobi-Dunkl operator are discussed. An inversion formula and certain convergence with respect to \(\delta\) and \(\Delta\) convergences are also introduced.Sequentially right-like properties on Banach spaceshttps://zbmath.org/1496.460032022-11-17T18:59:28.764376Z"Alikhani, Morteza"https://zbmath.org/authors/?q=ai:alikhani.mortezaSummary: In this paper, we study first the concept of \(p\)-sequentially Right property, which is \(p\)-version of the sequentially Right property. Also, we introduce a new class of subsets of Banach spaces which is called \(p\)-Right\(^\ast\) set and obtain the relationship between \(p\)-Right subsets and \(p\)-Right\(^\ast\) subsets of dual spaces. Furthermore, for \(1\leq p<q\leq\infty\), we introduce the concepts of properties \((SR)_{p,q}\) and \((SR^\ast)_{p,q}\) in order to find a condition such that every Dunford-Pettis \(q\)-convergent operator is Dunford-Pettis \(p\)-convergent. Finally, we apply these concepts and obtain some characterizations of the \(p\)-Dunford-Pettis relatively compact property of Banach spaces and their dual spaces.On the class of almost L-weakly and almost M-weakly compact operatorshttps://zbmath.org/1496.460052022-11-17T18:59:28.764376Z"Bouras, Khalid"https://zbmath.org/authors/?q=ai:bouras.khalid"Lhaimer, Driss"https://zbmath.org/authors/?q=ai:lhaimer.driss"Moussa, Mohammed"https://zbmath.org/authors/?q=ai:moussa.mohammedSummary: In this paper, we introduce and study new concepts of almost L-weakly and almost M-weakly compact operators.A norm inequality in James' space and stability of the fixed point propertyhttps://zbmath.org/1496.460122022-11-17T18:59:28.764376Z"Díaz-García, R."https://zbmath.org/authors/?q=ai:diaz-garcia.r"Jiménez-Melado, A."https://zbmath.org/authors/?q=ai:jimenez-melado.antonioSummary: In this paper we prove a norm inequality in James' space \(J\), and use it to show that the fixed point property for nonexpansive mappings is passed on from \(J\) to those Banach spaces \(X\) whose Banach-Mazur distance to \(J\) satisfies \(d(X,J)<\sqrt{\frac{17+\sqrt{97}}{12}}\).On the class of disjoint limited completely continuous operatorshttps://zbmath.org/1496.460152022-11-17T18:59:28.764376Z"H'michane, Jawad"https://zbmath.org/authors/?q=ai:hmichane.jawad"Hafidi, Noufissa"https://zbmath.org/authors/?q=ai:hafidi.noufissa"Zraoula, Larbi"https://zbmath.org/authors/?q=ai:zraoula.larbiSummary: We introduce and study new class of sets (almost L-limited sets). Also, we introduce new concept of property in Banach lattice (almost Gelfand-Phillips property) and we characterize this property using almost L-limited sets. On the other hand, we introduce the class of disjoint limited completely continuous operators which is a largest class than that of limited completely continuous operators, we characterize this class of operators and we study some of its properties.Two classes of de Branges spaces that are really onehttps://zbmath.org/1496.460192022-11-17T18:59:28.764376Z"Arov, Damir Z."https://zbmath.org/authors/?q=ai:arov.damir-zyamovich"Dym, Harry"https://zbmath.org/authors/?q=ai:dym.harrySummary: It is well known that if \(J\) is an \(m\times m\) signature matrix and \(U\) is \(J\)-inner with respect to the open upper half-plane \(\mathbb{C}_+\), then the kernel
\[
K_\omega^U(\lambda)=\frac{J-U(\lambda)JU(\omega)^\ast}{-2\pi i(\lambda-\overline{\omega})}
\]
is positive and hence is the reproducing kernel of a reproducing kernel Hilbert space \(\mathcal{H}(U)\) of a space of \(m\times 1\) vector valued functions that are holomorphic in the domain of holomorphy of \(U\).
It seems, however, to be not so well known that this reproducing kernel Hilbert space coincides with the de Branges space \(\mathcal{B}(\mathfrak{E})\) based on an appropriately defined de Branges matrix \(\mathfrak{E}=[E_-\;\; E_+]\) with \(m\times m\) components and reproducing kernel
\[
K_\omega^{\mathfrak{E}}(\lambda)=\frac{E_+(\lambda)E_+(\omega)^\ast-E_-(\lambda)E_-(\omega)^\ast}{-2\pi i(\lambda-\overline{\omega})}.
\]
This connection is significant, because it yields a recipe for the inner product in \(\mathcal{H}(U)\) that is not available from Aronszjan's theorem.
Enroute, a pleasing characterization of a class of finite dimensional de Branges spaces \(\mathcal{B}(\mathfrak{E})\) is developed.Nonlocal trace spaces and extension results for nonlocal calculushttps://zbmath.org/1496.460312022-11-17T18:59:28.764376Z"Du, Qiang"https://zbmath.org/authors/?q=ai:du.qiang"Tian, Xiaochuan"https://zbmath.org/authors/?q=ai:tian.xiaochuan"Wright, Cory"https://zbmath.org/authors/?q=ai:wright.cory-d"Yu, Yue"https://zbmath.org/authors/?q=ai:yu.yueSummary: For a given Lipschitz domain \(\Omega \), it is a classical result that the trace space of \(W^{1 , p}(\Omega)\) is \(W^{1 - 1 / p , p}(\partial \Omega)\), namely any \(W^{1 , p}(\Omega)\) function has a well-defined \(W^{1 - 1 / p , p}(\partial \Omega)\) trace on its codimension-1 boundary \(\partial \Omega\) and any \(W^{1 - 1 / p , p}(\partial \Omega)\) function on \(\partial \Omega\) can be extended to a \(W^{1 , p}(\Omega)\) function. In this work, we study function spaces for nonlocal Dirichlet problems involving integrodifferential operators with a finite range of nonlocal interactions, and provide a characterization of their trace spaces. For these nonlocal Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical \(W^{1 - 1 / p , p}(\partial \Omega)\) space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.Functional calculus on BMO-type spaces of Bourgain, Brezis and Mironescuhttps://zbmath.org/1496.460322022-11-17T18:59:28.764376Z"Liu, Liguang"https://zbmath.org/authors/?q=ai:liu.liguang"Yang, Dachun"https://zbmath.org/authors/?q=ai:yang.dachun"Yuan, Wen"https://zbmath.org/authors/?q=ai:yuan.wenSummary: A nonlinear superposition operator \(T_g\) related to a Borel measurable function \(g: \mathbb{C}\to\mathbb{C}\) is defined via \(T_g(f):=g\circ f\) for any complex-valued function \(f\) on \(\mathbb{R}^n\). This article is devoted to investigating the mapping properties of \(T_g\) on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [\textit{J.~Bourgain} et al., J. Eur. Math. Soc. (JEMS) 17, No.~9, 2083--2101 (2015; Zbl 1339.46028)],
as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion and the continuity properties of \(T_g\) on these spaces are obtained.Hilbert Arens algebra-modules and semigrouphttps://zbmath.org/1496.460442022-11-17T18:59:28.764376Z"Tao, Jicheng"https://zbmath.org/authors/?q=ai:tao.jicheng"Ai, Ying"https://zbmath.org/authors/?q=ai:ai.yingSummary: In this paper, we consider the semigroup of Hilbert Arens algebra-modules by using the semigroup theory and Hilbert Arens algebra-modules from [\textit{S. Cerreia-Vioglio} et al., J. Math. Anal. Appl. 446, No. 1, 970--1017 (2017; Zbl 1364.46044)]. We show when \(A\) is a finite dimensional Arens algebra and \(H\) is a Hilbert \(A\)-module, the semigroup \(\{(tT)\}_{t\geq 0}\subset L(H)\) is an \(m\)-continuous semigroup if and only if it is an \(H\)-continuous semigroup if and only if it is a \(\varphi\)-continuous semigroup.Almost multiplicative maps and \(\varepsilon\)-spectrum of an element in Fréchet \(Q\)-algebrahttps://zbmath.org/1496.460452022-11-17T18:59:28.764376Z"Farajzadeh, A. P."https://zbmath.org/authors/?q=ai:farajzadeh.ali-p"Omidi, M. R."https://zbmath.org/authors/?q=ai:omidi.mohammad-rezaSummary: Let \((A,(p_k))\) be a Fréchet \(Q\)-algebra with unit \(e_A\). The \(\varepsilon\)-spectrum of an element \(x\) in \(A\) is defined by
\[
\sigma_\varepsilon(x)=\left\{\lambda\in\mathbb{C}:p_{k_0}(\lambda e_A-x)p_{k_0}(\lambda e_A-x)^{-1}\geq \frac{1}{\varepsilon}\right\}
\]
for \(0<\varepsilon<1\). We show that there is a close relation between the
\(\varepsilon\)-spectrum and almost multiplicative maps. It is also shown that
\[
\{\varphi(x):\varphi\in M^\varepsilon_{alm}(A),\varphi(e_A)=1\}\subseteq\sigma_\varepsilon(x)
\]
for every \(x\in A\), where \(M^\varepsilon_{alm}(A)\) is the set of all \(\varepsilon\)-multiplicative maps from \(A\) to \(\mathbb{C}\).Certain properties of Jordan homomorphisms, \(n\)-Jordan homomorphisms and \(n\)-homomorphisms on rings and Banach algebrashttps://zbmath.org/1496.460462022-11-17T18:59:28.764376Z"Honary, Taher Ghasemi"https://zbmath.org/authors/?q=ai:ghasemi-honary.taherSummary: We investigate under what conditions \(n\)-Jordan homomorphisms between rings are \(n\)-homomorphism, or homomorphism; and under what conditions, \(n\)-Jordan homomorphisms are continuous.
One of the main goals in this work is to show that every \(n\)-Jordan homomorphism \(f : A \rightarrow B\), from a unital ring \(A\) into a ring \(B\) with characteristic greater than \(n\), is a multiple of a Jordan homomorphism and hence, it is an \(n\)-homomorphism if every Jordan homomorphism from \(A\) into \(B\) is a homomorphism. In particular, if \(B\) is an integral domain whose characteristic is greater than \(n\), then every \(n\)-Jordan homomorphism \(f : A \rightarrow B\) is an \(n\)-homomorphism.
Along with some other results, we show that if \(A\) and \(B\) are unital rings such that the characteristic of \(B\) is greater than \(n\), then every unital \(n\)-Jordan homomorphism \(f : A \rightarrow B\) is a Jordan homomorphism and hence, it is an \(m\)-Jordan homomorphism for any positive integer \(m \geq 2\).
We also investigate the automatic continuity of \(n\)-Jordan homomorphisms from a unital Banach algebra either into a semisimple commutative Banach algebra, onto a semisimple Banach algebra, or into a strongly semisimple Banach algebra whenever the \(n\)-Jordan homomorphism has dense range.Positivity and Schwarz inequality for Banach \(*\)-algebrashttps://zbmath.org/1496.460472022-11-17T18:59:28.764376Z"El Harti, Rachid"https://zbmath.org/authors/?q=ai:el-harti.rachid"Pinto, Paulo R."https://zbmath.org/authors/?q=ai:pinto.paulo-r-fSummary: We establish the Schwarz inequality for a class of Banach \(*\)-algebras, and use it to derive some consequences, for example when a linear map between Banach \(*\)-algebras is a Jordan homomorphism. This applies to the class of Banach \(*\)-algebras \(\ell^1 (G,A;\alpha)\) arising from \(C^*\)-dynamical systems \((A,G,\alpha)\) with \(\alpha\) an action of a discrete group \(G\) on a separable \(C^*\)-algebra \(A\).Strongly peaking representations and compressions of operator systemshttps://zbmath.org/1496.460532022-11-17T18:59:28.764376Z"Davidson, Kenneth R."https://zbmath.org/authors/?q=ai:davidson.kenneth-r"Passer, Benjamin"https://zbmath.org/authors/?q=ai:passer.benjamin-wSummary: We use Arveson's notion of strongly peaking representation to generalize uniqueness theorems for free spectrahedra and matrix convex sets that admit minimal presentations. A fully compressed separable operator system necessarily generates the \(C^*\)-envelope and is such that the identity is the direct sum of strongly peaking representations. In particular, a fully compressed presentation of a separable operator system is unique up to unitary equivalence. Under various additional assumptions, minimality conditions are sufficient to determine a separable operator system uniquely.Noncommutative Choquet simpliceshttps://zbmath.org/1496.460542022-11-17T18:59:28.764376Z"Kennedy, Matthew"https://zbmath.org/authors/?q=ai:kennedy.matthew"Shamovich, Eli"https://zbmath.org/authors/?q=ai:shamovich.eliSummary: We introduce a notion of noncommutative Choquet simplex, or briefly an nc simplex, that generalizes the classical notion of a simplex. While every simplex is an nc simplex, there are many more nc simplices. They arise naturally from \(C^*\)-algebras and in noncommutative dynamics. We characterize nc simplices in terms of their geometry and in terms of structural properties of their corresponding operator systems. There is a natural definition of nc Bauer simplex that generalizes the classical definition of a Bauer simplex. We show that a compact nc convex set is an nc Bauer simplex if and only if it is affinely homeomorphic to the nc state space of a unital \(C^*\)-algebra, generalizing a classical result of Bauer for unital commutative \(C^*\)-algebras. We obtain several applications to noncommutative dynamics. We show that the set of nc states of a \(C^*\)-algebra that are invariant with respect to the action of a discrete group is an nc simplex. From this, we obtain a noncommutative ergodic decomposition theorem with uniqueness. Finally, we establish a new characterization of discrete groups with Kazhdan's property (T) that extends a result of \textit{E. Glasner} and \textit{B. Weiss} [Geom. Funct. Anal. 7, No. 5, 917--935 (1997; Zbl 0899.22006)]. Specifically, we show that a discrete group has property (T) if and only if for every action of the group on a unital \(C^*\)-algebra, the set of invariant states is affinely homeomorphic to the state space of a unital \(C^*\)-algebra.Corrigendum to: ``Dual spaces of operator systems''https://zbmath.org/1496.460552022-11-17T18:59:28.764376Z"Ng, Chi-Keung"https://zbmath.org/authors/?q=ai:ng.chi-keungFrom the text: All the operator systems considered in the author's paper [ibid. 508, No. 2, Article ID 125890, 23 p. (2022; Zbl 1492.46051)] need be assumed to be complete. The precise changes can be found in the most updated arXiv version [\textit{C. K. Ng}, ``Duality of operator systems'''', Preprint (2022), \url{arXiv:2105.11112v3}].A generalized powers averaging property for commutative crossed productshttps://zbmath.org/1496.460572022-11-17T18:59:28.764376Z"Amrutam, Tattwamasi"https://zbmath.org/authors/?q=ai:amrutam.tattwamasi"Ursu, Dan"https://zbmath.org/authors/?q=ai:ursu.danPowers' averaging property for discrete groups has played an important role in questions about simplicity related to reduced group \(C^\ast\)-algebras and reduced crossed products. In the present paper, the authors introduce a generalized version of Powers' averaging property for reduced crossed products of group \(C^\ast\)-algebras, and prove that it is equivalent to simplicity of the crossed product.
Reviewer: Luoyi Shi (Tianjin)Essentially invertible measurable operators affiliated to a semifinite von Neumann algebra and commutatorshttps://zbmath.org/1496.460622022-11-17T18:59:28.764376Z"Bikchentaev, A. M."https://zbmath.org/authors/?q=ai:bikchentaev.airat-mSummary: Suppose that a von Neumann operator algebra \({\mathcal{M}}\) acts on a Hilbert space \({\mathcal{H}}\) and \(\tau\) is a faithful normal semifinite trace on \({\mathcal{M}} \). If Hermitian operators \(X,Y\in S({\mathcal{M}},\tau)\) are such that \(-X\leq Y\leq X\) and \(Y\) is \(\tau \)-essentially invertible then so is \(X \). Let \(0<p\leq 1 \). If a \(p \)-hyponormal operator \(A\in S({\mathcal{M}},\tau)\) is right \(\tau \)-essentially invertible then \(A\) is \(\tau \)-essentially invertible. If a \(p \)-hyponormal operator \(A\in{\mathcal{B}}({\mathcal{H}})\) is right invertible then \(A\) is invertible in \({\mathcal{B}}({\mathcal{H}}) \). If a hyponormal operator \(A\in S({\mathcal{M}},\tau)\) has a right inverse in \(S({\mathcal{M}},\tau)\) then \(A\) is invertible in \(S({\mathcal{M}},\tau) \). If \(A,T\in{\mathcal{M}}\) and \(\mu_t(A^n)^{\frac{1}{n}}\to 0\) as \(n\to\infty\) for every \(t>0\) then \(AT ( TA )\) has no right (left) \( \tau \)-essential inverse in \(S({\mathcal{M}},\tau) \). Suppose that \({\mathcal{H}}\) is separable and \(\dim{\mathcal{H}}=\infty \). A right (left) essentially invertible operator \(A\in{\mathcal{B}}({\mathcal{H}})\) is a commutator if and only if the right (left) essential inverse of \(A\) is a commutator.Hyperstability of orthogonally 3-Lie homomorphism: an orthogonally fixed point approachhttps://zbmath.org/1496.460782022-11-17T18:59:28.764376Z"Keshavarz, Vahid"https://zbmath.org/authors/?q=ai:keshavarz.vahid"Jahedi, Sedigheh"https://zbmath.org/authors/?q=ai:jahedi.sedighehSummary: In this chapter, by using the orthogonally fixed point method, we prove the Hyers-Ulam stability and the hyperstability of orthogonally 3-Lie homomorphisms for additive \(\rho\)-functional equation in 3-Lie algebras. Indeed, we investigate the stability and the hyperstability of the system of functional equations
\[
\begin{cases}
f(x+y)-f(x)-f(y)= \rho (2f(\frac{x+y}{2})+ f(x)+ f(y)),\\
f([[u,v],w])=[[f(u),f(v)],f(w)]
\end{cases}
\]
in 3-Lie algebras where \(\rho \neq 1\) is a fixed real number.
For the entire collection see [Zbl 1485.65002].Fuzzy bounded operators with application to Radon transformhttps://zbmath.org/1496.460792022-11-17T18:59:28.764376Z"Bînzar, Tudor"https://zbmath.org/authors/?q=ai:binzar.tudor"Pater, Flavius"https://zbmath.org/authors/?q=ai:pater.flavius-lucian"Nădăban, Sorin"https://zbmath.org/authors/?q=ai:nadaban.sorin-florinSummary: This paper is focused on developing the means to extend the range of application of the inverse Radon transform by enlarging the domain of definition of the Radon operator, namely from a specific Banach space to a more general fuzzy normed linear space. This is done by studying different types of fuzzy bounded linear operators acting between fuzzy normed linear spaces. The motivation for considering this type of spaces comes from the existence of an equivalence between the probabilistic metric spaces and fuzzy metric spaces, in particular fuzzy normed linear spaces. We mention that many notions and results belonging to classical metric spaces could also be found in this general context. Moreover, this setup allows to develop applications as diverse as: image processing, data compression, signal processing, computer graphics, etc. The class of operators that best fits the intended purpose is the class of strongly fuzzy bounded linear operators. The main results about this family of operators use the fact that the space of such operators becomes a normed algebra. An extension of the classical norm of a bounded linear operator between two normed spaces to the norm of strongly fuzzy bounded linear operators acting between fuzzy normed linear spaces is proved. A version of the classical Banach-Steinhaus theorem for strongly fuzzy bounded linear operators is given. A sufficient condition for the limit of a sequence of strongly fuzzy bounded linear operators to be strongly fuzzy bounded is shown. The adjoint operator of a strongly fuzzy bounded linear operator is a classic bounded linear operator. The class of neighborhood fuzzy bounded linear operators are studied as well, being established connections with two other classes of operator, namely the class of fuzzy bounded linear operators and strongly fuzzy bounded linear operators.Attractors for semigroups and evolution equations. With an introduction by Gregory A. Seregin, Varga K. Kalantarov and Sergey V. Zelikhttps://zbmath.org/1496.470012022-11-17T18:59:28.764376Z"Ladyzhenskaya, Olga A."https://zbmath.org/authors/?q=ai:ladyzhenskaya.olga-aleksandrovnaSee the review of the 1991 edition in [Zbl 0755.47049].On some essential spectra of off-diagonal block operator matrix with applicationhttps://zbmath.org/1496.470022022-11-17T18:59:28.764376Z"He, Ruxia"https://zbmath.org/authors/?q=ai:he.ruxia"Wu, Deyu"https://zbmath.org/authors/?q=ai:wu.deyuSummary: Let \(\mathcal{H}=\begin{pmatrix}0 & B \\ C & 0 \end{pmatrix}\) be an off-diagonal \(2\times 2\) block operator matrix on the product of Hilbert spaces \(X\times X\). In this paper, the essential spectra of \(\mathcal{H}\) is characterized by the essential spectra of \(BC\) and \(CB\). Furthermore, we give an application to infinite dimensional Hamiltonian operator.On the spectral radius of antidiagonal block operator matriceshttps://zbmath.org/1496.470032022-11-17T18:59:28.764376Z"Ipek Al, Pembe"https://zbmath.org/authors/?q=ai:ipek-al.pembe"Ismailov, Zameddin I."https://zbmath.org/authors/?q=ai:ismailov.zameddin-ismailovichSummary: In this paper, the difference between operator norm and spectral radius for the antidiagonal block operator matrix in the direct sum of Hilbert spaces is investigated. Also, the necessary and sufficient conditions for these operators belong to Schatten-von Neumann classes are given.Local spectral property of \(2 \times 2\) operator matriceshttps://zbmath.org/1496.470042022-11-17T18:59:28.764376Z"Ko, Eungil"https://zbmath.org/authors/?q=ai:ko.eungilSummary: In this paper we study the local spectral properties of \(2 \times 2\) operator matrices. In particular, we show that every \(2 \times 2\) operator matrix with three scalar entries has the single valued extension property. Moreover, we consider the spectral properties of such operator matrices. Finally, we show that some of such operator matrices are decomposable.Semigroup generations of unbounded block operator matrices based on the space decompositionhttps://zbmath.org/1496.470052022-11-17T18:59:28.764376Z"Liu, Jie"https://zbmath.org/authors/?q=ai:liu.jie|liu.jie.1|liu.jie.3|liu.jie.4|liu.jie.2|liu.jie.7|liu.jie.5"Huang, Junjie"https://zbmath.org/authors/?q=ai:huang.junjie"Chen, Alatancang"https://zbmath.org/authors/?q=ai:chen.alatancangThe authors find necessary and sufficient conditions under which an unbounded block operator matrix
\[
M = \begin{pmatrix} A & B \\
C & D \end{pmatrix}
\]
with natural domain \(\mathscr{D}(M)=(\mathscr{D}(A)\cap \mathscr{D}(C))\oplus (\mathscr{D}(B)\cap \mathscr{D}(D))\) generates a \(C_0\)-semigroup. Usually, in the literature, most of the results are presented using the diagonal domain \(\mathscr{D}(M) = \mathscr{D}(A)\oplus \mathscr{D}(D)\), using standard perturbation theorems. To prove the results in the natural domain, the authors characterize the right boundedness of \(M\) with the quadratic numerical range of \(M\), and consider the residual spectrum based on the space decomposition and quadratic complements.
Reviewer: Matheus Cheque Bortolan (Florianópolis)The point spectrum and residual spectrum of upper triangular operator matriceshttps://zbmath.org/1496.470062022-11-17T18:59:28.764376Z"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: The point and residual spectra of an operator are, respectively, split into \(1,2\)-point spectrum and \(1,2\)-residual spectrum, based on the denseness and closedness of its range. Let \(\mathcal{H,K}\) be infinite dimensional complex separable Hilbert spaces and write
\(M_X = \begin{pmatrix} A & X \\ 0 & B \end{pmatrix} \in \mathcal{B}(\mathcal{H} \oplus \mathcal{K})\).
For given operators \(A \in \mathcal{B(H)}\) and \(B \in \mathcal{B(K)}\), the sets \(\bigcup\limits_{X \in \mathcal{B}(\mathcal{K},\mathcal{H})} \sigma_{\ast,i}(M_X) \) (\(* = p,r\); \(i=1,2\)) are characterized.
Moreover, we obtain some necessary and sufficient condition such that \(\sigma_{\ast,i}(M_X) = \sigma_{*,i}(A) \cup \sigma_{*,i}(B)\) (\(* = p,r\); \(i=1,2\)) for every \( X \in \mathcal{B}(\mathcal{K},\mathcal{H})\).On the essential numerical spectrum of operators on Banach spaceshttps://zbmath.org/1496.470072022-11-17T18:59:28.764376Z"Abdelhedi, Bouthaina"https://zbmath.org/authors/?q=ai:abdelhedi.bouthaina"Boubaker, Wissal"https://zbmath.org/authors/?q=ai:boubaker.wissal"Moalla, Nedra"https://zbmath.org/authors/?q=ai:moalla.nedraSummary: The purpose of this paper is to define and develop a new notion of the essential numerical spectrum \(\sigma_{en}(.)\) of an operator on a Banach space \(X\) and to study its properties. Our definition is closely related to the essential numerical range \(W_e(.)\).The spectrum of the restriction to an invariant subspacehttps://zbmath.org/1496.470082022-11-17T18:59:28.764376Z"Drivaliaris, Dimosthenis"https://zbmath.org/authors/?q=ai:drivaliaris.dimosthenis"Yannakakis, Nikos"https://zbmath.org/authors/?q=ai:yannakakis.nikolaosLet \(A\) be a bounded linear operator on a Banach space, \(\rho(A)\) (resp., \(\sigma(A)\)) its resolvent (resp., spectrum). Let \(M\) be a closed invariant subspace of \(A\) and \(D\) be a connected component of \(\rho(A)\). The authors state that, if \(D\cap\sigma(A_{\restriction_M})\neq\emptyset\), then \(D\subset \sigma(A_{\restriction_M})\).
Reviewer: Mohammed El Aïdi (Bogotá)Pseudospectrum enclosures by discretizationhttps://zbmath.org/1496.470092022-11-17T18:59:28.764376Z"Frommer, Andreas"https://zbmath.org/authors/?q=ai:frommer.andreas"Jacob, Birgit"https://zbmath.org/authors/?q=ai:jacob.birgit"Vorberg, Lukas"https://zbmath.org/authors/?q=ai:vorberg.lukas-a"Wyss, Christian"https://zbmath.org/authors/?q=ai:wyss.christian"Zwaan, Ian"https://zbmath.org/authors/?q=ai:zwaan.ian-nLet \(A\) be a matrix (or operator) of finite or infinite dimension. For \(\lambda \in \mathbb{C}\), the resolvent matrix is \(R_\lambda(A)=(A-\lambda)^{-1}\). To understand more about an object represented by a matrix \(A\), we have to analyze not only the eigenvalues, spectrum and resolvent matrix, but also the pseudospectrum. The concept of pseudospectrum of matrices has a number of applications in different fields: dynamical systems, hydrodynamic stability, Markov chains, and non-Hermitian quantum mechanics. Note that the eigenvalues of the resolvent matrix of \(A\) never coincide with the eigenvalues of \(R_\lambda(A)\).
It is well known that for \(\epsilon>0,\) pseudospectrum of \(A\) is given by
\[\sigma_\epsilon(A)=\{\lambda\in \mathbb{C}:\|(A-\lambda)^{-1}\|< \epsilon^{-1}\}.\]
In this paper, a new method is employed to enclose the pseudospectrum via the numerical range of the inverse of the matrix or linear operator.
The results (Theorem 2.2, Theorem 2.5, Theorem 3.6 and Lemma 4.5) will have a significant impact in the area of studying spectral analysis of matrices or linear operators. Also, there are computations given in Example 7.1 and Example 7.2.
Reviewer: Ali Shukur (Minsk)Spectral \(\zeta\)-functions and \(\zeta\)-regularized functional determinants for regular Sturm-Liouville operatorshttps://zbmath.org/1496.470102022-11-17T18:59:28.764376Z"Fucci, Guglielmo"https://zbmath.org/authors/?q=ai:fucci.guglielmo"Gesztesy, Fritz"https://zbmath.org/authors/?q=ai:gesztesy.fritz"Kirsten, Klaus"https://zbmath.org/authors/?q=ai:kirsten.klaus"Stanfill, Jonathan"https://zbmath.org/authors/?q=ai:stanfill.jonathanSummary: The principal aim in this paper is to employ a recently developed unified approach to the computation of traces of resolvents and \(\zeta\)-functions to efficiently compute values of spectral \(\zeta\)-functions at positive integers associated with regular (three-coefficient) self-adjoint Sturm-Liouville differential expressions \(\tau\). Depending on the underlying boundary conditions, we express the \(\zeta\)-function values in terms of a fundamental system of solutions of \(\tau y=zy\) and their expansions about the spectral point \(z=0\). Furthermore, we give the full analytic continuation of the \(\zeta\)-function through a Liouville transformation and provide an explicit expression for the \(\zeta\)-regularized functional determinant in terms of a particular set of this fundamental system of solutions. An array of examples illustrating the applicability of these methods is provided, including regular Schrödinger operators with zero, piecewise constant, and a linear potential on a compact interval.Branching form of the resolvent at thresholds for multi-dimensional discrete Laplacianshttps://zbmath.org/1496.470112022-11-17T18:59:28.764376Z"Ito, Kenichi"https://zbmath.org/authors/?q=ai:ito.kenichi.1"Jensen, Arne"https://zbmath.org/authors/?q=ai:jensen.arneSummary: We consider the discrete Laplacian on \(\mathbb{Z}^d\), and compute asymptotic expansions of its resolvent around thresholds embedded in continuous spectrum as well as those at end points. We prove that the resolvent has a square-root branching if \(d\) is odd, and a logarithm branching if \(d\) is even, and, moreover, obtain explicit expressions for these branching parts involving the Lauricella hypergeometric function. In order to analyze a non-degenerate threshold of general form we use an elementary step-by-step expansion procedure, less dependent on special functions.Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensionshttps://zbmath.org/1496.470122022-11-17T18:59:28.764376Z"Ito, Kenichi"https://zbmath.org/authors/?q=ai:ito.kenichi.2|ito.kenichi.1"Jensen, Arne"https://zbmath.org/authors/?q=ai:jensen.arne-m|jensen.arne-skov|jensen.arneThe authors have obtained some closed formulae for lattice Green functions of the form \[ G(z,n)=(2\pi)^{-d}\int_{\mathbb{T}^d}\dfrac{e^{in\theta}}{2d-2\cos(\theta_1)-\dots-2\cos(\theta_d)-z}d\theta.\]
Such investigation was mainly restricted to dimensions \(d=1,2\). In the Introduction, they start to depict that \(2dG(0,n)\) (\(z=0\)) shall be represented as the expectation value \( \mathbb{E}[n]=\sum_{k=0}^\infty P(X_k=n)\) that counts the number of times that a walker visits \(n\in \mathbb{Z}^d\). To get rid of the fact that \(\mathbb{E}[n]\) is divergent for dimensions \(d=1,2\) (see also Appendix B), they propose a renormalization technique to approximate \(\mathbb{E}[n]\) by \(\mathbb{E}[\epsilon,n]=\frac{2d}{1-\epsilon}G(\frac{-2d\epsilon}{1-\epsilon},n)\), for values of \(\epsilon\in (0,1]\).
In this way, they succeed in representing \(G(z,n)\) as a convergent series (see, e.g., Theorem 2.2. and Theorem 2.3.). Such analysis goes far beyond the asymptotic analysis, in the limit \(z\rightarrow 0\), considered by so many authors in the past.
As a whole, this paper is complementary to the thors' previous paper [J. Funct. Anal. 277, No. 4, 965--993 (2019; Zbl 1496.47011)] in which the authors have shown that \(G(z,n)\) admits, for each threshold \(4q\), \(q=0,\dots,n\), the splitting formula \[ G(z,n)=\mathcal{E}_q(z,n)+f_q(z)\mathcal{F}_q(z,n), \] whereby \(\mathcal{F}_q(z,n)\) -- the singular part of \(G(z,n)\) -- was represented in terms of the so-called Appell-Lauricella hypergeometric function of type \(B\), \(F_B^{(d)}\). Further comparisons between both approaches may be found in Appendix~A.
Reviewer: Nelson Faustino (Alfeizerão)On the numerical range and operator norm of \(V^2\)https://zbmath.org/1496.470132022-11-17T18:59:28.764376Z"Khadkhuu, L."https://zbmath.org/authors/?q=ai:khadkhuu.lkhamzhav|khadkhuu.lkhamjav"Tsedenbayar, D."https://zbmath.org/authors/?q=ai:tsedenbayar.dashdondogThe authors consider the numerical range and operator norm of \(V^2\), the square of the Voltera operator. Also, they get the numerical range, numerical radius and norm of the real (resp., imaginary) part of \(V^2\).
Reviewer: Mohammed El Aïdi (Bogotá)Some refinements of numerical radius inequalities for Hilbert space operatorshttps://zbmath.org/1496.470142022-11-17T18:59:28.764376Z"Alizadeh, Ebrahim"https://zbmath.org/authors/?q=ai:alizadeh.ebrahim"Farokhinia, Ali"https://zbmath.org/authors/?q=ai:farokhinia.aliSummary: The main goal of this paper is to obtain some refinements of numerical radius inequalities for Hilbert space operators.On the numerical range of some weighted shift operatorshttps://zbmath.org/1496.470152022-11-17T18:59:28.764376Z"Chakraborty, Bikshan"https://zbmath.org/authors/?q=ai:chakraborty.bikshan"Ojha, Sarita"https://zbmath.org/authors/?q=ai:ojha.sarita"Birbonshi, Riddhick"https://zbmath.org/authors/?q=ai:birbonshi.riddhickThe paper under review is devoted to computing the numerical radius of a weighted shift operator \(T\) on a complex Hilbert space with weights \((h,k,ab,ab,\ldots)\) where \(h,k,a,b\) are positive numbers and satisfy \(bh^2+(a+b)k^2>(a+b)^2b\). This is done viewing \(T\) as acting on a Hardy space \(H^2\) for appropriate weights related to \(T\), and studying the properties of an eigenfunction \(f\) of the operator \(\frac{T+T^*}{2}\) for an eigenvalue \(\alpha\) satisfying \(\alpha=\big\|\frac{T+T^*}{2}\big\|=w\big(\frac{T+T^*}{2}\big)=w(T)\). As an application, the authors obtain a lower bound of the numerical radius of some tridiagonal operators.
Reviewer: Javier Merí (Granada)Some new refinements of generalized numerical radius inequalities for Hilbert space operatorshttps://zbmath.org/1496.470162022-11-17T18:59:28.764376Z"Feki, Kais"https://zbmath.org/authors/?q=ai:feki.kais"Kittaneh, Fuad"https://zbmath.org/authors/?q=ai:kittaneh.fuadSummary: Let \(A\) be a positive (semi-definite) bounded linear operator on a complex Hilbert space \((\mathcal{H}, \langle \cdot, \cdot \rangle)\). Let \(\omega_A(T)\) and \(\Vert T\Vert_A\) denote the \(A\)-numerical radius and the \(A\)-operator seminorm of an operator \(T\) acting on the semi-Hilbert space \((\mathcal{H}, \langle \cdot, \cdot \rangle_A)\) respectively, where \(\langle x, y\rangle_A :=\langle Ax, y\rangle\) for all \(x, y\in\mathcal{H}\). It is well known that
\[
\frac{1}{4}\Vert T^{\sharp_A} T+TT^{\sharp_A}\Vert_A\le \omega_A^2(T) \le \frac{1}{2}\Vert T^{\sharp_A} T+TT^{\sharp_A}\Vert_A,
\]
where \(T^{\sharp_A}\) denotes a distinguished \(A\)-adjoint operator of \(T\). In this paper, we aim to give some new refinements of the above inequalities. Furthermore, we establish an \(\mathbb{A}\)-seminorm inequality involving \(2\times 2\) operator matrices, where \(\mathbb{A}=\mathrm{diag}(A, A)\). This generalizes a recent result of \textit{W. Bani-Domi} and \textit{F. Kittaneh} [Linear Multilinear Algebra 69, No. 5, 934--945 (2021; Zbl 07333203)]. As an application, a refinement of the triangle inequality related to \(\Vert \cdot \Vert_A\) is given.Some numerical radius inequalities for products of Hilbert space operatorshttps://zbmath.org/1496.470172022-11-17T18:59:28.764376Z"Hosseini, Mohsen Shah"https://zbmath.org/authors/?q=ai:hosseini.mohsen-shah"Moosavi, Baharak"https://zbmath.org/authors/?q=ai:moosavi.baharakSummary: We prove several numerical radius inequalities for products of two Hilbert space operators. Some of our inequalities improve well-known ones. More precisely, we prove that, if \(A,B \in \mathbb{B}(\mathscr{H})\) such that \(A\) is self-adjoint with \(\lambda_1 = \min \lambda_i \in \sigma (A)\) (the spectrum of \(A\)) and \(\lambda_2 = \max \lambda_i \in \sigma (A)\). Then
\[\omega(AB) \leq \|A\| \omega(B) + \left( \|A\| - \frac{| \lambda_1 + \lambda_2|}{2} \right) D_B\]
where \(D_B = \inf\limits_{\lambda \in \mathbb{C}} \|B - \lambda I \|\). In particular, if \( A>0\) and \( \sigma \subseteq [k\|A\|,\|A\|]\), then
\[\omega(AB) \leq (2-k) \|A\| \omega(B).\]Inverse continuity of the numerical range map for Hilbert space operatorshttps://zbmath.org/1496.470182022-11-17T18:59:28.764376Z"Lins, Brian"https://zbmath.org/authors/?q=ai:lins.brian"Spitkovsky, Ilya M."https://zbmath.org/authors/?q=ai:spitkovsky.ilya-matveyGiven a Hilbert space \(\mathcal{H}\) with inner product \(\langle\cdot,\cdot\rangle\), the numerical range \(W(A)\) of a bounded linear operator \(A\) on \(\mathcal{H}\) is the set
\[
W(A)=\{\langle Ax,x\rangle: x\in\mathcal{H}, \langle x,x\rangle=1\},
\]
while the numerical range map \(f_A\) associated with \(A\) is
\[
f_A: \{x\in\mathcal{H}: \langle x,x\rangle=1\}\rightarrow \mathbb{C},\quad x\mapsto \langle Ax,x\rangle.
\]
To describe the contents of the article, we quote its abstract:
``We describe continuity properties of the multivalued inverse of the numerical range map \(f_A: x\mapsto \langle Ax,x\rangle\) associated with a linear operator \(A\) defined on a complex Hilbert space \(\mathcal{H}\). We prove in particular that \(f_A^{-1}\) is strongly continuous at all points of the interior of the numerical range \(W(A)\). We give examples where strong and weak continuity fail on the boundary and address special cases such as normal and compact operators.''
Reviewer: Agnes Radl (Fulda)Almost invariant subspaces of the shift operator on vector-valued Hardy spaceshttps://zbmath.org/1496.470192022-11-17T18:59:28.764376Z"Chattopadhyay, Arup"https://zbmath.org/authors/?q=ai:chattopadhyay.arup"Das, Soma"https://zbmath.org/authors/?q=ai:das.soma"Pradhan, Chandan"https://zbmath.org/authors/?q=ai:pradhan.chandanThe authors characterize nearly invariant subspaces of finite defect for the backward shift operator acting on vector-valued Hardy spaces \(H^2_{\mathbb C^m}(\mathbb D)\), generalizing the scalar-valued result by \textit{I. Chalendar} et al. [J. Oper. Theory 83, No. 2, 321--331 (2020; Zbl 1463.47096)]. Given a bounded analytic function \(\Theta\) with values in the space of linear operators \(\mathcal L(\mathbb C^r,\mathbb C^m)\), we can induce the multiplier \(T_\theta F(z)=\Theta(z)F(z)\) from \(H^2_{\mathbb C^r}(\mathbb D)\) into \(H^2_{\mathbb C^m}(\mathbb D)\). They are determined by the condition \(ST_\Theta=T_\Theta S\) where \(S\) denotes the forward shift operator \(SF(z)=zF(z)\) which acts on the corresponding space in each case. We write, as usual, the backward shift \(S^* F(z)=\frac{F(z)-F(0)}{z}\). A closed subspace \(\mathcal M\subset H^2_{\mathbb C^m}(\mathbb D)\) is called almost-invariant for \(S\) if there exists a finite-dimensional subspace \(\mathcal F\subset H^2_{\mathbb C^m}(\mathbb D)\) such that \(S(\mathcal M)\subset \mathcal M\oplus\mathcal F\). Similarly, a closed subspace \(\mathcal M\subset H^2_{\mathbb C^m}(\mathbb D)\) is called nearly invariant for \(S^*\) if any \(F\in \mathcal M\) with \(F(0)=0\) satisfies that \(S^*F\in \mathcal M\) and it is called nearly \(S^*\)-invariant with defect \(p\) if there exists a \(p\)-dimensional subspace \(\mathcal F\subset H^2_{\mathbb C^m}(\mathbb D)\) such that, if \(F\in \mathcal M\) with \(F(0)=0\), then \(S^*F\in \mathcal M \oplus \mathcal F\). Also, \(\mathcal M\) is called \(S^*\)-almost invariant with defect \(p\) if \(S^*\mathcal M\subset\mathcal M \oplus \mathcal F\) and \(\dim \mathcal F=p\).
In the paper under review, the authors present a characterization of nearly invariant subspaces for \(S^*\) with finite defect in the vector-valued Hardy spaces. Using such a result, they also manage to obtain the description of almost invariant subspaces for the shift and its adjoint acting on vector-valued Hardy spaces.
Reviewer: Oscar Blasco (València)Diskcyclic \(C_0\)-semigroups and diskcyclicity criteriahttps://zbmath.org/1496.470202022-11-17T18:59:28.764376Z"Moosapoor, Mansooreh"https://zbmath.org/authors/?q=ai:moosapoor.mansoorehSummary: In this article, we prove that diskcyclic \(C_0\)-semigroups exist on any infinite-dimensional Banach space. We show that a \(C_0\)-semigroup \((T_t)_{t \geq 0}\) satisfies the diskcyclicity criterion if and only if any of \(T_t\)'s satisfies the diskcyclicity criterion for operators. Moreover, we show that there are diskcyclic \(C_0\)-semigroups that do not satisfy the diskcyclicity criterion. Also, we state various criteria for diskcyclicity of \(C_0\)-semigrous based on dense sets and \(d\)-dense orbits.Equivalence of semi-norms related to super weakly compact operatorshttps://zbmath.org/1496.470212022-11-17T18:59:28.764376Z"Tu, Kun"https://zbmath.org/authors/?q=ai:tu.kunLet $X$ and $Y$ be real infinite-dimensional Banach spaces. A subset $A$ of $X$ is said to be relatively super weakly compact if $A_{\mathcal{U}}$ is relatively weakly compact in $X_{\mathcal{U}}$ for any free ultrafilter $\mathcal{U}$. $A$ is said to be super weakly compact if it is weakly closed and relatively super weakly compact. The measure of super weak noncompactness of a bounded subset $A$ of $X$, $\sigma(A)$ is defined as
\[
\sigma(A)=\inf\{t > 0 : A\subset S + tB_X,\ S\text{ is relatively super weakly compact}\}.
\]
A bounded linear operator $T:X\to Y$ is called super weakly compact if $T(B_X)$ is relatively super weakly compact. Equivalently, $T_{\mathcal{U}}$ is weakly compact for any free ultrafilter $\mathcal{U}$. $A$ is called weakly compact if $T(B_X)$ is relatively weakly compact where $B_X$ is the closed unit ball in $X$.
Let $L(X,Y)$ denote the collection of all bounded linear operators mapping $X$ to $Y$ and $S(X,Y)$ represent the collection of all super weakly compact operators. The super weak essential norm $\|\cdot\|_s$ of $T\in L(X,Y)$ is the semi-norm induced from the quotient space $L(X,Y)/S(X,Y)$, that is,
\[
\|T\|_s=\inf\{\|T -S\|:S\in S(X,Y)\}.
\]
The space $X$ is said to have the super weakly compact approximation property (SWAP) if there is a real number $\lambda>0$ such that for any super weakly compact set $A\subset X$ and any $\varepsilon>0$, there is a super weakly compact operator $R:X\to X$ with $\sup_{x\in A} \|x- Rx\|\leq\varepsilon$ and $\|R\|\leq\lambda$.
In this paper, super weakly compact operators are discussed through a quantative method. By introducing the semi-norm $\sigma(T)$ of the operator $T:X\to Y$, which measures how far $T$ is from the family of super weakly compact operators, the following equivalence of the measure $\sigma(T)$ and the super weak essential norm $\|T\|_s$ of $T$ is proved:
Theorem. A Banach space $Y$ has the (SWAP) if only if the semi-norms $\sigma$ and ${\| \cdot\|}_s$ are equivalent in $L(X,Y)$ for any Banach space $X$.
In order to give an application of this theorem, some basic properties of Banach spaces having the SWAP are studied in Section~4 of the paper and then an example is constructed to show that the measures of $T$ and its dual $T^*$ are not always equivalent. Moreover, some examples of Banach spaces which have and which do not have the SWAP are given in this paper.
Reviewer: T. D. Narang (Amritsar)On two-dimensional model representations of one class of commuting operatorshttps://zbmath.org/1496.470222022-11-17T18:59:28.764376Z"Hatamleh, R."https://zbmath.org/authors/?q=ai:hatamleh.raed-m"Zolotarev, V. A."https://zbmath.org/authors/?q=ai:zolotarev.vladimir-alekseevichSummary: In the work by \textit{V. A. Zolotarev} [Dokl. Akad. Nauk Arm. SSR 63, No. 3, 136--140 (1976; Zbl 0351.47010)], a triangular model is constructed for a system of twice-commuting linear bounded completely nonself-adjoint operators \(\{A_1,A_2\}\) (\([A_1,A_2]=0\), \([A_1^\ast,A_2]=0\)) such that rank (\(A_1)_I(A_2)_I=1\) (\(2i(A_k)_I=A_k-A_k^\ast\), \(k=1,2\)) and the spectrum of each operator \(A_k\), \(k=1,2\), is concentrated at zero. The indicated triangular model has the form of a system of operators of integration over the independent variable in \(L_\Omega^2\) where the domain \(\Omega=[0,a]\times [0,b]\) is a compact set in \(\mathbb R^2\) bounded by the lines \(x=a\) and \(y=b\) and a decreasing smooth curve \(L\) connecting the points \((0,b)\) and \((a,0)\).Spectral theory for polynomially demicompact operatorshttps://zbmath.org/1496.470232022-11-17T18:59:28.764376Z"Brahim, Fatma Ben"https://zbmath.org/authors/?q=ai:brahim.fatma-ben"Jeribi, Aref"https://zbmath.org/authors/?q=ai:jeribi.aref"Krichen, Bilel"https://zbmath.org/authors/?q=ai:krichen.bilelSummary: In this article, we introduce the notion of polynomial demicompactness and we use it to give some results on Fredholm operators and to establish a fine description of some essential spectra of a closed densely defined linear operator. Our work is a generalization of many known ones in the literature.Essential pseudospectra involving demicompact and pseudodemicompact operators and some perturbation resultshttps://zbmath.org/1496.470242022-11-17T18:59:28.764376Z"Brahim, Fatma Ben"https://zbmath.org/authors/?q=ai:brahim.fatma-ben"Jeribi, Aref"https://zbmath.org/authors/?q=ai:jeribi.aref"Krichen, Bilel"https://zbmath.org/authors/?q=ai:krichen.bilelSummary: In this paper, we study the essential and the structured essential pseudospectra of closed densely defined linear operators acting on a Banach space \(X\). We start by giving a refinement and investigating the stability of these essential pseudospectra by means of the class of demicompact linear operators. Moreover, we introduce the notion of pseudo demicompactness and we study its relationship with pseudo upper semi-Fredholm operators. Some stability results for the Gustafson essential pseudospectrum involving pseudo demicompact operators is given.Property \((R)\) for functions of operators and its perturbationshttps://zbmath.org/1496.470252022-11-17T18:59:28.764376Z"Yang, Lili"https://zbmath.org/authors/?q=ai:yang.lili"Cao, Xiaohong"https://zbmath.org/authors/?q=ai:cao.xiaohongSummary: Let \(\mathcal{H}\) be a complex separable infinite dimensional Hilbert space and \(\mathcal{B(H)}\) be the algebra of all bounded linear operators on \(\mathcal{H}\). \(T\in \mathcal{B(H)}\) is said to satisfy property \((R)\) if \(\sigma_a(T)\backslash\sigma_{ab}(T)=\pi_{00}(T)\), where \(\sigma_a(T)\) and \(\sigma_{ab}(T)\) denote the approximate point spectrum and the Browder essential approximate point spectrum of \(T\), respectively, and \(\pi_{00}(T)=\{\lambda \in \operatorname{iso}\sigma (T): 0 < \dim N(T-\lambda I)<\infty\}\). In this paper, using a new spectrum, we talk about the property \((R)\) for functions of operators as well as its stability.Interpolating matriceshttps://zbmath.org/1496.470262022-11-17T18:59:28.764376Z"Dayan, Alberto"https://zbmath.org/authors/?q=ai:dayan.albertoThe main result is a matrix version of the Carleson theorem [\textit{L. Carleson}, Am. J. Math. 80, 921--930 (1958; Zbl 0085.06504)] which says that a sequence of points \(\Lambda=(\lambda_k)_{n\in\mathbb{N}}\) in the open unit disk \(\mathbb{D}\) is an interpolation sequence in \(H^\infty\) if and only if it is strongly separated. Here, \(\Lambda\) is replaced by a matrix sequence \(A=(A_n)_{n\in\mathbb{N}}\) (possibly with varying dimensions) with spectra in \(\mathbb{D}\) and interpolating means that for any bounded sequence \((\phi_n)_{n\in\mathbb{N}}\) in \(H^\infty\) there is a \(\varphi\in H^\infty\) such that \(\varphi(A_n)=\phi_n(A_n)\), \(n\in\mathbb{N}\). Strong separated means \(\inf_{z\in\mathbb{D}}\sup_{n\in\mathbb{N}}\prod_{k\ne n}|B_{A_k}(z)|>0\) in \(\mathbb{D}\) where \(B_M(z)\) is a Blaschke product whose zeros are eigenvalues of \(M\) (including multiplicity). An equivalent formulation is that the model spaces \(H_n=H^2\ominus B_{A_n}H^2\) are strongly separated. If the dimensions of the matrices \(A_n\) are uniformly bounded, then \((H_n)_{n\in\mathbb{N}}\) being a weakly separated Bessel system is an alternative equivalent condition for \(A\) to be interpolating.
Reviewer: Adhemar Bultheel (Leuven)New properties of the multivariable \(H^\infty\) functional calculus of sectorial operatorshttps://zbmath.org/1496.470272022-11-17T18:59:28.764376Z"Arrigoni, Olivier"https://zbmath.org/authors/?q=ai:arrigoni.olivier"Le Merdy, Christian"https://zbmath.org/authors/?q=ai:le-merdy.christianLet \(A : \operatorname{dom}A \rightarrow X\) be a closed and densely defined operator on a Banach space \(X\). We say that \(A\) is sectorial of type \(\omega \in(0, \pi)\) if its spectrum \(\sigma(A)\) satisfies \(\sigma(A) \subseteq \overline{\Sigma_\omega}\) and for any \(\theta \in (\omega, \pi)\), there exists a constant \(C_\theta \geq 0\) such that
\[
\|z R(z,A)\| \leq C_\theta \qquad (z \in \mathbb{C} \setminus \overline{\Sigma_\theta}),
\]
where \(R(z,A) = (z I_X - A)^{-1}\), and
\[
\Sigma_\theta = \{z \in \mathbb{C}^*: |\!\operatorname{Arg} z| < \theta\}.
\]
The paper under review concerns a multivariable \(H^\infty\)-functional calculus associated with commuting families of sectorial operators on Banach spaces.
One of the main results of this paper extends a result of \textit{N. J. Kalton} and \textit{L. Weis} [Math. Ann. 321, No. 2, 319--345 (2001; Zbl 0992.47005)] on the optimal angle of bounded \(H^\infty\)-functional calculus into the setting of tuples of operators. Another important result states: If \(X\) is reflexive and \(K\)-convex, then a tuple \((A_1, \dots, A_d)\) admitting a bounded \(H^\infty(\Sigma_{\theta_1} \times \dots \times \Sigma_{\theta_d})\) joint functional calculus for some \(\theta_k < \frac{\pi}{2}\), \(k=1, \ldots, d\), dilates into a commuting tuple of sectorial operators \((B_1, \dots, B_d)\) on a Bochner space admitting bounded \(H^\infty(\Sigma_{\frac{\pi}{2}} \times \dots \times \Sigma_{\frac{\pi}{2}})\) joint functional calculus, where each \(B_k\) generates a bounded \(C_0\)-group.
Reviewer: Jaydeb Sarkar (Bangalore)On a class of operator equations in locally convex spaceshttps://zbmath.org/1496.470282022-11-17T18:59:28.764376Z"Mishin, Sergeĭ N."https://zbmath.org/authors/?q=ai:mishin.sergey-nSummary: We consider a general method of solving equations whose left-hand side is a series by powers of a linear continuous operator acting in a locally convex space. Obtained solutions are given by operator series by powers of the same operator as the left-hand side of the equation. The research is realized by means of characteristics (of order and type) of operator as well as operator characteristics (of operator order and operator type) of vector relatively of an operator. In research we also use a convergence of operator series on equicontinuous bornology.Numerical interpretation of the Gurov-Reshetnyak inequality on the real axishttps://zbmath.org/1496.470292022-11-17T18:59:28.764376Z"Didenko, V. D."https://zbmath.org/authors/?q=ai:didenko.victor-d"Korenovskiĭ, A. A."https://zbmath.org/authors/?q=ai:korenovskii.anatolii-a"Tuah, N. J."https://zbmath.org/authors/?q=ai:tuah.nor-jaidiSummary: We find the ``norm'' of a power function in the Gurov-Reshetnyak class on the real line. Moreover, as a result of numerical experiments, we establish a lower bound for the norm of the operator of even extension of a function from the Gurov-Reshetnyak class from the semiaxis onto the entire real line.Applications of Kato's inequality for \(n\)-tuples of operators in Hilbert spaces. IIhttps://zbmath.org/1496.470302022-11-17T18:59:28.764376Z"Dragomir, Sever S."https://zbmath.org/authors/?q=ai:dragomir.sever-silvestru"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-je"Kim, Young-Ho"https://zbmath.org/authors/?q=ai:kim.youngho.1Summary: In this paper, by the use of famous Kato's inequality for bounded linear operators, we establish some new inequalities for \(n\)-tuples of operators and apply them to functions of normal operators defined by power series as well as to some norms and numerical radii that arise in multivariate operator theory. They provide a natural continuation of the results in Part I [\textit{S. S. Dragomir} et al., J. Inequal. Appl. 2013, Paper No. 21, 16 p. (2013; Zbl 1294.47028)].Čebyšev's type inequalities and power inequalities for the Berezin number of operatorshttps://zbmath.org/1496.470312022-11-17T18:59:28.764376Z"Garayev, Mubariz T."https://zbmath.org/authors/?q=ai:garayev.mubariz-tapdigoglu"Yamancı, Ulaş"https://zbmath.org/authors/?q=ai:yamanci.ulasSummary: We give operator analogues of some classical inequalities, including Čebyšev type inequality for synchronous and convex functions of selfadjoint operators in Reproducing Kernel Hilbert Spaces (RKHSs). We obtain some Berezin number inequalities for the product of operators. Also, we prove the Berezin number inequality for the commutator of two operators.Stability of versions of the Weyl-type theorems under the tensor producthttps://zbmath.org/1496.470322022-11-17T18:59:28.764376Z"Rashid, M. H. M."https://zbmath.org/authors/?q=ai:rashid.mohammad-hussein-mohammad|rashid.malik-h-m"Prasad, T."https://zbmath.org/authors/?q=ai:prasad.t-jayachandra|prasad.t-v-s-r-k|prasad.t-ram|prasad.tribhuan|prasad.t-b-arunaSummary: We study the transformation versions of the Weyl-type theorems for operators \(T\) and \(S\) and their tensor product \(T \otimes S\) in the infinite-dimensional space setting.Example of a quasianalytic contraction whose spectrum is a proper subarc of the unit circlehttps://zbmath.org/1496.470332022-11-17T18:59:28.764376Z"Gamal', Maria F."https://zbmath.org/authors/?q=ai:gamal.maria-fSummary: A partial answer to a question of \textit{L. Kérchy} and \textit{A. Szalai} [Proc. Am. Math. Soc. 143, No. 6, 2579--2584 (2015; Zbl 1321.47012)] is given. Namely, it is proved that there exists a quasianalytic contraction whose quasianalytic spectral set is equal to its spectrum and is a proper subarc of the unit circle, but no estimates of the norm of its inverse are given.Drazin-star and star-Drazin inverses of bounded finite potent operators on Hilbert spaceshttps://zbmath.org/1496.470342022-11-17T18:59:28.764376Z"Pablos Romo, Fernando"https://zbmath.org/authors/?q=ai:pablos-romo.fernandoThe notions of the Drazin-star and the star-Drazin of matrices were introduced by \textit{D. Mosić} [Result. Math. 75, No. 2, Paper No. 61, 21 p. (2020; Zbl 1437.15004)]. In the paper under review, the author extends the above notions to bounded finite potent endomorphisms on arbitrary Hilbert spaces. He demonstrates the existence, structure and main properties of these operators. Moreover, for bounded finite potent endomorphisms with index less or equal to~1, he studies the group-star and the star-group inverses. He also applies the properties of the Drazin-star inverse of a bounded finite potent endomorphism for studying the consistence and the general solutions of linear systems on Hilbert spaces.
Reviewer: Mohammad Sal Moslehian (Mashhad)Subexponential decay and regularity estimates for eigenfunctions of localization operatorshttps://zbmath.org/1496.470352022-11-17T18:59:28.764376Z"Bastianoni, Federico"https://zbmath.org/authors/?q=ai:bastianoni.federico"Teofanov, Nenad"https://zbmath.org/authors/?q=ai:teofanov.nenadInspired by the recent work [\textit{D. Bayer} and \textit{K. Gröchenig}, Integral Equations Oper. Theory 82, No. 1, 95--117 (2015; Zbl 1337.47029)], the authors consider the properties of eigenfunctions of compact localization operators. They extend the framework of the Schwartz space of test functions and its dual space of tempered distributions given in the previous reference by replacing it with a more subtle family of Gelfand-Shilov spaces and their duals, spaces of ultra-distributions. As an important technical tool, the authors consider a class of weights which contains the weights of subexponential growth, apart from polynomial type weights. Among the main tools is the \(\tau \)-Wigner distribution \(W_{\tau }(f,g)\) with \(f,g\in L^{2}(\mathbb{R}^{2}).\)
Reviewer: Elhadj Dahia (Bou Saâda)Extrapolation theorems for \((p,q)\)-factorable operatorshttps://zbmath.org/1496.470362022-11-17T18:59:28.764376Z"Galdames-Bravo, Orlando"https://zbmath.org/authors/?q=ai:galdames-bravo.orlandoSummary: The operator ideal of \((p,q)\)-factorable operators can be characterized as the class of operators that factors through the embedding \(L^{q'}(\mu)\hookrightarrow L^{p}(\mu)\) for a finite measure \(\mu\), where \(p,q\in[1,\infty)\) are such that \(1/p+1/q\geq1\). We prove that this operator ideal is included into a Banach operator ideal characterized by means of factorizations through \(r\)th and \(s\)th power factorable operators, for suitable \(r,s\in[1,\infty)\). Thus, they also factor through a positive map \(L^{s}(m_{1})^{\ast}\to L^{r}(m_{2})\), where \(m_{1}\) and \(m_{2}\) are vector measures. We use the properties of the spaces of \(u\)-integrable functions with respect to a vector measure and the \(u\)th power factorable operators to obtain a characterization of \((p,q)\)-factorable operators and conditions under which a \((p,q)\)-factorable operator is \(r\)-summing for \(r\in[1,p]\).Some properties of \((m, C)\)-isometric operatorshttps://zbmath.org/1496.470372022-11-17T18:59:28.764376Z"Li, Haiying"https://zbmath.org/authors/?q=ai:li.haiying"Wang, Yaru"https://zbmath.org/authors/?q=ai:wang.yaruSummary: In this paper, we study if \(T\) is an \((m,C)\)-isometric operator and \(CT^\ast C\) commutes with \(T\), then \(T^\ast\) is an \((m,C)\)-isometric operator. We also give local spectral properties and spectral relations of \((m,C)\)-isometric operators, such as property (\( \beta \)), decomposability, the single-valued extension property and Dunford's boundedness. We also investigate perturbation of \((m,C)\)-isometric operators by nilpotent operators and by algebraic operators and give some properties.Joint \(m\)-quasihyponormal operators on a Hilbert spacehttps://zbmath.org/1496.470382022-11-17T18:59:28.764376Z"Mahmoud, Sid Ahmed Ould Ahmed"https://zbmath.org/authors/?q=ai:sid-ahmed.ould-ahmed-mahmoud"Alshammari, Hadi Obaid"https://zbmath.org/authors/?q=ai:alshammari.hadi-obaidSummary: In this paper, We introduce a new class of multivariable operators known as joint \(m\)-quasihyponormal tuple of operators. It is a natural extension of joint normal and joint hyponormal tuples of operators. An \(m\)-tuple of operators \(\mathbf{S}=(S_1,\dots,S_m)\in\mathcal{B}(\mathcal{H})^m\) is said to be joint \(m\)-quasihyponormal tuple if \(\mathbf{S}\) satisfying
\[
\sum\limits_{1\le l,k\le m}\big \langle S_k^*\big [S_k^*,\;\; S_l\big ]S_lu_k\;|\;u_l\big \rangle \ge 0,
\]
for each finite collections \((u_l)_{1\le l\le m}\in\mathcal{H}\). Some properties of this class of multivariable operators are studied.Weyl-type theorems and \(k\)-quasi-\(M\)-hyponormal operatorshttps://zbmath.org/1496.470392022-11-17T18:59:28.764376Z"Zuo, Fei"https://zbmath.org/authors/?q=ai:zuo.fei"Zuo, Hongliang"https://zbmath.org/authors/?q=ai:zuo.hongliangSummary: In this paper, we show that if \(E\) is the Riesz idempotent for a non-zero isolated point \(\lambda\) of the spectrum of a \(k\)-quasi-\(M\)-hyponormal operator \(T\), then \(E\) is self-adjoint, and \(R(E) = N(T - \lambda) = N(T- \lambda)^\ast\). Also, we obtain that Weyl-type theorems hold for algebraically \(k\)-quasi-\(M\)-hyponormal operators.Perspectives on general left-definite theoryhttps://zbmath.org/1496.470402022-11-17T18:59:28.764376Z"Frymark, Dale"https://zbmath.org/authors/?q=ai:frymark.dale"Liaw, Constanze"https://zbmath.org/authors/?q=ai:liaw.constanzeSummary: In 2002, \textit{L. L. Littlejohn} and \textit{R. Wellman} [J. Differ. Equations 181, No. 2, 280--339 (2002; Zbl 1008.47029)] developed a celebrated general left-definite theory for semi-bounded self-adjoint operators with many applications to differential operators. The theory starts with a semi-bounded self-adjoint operator and constructs a continuum of related Hilbert spaces and self-adjoint operators that are intimately related with powers of the initial operator. The development spurred a flurry of activity in the field that is still ongoing today. The main goal of this expository (with the exception of Proposition~1) manuscript is to compare and contrast the complementary theories of general left-definite theory, the Birman-Krein-Vishik (BKV) theory of self-adjoint extensions and singular perturbation theory. In this way, we hope to encourage interest in left-definite theory as well as point out directions of potential growth where the fields are interconnected. We include several related open questions to further these goals.
For the entire collection see [Zbl 1479.47003].On extensions of symmetric operatorshttps://zbmath.org/1496.470412022-11-17T18:59:28.764376Z"Guliyev, Namig J."https://zbmath.org/authors/?q=ai:guliyev.namig-jThe author considers a symmetric operator \(A\) in a Hilbert space \(H\) and its selfadjoint extensions \(\tilde{A}\) in a wider space \(\tilde{H}\supset H\). An extension is called minimal if no nontrivial subspace of \(\tilde{H}\ominus H\) is reducing for \(\tilde{A}\). A description of minimal extensions is obtained for the case where \(A\) has the deficiency index \((1,1)\).
Reviewer: Anatoly N. Kochubei (Kyïv)Essential spectrum of a weighted geometric realizationhttps://zbmath.org/1496.470422022-11-17T18:59:28.764376Z"Hatim, Khalid"https://zbmath.org/authors/?q=ai:hatim.khalid"Baalal, Azeddine"https://zbmath.org/authors/?q=ai:baalal.azeddineSummary: In this present article, we construct a new framework that's we call the weighted geometric realization of 2 and 3-simplexes. On this new weighted framework, we construct a nonself-adjoint 2-simplex Laplacian \(L\) and a self-adjoint 2-simplex Laplacian \(N\). We propose general conditions to ensure sectoriality for our new nonself-adjoint 2-simplex Laplacian \(L\). We show the relation between the essential spectra of \(L\) and \(N\). Finally, we prove the absence of the essential spectrum for our 2-simplex Laplacians \(L\) and \(N\).On couplings of symmetric operators with possibly unequal and infinite deficiency indiceshttps://zbmath.org/1496.470432022-11-17T18:59:28.764376Z"Mogilevskii, V. I."https://zbmath.org/authors/?q=ai:mogilevskii.vadimThe known results on couplings of symmetric operators \(A_j\), \(j\in\{1,2\}\), defined on the orthogonal sum of the Hilbert spaces, introduced by \textit{A. V. Shtraus} [Sov. Math., Dokl. 3, 779--782 (1962; Zbl 0151.19501); translation from Dokl. Akad. Nauk SSSR 144, 512--515 (1962)] are extended to the case of operators \(A_j\) with arbitrary (possibly unequal and infinite) deficiency indices. In particular, the coupling method based on the theory of boundary triplets is generalized. This makes it possible to obtain the abstract Titchmarsh formula, which gives the representation of the Weyl function of the coupling in terms of Weyl functions of boundary triplets for \(A_1\) and \(A_2\). \par Applications to ordinary differential operators are given.
Reviewer: Anatoly N. Kochubei (Kyïv)Liouville operators over the Hardy spacehttps://zbmath.org/1496.470442022-11-17T18:59:28.764376Z"Russo, Benjamin P."https://zbmath.org/authors/?q=ai:russo.benjamin-p"Rosenfeld, Joel A."https://zbmath.org/authors/?q=ai:rosenfeld.joel-aSummary: The role of Liouville operators in the study of dynamical systems through the use of occupation measures has been an active area of research in control theory over the past decade. This manuscript investigates Liouville operators over the Hardy space, which encode complex ordinary differential equations in an operator over a reproducing kernel Hilbert space.Generalized weighted composition operators from the Bloch-type spaces to the weighted Zygmund spaceshttps://zbmath.org/1496.470452022-11-17T18:59:28.764376Z"Abbasi, Ebrahim"https://zbmath.org/authors/?q=ai:abbasi.ebrahim"Vaezi, Hamid"https://zbmath.org/authors/?q=ai:vaezi.hamidSummary: The boundedness and compactness of generalized weighted composition operators from Bloch-type spaces and little Bloch-type spaces into weighted Zygmund spaces on the unit disc are characterized in this paper.A class of \(C\)-normal weighted composition operators on Fock space \(\mathcal{F}^2 (\mathbb{C})\)https://zbmath.org/1496.470462022-11-17T18:59:28.764376Z"Bhuia, Sudip Ranjan"https://zbmath.org/authors/?q=ai:bhuia.sudip-ranjanSummary: In this paper we study a class of \(C\)-normal weighted composition operators \(W_{\psi, \varphi}\) on the Fock space \(\mathcal{F}^2 (\mathbb{C})\). We provide some properties of \(\psi\) and \(\varphi\) when a weighted composition operator \(W_{\psi, \varphi}\) is \(C\)-normal with a conjugation \(C\) defined on \(\mathcal{F}^2 (\mathbb{C})\). We also show that hyponormal \(C\)-normal operators are normal. We investigate \(C\)-normality of \(W_{\psi, \varphi}\) with weighted composition conjugation and the weight function as a kernel function. Alongside we give eigenvalues and eigenvectors of \(C\)-normal \(W_{\psi, \varphi}\). We establish a condition on the symbols for a class of \(C\)-normal weighted composition operators to be normal.Topological structure of the space of composition operators between different Fock spaceshttps://zbmath.org/1496.470472022-11-17T18:59:28.764376Z"Le Hai Khoi"https://zbmath.org/authors/?q=ai:le-hai-khoi."Le Thi Hong Thom"https://zbmath.org/authors/?q=ai:le-thi-hong-thom."Pham Trong Tien"https://zbmath.org/authors/?q=ai:pham-trong-tien.Composition operators \(C_\varphi\) have been intensively investigated on various Banach spaces of holomorphic functions during the past several decades in different directions. In particular, one of the main problems is to characterize (path) components and isolated points in the space of composition operators endowed with the operator norm topology. It is worth mentioning that such problems have not been solved completely on many spaces.
Let \(\mathcal{F}^\infty(\mathbb{C}^n)\) and \(\mathcal{F}^p(\mathbb{C}^n)\) with \(p\in(0,\infty)\) denote the Fock spaces in several variables. In the paper under review, the authors solve the topological problem, including path components and isolated points, for the space of all composition operators between Fock spaces \(\mathcal{F}^p(\mathbb{C}^n)\) and \(\mathcal{F}^q(\mathbb{C}^n)\) with \(0<p,q\leq\infty\) and they list an example simultaneously. They extend the corresponding results to all general Fock spaces in several variables. It should be noted that the previous methods are no longer suitable and the authors develop a new approach to overcome such difficulities.
Reviewer: Zehua Zhou (Tianjin)On double difference of composition operators from a space generated by the Cauchy kernel and a special measurehttps://zbmath.org/1496.470482022-11-17T18:59:28.764376Z"Sharma, Mehak"https://zbmath.org/authors/?q=ai:sharma.mehak"Sharma, Ajay K."https://zbmath.org/authors/?q=ai:sharma.ajay-kumar|sharma.ayay-k"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad-ayman|mursaleen.mohammadSummary: In this paper, compact double difference of composition operators acting from a space generated by the Cauchy kernel and a special measure to analytic Besov spaces is characterized. Moreover, operator norm of these operators acting from Cauchy transforms to analytic Besov spaces is obtained explicitly.Differences of weighted differentiation composition operators from \(\alpha\)-Bloch space to \(H^\infty\) spacehttps://zbmath.org/1496.470492022-11-17T18:59:28.764376Z"Wang, Cui"https://zbmath.org/authors/?q=ai:wang.cui"Zhou, Ze-Hua"https://zbmath.org/authors/?q=ai:zhou.zehuaSummary: This paper characterizes the boundedness and compactness of the differences of weighted differentiation composition operators acting from the \(\alpha\)-Bloch space \(B^\alpha\) to the space \(H^\infty\) of bounded holomorphic functions on the unit disk \(\mathbb D\).Conjugations in \(L^2(\mathcal{H})\)https://zbmath.org/1496.470502022-11-17T18:59:28.764376Z"Câmara, M. Cristina"https://zbmath.org/authors/?q=ai:camara.m-cristina"Kliś-Garlicka, Kamila"https://zbmath.org/authors/?q=ai:klis-garlicka.kamila"Łanucha, Bartosz"https://zbmath.org/authors/?q=ai:lanucha.bartosz"Ptak, Marek"https://zbmath.org/authors/?q=ai:ptak.marekLet \(L_2=L_2(\mathbb T,\mathfrak{m})\) (resp., \(L_\infty=L_\infty(\mathbb T,\mathfrak{m})\)) be the space of square-integrable (resp., bounded) functions on \(\mathbb T\) (the unit circle) w.r.t the normalized Lebesgue measure \(\mathfrak{m}\). Let \(\mathbf{M}_z=\begin{bmatrix} M_z & 0 \\
0 & M_z \end{bmatrix}\) where \(M_z\) is the operator defined on \(L_2\) of multiplication by \(z\in L_\infty\). Let \((\mathcal{H},\langle\cdot\rangle_{\mathcal{H}})\) be a complex Hilbert space. We say that \(C:\mathcal{H}\to\mathcal{H}\) is a conjugation in \(\mathcal{H}\) if it is an antilinear isometric involution and \(\langle Cf,Cg\rangle_{\mathcal{H}}=\langle g,f\rangle_{\mathcal{H}}\) for all \((f,g)\in\mathcal{H}^2\) and we denote \(\mathbf{C}=\begin{bmatrix} D_1& D_2 \\
D_3 & D_4 \end{bmatrix}\) where \((D_j)_{1\le j\le 4}\) are antilinear operators on \(\mathcal{H}.\) The purpose of the authors is to characterize all \(\mathbf{M}_z\)-conjugations \(\mathbf{C}\) in \(L_2(\mathcal{H})\) and all \(\mathbf{M}_z\)-commuting conjugations in \(L_2(\mathcal{H})\). Also, the authors study these conjugations for which vector valued model spaces are invariant.
Reviewer: Mohammed El Aïdi (Bogotá)Generalized kernels of the Toeplitz type for exponentially convex functionshttps://zbmath.org/1496.470512022-11-17T18:59:28.764376Z"Chernobai, O. B."https://zbmath.org/authors/?q=ai:chernobai.o-bSummary: We prove an integral representation for the generalized kernels of the Toeplitz type connected with exponentially convex functions but not with positive-definite functions.Reducing subspaces for a class of nonanalytic Toeplitz operatorshttps://zbmath.org/1496.470522022-11-17T18:59:28.764376Z"Deng, Jia"https://zbmath.org/authors/?q=ai:deng.jia"Lu, Yufeng"https://zbmath.org/authors/?q=ai:lu.yufeng"Shi, Yanyue"https://zbmath.org/authors/?q=ai:shi.yanyue"Hu, Yinyin"https://zbmath.org/authors/?q=ai:hu.yinyinSummary: In this paper, we give a uniform characterization for the reducing subspaces for \(T_{\varphi}\) with the symbol \(\varphi(z)=z^{k}+\bar{z}^{l}\) (\(k,l\in\mathbb{Z}_{+}^{2}\)) on the Bergman spaces over the bidisk, including the known cases that \(\varphi(z_{1},z_{2})=z_{1}^{N}z_{2}^{M}\) and \(\varphi(z_{1},z_{2})=z_{1}^{N}+\overline{z}_{2}^{M}\) with \(N,M\in\mathbb{Z}_{+}\). Meanwhile, the reducing subspaces for \(T_{z^{N}+\overline{z}^{M}}\) (\(N,M\in \mathbb{Z}_{+}\)) on the Bergman space over the unit disk are also described. Finally, we state these results in terms of the commutant algebra \(\mathcal{V}^{*}(\varphi)\).Commuting Toeplitz operators on the Fock-Sobolev spacehttps://zbmath.org/1496.470532022-11-17T18:59:28.764376Z"Fan, Junmei"https://zbmath.org/authors/?q=ai:fan.junmei"Liu, Liu"https://zbmath.org/authors/?q=ai:liu.liu"Lu, Yufeng"https://zbmath.org/authors/?q=ai:lu.yufengSummary: The purpose of this paper is to study the commutator and the semi-commutator of Toeplitz operators on the Fock-Sobolev space \(F^{2,m}({\mathbb{C}})\) in a different function theoretic way instead of Berezin transform. We determine conditions for \((T_f, T_{\overline{g}}]=0\) and \([T_f, T_{\overline{g}}]=0\) in the cases when the symbol functions \(f\) and \(g\) are both polynomials or when \(f\) is a finite linear combination of reproducing kernels and \(g\) is a polynomial. We also determine the boundedness of Hankel products on the Fock-Sobolev space for some symbol classes.A generalized Hilbert operator acting on conformally invariant spaceshttps://zbmath.org/1496.470542022-11-17T18:59:28.764376Z"Girela, Daniel"https://zbmath.org/authors/?q=ai:girela.daniel"Merchán, Noel"https://zbmath.org/authors/?q=ai:merchan.noelSummary: If \(\mu\) is a positive Borel measure on the interval \([0,1)\), we let \(\mathcal{H}_{\mu}\) be the Hankel matrix \(\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq0}\) with entries \(\mu_{n,k}=\mu_{n+k}\), where, for \(n=0,1,2,\dots\), \(\mu_{n}\) denotes the moment of order \(n\) of \(\mu\). This matrix formally induces the operator
\[
\mathcal{H}_{\mu}(f)(z)=\sum_{n=0}^{\infty}(\sum_{k=0}^{\infty}\mu_{n,k}{a_{k}})z^{n}
\]
on the space of all analytic functions \(f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}\), in the unit disk \(\mathbb{D}\). This is a natural generalization of the classical Hilbert operator. The action of the operators \(H_{\mu}\) on Hardy spaces has been recently studied (cf. [\textit{C. Chatzifountas} et al., J. Math. Anal. Appl. 413, No. 1, 154--168 (2014; Zbl 1308.42021)]). This article is devoted to a study of the operators \(H_{\mu}\) acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the \(Q_{s}\)-spaces.Toeplitz operators on weighted pluriharmonic Bergman spacehttps://zbmath.org/1496.470552022-11-17T18:59:28.764376Z"Kong, Linghui"https://zbmath.org/authors/?q=ai:kong.linghui"Lu, Yufeng"https://zbmath.org/authors/?q=ai:lu.yufengSummary: In this article, we consider some algebraic properties of Toeplitz operators on weighted pluriharmonic Bergman space on the unit ball. We characterize the commutants of Toeplitz operators whose symbols are certain separately radial functions or holomorphic monomials, and then give a partial answer to the finite-rank product problem of Toeplitz operators.Hardy classes and symbols of Toeplitz operatorshttps://zbmath.org/1496.470562022-11-17T18:59:28.764376Z"López-García, Marco"https://zbmath.org/authors/?q=ai:lopez-garcia.marco"Pérez-Esteva, Salvador"https://zbmath.org/authors/?q=ai:perez-esteva.salvadorSummary: The purpose of this paper is to study functions in the unit disk \(\mathbb D\) through the family of Toeplitz operators \(\{T_{\phi d\sigma_{t}}\}_{t\in[0,1)}\), where \(T_{\phi d\sigma_{t}}\) is the Toeplitz operator acting the Bergman space of \(\mathbb D\) and where \(d\sigma_t\) is the Lebesgue measure in the circle \(tS^1\). In particular for \(1\leq p < \infty\) we characterize the harmonic functions \(\phi\) in the Hardy space \(h^{p}(\mathbb D)\) by the growth in \(t\) of the \(p\)-Schatten norms of \(T_{\phi d\sigma_{t}}\). We also study the dependence in \(t\) of the norm operator of \(T_{ad\sigma_{t}}\) when \(a\in H^p_{at}\), the atomic Hardy space in the unit circle with \(1/2 < p \leq 1\).Polynomial birth-death processes and the 2nd conjecture of Valenthttps://zbmath.org/1496.470572022-11-17T18:59:28.764376Z"Bochkov, Ivan"https://zbmath.org/authors/?q=ai:bochkov.ivanSummary: The conjecture of \textit{G. Valent} [ISNM, Int. Ser. Numer. Math. 131, 227--237 (1999; Zbl 0935.30025)] about the type of Jacobi matrices with polynomially growing weights is proved.Ballistic transport for periodic Jacobi operators on \(\mathbb{Z}^d\)https://zbmath.org/1496.470582022-11-17T18:59:28.764376Z"Fillman, Jake"https://zbmath.org/authors/?q=ai:fillman.jakeSummary: In this expository work, we collect some background results and give a short proof of the following theorem: periodic Jacobi matrices on \(\mathbb{Z}^d\) exhibit strong ballistic motion.
For the entire collection see [Zbl 1479.47003].Reflexive operators on analytic function spaceshttps://zbmath.org/1496.470592022-11-17T18:59:28.764376Z"Ershad, F."https://zbmath.org/authors/?q=ai:ershad.fariba"Khorami, M. M."https://zbmath.org/authors/?q=ai:khorami.mohammad-morad"Yousefi, B."https://zbmath.org/authors/?q=ai:yousefi.bahmannSummary: Following the recent work done in [\textit{B. Yousefi} and \textit{F. Zangeneh}, Proc. Indian Acad. Sci., Math. Sci. 128, No. 3, Paper No. 38, 9 p. (2018; Zbl 06911924)], we give various other conditions to ensure that the powers of the multiplication operator \(M_z\) are reflexive on a Banach space \({\mathcal{X}}\) of functions analytic on a plane domain. Also, some examples of function spaces satisfying the given conditions are considered.Compactness of multiplication operators on Riesz bounded variation spaceshttps://zbmath.org/1496.470602022-11-17T18:59:28.764376Z"Guzmán-Partida, Martha"https://zbmath.org/authors/?q=ai:guzman-partida.marthaSummary: We prove compactness of the operator \(M_h C_g\) on a subspace of the space of \(2 \pi\)-periodic functions of Riesz bounded variation on \([-\pi,\pi]\), for appropriate functions \(g\) and \(h\). Here, \(M_h\) denotes multiplication by \(h\) and \(C_g\) convolution by \(g\).On the equivalence of some perturbations of the operator of multiplication by independent variablehttps://zbmath.org/1496.470612022-11-17T18:59:28.764376Z"Linchuk, Yu. S."https://zbmath.org/authors/?q=ai:linchuk.yu-sSummary: We study the conditions of equivalence of two operators obtained as perturbations of the operator of multiplication by independent variable by certain Volterra operators in the space of functions analytic in an arbitrary domain of the complex plane starlike with respect to the origin.Grüss-Landau inequalities for elementary operators and inner product type transformers in \(\mathrm{Q}\) and \(\mathrm{Q}^*\) norm ideals of compact operatorshttps://zbmath.org/1496.470622022-11-17T18:59:28.764376Z"Lazarević, Milan"https://zbmath.org/authors/?q=ai:lazarevic.milanSummary: For a probability measure $\mu$ on $\Omega$ and square integrable (Hilbert space) operator valued functions $\{A^*_t\}_{t\in \Omega}$, $\{B_t\}_{t\in\Omega}$, we prove Grüss-Landau type operator inequality for inner product type transformers
$$
\begin{multlined}
\left| \int_\Omega A_t X B_t \,d\mu(t) - \int_\Omega A_t\,d\mu(t) X \int_\Omega B_t \,d\mu(t) \right|^{2\eta} \\
\leqslant
\left\Vert \int_\Omega A_t A^*_t\,d\mu(t) - \left| \int_\Omega A^*_t \,d\mu(t) \right|^2 \right\Vert^\eta \left( \int_\Omega B^*_t X^* X B_t \,d\mu(t) - \left| X \int_\Omega B_t \,d\mu(t)\right|^2 \right)^\eta,
\end{multlined}
$$
for all $X \in \mathcal{B(H)}$ and for all $\eta \in [0,1]$.
Let $p\geqslant2$, $\Phi$ to be a symmetrically norming (s.n.) function, $\Phi^{(p)}$ to be its $p$-modification, $\Phi^{(p)^*}$ is a s.n. function adjoint to $\Phi^{(p)}$ and $\Vert\cdot\Vert_{\Phi^{(p)^*}}$ to be a norm on its associated ideal $\mathcal{C}_{\Phi^{(p)^*}}(\mathcal{H})$ of compact operators. If $X\in \mathcal{C}_{\Phi^{(p)^*}}(\mathcal{H})$ and $\{\alpha_n\}^\infty_{n=1}$ is a sequence in $(0,1]$, such that $\sum^\infty_{n=1}\alpha_n=1$ and $\sum^\infty_{n=1}\Vert\alpha ^{-1/2}_n A_n f \Vert^2 + \Vert \alpha^{-1/2}_n B^*_n f \Vert^2<+\infty$ for some families $\{A_n\}^\infty_{n=1}$ and $\{B_n\}^\infty_{n=1}$ of bounded operators on Hilbert space $\mathcal{H}$ and for all $f\in \mathcal{H}$, then
$$
\begin{multlined}
\left\Vert\sum^\infty_{n=1} \alpha^{-1}_n A_n X B_n - \sum^\infty_{n=1} A_n X \sum^\infty_{n=1} B_n \right\Vert _{\Phi^{(p)^*}} \\
\leqslant
\left\Vert \sqrt{ \sum^\infty_{n=1} \alpha^{-1}_n |A_n|^2 - \left| \sum^\infty_{n=1} A_n \right|^2} X \sqrt{ \sum^\infty_{n=1} \alpha^{-1}_n |B^*_n|^2 - \left| \sum^\infty_{n=1} B^*_n\right|^2} \right\Vert_{\Phi^{(p)^*}},
\end{multlined}
$$
if at least one of those operator families consists of mutually commuting normal operators.
The related Grüss-Landau type $\Vert\cdot\Vert_{\Phi^{(p)}}$ norm inequalities for inner product type transformers are also provided.Spectrally additive group homomorphisms on Banach algebrashttps://zbmath.org/1496.470632022-11-17T18:59:28.764376Z"Askes, Miles"https://zbmath.org/authors/?q=ai:askes.miles"Brits, Rudi"https://zbmath.org/authors/?q=ai:brits.rudi-m"Schulz, Francois"https://zbmath.org/authors/?q=ai:schulz.francoisSummary: Let \(A\) and \(B\) be unital complex Banach algebras. A surjective map \(\phi : A \to B\) is called a spectrally additive group homomorphism if the spectrum of \(x \pm y\) is equal to the spectrum of \(\phi (x) \pm \phi (y)\) for each \(x, y \in A\). If \(A\) is semisimple and either \(A\) or \(B\) has an essential socle, then we prove that a spectrally additive group homomorphism \(\phi : A \to B\) is a continuous Jordan-isomorphism. If, in addition, either \(A\) or \(B\) is prime, then we conclude that \(\phi\) is either a continuous algebra isomorphism or anti-isomorphism. It is noteworthy that the continuity and linearity (or even additivity) of the map \(\phi\) does not form part of the hypothesis, but is rather obtained in the conclusion. The techniques employed in the proof of these results utilize the spectral rank, trace and determinant, and yield a new additive characterization of finite rank elements in a Banach algebra which is of independent interest.An indefinite range inclusion theorem for triplets of bounded linear operators on a Hilbert spacehttps://zbmath.org/1496.470642022-11-17T18:59:28.764376Z"Seto, Michio"https://zbmath.org/authors/?q=ai:seto.michio"Uchiyama, Atsushi"https://zbmath.org/authors/?q=ai:uchiyama.atsushiThe authors study triplets \((T_2,T_2,T_3)\) of bounded operators on a Hilbert space \(\mathcal{H}\) satisfying the following inequality
\[
0 \le T_1T_1^* + T_2T_2^* + T_3T_3^* \le I.
\]
Let \(T = ( T_1T_1^* + T_2T_2^* + T_3T_3^* )^{1/2}\), and let \(\mathcal{M}(T)\) denote the Hilbert space \((R(T), (\cdot, \cdot)_{\mathcal{M}(T)})\), where \(R(T)\) is the range of \(T\), and \((u, v)_{\mathcal{M}(T)} = (Q\tilde u,Q\tilde v)\), where \(Q\) is the orthogonal projector onto \( \mathop{\mathrm{Ker}}(T)^\perp\) and \(\tilde u, \tilde v\) satisfy \(u = T\tilde u\), \(v = T\tilde v\).
As the main result, the authors prove that for any vector \(u \in \mathcal{M}(T)\) and any \(\varepsilon >0\), there exists \(z_{\varepsilon}=(z_1(\varepsilon),z_2(\varepsilon),z_3(\varepsilon)) \in \mathcal{H}\oplus \mathcal{H} \oplus \mathcal{H}\) such that
\begin{itemize}
\item[(i)] \(T_1z_1(\varepsilon) + T_2z_2(\varepsilon) - T_3z_3(\varepsilon) \to u\) as \(\varepsilon \to 0\),
\item[(ii)] \( 0 \le \| z_1(\varepsilon) \|^2 + \| z_2(\varepsilon) \|^2 - \| z_3(\varepsilon) \|^2 \to \|u\|^2_{\mathcal{M}(T)}\) as \(\varepsilon \to 0\).
\end{itemize}
The authors apply the obtained results on an indefinite Toeplitz corona problem.
Reviewer: Ivica Nakić (Zagreb)The Kalton and Rosenthal type decomposition of operators in Köthe-Bochner spaceshttps://zbmath.org/1496.470652022-11-17T18:59:28.764376Z"Pliev, Marat"https://zbmath.org/authors/?q=ai:pliev.marat-amurkhanovich"Sukochev, Fedor"https://zbmath.org/authors/?q=ai:sukochev.fedor-aThe paper under review is partly motivated by a classical result of \textit{N. J. Kalton} [Indiana Univ. Math. J. 27, 353--381 (1978; Zbl 0403.46032)] which in particular implies that every operator \(T:L_1(\mu)\rightarrow L_1(\mu)\) admits a decomposition \(T=T_a+T_d\) where \(T_a\) is a pseudo-integral operator with respect to a family of absolutely continuous measures and \(T_d\) is an atomic operator. This decomposition is here extended to a certain class of operators (dominated operators) from a certain family of lattice-normed spaces (decomposable spaces) into order continuous Banach lattices. Here, a lattice-normed space is a triple \((E,V,p)\), where \(E\) is a vector space, \(V\) is an Archimedean vector lattice, and \(p:E\rightarrow V\) is a map satisfying: (1) \(p(x)\geq0\), with \(p(x)=0\) if and only if \(x=0\); (2) \(p(x_1+x_2)\leq p(x_1)+p(x_2)\); (3) \(p(\lambda x)=|\lambda|p(x)\). In this setting, several interesting results are obtained concerning the class of disjointness preserving and narrow operators. Moreover, the results are in particular applied to the situation of operators on Köthe-Bochner spaces. We refer the interested reader to the paper for details.
Reviewer: Pedro Tradacete (Madrid)Conditions of invertibility for functional operators with shift in weighted Hölder spaceshttps://zbmath.org/1496.470662022-11-17T18:59:28.764376Z"Tarasenko, G."https://zbmath.org/authors/?q=ai:tarasenko.george-s"Karelin, O."https://zbmath.org/authors/?q=ai:karelin.oleksandrSummary: We consider functional operators with shift in weighted Hölder spaces. The main result of the work is the proof of the conditions of invertibility for these operators. We also indicate the forms of the inverse operators. As an application, we propose to use these results for the solution of equations with shift encountered in the study of cyclic models for natural systems with renewable resources.Improving semigroup bounds with resolvent estimateshttps://zbmath.org/1496.470672022-11-17T18:59:28.764376Z"Helffer, B."https://zbmath.org/authors/?q=ai:helffer.bernard"Sjöstrand, J."https://zbmath.org/authors/?q=ai:sjostrand.johannesThe authors revisit the proof of the Gearhart-Prüss-Huang-Greiner-theorem for a semigroup \((S(t))_{t\geq 0}\), following the general idea of the proofs that have been given in the literature and to get an explicit estimate on \(\|S(t)\|\) in terms of bounds on the resolvent of the generator. A~first version of this paper was presented by the two authors in [``From resolvent bounds to semigroup bounds'', Preprint (2010), \url{arXiv:1001.4171}] together with applications in semi-classical analysis and some of these results has been subsequently published in two books written by the authors. Their aim in the paper under review is to present new improvements, partially motivated by a paper of \textit{D.-Y. Wei} [Sci. China, Math. 64, No. 3, 507--518 (2021; Zbl 1464.35260)]. Along the way, they discuss optimization problems confirming the optimality of their results.
Reviewer: Sven-Ake Wegner (Hamburg)Intrinsic properties of strongly continuous fractional semigroups in normed vector spaceshttps://zbmath.org/1496.470682022-11-17T18:59:28.764376Z"Jones, Tiffany Frugé"https://zbmath.org/authors/?q=ai:fruge-jones.tiffany"Padgett, Joshua Lee"https://zbmath.org/authors/?q=ai:padgett.joshua-lee"Sheng, Qin"https://zbmath.org/authors/?q=ai:sheng.qinSummary: Norm estimates for strongly continuous semigroups have been successfully studied in numerous settings, but at the moment there are no corresponding studies in the case of solution operators of singular integral equations. Such equations have recently garnered a large amount of interest due to their potential to model numerous physically relevant phenomena with increased accuracy by incorporating so-called non-local effects. In this article, we provide the first step in the direction of providing such estimates for a particular class of operators which serve as solutions to certain integral equations. The provided results hold in arbitrary normed vector spaces and include the classical results for strongly continuous semigroups as a special case.
For the entire collection see [Zbl 1479.47003].Multiplicative operator functions and abstract Cauchy problemshttps://zbmath.org/1496.470692022-11-17T18:59:28.764376Z"Früchtl, Felix"https://zbmath.org/authors/?q=ai:fruchtl.felixSummary: We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including \(C_{0}\)-groups and cosine operator functions, and more generally, Sturm-Liouville operator functions.A refinement of Baillon's theorem on maximal regularityhttps://zbmath.org/1496.470702022-11-17T18:59:28.764376Z"Jacob, Birgit"https://zbmath.org/authors/?q=ai:jacob.birgit"Schwenninger, Felix L."https://zbmath.org/authors/?q=ai:schwenninger.felix-l"Wintermayr, Jens"https://zbmath.org/authors/?q=ai:wintermayr.jensSummary: By Baillon's theorem, it is known that maximal regularity with respect to the space of continuous functions is rare; it implies that either the semigroup generator involved is a bounded operator or the space considered contains \(c_0\). We show that the latter alternative can be excluded under a refined condition resembling maximal regularity with respect to \(\text{L}^{\infty }\).Global right inverses for Euler type differential operators on the space of smooth functionshttps://zbmath.org/1496.470712022-11-17T18:59:28.764376Z"Langenbruch, Michael"https://zbmath.org/authors/?q=ai:langenbruch.michaelSummary: We study Euler type partial differential operators \(P(\theta)\) admitting a continuous linear right inverse on the space of smooth functions defined on \(\mathbb{R}^d\). Specifically, if the canonical unit vectors \(\{ e_j | j \leq d \}\) are non characteristic for \(P\) then \(P(\theta)\) admits a continuous linear right inverse iff \(P(\partial)\) is hyperbolic w.r.t.\ \( e_j\) for any \(j \leq d\).Conditions for discreteness of the spectrum to Schrödinger operator via non-increasing rearrangement, Lagrangian relaxation and perturbationshttps://zbmath.org/1496.470722022-11-17T18:59:28.764376Z"Zelenko, Leonid"https://zbmath.org/authors/?q=ai:zelenko.leonidConsider the Schrödinger operator \(H=-\Delta +V(x)\) on \(L_2(\mathbb{R}^d)\) with \(d\ge3\). This paper investigates sufficient conditions for which \(H\) has discrete spectrum. Unlike its related results in other literature, the author intends to obtain practical conditions for having discrete spectrum that can be easily verified in practice. In his previous work, the author presented sufficient conditions for discrete spectrum in terms of non-increasing rearrangement of the potential \(V(x)\). In this paper, the author continues to look for further sufficient conditions for the same matter. The method of Lagrangian relaxation for optimization is applied to formulate a new result in terms of expectation and deviation of \(V(x)\). The paper also includes some results on perturbations of \(V(x)\) that preserve discreteness of the spectrum of \(H\).
Reviewer: Miyeon Kwon (Platteville)New \(L^p\)-inequalities for hyperbolic weights concerning the operators with complex Gaussian kernelshttps://zbmath.org/1496.470732022-11-17T18:59:28.764376Z"González, Benito J."https://zbmath.org/authors/?q=ai:gonzalez.benito-juan"Negrín, Emilio R."https://zbmath.org/authors/?q=ai:negrin.emilio-rSummary: In this article, the authors present a systematic study of several new \(L^{p}\)-boundedness properties and Parseval-type relations concerning the operators with complex Gaussian kernels over the spaces \(L^{p}(\mathbb{R},\cosh(\alpha x)\,dx)\) and \(L^{p}(\mathbb{R},\cosh(\alpha x^{2})\,dx)\), \(1\leq p\leq\infty\), \(\alpha\in\mathbb{R}\). Relevant connections with various earlier related results are also pointed out.Erdélyi-Kober fractional integral operators on ball Banach function spaceshttps://zbmath.org/1496.470742022-11-17T18:59:28.764376Z"Ho, Kwok-Pun"https://zbmath.org/authors/?q=ai:ho.kwok-punThe author studies ball Banach function spaces and the Erdélyi-Kober fractional integral operators. The boundedness of the above operators on ball Banach function spaces is derived and proved. Furthermore, the boundedness of Erdélyi-Kober fractional integral operators on amalgan and Morrey spaces is also derived and proved.
Reviewer: James Adedayo Oguntuase (Abeokuta)Characterizations of pseudo-differential operators on \(\mathbb{S}^1\) based on separation-preserving operatorshttps://zbmath.org/1496.470752022-11-17T18:59:28.764376Z"Faghih, Zahra"https://zbmath.org/authors/?q=ai:faghih.zahra"Ghaemi, M. B."https://zbmath.org/authors/?q=ai:ghaemi.mohammad-bagherSummary: In this paper, we prove that a bounded pseudo-differential operator \(T_{\sigma}:L^p(\mathbb{S}^1)\rightarrow L^p(\mathbb{S}^1)\) for \(1\leq p<\infty\), is a separation-preserving operator and give a formula for its symbols \(\sigma\). Using these formulas, we give a new representation for the symbol of adjoint and products of two pseudo-differential operators.Minimal and maximal extensions of \(M\)-hypoelliptic proper uniform pseudo-differential operators in \(L^p\)-spaces on non-compact manifoldshttps://zbmath.org/1496.470762022-11-17T18:59:28.764376Z"Milatovic, Ognjen"https://zbmath.org/authors/?q=ai:milatovic.ognjenThe paper concerns pseudo-differential operators on manifolds of bounded geometry. In [\textit{Yu. A. Kordyukov}, Acta Appl. Math. 23, No. 3, 223--260 (1991; Zbl 0743.58030)] and [\textit{M. A. Shubin}, in: Méthodes semi-classiques. Vol. 1. École d'été (Nantes, juin 1991). Paris: Société Mathématique de France. 35--108 (1992; Zbl 0793.58039)], proper uniform elliptic operators were considered in this setting with action on the \(p\)-Lebesgue spaces. Willing to pass to more general pseudo-differential operators, the author here addresses to the classes of \textit{G. Garello} and \textit{A. Morando} [Integral Equations Oper. Theory 51, No. 4, 501--517 (2005; Zbl 1082.35175)]. For them, action on Lebesgue spaces is granted and the classical ellipticity condition is generalized by the notion of multi-quasi-ellipticity. The author introduces in this context a related definition of proper uniform operator on a manifold with bounded geometry. As an application, equality of minimal and maximal extension of the operator is proved. The paper is a very interesting contribution to the theory of pseudo-differential operators on non-compact manifolds. Similar results for the same classes in Euclidean spaces were proved by \textit{M. W. Wong} [Math. Nachr. 279, No. 3, 319--326 (2006; Zbl 1102.47038)] and \textit{V. Catană} [Appl. Anal. 87, No. 6, 657--666 (2008; Zbl 1158.47031)].
Reviewer: Luigi Rodino (Torino)On the construction of maximal \(p\)-cyclically monotone operatorshttps://zbmath.org/1496.470772022-11-17T18:59:28.764376Z"Bueno, Orestes"https://zbmath.org/authors/?q=ai:bueno.orestes"Cotrina, John"https://zbmath.org/authors/?q=ai:cotrina.johnThe authors describe a method for the construction of explicit examples of maximal \(p\)-cyclically monotone operators. Some new examples of such operators with \(p=2\) and \(p=3\) are also presented.
Reviewer: Rodica Luca (Iaşi)Choquet operators associated to vector capacitieshttps://zbmath.org/1496.470782022-11-17T18:59:28.764376Z"Gal, Sorin G."https://zbmath.org/authors/?q=ai:gal.sorin-gheorghe"Niculescu, Constantin P."https://zbmath.org/authors/?q=ai:niculescu.constantin-pGiven a Hausdorff topological space \(X\), let \(\mathcal{F}(X)\) denote the vector lattice of all real-valued functions defined on \(X\) endowed with the pointwise ordering. Now, let \(X, Y\) be Hausdorff topological spaces and let \(E, F\) be ordered vector subspaces of \(\mathcal{F}(X)\) and \(\mathcal{F}(Y)\), respectively. An operator \(T:E \rightarrow F\) is called a Choquet operator if it is sublinear, comonotically additive and monotonic. It is known that linear Choquet operators acting on ordered Banach spaces are nothing but linear and positive operators acting on these spaces. The authors establish an integral representation of Choquet operators defined on the space \(C(X)\), by using the Choquet-Bochner integral of a real-valued function with respect to a vector capacity.
Reviewer: K. C. Sivakumar (Chennai)On generalized \((\alpha,\beta)\)-nonexpansive mappings in Banach spaces with applicationshttps://zbmath.org/1496.470792022-11-17T18:59:28.764376Z"Akutsah, F."https://zbmath.org/authors/?q=ai:akutsah.francis"Narain, O. K."https://zbmath.org/authors/?q=ai:narain.ojen-kumarSummary: In this paper, we present some fixed point results for a general class of nonexpansive mappings in the framework of Banach space and also proposed a new iterative scheme for approximating the fixed point of this class of mappings in the framework of uniformly convex Banach spaces. Furthermore, we establish some basic properties and convergence results for our new class of mappings in uniformly convex Banach spaces. Finally, we present an application to nonlinear integral equation and also, a numerical example to illustrate our main result and then display the efficiency of the proposed algorithm compared to different iterative algorithms in the literature with different choices of parameters and initial guesses. The results obtained in this paper improve, extend and unify some related results in the literature.Tripled fixed point theorems and applications to a fractional differential equation boundary value problemhttps://zbmath.org/1496.470802022-11-17T18:59:28.764376Z"Afshari, Hojjat"https://zbmath.org/authors/?q=ai:afshari.hojjat"Kheiryan, Alireza"https://zbmath.org/authors/?q=ai:kheiryan.alirezaCoupled fixed point theorems on FLM algebrashttps://zbmath.org/1496.470812022-11-17T18:59:28.764376Z"Amini, Kheghat"https://zbmath.org/authors/?q=ai:amini.kheghat"Hosseinzadeh, Hasan"https://zbmath.org/authors/?q=ai:hosseinzadeh.hasan"Vakilabad, Ali Bagheri"https://zbmath.org/authors/?q=ai:vakilabad.ali-bagheri"Abazari, Rasoul"https://zbmath.org/authors/?q=ai:abazari.rasoulThe authors establish some couple fixed point results for fundamental locally multiplicative (FLM) algebras. Some basic results on FLM algebras are established in Section 2. The main result contained in this article is Theorem 3.1, and the coupled fixed point result is proved on a unital complete semi-simple metrizable FLM algebra. Finally, a characterization of couple fixed point
of holomorphic function on FlM algebra is given in Theorem 3.4.
Reviewer: Mewomo Oluwatosin Temitope (Durban)A new approach to the generalization of Darbo's fixed point problem by using simulation functions with application to integral equationshttps://zbmath.org/1496.470822022-11-17T18:59:28.764376Z"Asadi, Mehdi"https://zbmath.org/authors/?q=ai:asadi.mehdi"Gabeleh, Moosa"https://zbmath.org/authors/?q=ai:gabeleh.moosa"Vetro, Calogero"https://zbmath.org/authors/?q=ai:vetro.calogeroSummary: We investigate the existence of fixed points of self-mappings via simulation functions and measure of noncompactness. We use different classes of additional functions to get some general contractive inequalities. As an application of our main conclusions, we survey the existence of a solution for a class of integral equations under some new conditions. An example will be given to support our results.Two abstract approaches in vectorial fixed point theoryhttps://zbmath.org/1496.470832022-11-17T18:59:28.764376Z"Cardinali, Tiziana"https://zbmath.org/authors/?q=ai:cardinali.tiziana"Precup, Radu"https://zbmath.org/authors/?q=ai:precup.radu"Rubbioni, Paola"https://zbmath.org/authors/?q=ai:rubbioni.paolaSummary: In this paper a fixed point theory is established for operators defined on Cartesian product spaces. Two abstract approaches are presented in terms of closure operators and of general functionals called measures of deviations from zero resembling the measures of noncompactness. In particular, we give vectorial versions to Mönch's fixed point theorems. An application is included to illustrate the theory.Near fixed point theorems in hyperspaceshttps://zbmath.org/1496.470842022-11-17T18:59:28.764376Z"Wu, Hsien-Chung"https://zbmath.org/authors/?q=ai:wu.hsien-chungSummary: The hyperspace consists of all the subsets of a vector space. It is well-known that the hyperspace is not a vector space because it lacks the concept of inverse element. This also says that we cannot consider its normed structure, and some kinds of fixed point theorems cannot be established in this space. In this paper, we shall propose the concept of null set that will be used to endow a norm to the hyperspace. This normed hyperspace is clearly not a conventional normed space. Based on this norm, the concept of Cauchy sequence can be similarly defined. In addition, a Banach hyperspace can be defined according to the concept of Cauchy sequence. The main aim of this paper is to study and establish the so-called near fixed point theorems in Banach hyperspace.Approximating fixed points for a reversible semigroup of Lipschitzian mappings in a smooth Banach spacehttps://zbmath.org/1496.470852022-11-17T18:59:28.764376Z"Piri, Hossein"https://zbmath.org/authors/?q=ai:piri.hossein"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poomSummary: In this paper, we approximate a fixed point of the semigroup \(\varphi = \{T_s : s \in S\}\) of Lipschitzian mappings from a nonempty compact convex subset \(C\) of a smooth Banach space \(E\) into \(C\) with a uniform Lipschitzian condition and with respect to a finite family of sequences \(\{\mu_{i,n}\}^{m, \infty}_{i=1, n=1}\) of left strong regular invariant means defined on an appropriate invariant subspace of \(l^\infty(S)\). Our result extends the main results announced by several others.Cohen summing multilinear multiplication operatorshttps://zbmath.org/1496.470862022-11-17T18:59:28.764376Z"Popa, Dumitru"https://zbmath.org/authors/?q=ai:popa.dumitruA continuous linear operator between Banach spaces \(T\in\mathcal L(X,Y)\) is strongly \(p\)-summing in the sense of \textit{J. S. Cohen} [Math. Ann. 201, 177--200 (1973; Zbl 0233.47019)] if and only if its adjoint \(T^*\in\mathcal L(Y^*,X^*)\) is absolutely \(p\)-summing. As an easy consequence of a classical result about absolutely \(p\)-summing maps with range on cotype 2 spaces, the following statement is derived: If \(X\) has type 2, then, for every \(1<p<2\), any strongly \(p\)-summing operator \(T\in\mathcal L(X,Y)\) is strongly 2-summing.
The notion of strongly \(p\)-summability in the sense of Cohen has its multilinear counterpart. A bounded \(k\)-linear operator \(U:X_1\times\cdots\times X_k\to Y\) is \textit{Cohen} \(p\)-\textit{summing} if the linear adjoint \(U^*\in\mathcal L(Y^*, \mathcal L(X_1,\dots, X_k))\) is absolutely \(p\)-summing. \textit{Q. Bu} and \textit{Z. Shi} [J. Math. Anal. Appl. 401, No. 1, 174--181 (2013; Zbl 1275.47117)] commented that they did not know whether the multilinear version of the above result holds. That is, if \(X_1,\dots, X_k\) have type \(p\), is it true that every Cohen \(p\)-summing operator from \(X_1\times\cdots\times X_k\) to \(Y\) is Cohen 2-summing, for every \(1<p<2\)?
Motivated by this question, the author of the paper under review examines Cohen \(p\)-summability for multilinear \textsl{multiplication} operators. As a result of this study, several examples are presented showing that the answear to Bu and Shi's question is negative.
Reviewer: Verónica Dimant (Victoria)Completely rank-nonincreasing multilinear mapshttps://zbmath.org/1496.470872022-11-17T18:59:28.764376Z"Yousefi, Hassan"https://zbmath.org/authors/?q=ai:yousefi.hassanSummary: We extend the notion of completely rank-nonincreasing (CRNI) linear maps to include the multilinear maps. We show that a bilinear map on a finite-dimensional vector space on any field is CRNI if and only if it is a skew-compression bilinear map. We also characterize CRNI continuous bilinear maps defined on the set of compact operators.On solution sets of nonlinear equations with nonsmooth operators in Hilbert space and the quasi-solution methodhttps://zbmath.org/1496.470882022-11-17T18:59:28.764376Z"Kokurin, Mikhail Yu."https://zbmath.org/authors/?q=ai:kokurin.mihail-yu|kokurin.mikhail-yurjevichThe author investigates nonlinear irregular equations in Hilbert space with a~priori constraints, without assuming differentiability for the governing operator. The constraints are described by a bounded closed set that is part of an extended source representation class expressed in terms of a given linear operator. The unique solvability of the problem is not assumed. It is established that solutions to the problem form a cluster of diameter strictly less than the one of the constraints set. The approximation properties of the quasi-solution method are addressed in relation to the solution set of the original problem.
Reviewer: Radu Ioan Boţ (Wien)Some remarks on regularized nonconvex variational inequalitieshttps://zbmath.org/1496.470892022-11-17T18:59:28.764376Z"Balooee, Javad"https://zbmath.org/authors/?q=ai:balooee.javad"Kim, Jong Kyu"https://zbmath.org/authors/?q=ai:kim.jong-kyuSummary: In this paper, we investigate and analyze the nonconvex variational inequalities introduced by \textit{M. A. Noor} in [Optim. Lett. 3, No. 3, 411--418 (2009; Zbl 1171.58307)] and [Comput. Math. Model. 21, No. 1, 97--108 (2010; Zbl 1201.65114)] and prove that the algorithms and results in the above mentioned papers are not valid. To overcome the problems in the above cited papers, we introduce and consider a new class of variational inequalities, named regularized nonconvex variational inequalities, instead of the class of nonconvex variational inequalities introduced in the above mentioned papers. We also consider a class of nonconvex Wiener-Hopf equations and establish the equivalence between the regularized nonconvex variational inequalities and the fixed point problems as well as the nonconvex Wiener-Hopf equations. By using the obtained equivalence formulations, we prove the existence of a unique solution for the regularized nonconvex variational inequalities and propose some projection iterative schemes for solving the regularized nonconvex variational inequalities. We also study the convergence analysis of the suggested iterative schemes under some certain conditions.Set-valued mixed quasi-equilibrium problems with operator solutionshttps://zbmath.org/1496.470902022-11-17T18:59:28.764376Z"Ram, Tirth"https://zbmath.org/authors/?q=ai:ram.tirth"Khanna, Anu Kumari"https://zbmath.org/authors/?q=ai:khanna.anu-kumari"Kour, Ravdeep"https://zbmath.org/authors/?q=ai:kour.ravdeepSummary: In this paper, we introduce and study generalized mixed operator quasi-equilibrium problems (GMQOEP) in Hausdorff topological vector spaces and prove the existence results for the solution of (GMQOEP) in compact and noncompact settings by employing 1-person game theorems. Moreover, using coercive condition, hemicontinuity of the functions and KKM theorem, we prove new results on the existence of solution for the particular case of (GMQOEP), that is, generalized mixed operator equilibrium problem (GMOEP).Parametric generalized multi-valued nonlinear quasi-variational inclusion problemhttps://zbmath.org/1496.470912022-11-17T18:59:28.764376Z"Khan, F. A."https://zbmath.org/authors/?q=ai:khan.faizan-ahmad"Alanazi, A. M."https://zbmath.org/authors/?q=ai:alanazi.abdulaziz-m"Ali, Javid"https://zbmath.org/authors/?q=ai:ali.javid"Alanazi, Dalal J."https://zbmath.org/authors/?q=ai:alanazi.dalal-jSummary: In this paper, we investigate the behavior and sensitivity analysis of a solution set for a parametric generalized multi-valued nonlinear quasi-variational inclusion problem in a real Hilbert space. For this study, we utilize the technique of resolvent operator and the property of a fixed-point set of a multi-valued contractive mapping. We also examine Lipschitz continuity of the solution set with respect to the parameter under some appropriate conditions.A random generalized nonlinear implicit variational-like inclusion with random fuzzy mappingshttps://zbmath.org/1496.470922022-11-17T18:59:28.764376Z"Khan, F. A."https://zbmath.org/authors/?q=ai:khan.faizan-ahmad"Aljohani, A. S."https://zbmath.org/authors/?q=ai:aljohani.a-s"Alshehri, M. G."https://zbmath.org/authors/?q=ai:alshehri.maryam-gharamah-ali"Ali, J."https://zbmath.org/authors/?q=ai:ali.javidSummary: In this paper, we introduce and study a new class of random generalized nonlinear implicit variational-like inclusions with random fuzzy mappings in a real separable Hilbert space and give its fixed point formulation. Using the fixed point formulation and the proximal mapping technique for strongly maximal monotone mapping, we suggest and analyze a random iterative scheme for finding the approximate solution of this class of inclusion. Further, we prove the existence of solution and discuss the convergence analysis of iterative scheme of this class of inclusion. Our results in this paper improve and generalize several known results in the literature.General proximal-point algorithm for monotone operatorshttps://zbmath.org/1496.470932022-11-17T18:59:28.764376Z"Eslamian, M."https://zbmath.org/authors/?q=ai:eslamian.mohammad"Vahidi, J."https://zbmath.org/authors/?q=ai:vahidi.javadSummary: We introduce a new general proximal-point algorithm for an infinite family of monotone operators in a real Hilbert space and establish strong convergence of the iterative process to a common null point of the infinite family of monotone operators. Our result generalizes and improves numerous results in the available literature.General convergence analysis of projection methods for a system of variational inequalities in \(q\)-uniformly smooth Banach spaceshttps://zbmath.org/1496.470942022-11-17T18:59:28.764376Z"Gong, Qian-Fen"https://zbmath.org/authors/?q=ai:gong.qianfen"Wen, Dao-Jun"https://zbmath.org/authors/?q=ai:wen.daojunSummary: In this paper, we introduce and consider a system of variational inequalities involving two different operators in \(q\)-uniformly smooth Banach spaces. We suggest and analyze a new explicit projection method for solving the system under some more general conditions. Our results extend and unify the results of \textit{R. U. Verma} [Appl. Math. Lett. 18, No. 11, 1286--1292 (2005; Zbl 1099.47054)] and Yao, Liou and Kang [\textit{Y.-H. Yao} et al., J. Glob. Optim. 55, No. 4, 801--811 (2013; Zbl 1260.47085)] and some other previously known results.Ishikawa iterative algorithms with errors for nonlinear mappings in Banach spaceshttps://zbmath.org/1496.470952022-11-17T18:59:28.764376Z"He, Guozhu"https://zbmath.org/authors/?q=ai:he.guozhuSummary: In this paper, we present some further findings about Ishikawa iterative algorithms with errors for nonlinear mappings. We prove a new convergence theorem of Ishikawa iteration sequences with errors to approximate the solution of nonlinear equation $f=Hx+Tx$ in Banach space $X$, where $H$ is a Lipschitzian continuous operator and $T$ is a uniformly continuous mapping such that $T+H$ is strongly accretive. Related results deal with approximation to the fixed point of a strongly pseudo-contractive mapping and the solution of a nonlinear equation involving an $m$-accretive mapping, respectively.Strong convergence of a selection of Ishikawa-Reich-Sabach-type algorithmhttps://zbmath.org/1496.470962022-11-17T18:59:28.764376Z"Isiogugu, Felicia Obiageli"https://zbmath.org/authors/?q=ai:isiogugu.felicia-obiageli"Pillay, Paranjothi"https://zbmath.org/authors/?q=ai:pillay.paranjothi"Uzoma-Oguguo, Osuo-Siseken"https://zbmath.org/authors/?q=ai:uzoma-oguguo.osuo-sisekenSummary: We establish the strong convergence of a selection of an Ishikawa-Reich-Sabach iteration scheme for approximating the common elements of the set of fixed points \(F(T)\) of a multi-valued (or single-valued) pseudocontractive-type mapping \(T\) and the set of solutions \(EP(F)\) of an equilibrium problem for a bifunction \(F\) in a real Hilbert space \(H\). This work is a contribution to the study on the computability and applicability of algorithms for approximating the solutions of equilibrium problems for bifunctions involving the construction of the sequence \(\{K_n\}_{n=1}^\infty\) of closed convex subsets of \(H\) from an arbitrary \(x_0 \in H\) and the sequence \(\{x_n\}_{n=1}^\infty\) of the metric projections of \(x_0\) into \(K_n\). The results obtained are contributions to the resolution of the controversy over the computability and applicability of such algorithms in the contemporary literature.Convergence and stability of iterative algorithm of system of generalized implicit variational-like inclusion problems using \((\theta,\phi,\gamma)\)-relaxed cocoercivityhttps://zbmath.org/1496.470972022-11-17T18:59:28.764376Z"Kim, Jong Kyu"https://zbmath.org/authors/?q=ai:kim.jong-kyu|kim.jongkyu"Bhat, Mohd Iqbal"https://zbmath.org/authors/?q=ai:bhat.mohd-iqbal"Shafi, Sumeera"https://zbmath.org/authors/?q=ai:shafi.sumeeraSummary: In this paper, we give the notion of \(M(.,.)\)-\( \eta \)-proximal mapping for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. As an application, we introduce and investigate a new system of variational-like inclusions in Banach spaces. By means of \(M(.,.)\)-\( \eta \)-proximal mapping method, we give the existence of solution for the system of variational inclusions. Further, propose an iterative algorithm for finding the approximate solution of this class of variational inclusions. Furthermore, we discuss the convergence and stability analysis of the iterative algorithm. The results presented in this paper may be further expolited to solve some more important classes of problems in this direction.Viscosity approximation methods for hierarchical optimization problems in \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1496.470982022-11-17T18:59:28.764376Z"Liu, Xin-Dong"https://zbmath.org/authors/?q=ai:liu.xindong"Chang, Shih-Sen"https://zbmath.org/authors/?q=ai:chang.shih-senSummary: This paper aims at investigating viscosity approximation methods for solving a system of variational inequalities in a \(\mathrm{CAT}(0)\) space. Two algorithms are given. Under certain appropriate conditions, we prove that the iterative schemes converge strongly to the unique solution of the hierarchical optimization problem. The result presented in this paper mainly improves and extends the corresponding results of \textit{L. Y. Shi} and \textit{R. D. Chen} [J. Appl. Math. 2012, Article ID 421050, 11 p. (2012; Zbl 1281.47059)], \textit{R. Wangkeeree} and \textit{P. Preechasilp} [J. Inequal. Appl. 2013, Paper No. 93, 15 p. (2013; Zbl 1292.47056)] and others.A new explicit extragradient method for solving equilibrium problems with convex constraintshttps://zbmath.org/1496.470992022-11-17T18:59:28.764376Z"Muangchoo, Kanikar"https://zbmath.org/authors/?q=ai:muangchoo.kanikarSummary: The purpose of this research is to formulate a new proximal-type algorithm to solve the equilibrium problem in a real Hilbert space. A~new algorithm is analogous to the famous two-step extragradient algorithm that was used to solve variational inequalities in the Hilbert spaces previously. The proposed iterative scheme uses a new step size rule based on local bifunction details instead of Lipschitz constants or any line search scheme. The strong convergence theorem for the proposed algorithm is well-proven by letting mild assumptions about the bifunction. Applications of these results are presented to solve the fixed point problems and the variational inequality problems. Finally, we discuss two test problems and computational performance is explicating to show the efficiency and effectiveness of the proposed algorithm.Strong convergence of projection methods for a countable family of nonexpansive mappings and applications to constrained convex minimization problemshttps://zbmath.org/1496.471002022-11-17T18:59:28.764376Z"Naraghirad, Eskandar"https://zbmath.org/authors/?q=ai:naraghirad.eskandarSummary: In this paper, we introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a real Hilbert space, which solves a corresponding variational inequality. Furthermore, we propose explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem. Our results improve and generalize some known results in the current literature.A Tseng extragradient method for solving variational inequality problems in Banach spaceshttps://zbmath.org/1496.471012022-11-17T18:59:28.764376Z"Oyewole, O. K."https://zbmath.org/authors/?q=ai:oyewole.olawale-kazeem|oyewole.olalwale-k"Abass, H. A."https://zbmath.org/authors/?q=ai:abass.hammad-anuoluwapo|abass.hammed-anuoluwapo"Mebawondu, A. A."https://zbmath.org/authors/?q=ai:mebawondu.akindele-adebayo"Aremu, K. O."https://zbmath.org/authors/?q=ai:aremu.kazeem-olalekanSummary: This paper presents an inertial Tseng extragradient method for approximating a solution of the variational inequality problem. The proposed method uses a single projection onto a half space which can be easily evaluated. The method considered in this paper does not require the knowledge of the Lipschitz constant as it uses variable stepsizes from step to step which are updated over each iteration by a simple calculation. We prove a strong convergence theorem of the sequence generated by this method to a solution of the variational inequality problem in the framework of a 2-uniformly convex Banach space which is also uniformly smooth. Furthermore, we report some numerical experiments to illustrate the performance of this method. Our result extends and unifies corresponding results in this direction in the literature.S-iteration process of Halpern-type for common solutions of nonexpansive mappings and monotone variational inequalitieshttps://zbmath.org/1496.471022022-11-17T18:59:28.764376Z"Sahu, D. R."https://zbmath.org/authors/?q=ai:sahu.daya-ram"Kumar, Ajeet"https://zbmath.org/authors/?q=ai:kumar.ajeet"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfengSummary: This paper is devoted to the strong convergence of the S-iteration process of Halpern-type for approximating a common element of the set of fixed points of a nonexpansive mapping and the set of common solutions of variational inequality problems formed by two inverse strongly monotone mappings in the framework of Hilbert spaces. We also give some numerical examples in support of our main result.Strong convergence theorems under shrinking projection methods for split common fixed point problems in two Banach spaceshttps://zbmath.org/1496.471032022-11-17T18:59:28.764376Z"Takahashi, Wataru"https://zbmath.org/authors/?q=ai:takahashi.wataru"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: We deal with the split common fixed point problem in two Banach spaces. Using the resolvents of maximal monotone operators, demimetric mappings, demigeneralized mappings in Banach spaces, we prove strong convergence theorems under shrinking projection methods for finding solutions of split common fixed point problems with zero points of maximal monotone operators in two Banach spaces. Using these results, we get new results which are connected with the split feasibility problem, the split common null point problem and the split common fixed point problem in Hilbert spaces and Banach spaces.Modified inertial projection and contraction algorithms for solving variational inequality problems with non-Lipschitz continuous operatorshttps://zbmath.org/1496.471042022-11-17T18:59:28.764376Z"Tan, Bing"https://zbmath.org/authors/?q=ai:tan.bing.1"Qin, Xiaolong"https://zbmath.org/authors/?q=ai:qin.xiaolongSummary: In this paper, we present four modified inertial projection and contraction methods to solve the variational inequality problem with a pseudo-monotone and non-Lipschitz continuous operator in real Hilbert spaces. Strong convergence theorems of the proposed algorithms are established without the prior knowledge of the Lipschitz constant of the operator. Several numerical experiments and the applications to optimal control problems are provided to verify the advantages and efficiency of the proposed algorithms.General iterative scheme based on the regularization for solving a constrained convex minimization problemhttps://zbmath.org/1496.471052022-11-17T18:59:28.764376Z"Tian, Ming"https://zbmath.org/authors/?q=ai:tian.mingSummary: It is well known that the regularization method plays an important role in solving a constrained convex minimization problem. In this article, we introduce implicit and explicit iterative schemes based on the regularization for solving a constrained convex minimization problem. We establish results on the strong convergence of the sequences generated by the proposed schemes to a solution of the minimization problem. Such a point is also a solution of a variational inequality. We also apply the algorithm to solve a split feasibility problem.Halpern Tseng's extragradient methods for solving variational inequalities involving semistrictly quasimonotone operatorhttps://zbmath.org/1496.471062022-11-17T18:59:28.764376Z"Wairojjana, Nopparat"https://zbmath.org/authors/?q=ai:wairojjana.nopparat"Pakkaranang, Nuttapol"https://zbmath.org/authors/?q=ai:pakkaranang.nuttapolSummary: In this paper, we study the strong convergence of new methods for solving classical variational inequalities problems involving semistrictly quasimonotone and Lipschitz-continuous operators in a real Hilbert space. Three proposed methods are based on Tseng's extragradient method and use a simple self-adaptive step size rule that is independent of the Lipschitz constant. The step size rule is built around two techniques: the monotone and the non-monotone step size rule. We establish strong convergence theorems for the proposed methods that do not require any additional projections or knowledge of an involved operator's Lipschitz constant. Finally, we present some numerical experiments that demonstrate the efficiency and advantages of the proposed methods.On strong convergence theorems for a viscosity-type Tseng's extragradient methods solving quasimonotone variational inequalitieshttps://zbmath.org/1496.471072022-11-17T18:59:28.764376Z"Wairojjana, Nopparat"https://zbmath.org/authors/?q=ai:wairojjana.nopparat"Pholasa, Nattawut"https://zbmath.org/authors/?q=ai:pholasa.nattawut"Pakkaranang, Nuttapol"https://zbmath.org/authors/?q=ai:pakkaranang.nuttapolSummary: The main goal of this research is to solve variational inequalities involving quasi-monotone operators in infinite-dimensional real Hilbert spaces numerically. The main advantage of these iterative schemes is the ease with which step size rules can be designed based on an operator explanation rather than the Lipschitz constant or another line search method. The proposed iterative schemes use a monotone and non-monotone step size strategy based on mapping (operator) knowledge as a replacement for the Lipschitz constant or another line search method. The strong convergence is demonstrated to correspond well to the proposed methods and to settle certain control specification conditions. Finally, we propose some numerical experiments to assess the effectiveness and influence of iterative methods.An algorithm for a common minimum-norm zero of a finite family of monotone mappings in Banach spaceshttps://zbmath.org/1496.471082022-11-17T18:59:28.764376Z"Zegeye, Habtu"https://zbmath.org/authors/?q=ai:zegeye.habtu"Shahzad, Naseer"https://zbmath.org/authors/?q=ai:shahzad.naseerSummary: We introduce an iterative process which converges strongly to a common minimum-norm point of solutions of a finite family of monotone mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.Approximation of fixed points for mean nonexpansive mappings in Banach spaceshttps://zbmath.org/1496.471092022-11-17T18:59:28.764376Z"Ahmad, Junaid"https://zbmath.org/authors/?q=ai:ahmad.junaid"Ullah, Kifayat"https://zbmath.org/authors/?q=ai:ullah.kifayat"Arshad, Muhammad"https://zbmath.org/authors/?q=ai:arshad.muhammad-junaid|arshad.muhammad-sarmad"de la Sen, Manuel"https://zbmath.org/authors/?q=ai:de-la-sen.manuelSummary: In this paper, we establish weak and strong convergence theorems for mean nonexpansive maps in Banach spaces under the Picard-Mann hybrid iteration process. We also construct an example of mean nonexpansive mappings and show that it exceeds the class of nonexpansive mappings. To show the numerical accuracy of our main outcome, we show that Picard-Mann hybrid iteration process of this example is more effective than all of the Picard, Mann, and Ishikawa iterative processes.Improved generalized \(M\)-iteration for quasi-nonexpansive multivalued mappings with application in real Hilbert spaceshttps://zbmath.org/1496.471102022-11-17T18:59:28.764376Z"Akutsah, Francis"https://zbmath.org/authors/?q=ai:akutsah.francis"Narain, Ojen Kumar"https://zbmath.org/authors/?q=ai:narain.ojen-kumar"Kim, Jong Kyu"https://zbmath.org/authors/?q=ai:kim.jong-kyu|kim.jongkyuSummary: In this paper, we present a modified (improved) generalized \(M\)-iteration with the inertial technique for three quasi-nonexpansive multivalued mappings in a real Hilbert space. In addition, we obtain a weak convergence result under suitable conditions and the strong convergence result is achieved using the hybrid projection method with our modified generalized \(M\)-iteration. Finally, we apply our convergence results to certain optimization problem, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other improved iterative methods (modified SP-iterative scheme) in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.Approximation of fixed points and the solution of a nonlinear integral equationhttps://zbmath.org/1496.471112022-11-17T18:59:28.764376Z"Ali, Faeem"https://zbmath.org/authors/?q=ai:ali.faeem"Ali, Javid"https://zbmath.org/authors/?q=ai:ali.javid"Rodríguez-López, Rosana"https://zbmath.org/authors/?q=ai:rodriguez-lopez.rosanaSummary: In this article, we define Picard's three-step iteration process for the approximation of fixed points of Zamfirescu operators in an arbitrary Banach space. We prove a convergence result for Zamfirescu operator using the proposed iteration process. Further, we prove that Picard's three-step iteration process is almost \(T\)-stable and converges faster than all the known and leading iteration processes. To support our results, we furnish an illustrative numerical example. Finally, we apply the proposed iteration process to approximate the solution of a mixed Volterra-Fredholm functional nonlinear integral equation.On the strong and \(\delta\)-convergence of new multi-step and \(S\)-iteration processes in a \(\mathrm{CAT}(0)\) spacehttps://zbmath.org/1496.471122022-11-17T18:59:28.764376Z"Başarır, Metin"https://zbmath.org/authors/?q=ai:basarir.metin"Şahin, Aynur"https://zbmath.org/authors/?q=ai:sahin.aynurSummary: In this paper, we introduce a new class of mappings and prove the demiclosedness
principle for mappings of this type in a \(\mathrm{CAT}(0)\) space. Also, we obtain the strong and \(\Delta\)-convergence theorems of new multi-step and \(S\)-iteration processes in a \(\mathrm{CAT}(0)\) space. Our results extend and improve the corresponding recent results announced by many authors in the literature.Weak and strong convergence theorems for the modified Ishikawa iteration for two hybrid multivalued mappings in Hilbert spaceshttps://zbmath.org/1496.471132022-11-17T18:59:28.764376Z"Cholamjiak, Watcharaporn"https://zbmath.org/authors/?q=ai:cholamjiak.watcharaporn"Chutibutr, Natchaphan"https://zbmath.org/authors/?q=ai:chutibutr.natchaphan"Weerakham, Siwanat"https://zbmath.org/authors/?q=ai:weerakham.siwanatSummary: In this paper, we introduce new iterative schemes by using the modified Ishikawa iteration for two hybrid multivalued mappings in a Hilbert space. We then obtain weak convergence theorem under suitable conditions. We use CQ and shrinking projection methods with Ishikawa iteration for obtaining strong convergence theorems. Furthermore, we give examples and numerical results for supporting our main results.Inertial Picard normal S-iteration processhttps://zbmath.org/1496.471142022-11-17T18:59:28.764376Z"Dashputre, Samir"https://zbmath.org/authors/?q=ai:dashputre.samir"Padmavati"https://zbmath.org/authors/?q=ai:padmavati.b-sri|padmavati.m-v"Sakure, Kavita"https://zbmath.org/authors/?q=ai:sakure.kavitaSummary: Many iterative algorithms like that Picard, Mann, Ishikawa and S-iteration are very useful to elucidate the fixed point problems of a nonlinear operators in various topological spaces. The recent trend for elucidate the fixed point via inertial iterative algorithm, in which next iterative depends on more than one previous terms. The purpose of the paper is to establish convergence theorems of new inertial Picard normal S-iteration algorithm for nonexpansive mapping in Hilbert spaces. The comparison of convergence of InerNSP and InerPNSP is done with InerSP (introduced by \textit{A. Phon-on} et al. [Fixed Point Theory Appl. 2019, Paper No. 4, 14 p. (2019; Zbl 1467.47042)]) and MSP (introduced by \textit{R. Suparatulatorn} et al. [Numer. Algorithms 77, No. 2, 479--490 (2018; Zbl 1467.47037)]) via numerical example.Convergence analysis of Agarwal et al. iterative scheme for Lipschitzian hemicontractive mappingshttps://zbmath.org/1496.471152022-11-17T18:59:28.764376Z"Kang, Shin Min"https://zbmath.org/authors/?q=ai:kang.shin-min"Rafiq, Arif"https://zbmath.org/authors/?q=ai:rafiq.arif"Ali, Faisal"https://zbmath.org/authors/?q=ai:ali.faisal"Kwun, Young Chel"https://zbmath.org/authors/?q=ai:kwun.young-chelSummary: In this paper, we establish strong convergence for the \textit{R. P. Agarwal} et al. iterative scheme [J. Nonlinear Convex Anal. 8, No. 1, 61--79 (2007; Zbl 1134.47047)] associated with Lipschitzian hemicontractive mappings in Hilbert spaces.Mean convergence theorems using hybrid methods to find common fixed points for noncommutative nonlinear mappings in Hilbert spaceshttps://zbmath.org/1496.471162022-11-17T18:59:28.764376Z"Kondo, Atsumasa"https://zbmath.org/authors/?q=ai:kondo.atsumasaSummary: This paper considers approximation methods to find common fixed points for two general nonlinear mappings that contain generalized hybrid mappings or normally 2-generalized hybrid mappings. First, we combine Nakajo and Takahashi's hybrid method [\textit{K. Nakajo} and \textit{W. Takahashi}, J. Math. Anal. Appl. 279, No. 2, 372--379 (2003; Zbl 1035.47048)] with a mean iteration method and prove three strong convergence theorems that approximate common fixed points for nonlinear mappings. We then develop Takahashi, Takeuchi and Kubota's shrinking projection method [\textit{W. Takahashi} et al., J. Math. Anal. Appl. 341, No. 1, 276--286 (2008; Zbl 1134.47052)] and prove three strong convergence theorems. Nonlinear mappings are not necessarily continuous or commutative.Convergence theorems using Ishikawa iteration for finding common fixed points of demiclosed and 2-demiclosed mappings in Hilbert spaceshttps://zbmath.org/1496.471172022-11-17T18:59:28.764376Z"Kondo, Atsumasa"https://zbmath.org/authors/?q=ai:kondo.atsumasaSummary: This paper presents weak and strong convergence theorems for finding common fixed points of two nonlinear mappings, where one mapping is demiclosed, and the other is 2-demiclosed. For this purpose, we use Ishikawa type iteration and obtain weak convergence theorems. Nakajo and Takahashi's hybrid method [\textit{K. Nakajo} and \textit{W. Takahashi}, J. Math. Anal. Appl. 279, No. 2, 372--379 (2003; Zbl 1035.47048)] and Takahashi, Takeuchi, and Kubota's shrinking projection method [\textit{W. Takahashi} et al., J. Math. Anal. Appl. 341, No. 1, 276--286 (2008; Zbl 1134.47052)] are also employed alongside Ishikawa iteration to derive strong convergence. Our proofs do not require the mappings to be commutative or continuous, and the results obtained in this paper extend many theorems in the literature.Implicit iteration scheme for two \(\phi\)-hemicontractive operators in arbitrary Banach spaceshttps://zbmath.org/1496.471182022-11-17T18:59:28.764376Z"Lv, Guiwen"https://zbmath.org/authors/?q=ai:lv.guiwen"Rafiq, Arif"https://zbmath.org/authors/?q=ai:rafiq.arif"Xue, Zhiqun"https://zbmath.org/authors/?q=ai:xue.zhiqunSummary: The purpose of this paper is to characterize conditions for the convergence of the implicit Ishikawa iterative scheme with errors in the sense of \textit{R. P. Agarwal} et al. [J. Math. Anal. Appl. 272, No. 2, 435--447 (2002; Zbl 1012.65051)] to a common fixed point of two \(\phi\)-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.Some results on continuous pseudo-contractions in a reflexive Banach spacehttps://zbmath.org/1496.471192022-11-17T18:59:28.764376Z"Lv, Songtao"https://zbmath.org/authors/?q=ai:lv.songtao"Hao, Yan"https://zbmath.org/authors/?q=ai:hao.yanSummary: In this paper, we investigate fixed point problems of a continuous pseudo-contraction based on a viscosity iterative scheme. Strong convergence theorems are established in a reflexive Banach space which also enjoys a weakly continuous duality mapping.Strong and \(\Delta \)-convergence theorems for a countable family of multi-valued demicontractive maps in Hadamard spaceshttps://zbmath.org/1496.471202022-11-17T18:59:28.764376Z"Minjibir, Ma'aruf Shehu"https://zbmath.org/authors/?q=ai:minjibir.maaruf-shehu"Salisu, Sani"https://zbmath.org/authors/?q=ai:salisu.saniSummary: In this paper, iterative algorithms for approximating a common fixed point of a countable family of multi-valued demicontractive maps in the setting of Hadamard spaces are presented. Under different mild conditions, the sequences generated are shown to strongly convergent and \(\Delta \)-convergent to a common fixed point of the considered family, accordingly. Our theorems complement many results in the literature.Some convergence results for monotone nonexpansive mappings in ordered \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1496.471212022-11-17T18:59:28.764376Z"Uddin, Izhar"https://zbmath.org/authors/?q=ai:uddin.izhar"Garodia, Chanchal"https://zbmath.org/authors/?q=ai:garodia.chanchal"Yildirim, İsa"https://zbmath.org/authors/?q=ai:yildirim.isaSummary: In this paper, we study Picard-S iteration scheme for monotone nonexpansive mappings in the setting of \(\mathrm{CAT}(0)\) spaces and establish some convergence results. We prove \(\Delta\) and strong convergence results. Further, we provide a non trivial numerical example to illustrate the convergence of our iteration scheme and its stability with respect to the different parameters and initial values.\(\triangle\)-convergence for mixed-type total asymptotically nonexpansive mappings in hyperbolic spaceshttps://zbmath.org/1496.471222022-11-17T18:59:28.764376Z"Wan, Li-Li"https://zbmath.org/authors/?q=ai:wan.liliSummary: In this paper, we prove some \(\triangle\)-convergence theorems in a hyperbolic space. A~mixed Agarwal-O'Regan-Sahu type iterative scheme for approximating a common fixed point of total asymptotically nonexpansive mappings is constructed. Our results extend some results in the literature.Strong and \(\triangle\)-convergence theorems for total asymptotically nonexpansive nonself mappings in \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1496.471232022-11-17T18:59:28.764376Z"Yang, Li"https://zbmath.org/authors/?q=ai:yang.li.2"Zhao, Fu Hai"https://zbmath.org/authors/?q=ai:zhao.fuhaiSummary: The purpose of this paper is to study the existence theorems of fixed points, \(\triangle\)-convergence and strong convergence theorems for total asymptotically nonexpansive nonself mappings in the framework of \(\mathrm{CAT}(0)\) spaces. The convexity and closedness of a fixed point set of such mappings are also studied. Our results generalize, unify and extend several comparable results in the existing literature.Convergence rate of implicit iteration process and a data dependence resulthttps://zbmath.org/1496.471242022-11-17T18:59:28.764376Z"Yildirim, Isa"https://zbmath.org/authors/?q=ai:yildirim.isa"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahidSummary: The aim of this paper is to introduce an implicit S-iteration process and study its convergence in the framework of W-hyperbolic spaces. We show that the implicit S-iteration process has higher rate of convergence than implicit Mann type iteration and implicit Ishikawa-type iteration processes. We present a numerical example to support the analytic result proved herein. Finally, we prove a data dependence result for a contractive type mapping using implicit S-iteration process.Convex contractions of order \(n\) in \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1496.471252022-11-17T18:59:28.764376Z"Yildirim, Isa"https://zbmath.org/authors/?q=ai:yildirim.isa"Tekmanli, Yücel"https://zbmath.org/authors/?q=ai:tekmanli.yucel"Khan, Safeer Hussain"https://zbmath.org/authors/?q=ai:khan.safeer-hussainSummary: In this paper, we work on convex contraction of order \(n\). Our first
result in general metric spaces shows that each convex contraction of order \(n\) is a Bessaga mapping. We then turn our attention to \(\mathrm{CAT}(0)\) spaces. We prove a demiclosedness principle for such mappings in this setting. Next, we consider modified Mann iteration process and prove some convergence theorems for fixed points of such mappings in \(\mathrm{CAT}(0)\) spaces. Our results are new in \(\mathrm{CAT}(0)\) setting. Our results remain true in linear spaces like Hilbert and Banach spaces. Finally, we give an example in order to support our main results and to demonstrate the efficiency of modified Mann iteration process.Gamma-convergence of generalized gradient flows with conjugate typehttps://zbmath.org/1496.471262022-11-17T18:59:28.764376Z"Chang, Mao-Sheng"https://zbmath.org/authors/?q=ai:chang.mao-sheng"Liao, Jian-Tong"https://zbmath.org/authors/?q=ai:liao.jian-tongSummary: In this paper we establish the Gamma-convergence of generalized gradient flows with conjugate type. It provides a criteria for obtaining the convergence of generalized gradient flows that correspond to a sort of \(C^1\)-order \(\Gamma \)-convergence of energy functionals and a kind of bounded symmetric positive definite linear operators.The SHAI property for the operators on \(L^p\)https://zbmath.org/1496.471272022-11-17T18:59:28.764376Z"Johnson, W. B."https://zbmath.org/authors/?q=ai:johnson.william-b"Phillips, N. C."https://zbmath.org/authors/?q=ai:phillips.nicholas-c|phillips.n-c-k|phillips.n-christopher"Schechtman, G."https://zbmath.org/authors/?q=ai:schechtman.gideonThis work is motivated by the problem mentioned by \textit{B. Horváth} [Stud. Math. 253, No. 3, 259--282 (2020; Zbl 1460.46036)] whether \(L_{p}(0,1)\) (\(1<p<\infty\)) has the SHAI (surjective homomorphisms are injective) property provided that for every Banach space \(Y\), every continuous surjective algebra homomorphism from the bounded linear operators on \(L_{p}(0,1)\) onto the bounded linear operators on \(Y\) is injective. This problem is solved in the present paper (Corollary 1.6).
Reviewer: Elhadj Dahia (Bou Saâda)Reinhardt free spectrahedrahttps://zbmath.org/1496.471282022-11-17T18:59:28.764376Z"McCullough, Scott"https://zbmath.org/authors/?q=ai:mccullough.scott-a"Tuovila, Nicole"https://zbmath.org/authors/?q=ai:tuovila.nicoleSummary: Free spectrahedra are natural objects in the theories of operator systems and spaces and completely positive maps. They also appear in various engineering applications. In this paper, free spectrahedra satisfying a Reinhardt symmetry condition are characterized graph theoretically. It is also shown that, for a simple class of such spectrahedra, automorphisms are trivial.On the \(k\) point density problem for band-diagonal \(M\)-baseshttps://zbmath.org/1496.471292022-11-17T18:59:28.764376Z"Pyshkin, Alexey"https://zbmath.org/authors/?q=ai:pyshkin.alexeyLet \(\mathcal{H}\) be a real infinite-dimensional Hilbert space having a sequence of vectors \(\{f_n\}\) which is complete. Consider the operator algebra \(\mathcal{A}=\{T\in \mathcal{B}(\mathcal{H})\mid Tf_n=\lambda_nf_n,\,\lambda_n\in \mathbb{R}\}\) and the algebra \(R_1(\mathcal{A})\) generated by rank one operators of \(\mathcal{A}\). The algebra \(R_1(A)\) is said to be \(k\) point dense in \(\mathcal{A}\) if for any \(x_1,x_2,\dots,x_k\in \mathcal{H}\) and \(\epsilon>0\), there exists \(R\in R_1(\mathcal{A})\) such that \(\Vert Rx_s-x_s\Vert<\epsilon\) for any \(1\leq s\leq k\). The algebra \(\mathcal{A}\) has rank one density property if the unit ball of \(R_1(\mathcal{A})\) is dense in the unit ball of \(\mathcal{A}\) in the strong operator topology. In [\textit{W. E. Longstaff}, Can. J. Math. 28, 19--23 (1976; Zbl 0317.46052)], an interesting question was raised: does one point density property imply rank one density property? The answer was given in the negative in [\textit{D. R. Larson} and \textit{W. R. Wogen}, J. Funct. Anal. 92, No. 2, 448--467 (1990; Zbl 0738.47045)].
The author considers band-diagonal systems similar to the one regarded by Larson and Wogen [loc.\,cit.]\ to determine the exact conditions for \(k\) point density property of such vector systems. Simpler proofs of results in Larson and Wogen's paper [loc.\,cit.] are provided. Finally, a similar theorem for a pentadiagonal system is proved.
Reviewer: Ajay Kumar (Delhi)On the \(C^\ast\)-algebra generated by the Bergman operator, Carleman second-order shift, and piecewise continuous coefficientshttps://zbmath.org/1496.471302022-11-17T18:59:28.764376Z"Mozel', V. A."https://zbmath.org/authors/?q=ai:mozel.v-aSummary: We study the \(C^\ast\)-algebra generated by the Bergman operator with piecewise continuous coefficients in the Hilbert space \(L_2\) and extended by the Carleman rotation by an angle \(\pi\). As a result, we obtain an efficient criterion for the operators from the indicated \(C^\ast\)-algebra to be Fredholm operators.Application measure of noncompactness and operator type contraction for solvability of an infinite system of differential equations in \(\ell_p\)-spacehttps://zbmath.org/1496.471312022-11-17T18:59:28.764376Z"Hazarikaa, Bipan"https://zbmath.org/authors/?q=ai:hazarikaa.bipan"Arab, Reza"https://zbmath.org/authors/?q=ai:arab.reza"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammadSummary: The aim of this paper is to obtain existence results for an infinite system of second order differential equations in the sequence space \(\ell_p\) for \(1 \leq p< \infty\) with the help of a technique associated with measures of noncompactness and contractive condition of operator type. We also provide some illustrative examples in support of our existence theorems.On the applications of a minimax theoremhttps://zbmath.org/1496.490052022-11-17T18:59:28.764376Z"Ricceri, Biagio"https://zbmath.org/authors/?q=ai:ricceri.biagioIn this paper, a minimax theorem is presented and four applications are given: uniquely remotal sets in normed spaces; multiple global minima for the integral functional of the Calculus of Variations; multiple periodic solutions for Lagrangian systems of relativistic oscillators; variational inequalities in balls of Hilbert spaces. A related challenging open problem is also pointed out.
Reviewer: Yang Yang (Wuxi)Applications of generalized fractional hemivariational inequalities in solid viscoelastic contact mechanicshttps://zbmath.org/1496.490072022-11-17T18:59:28.764376Z"Han, Jiangfeng"https://zbmath.org/authors/?q=ai:han.jiangfeng"Li, Changpin"https://zbmath.org/authors/?q=ai:li.changpin.1|li.changpin"Zeng, Shengda"https://zbmath.org/authors/?q=ai:zeng.shengdaSummary: This paper studies a generalized fractional hemivariational inequality in infinite-dimensional spaces. Under the suitable assumptions, the existence result is delivered by using the temporally semi-discrete scheme and the surjectivity result for multivalued pseudomonotone operator. As an illustrative application, we propose a frictional contact model that describes the quasi-static contact between a viscoelastic body and a solid foundation. The viscoelastic constitutive equation is modeled by fractional Kelvin-Voigt law with \(\psi\)-Caputo derivative, and the frictional contact conditions are expressed as the Clarke subdifferentials of the nonconvex and nonsmooth functionals. Finally, the weak solvability of the mechanical system is obtained by using our abstract mathematical result.Correction to: ``Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization''https://zbmath.org/1496.490112022-11-17T18:59:28.764376Z"Bianchi, M."https://zbmath.org/authors/?q=ai:bianchi.monica"Kassay, G."https://zbmath.org/authors/?q=ai:kassay.gabor"Pini, R."https://zbmath.org/authors/?q=ai:pini.ritaCorrection to the authors' paper [ibid. 82, No. 3, 483--498 (2022; Zbl 1484.49032)].Asymptotic properties of an optimal principal eigenvalue with spherical weight and Dirichlet boundary conditionshttps://zbmath.org/1496.490132022-11-17T18:59:28.764376Z"Ferreri, Lorenzo"https://zbmath.org/authors/?q=ai:ferreri.lorenzo"Verzini, Gianmaria"https://zbmath.org/authors/?q=ai:verzini.gianmariaSummary: We consider a weighted eigenvalue problem for the Dirichlet laplacian in a smooth bounded domain \(\Omega \subset \mathbb{R}^N\), where the bang-bang weight equals a positive constant \(\overline{m}\) on a ball \(B \subset \Omega\) and a negative constant \(- \underline{m}\) on \(\Omega \backslash B\). The corresponding positive principal eigenvalue provides a threshold to detect persistence/extinction of a species whose evolution is described by the heterogeneous Fisher-KPP equation in population dynamics. In particular, we study the minimization of such eigenvalue with respect to the position of \(B\) in \(\Omega \). We provide sharp asymptotic expansions of the optimal eigenpair in the singularly perturbed regime in which the volume of \(B\) vanishes. We deduce that, up to subsequences, the optimal ball concentrates at a point maximizing the distance from \(\partial \Omega \).A unified approach to collectively maximal elements in abstract convex spaceshttps://zbmath.org/1496.520022022-11-17T18:59:28.764376Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieThe author establishes a very KKM type theorem in abstract convex spaces from which he then obtains an abstract collectively maximal element theorem. Finally, the author shows that a large number of previous theorems on the existence of maximal element and of equilibrium can be derived from his result.
Reviewer: Mircea Balaj (Oradea)Generalized multivalued integral type contraction on weak partial metric spacehttps://zbmath.org/1496.540232022-11-17T18:59:28.764376Z"Acar, Özlem"https://zbmath.org/authors/?q=ai:acar.ozlem"Coşkun, Sümeyye"https://zbmath.org/authors/?q=ai:coskun.sumeyye(no abstract)Some common fixed point theorems in complex valued metric spaceshttps://zbmath.org/1496.540242022-11-17T18:59:28.764376Z"Adegani, Ebrahim Analouei"https://zbmath.org/authors/?q=ai:adegani.ebrahim-analouei"Motamednezad, Ahmad"https://zbmath.org/authors/?q=ai:motamednezad.ahmadSummary: In this work, some common fixed point results for the mappings satisfying rational expressions on a closed ball in complex valued metric spaces will be proposed. Presented theorems can be realized as extensions of some well-known results in the literature. Further, our result is well supported by nontrivial example which shows that the improvement is a actual.Fixed point theorems for \(F\)-contraction in generalized asymmetric metric spaceshttps://zbmath.org/1496.540442022-11-17T18:59:28.764376Z"Kari, Abdelkarim"https://zbmath.org/authors/?q=ai:kari.abdelkarim"Rossafi, Mohamed"https://zbmath.org/authors/?q=ai:rossafi.mohamed"Saffaj, Hamza"https://zbmath.org/authors/?q=ai:saffaj.hamza"Marhrani, El Miloudi"https://zbmath.org/authors/?q=ai:marhrani.elmiloudi"Aamri, Mohamed"https://zbmath.org/authors/?q=ai:aamri.mohamedSummary: Recently, a new type of mapping called \(F\)-contraction was introduced in the literature as a generalization of the concepts of contractive mappings. This present article extends the new notion in generalized asymmetric metric spaces and establishing the existence and uniqueness of fixed point for them. Non-trivial examples are further provided to support the hypotheses of our results.Some classes of Meir-Keeler contractionshttps://zbmath.org/1496.540502022-11-17T18:59:28.764376Z"Manolescu, Laura"https://zbmath.org/authors/?q=ai:manolescu.laura"Găvruţa, Paşc"https://zbmath.org/authors/?q=ai:gavruta.pasc"Khojasteh, Farshid"https://zbmath.org/authors/?q=ai:khojasteh.farshidSummary: In the present paper, we prove that \(\mathcal{Z}\)-contractions, weakly type contractions, and some type of \(F\)-contractions are actually Meir-Keeler contractions.
For the entire collection see [Zbl 1485.65002].Fuzzy fixed point results via simulation functionshttps://zbmath.org/1496.540512022-11-17T18:59:28.764376Z"Mohammed, Shehu Shagari"https://zbmath.org/authors/?q=ai:mohammed.shehu-shagari"Fulatan, Ibrahim Aliyu"https://zbmath.org/authors/?q=ai:fulatan.ibrahim-aliyuSummary: We inaugurate two concepts, admissible hybrid fuzzy \(\mathcal{Z} \)-contractions and hybrid fuzzy \(\mathcal{Z} \)-contractions in the bodywork of \(b\)-metric spaces and establish sufficient criteria for fuzzy fixed points for such contractions. Nontrivial illustrations are constructed to support the hypotheses of our main notions. From application point of view, a handful of fixed point results of \(b\)-metric spaces endowed with partial ordering and graph are deduced. The ideas established herein unify and complement several well-known crisp and fuzzy fixed point theorems in the framework of both single-valued and set-valued mappings involving either linear or nonlinear contractions. A few important consequences of our main theorem are highlighted and analysed by using various forms of simulation functions.Coincidence and common fixed point results in \(G\)-metric spaces using generalized cyclic contractionhttps://zbmath.org/1496.540582022-11-17T18:59:28.764376Z"Puvar, Sejal V."https://zbmath.org/authors/?q=ai:puvar.sejal-v"Vyas, Rajendra G."https://zbmath.org/authors/?q=ai:vyas.rajendra-gSummary: Here, we have established the generalized cyclic contractive condition in \(G\)-metric spaces which can't be reduced to the contractive condition in standard metric spaces. The coincidence and common fixed point results are obtained for the pair of \((A,B)\)-weakly increasing mappings in \(G\)-metric spaces.A new kind of \(F\)-contraction and some best proximity point results for such mappings with an applicationhttps://zbmath.org/1496.540612022-11-17T18:59:28.764376Z"Şahin, Hakan"https://zbmath.org/authors/?q=ai:sahin.hakanSummary: In this paper, we aim to present a new and unified way, including the previously mentioned solution methods, to overcome the problem in [\textit{I. Altun} et al., J. Nonlinear Convex Anal. 16, No. 4, 659--666 (2015; Zbl 1315.54032)] for closed and bounded valued \(F\)-contraction mappings. We also want to obtain a real generalization of fixed point results existing in the literature by using best proximity point theory. Further, considering the strong relationship between homotopy theory and various branches of mathematics such as category theory, topological spaces, and Hamiltonian manifolds in quantum mechanics, our objective is to present an application to homotopy theory of our best proximity point results obtained in the paper. In this sense, we first introduce a new family, which is larger than \(\mathcal{F}^\ast\) that has often been used to give a positive answer to the problem. Then, we prove some best proximity point results for the new kind of \(F\)-contractions on quasi metric spaces via the new family. Additionally, we show that the note given by \textit{A. Almeida} et al. [Abstr. Appl. Anal. 2014, Article ID 716825, 4 p. (2014; Zbl 1478.54036)] is not valid for our results. Therefore, our results are real generalizations of fixed point results in the literature. Moreover, we give comparative examples to demonstrate that our results unify and generalize some well-known results in the literature. As an application, we show that each homotopic mapping to \(\varphi\) satisfying all the hypotheses of our best proximity point result has also a best proximity point.Best proximity coincidence point theorem for \(G\)-proximal generalized Geraghty auxiliary function in a metric space with graph \(G\)https://zbmath.org/1496.540652022-11-17T18:59:28.764376Z"Sinsongkham, Khamsanga"https://zbmath.org/authors/?q=ai:sinsongkham.khamsanga"Atiponrat, Watchareepan"https://zbmath.org/authors/?q=ai:atiponrat.watchareepanSummary: In a complete metric space endowed with a directed graph \(G\), we investigate the best proximity coincidence points of a pair of mappings that is \(G\)-proximal generalized auxiliary function. We show that the best proximity coincidence point is unique if any pair of two best proximity coincidence points is an edge of the graph \(G\). In addition, we provide an example as well as corollaries that are pertinent to our main theorem.Meir-Keeler sequential contractions and Pata fixed point resultshttps://zbmath.org/1496.540702022-11-17T18:59:28.764376Z"Turinici, Mihai"https://zbmath.org/authors/?q=ai:turinici.mihaiSummary: The (contractive) maps introduced by \textit{V. Pata} [J. Fixed Point Theory Appl. 10, No. 2, 299--305 (2011; Zbl 1264.54065)] are in fact Meir-Keeler sequential maps. This allows us treating in a unitary manner all fixed point results of this type.
For the entire collection see [Zbl 1483.00042].Smoothing effect and derivative formulas for Ornstein-Uhlenbeck processes driven by subordinated cylindrical Brownian noiseshttps://zbmath.org/1496.600142022-11-17T18:59:28.764376Z"Bondi, Alessandro"https://zbmath.org/authors/?q=ai:bondi.alessandroSummary: We investigate the concept of cylindrical Wiener process subordinated to a strictly \(\alpha\)-stable Lévy process, with \(\alpha \in (0, 1)\), in an infinite-dimensional, separable Hilbert space, and consider the related stochastic convolution. We then introduce the corresponding Ornstein-Uhlenbeck process, focusing on the regularizing properties of the Markov transition semigroup defined by it. In particular, we provide an explicit, original formula -- which is not of Bismut-Elworthy-Li's type -- for the Gateaux derivatives of the functions generated by the operators of the semigroup, as well as an upper bound for the norm of their gradients. In the case \(\alpha \in (\frac{1}{2}, 1)\), this estimate represents the starting point for studying the Kolmogorov equation in its mild formulation.Nonlinear parabolic stochastic evolution equations in critical spaces. I: Stochastic maximal regularity and local existencehttps://zbmath.org/1496.600682022-11-17T18:59:28.764376Z"Agresti, Antonio"https://zbmath.org/authors/?q=ai:agresti.antonio"Veraar, Mark"https://zbmath.org/authors/?q=ai:veraar.mark-cAuthors' abstract: In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an \(L^p\)-setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers' equation, the Allen-Cahn equation, the Cahn-Hilliard equation, reaction-diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to \textit{J. Prüss} et al. [J. Differ. Equations 264, No. 3, 2028--2074 (2018; Zbl 1377.35176)]. Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.
Reviewer: Piotr Biler (Wrocław)Mild solution of stochastic partial differential equation with nonlocal conditions and noncompact semigroupshttps://zbmath.org/1496.600742022-11-17T18:59:28.764376Z"Zhang, Xuping"https://zbmath.org/authors/?q=ai:zhang.xuping"Chen, Pengyu"https://zbmath.org/authors/?q=ai:chen.pengyu"Abdelmonem, Ahmed"https://zbmath.org/authors/?q=ai:abdelmonem.ahmed"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: The aim of this paper is to discuss the existence of mild solutions for a class of semilinear stochastic partial differential equation with nonlocal initial conditions and noncompact semigroups in a real separable Hilbert space. Combined with the theory of stochastic analysis and operator semigroups, a generalized Darbo's fixed point theorem and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and a noncompactness measure condition. In addition, the condition of uniformly continuity of the nonlinearity is not required and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example to illustrate the feasibility of the main results is also given.Decay of harmonic functions for discrete time Feynman-Kac operators with confining potentialshttps://zbmath.org/1496.600872022-11-17T18:59:28.764376Z"Cygan, Wojciech"https://zbmath.org/authors/?q=ai:cygan.wojciech"Kaleta, Kamil"https://zbmath.org/authors/?q=ai:kaleta.kamil"Śliwiński, Mateusz"https://zbmath.org/authors/?q=ai:sliwinski.mateuszSummary: We propose and study a certain discrete time counterpart of the classical Feynman-Kac semigroup with a confining potential in a countably infinite space. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman-Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman-Kac operators. We include such examples as non-local discrete Schrödinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.Green function for gradient perturbation of unimodal Lévy processes in the real linehttps://zbmath.org/1496.601032022-11-17T18:59:28.764376Z"Grzywny, T."https://zbmath.org/authors/?q=ai:grzywny.tomasz"Jakubowski, T."https://zbmath.org/authors/?q=ai:jakubowski.tomasz"Żurek, G."https://zbmath.org/authors/?q=ai:zurek.grzegorzSummary: We prove that the Green function of the generator of symmetric unimodal Lévy process with the weak lower scaling order bigger than one and the Green functions of its gradient perturbations are comparable for bounded \(C^{1,1}\) subsets of the real line if the drift function is from an appropriate Kato class.The discrete analogue of the differential operator \( \frac{\mathrm{d}^{2m}}{\mathrm{d}\,x^{2m}}+2\omega^2\frac{\mathrm{d}^{2m-2}}{\mathrm{d}\,x^{2m-2}}+\omega^4\frac{\mathrm{d}^{2m-4}}{\mathrm{d}\,x^{2m-4}} \)https://zbmath.org/1496.650152022-11-17T18:59:28.764376Z"Hayotov, A. R."https://zbmath.org/authors/?q=ai:hayotov.abdullo-rakhmonovich(no abstract)Data smoothing with applications to edge detectionhttps://zbmath.org/1496.650572022-11-17T18:59:28.764376Z"Al-Jamal, Mohammad F."https://zbmath.org/authors/?q=ai:al-jamal.mohammad-f"Baniabedalruhman, Ahmad"https://zbmath.org/authors/?q=ai:baniabedalruhman.ahmad"Alomari, Abedel-Karrem"https://zbmath.org/authors/?q=ai:alomari.abedel-karremSummary: The aim of this paper is to present a new stable method for smoothing and differentiating noisy data defined on a bounded domain \(\Omega \subset \mathbb{R}^N\) with \(N\geq 1\). The proposed method stems from the smoothing properties of the classical diffusion equation; the smoothed data are obtained by solving a diffusion equation with the noisy data imposed as the initial condition. We analyze the stability and convergence of the proposed method and we give optimal convergence rates. One of the main advantages of this method lies in multivariable problems, where some of the other approaches are not easily generalized. Moreover, this approach does not require strong smoothness assumptions on the underlying data, which makes it appealing for detecting data corners or edges. Numerical examples demonstrate the feasibility and robustness of the method even with the presence of a large amounts of noise.The selective projection method for convex feasibility and split feasibility problemshttps://zbmath.org/1496.650752022-11-17T18:59:28.764376Z"He, Songnian"https://zbmath.org/authors/?q=ai:he.songnian"Tian, Hanlin"https://zbmath.org/authors/?q=ai:tian.hanlin"Xu, Hong-Kun"https://zbmath.org/authors/?q=ai:xu.hong-kunSummary: A convex feasibility problem (CFP) is to find a point \(x^\ast\) with the property \(x^\ast\in\bigcap_{i=1}^mC^i \), where \(m\ge 1\) is an integer and \(\{C^i\}_{i=1}^m\) are closed convex subsets of a Hilbert space \(H\) with a common point. Projection methods are popular methods for solving CFP. Many projection algorithms in the existing literature have however to compute all the projections \(\{P_{C^i}\}_{i=1}^m\) in each iteration, which would be heavy computing workload and costly. It is therefore an interesting problem to invent new algorithms that can solve CFP efficiently and that compute as fewer projections as possible. In this paper we will introduce such new algorithms, which are called selective projection methods (SPM), for solving CFP in the case where each \(C^i\) is the level set of a convex function. In this case SPM computes only one projection in each iteration and also guarantees (weak) convergence. We also introduce relaxed SPM to project onto half-spaces to make SPM easily implementable. In addition we extend SPM to solve the multiple-sets split feasibility problems and the common fixed point problem of nonexpansive mappings.A second-order dynamical system for equilibrium problemshttps://zbmath.org/1496.650812022-11-17T18:59:28.764376Z"Le Van Vinh"https://zbmath.org/authors/?q=ai:le-van-vinh."Van Nam Tran"https://zbmath.org/authors/?q=ai:van-nam-tran."Phan Tu Vuong"https://zbmath.org/authors/?q=ai:vuong.phan-tuSummary: We consider a second-order dynamical system for solving equilibrium problems in Hilbert spaces. Under mild conditions, we prove existence and uniqueness of strong global solution of the proposed dynamical system. We establish the exponential convergence of trajectories under strong pseudo-monotonicity and Lipschitz-type conditions. We then investigate a discrete version of the second-order dynamical system, which leads to a fixed point-type algorithm with inertial effect and relaxation. The linear convergence of this algorithm is established under suitable conditions on parameters. Finally, some numerical experiments are reported confirming the theoretical results.Inclusion method of optimal constant with quadratic convergence for \(H_0^1\)-projection error estimates and its applicationshttps://zbmath.org/1496.652212022-11-17T18:59:28.764376Z"Kinoshita, Takehiko"https://zbmath.org/authors/?q=ai:kinoshita.takehiko"Watanabe, Yoshitaka"https://zbmath.org/authors/?q=ai:watanabe.yoshitaka"Yamamoto, Nobito"https://zbmath.org/authors/?q=ai:yamamoto.nobito"Nakao, Mitsuhiro T."https://zbmath.org/authors/?q=ai:nakao.mitsuhiro-tSummary: We present an interval inclusion method for optimal constants of second-order error estimates of \(H_0^1\)-projections to finite-degree polynomial spaces. These constants can be applied to error estimates of the Lagrange-type finite element method. Moreover, the proposed a priori error estimates are applicable to residual iteration techniques for the verification of solutions to nonlinear elliptic equations. Some numerical examples by the finite element method will be shown for comparison with other approaches, which confirm us the actual usefulness of the results in this paper for the numerical verification method for PDEs.Simultaneous vs. non-simultaneous measurements in quantum and classical mechanicshttps://zbmath.org/1496.810262022-11-17T18:59:28.764376Z"Chudak, N. O."https://zbmath.org/authors/?q=ai:chudak.n-o"Potiienko, O. S."https://zbmath.org/authors/?q=ai:potiienko.o-s"Sharph, I. V."https://zbmath.org/authors/?q=ai:sharph.i-vSummary: In a traditional implementation of the relativity principles, different observers consider \textit{the same} events and relate their space-time coordinates through the Lorentz transformation. In this paper we consider the problems where it is impossible to use the non-simultaneous measurements in any inertial reference frame. In such case different observers have to use \textit{different} sets of events, the space-time coordinates of which are impossible to relate through Lorentz transform. Therefore we suggest another way of implementing the relativity principles, and discuss some of its consequences and prospects.Bound state solutions and thermodynamic properties of modified exponential screened plus Yukawa potentialhttps://zbmath.org/1496.810482022-11-17T18:59:28.764376Z"Antia, Akaninyene D."https://zbmath.org/authors/?q=ai:antia.akaninyene-d"Okon, Ituen B."https://zbmath.org/authors/?q=ai:okon.ituen-b"Isonguyo, Cecilia N."https://zbmath.org/authors/?q=ai:isonguyo.cecilia-n"Akankpo, Akaninyene O."https://zbmath.org/authors/?q=ai:akankpo.akaninyene-o"Eyo, Nsemeke E."https://zbmath.org/authors/?q=ai:eyo.nsemeke-eSummary: In this research paper, the approximate bound state solutions and thermodynamic properties of Schrödinger equation with modified exponential screened plus Yukawa potential (MESPYP) were obtained with the help Greene-Aldrich approximation to evaluate the centrifugal term. The Nikiforov-Uvarov (NU) method was used to obtain the analytical solutions. The numerical bound state solutions of five selected diatomic molecules, namely mercury hydride (HgH), zinc hydride (ZnH), cadmium hydride (CdH), hydrogen bromide (HBr) and hydrogen fluoride (HF) molecules were also obtained. We obtained the energy eigenvalues for these molecules using the resulting energy eigenequation and total unnormalized wave function expressed in terms of associated Jacobi polynomial. The resulting energy eigenequation was presented in a closed form and applied to study partition function (Z) and other thermodynamic properties of the system such as vibrational mean energy (U), vibrational specific heat capacity (C), vibrational entropy (S) and vibrational free energy (F). The numerical bound state solutions were obtained from the resulting energy eigenequation for some orbital angular quantum number. The results obtained from the thermodynamic properties are in excellent agreement with the existing literature. All numerical computations were carried out using spectroscopic constants of the selected diatomic molecules with the help of MATLAB 10.0 version. The numerical bound state solutions obtained increases with an increase in quantum state.Evolution of energy and magnetic moment of a quantum charged particle in power-decaying magnetic fieldshttps://zbmath.org/1496.810522022-11-17T18:59:28.764376Z"Dodonov, V. V."https://zbmath.org/authors/?q=ai:dodonov.victor-v"Horovits, M. B."https://zbmath.org/authors/?q=ai:horovits.m-bSummary: We consider a quantum spinless nonrelativistic charged particle moving in the \(xy\) plane under the action of a homogeneous time-dependent magnetic field \(B(t) = B_0(1 + t/t_0)^{-1-g}\), directed along the \(z\)-axis and described by means of the vector potential \(\mathbf{A}(t) = B(t)[-y, x]/2\). Assuming that the particle was initially in the thermal equilibrium state with a negligible coupling to a reservoir, we obtain exact formulas for the time-dependent mean values of the energy and magnetic moment in terms of the Bessel functions. Different regimes of the evolution are discovered and illustrated in several figures. The energy goes asymptotically to a finite value if \(g > 0\) (``fast'' decay), while it goes asymptotically to zero if \(g \leq 0\) (``slow'' decay). The dependence on parameter \(t_0\) practically disappears when \(1 + g\) is close to zero value (``superslow'' decay). The mean magnetic moment goes to zero for \(g > 1\), while it grows unlimitedly if \(g < 1\).Analysis of solutions of time-dependent Schrödinger equation of a particle trapped in a spherical boxhttps://zbmath.org/1496.810532022-11-17T18:59:28.764376Z"Nath, Debraj"https://zbmath.org/authors/?q=ai:nath.debraj"Carbó-Dorca, Ramon"https://zbmath.org/authors/?q=ai:carbo-dorca.ramonSummary: Three sets of exact solutions of the time-dependent Schrödinger equation of a particle that is trapped in a spherical box with a moving boundary wall have been calculated analytically. For these solutions, some physical quantities such as time-dependent average energy, average force, disequilibrium, quantum similarity measures as well as quantum similarity index have been investigated. Moreover, these solutions are compared concerning these physical quantities. The time-correlation functions among these solutions are investigated.On the number of eigenvalues of the lattice model operator in one-dimensional casehttps://zbmath.org/1496.810562022-11-17T18:59:28.764376Z"Bozorov, I. N."https://zbmath.org/authors/?q=ai:bozorov.i-n"Khurramov, A. M."https://zbmath.org/authors/?q=ai:khurramov.abdumazhid-molikovichSummary: It is considered a model operator \(h_{\mu}(k),k\in\mathbb{T}\equiv(-\pi,\pi]\), corresponding to the Hamiltonian of systems of two arbitrary quantum particles on a one-dimensional lattice with a special dispersion function that describes the transfer of a particle from one site to another interacting by a some short-range attraction potential \(v_{\mu}, \mu=(\mu_0,\mu_1,\mu_2,\mu_3)\in\mathbb{R}^4_+ \). The number of eigenvalues of the operator \(h_{\mu}(k),k\in\mathbb{T}\) depending on the energy of the particle interaction vector \(\mu\in\mathbb{R}^4_+\) and the total quasi-momentum \(k\in\mathbb{T}\) is studied.On a generalized central limit theorem and large deviations for homogeneous open quantum walkshttps://zbmath.org/1496.810602022-11-17T18:59:28.764376Z"Carbone, Raffaella"https://zbmath.org/authors/?q=ai:carbone.raffaella"Girotti, Federico"https://zbmath.org/authors/?q=ai:girotti.federico"Hernandez, Anderson Melchor"https://zbmath.org/authors/?q=ai:hernandez.anderson-melchorSummary: We consider homogeneous open quantum walks on a lattice with finite dimensional local Hilbert space and we study in particular the position process of the quantum trajectories of the walk. We prove that the properly rescaled position process asymptotically approaches a mixture of Gaussian measures. We can generalize the existing central limit type results and give more explicit expressions for the involved asymptotic quantities, dropping any additional condition on the walk. We use deformation and spectral techniques, together with reducibility properties of the local channel associated with the open quantum walk. Further, we can provide a large deviation principle in the case of a fast recurrent local channel and at least lower and upper bounds in the general case.Energy shift of a uniformly moving two-level atom through a thermal reservoirhttps://zbmath.org/1496.810722022-11-17T18:59:28.764376Z"Cai, Huabing"https://zbmath.org/authors/?q=ai:cai.huabing"Wang, Li-Gang"https://zbmath.org/authors/?q=ai:wang.ligangSummary: We investigate the implications of an atomic constant velocity in the energy shift of a two-level atom inside the thermal bath of a quantum scalar field, which is described by the Bose-Einstein distribution. The use of DDC formalism shows that the contribution of thermal fluctuations on the atomic level shifts depends on the atomic velocity and the temperature of the heat reservoir but the contribution of radiation reaction is totally insusceptible. The resulting energy shifts are analyzed and examined in detail under different circumstances. The atomic uniform linear motion always broadens the atomic level spacing in the limit of low temperature but narrows down it in the limit of high temperature. Our work clearly indicates that the moving heat reservoir shifts the atomic levels in a way quite different from that of the static one.Differential recurrences for the distribution of the trace of the \(\beta\)-Jacobi ensemblehttps://zbmath.org/1496.810912022-11-17T18:59:28.764376Z"Forrester, Peter J."https://zbmath.org/authors/?q=ai:forrester.peter-j"Kumar, Santosh"https://zbmath.org/authors/?q=ai:kumar.santosh.2|kumar.santosh.1|kumar.santosh.4|kumar.santosh|kumar.santosh.3Summary: Examples of the \(\beta\)-Jacobi ensemble in random matrix theory specify the joint distribution of the transmission eigenvalues in scattering problems. For this application, the trace is of relevance as determining the conductance. Earlier, in the case \(\beta = 1\), the trace statistic was isolated in studies of covariance matrices in multivariate statistics. There, Davis showed that for \(\beta = 1\) the trace statistic could be characterised by \((N + 1) \times (N + 1)\) matrix differential equations, now understood for general \(\beta > 0\) as part of the theory of Selberg correlation integrals. However the characterisation provided was incomplete, as the connection problem of determining the linear combination of Frobenius type solutions that correspond to the statistic was not solved. We solve this connection problem for Jacobi parameters \(b\) and Dyson index \(\beta\) non-negative integers. The distribution then has the functional form of a series of piecewise power functions times a polynomial, and our characterisation gives a recurrence for the computation of the polynomials. For all \(\beta > 0\) we express the Fourier-Laplace transform of the trace statistic in terms of a generalised hypergeometric function based on Jack polynomials.The factorization method for inverse scattering by a two-layered cavity with conductive boundary conditionhttps://zbmath.org/1496.810922022-11-17T18:59:28.764376Z"Ye, Jianguo"https://zbmath.org/authors/?q=ai:ye.jianguo"Yan, Guozheng"https://zbmath.org/authors/?q=ai:yan.guozhengSummary: In this paper we consider the inverse scattering problem of determining the shape of a two-layered cavity with conductive boundary condition from sources and measurements placed on a curve inside the cavity. First, we show the well-posedness of the direct scattering problem by using the boundary integral equation method. Then, we prove that the factorization method can be applied to reconstruct the interface of the two-layered cavity from near-field data. Some numerical experiments are also presented to demonstrate the feasibility and effectiveness of the factorization method.Four-dimensional factorization of the fermion determinant in lattice QCDhttps://zbmath.org/1496.810942022-11-17T18:59:28.764376Z"Giusti, Leonardo"https://zbmath.org/authors/?q=ai:giusti.leonardo"Saccardi, Matteo"https://zbmath.org/authors/?q=ai:saccardi.matteoSummary: In the last few years it has been proposed a one-dimensional factorization of the fermion determinant in lattice QCD with Wilson-type fermions that leads to a block-local action of the auxiliary bosonic fields. Here we propose a four-dimensional generalization of this factorization. Possible applications are more efficient parallelizations of Monte Carlo algorithms and codes, master field simulations, and multi-level integration.Radiative (anti)neutrino energy spectra from muon, pion, and kaon decayshttps://zbmath.org/1496.811062022-11-17T18:59:28.764376Z"Tomalak, Oleksandr"https://zbmath.org/authors/?q=ai:tomalak.oleksandrSummary: To describe low-energy (anti)neutrino fluxes in modern coherent elastic neutrino-nucleus scattering experiments as well as high-energy fluxes in precision-frontier projects such as the Enhanced NeUtrino BEams from kaon Tagging (ENUBET) and the Neutrinos from STORed Muons (nuSTORM), we evaluate (anti)neutrino energy spectra from radiative muon (\(\mu^- \to e^- \bar{\nu}_e \nu_\mu(\gamma)\), \(\mu^+ \to e^+ \nu_e \bar{\nu}_\mu(\gamma)\)), pion \(\pi_{\ell2}\) (\(\pi^- \to \mu^- \bar{\nu}_\mu(\gamma)\), \(\pi^+ \to \mu^+ \nu_\mu(\gamma)\)), and kaon \(K_{\ell 2}\) (\(K^- \to \mu^- \bar{\nu}_\mu(\gamma)\), \(K^+ \to \mu^+ \nu_\mu(\gamma)\)) decays. We compare detailed \(\mathrm{O}(\alpha)\) distributions to the well-known tree-level results, investigate electron-mass corrections and provide energy spectra in analytical form. Radiative corrections introduce continuous and divergent spectral components near the endpoint, on top of the monochromatic tree-level meson-decay spectra, which can change the flux-averaged cross section at \(6 \times 10^{-5}\) level for the scattering on \(^{40}\mathrm{Ar}\) nucleus with (anti)neutrinos from the pion decay at rest. Radiative effects modify the expected (anti)neutrino fluxes from the muon decay around the peak region by 3--4 permille, which is a precision goal for next-generation artificial neutrino sources.Generalized intensity-dependent multiphoton Jaynes-Cummings modelhttps://zbmath.org/1496.811132022-11-17T18:59:28.764376Z"Bartzis, V."https://zbmath.org/authors/?q=ai:bartzis.v"Merlemis, N."https://zbmath.org/authors/?q=ai:merlemis.nikolaos"Serris, M."https://zbmath.org/authors/?q=ai:serris.m"Ninos, G."https://zbmath.org/authors/?q=ai:ninos.gSummary: In this chapter, we study the Jaynes-Cummings model under multiphoton excitation and in the general case of intensity-dependent coupling strength given by an arbitrary function \(f\). The Jaynes-Cummings theoretical model is of great interest to atomic physics, quantum optics, solid-state physics, and quantum information theory with several applications in coherent control and quantum information processing. As the initial state of the radiation mode, we consider a squeezed state, which is the most general Gaussian pure state. The time evolution of the mean photon number and the dispersions of the two quadrature components of the electromagnetic field are calculated for an arbitrary function \(f\). The mean value of the inversion operator of the atom is also calculated for some simple forms of the function \(f\).
For the entire collection see [Zbl 1485.65002].Very special linear gravity: a gauge-invariant graviton masshttps://zbmath.org/1496.830012022-11-17T18:59:28.764376Z"Alfaro, Jorge"https://zbmath.org/authors/?q=ai:alfaro.jorge"Santoni, Alessandro"https://zbmath.org/authors/?q=ai:santoni.alessandroSummary: Linearized gravity in the Very Special Relativity (VSR) framework is considered. We prove that this theory allows for a non-zero graviton mass \(m_g\) without breaking gauge invariance nor modifying the relativistic dispersion relation. We find the analytic solution for the new equations of motion in our gauge choice, verifying as expected the existence of only two physical degrees of freedom. Finally, through the geodesic deviation equation, we confront some results for classic gravitational waves (GW) with the VSR ones: we see that the ratios between VSR effects and classical ones are proportional to \((m_g/E)^2\), \(E\) being the energy of a graviton in the GW. For GW detectable by the interferometers LIGO and VIRGO this ratio is at most \(10^{-20}\). However, for GW in the lower frequency range of future detectors, like LISA, the ratio increases significantly to \(10^{-10}\), that combined with the anisotropic nature of VSR phenomena may lead to observable effects.Möbius mirrorshttps://zbmath.org/1496.830062022-11-17T18:59:28.764376Z"Good, Michael R. R."https://zbmath.org/authors/?q=ai:good.michael-r-r"Linder, Eric V."https://zbmath.org/authors/?q=ai:linder.eric-vHow not to extract information from black holes: cosmic censorship as a guiding principlehttps://zbmath.org/1496.830272022-11-17T18:59:28.764376Z"Di Gennaro, Sofia"https://zbmath.org/authors/?q=ai:di-gennaro.sofia"Ong, Yen Chin"https://zbmath.org/authors/?q=ai:ong.yen-chinSummary: Black holes in general relativity are commonly believed to evolve towards a Schwarzschild state as they gradually lose angular momentum and electrical charge under Hawking evaporation. However, when Kim and Wen applied quantum information theory to Hawking evaporation and argued that Hawking particles with maximum mutual information could dominate the emission process, they found that charged black holes tend towards extremality. In view of some evidence pointing towards extremal black holes being effectively singular, this would violate the cosmic censorship conjecture. Nevertheless, since the Kim-Wen model is too simplistic (e.g. it assumes a continuous spectrum of particles with arbitrary charge-to-mass ratio), one might hope that a more realistic model could avoid this problem. In this work, we show that having only a finite species of charged particles would actually worsen the situation, with some end states becoming a naked singularity. With this model as an example, we emphasize the need to study whether charged black holes can violate cosmic censorship under a given model of Hawking evaporation.Laplacian on fuzzy de Sitter spacehttps://zbmath.org/1496.830332022-11-17T18:59:28.764376Z"Brkić, Bojana"https://zbmath.org/authors/?q=ai:brkic.bojana"Burić, Maja"https://zbmath.org/authors/?q=ai:buric.maja"Latas, Duško"https://zbmath.org/authors/?q=ai:latas.duskoUniformly accelerated Brownian oscillator in (2+1)D: temperature-dependent dissipation and frequency shifthttps://zbmath.org/1496.830352022-11-17T18:59:28.764376Z"Moustos, Dimitris"https://zbmath.org/authors/?q=ai:moustos.dimitrisSummary: We consider an Unruh-DeWitt detector modeled as a harmonic oscillator that is coupled to a massless quantum scalar field in the (2+1)-dimensional Minkowski spacetime. We treat the detector as an open quantum system and employ a quantum Langevin equation to describe its time evolution, with the field, which is characterized by a frequency-independent spectral density, acting as a stochastic force. We investigate a point-like detector moving with constant acceleration through the Minkowski vacuum and an inertial one immersed in a thermal reservoir at the Unruh temperature, exploring the implications of the well-known non-equivalence between the two cases on their dynamics. We find that both the accelerated detector's dissipation rate and the shift of its frequency caused by the coupling to the field bath depend on the acceleration temperature. Interestingly enough this is not only in contrast to the case of inertial motion in a heat bath but also to any analogous quantum Brownian motion model in open systems, where dissipation and frequency shifts are not known to exhibit temperature dependencies. Nonetheless, we show that the fluctuating-dissipation theorem still holds for the detector-field system and in the weak-coupling limit an accelerated detector is driven at late times to a thermal equilibrium state at the Unruh temperature.Stability analysis of anisotropic Bianchi type-I cosmological model in teleparallel gravityhttps://zbmath.org/1496.830362022-11-17T18:59:28.764376Z"Koussour, M."https://zbmath.org/authors/?q=ai:koussour.m"Bennai, M."https://zbmath.org/authors/?q=ai:bennai.mohamedElectromagnetic effects on dynamics of string fluid and information paradox in rainbow gravityhttps://zbmath.org/1496.830372022-11-17T18:59:28.764376Z"Sheikh, Umber"https://zbmath.org/authors/?q=ai:sheikh.umber"Arshad, Sana"https://zbmath.org/authors/?q=ai:arshad.sanaSummary: This work is devoted to studying the effects of electric field intensity on collapsing anisotropic string fluid in Rainbow gravity. The Einstein field equations are modified and solved for the spherical symmetric spacetime. The physical parameters of fluid including energy density, pressure and string tension are obtained. Moreover, the time and radius of formation of the apparent horizon are estimated. All these quantities depend on the fluid's electric intensity. The graphical analysis of the physical existence of dynamical quantities depending on the energy of the probing particle is presented. It is found that the presence of an electric field decreases the mass density and increases fluid's pressure. The electric field increases the time and radius of apparent horizon formation resulting in slowing down the collapsing process.Interaction of inhomogeneous axions with magnetic fields in the early universehttps://zbmath.org/1496.830382022-11-17T18:59:28.764376Z"Dvornikov, Maxim"https://zbmath.org/authors/?q=ai:dvornikov.maximSummary: We study the system of interacting axions and magnetic fields in the early universe after the quantum chromodynamics phase transition, when axions acquire masses. Both axions and magnetic fields are supposed to be spatially inhomogeneous. We derive the equations for the spatial spectra of these fields, which depend on conformal time. In case of the magnetic field, we deal with the spectra of the energy density and the magnetic helicity density. The evolution equations are obtained in the closed form within the mean field approximation. We choose the parameters of the system and the initial condition which correspond to realistic primordial magnetic fields and axions. The system of equations for the spectra is solved numerically. We compare the cases of inhomogeneous and homogeneous axions. The evolution of the magnetic field in these cases is different only within small time intervals. Generally, magnetic fields are driven mainly by the magnetic diffusion. We find that the magnetic field instability takes place for the amplified initial wavefunction of the homogeneous axion. This instability is suppressed if we account for the inhomogeneity of the axion.Emergence of space and expansion of universehttps://zbmath.org/1496.830402022-11-17T18:59:28.764376Z"V T, Hassan Basari"https://zbmath.org/authors/?q=ai:v-t.hassan-basari"Krishna, P. B."https://zbmath.org/authors/?q=ai:krishna.p-b"K. V, Priyesh"https://zbmath.org/authors/?q=ai:k-v.priyesh"Mathew, Titus K."https://zbmath.org/authors/?q=ai:mathew.titus-kQuark condensate and chiral symmetry restoration in neutron starshttps://zbmath.org/1496.850012022-11-17T18:59:28.764376Z"Jin, Hao-Miao"https://zbmath.org/authors/?q=ai:jin.hao-miao"Xia, Cheng-Jun"https://zbmath.org/authors/?q=ai:xia.cheng-jun"Sun, Ting-Ting"https://zbmath.org/authors/?q=ai:sun.tingting"Peng, Guang-Xiong"https://zbmath.org/authors/?q=ai:peng.guang-xiongSummary: Based on an equivparticle model, we investigate the in-medium quark condensate in neutron stars. Carrying out a Taylor expansion of the nuclear binding energy to the order of \(\rho^3\), we obtain a series of EOSs for neutron star matter, which are confronted with the latest nuclear and astrophysical constraints. The in-medium quark condensate is then extracted from the constrained properties of neutron star matter, which decreases non-linearly with density. However, the chiral symmetry is only partially restored with non-vanishing quark condensates, which may vanish at a density that is out of reach for neutron stars.Abstract strongly convergent variants of the proximal point algorithmhttps://zbmath.org/1496.900612022-11-17T18:59:28.764376Z"Sipoş, Andrei"https://zbmath.org/authors/?q=ai:sipos.andreiSummary: We prove an abstract form of the strong convergence of the Halpern-type and Tikhonov-type proximal point algorithms in CAT(0) spaces. In addition, we derive uniform and computable rates of metastability (in the sense of Tao) for these iterations using proof mining techniques.Some problems of linear differential equations on abstract spaces and unbounded perturbations of linear operator semigrouphttps://zbmath.org/1496.930632022-11-17T18:59:28.764376Z"Xu, Genqi"https://zbmath.org/authors/?q=ai:xu.gen-qiSummary: This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the \(L^p\) linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system \((A, B)\). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an \(L^p\) well-posed system.