Recent zbMATH articles in MSC 47https://zbmath.org/atom/cc/472021-11-25T18:46:10.358925ZWerkzeugBook review of: U. Daepp et al., Finding ellipses. What Blaschke products, Poncelet's theorem, and the numerical range know about each other.https://zbmath.org/1472.000422021-11-25T18:46:10.358925Z"Zeytuncu, Yunus E."https://zbmath.org/authors/?q=ai:zeytuncu.yunus-ergynReview of [Zbl 1419.51001].Effective sup-norm bounds on average for cusp forms of even weighthttps://zbmath.org/1472.111192021-11-25T18:46:10.358925Z"Friedman, J. S."https://zbmath.org/authors/?q=ai:friedman.joshua-s"Jorgenson, J."https://zbmath.org/authors/?q=ai:jorgenson.jay-a"Kramer, J."https://zbmath.org/authors/?q=ai:kramer.jurgSummary: Let \(\Gamma \subset \operatorname{PSL}_2(\mathbb{R})\) be a Fuchsian subgroup of the first kind acting on the upper half-plane \(\mathbb{H}\). Consider the \(d_{2k}\)-dimensional space of cusp forms \(\mathcal{S}_{2k}^{\Gamma }\) of weight \(2k\) for \(\Gamma \), and let \(\{f_1,\ldots ,f_{d_{2k}}\}\) be an orthonormal basis of \(\mathcal{S}_{2k}^{\Gamma }\) with respect to the Petersson inner product. In this paper, we will give effective upper and lower bounds for the supremum of the quantity \(S_{2k}^{\Gamma }(z):=\sum _{j=1}^{d_{2k}}\vert f_j(z)\vert ^2\operatorname{Im}(z)^{2k}\) as \(z\) ranges through \(\mathbb{H}\).The distribution relation and inverse function theorem in arithmetic geometryhttps://zbmath.org/1472.112002021-11-25T18:46:10.358925Z"Matsuzawa, Yohsuke"https://zbmath.org/authors/?q=ai:matsuzawa.yohsuke"Silverman, Joseph H."https://zbmath.org/authors/?q=ai:silverman.joseph-hillelThe paper presents an arithmetic distribution relation as well as two versions of the inverse function theorem in terms of an arithmetic distance function. In sections 3-5, the field \(K\) denotes a field with a complete set of absolute values \(\mathcal{M}_K\) satisfying a product formula. If we fix an algebraic closure \(\bar{K}\) of \(K\), an \(\mathcal{M}_K\)-constant will be a function \(\gamma \colon \mathcal{M}_{\bar{K}} \longrightarrow \mathbb{R}_{\geq 0}\) such that \(\gamma(v)\) depends only on the restriction \(v|_K\) and the set \(\{ v|_K \, | \, \gamma(v) \neq 0 \}\) is finite. The notation \(O(\mathcal{M}_K)\) will be used to denote a relation that holds up to an \(\mathcal{M}_K\)-constant. For instance \(f \leq g +O(h) + O(\mathcal{M}_K)\) means that there exist a \(C>0\) and an \(\mathcal{M}_K\)-constant \(\gamma\) such that \(f \leq g + C|h| + \gamma\).
Definition: (Local height functions) Let \(K\) be a field with a set of absolute values \(\mathcal{M}_K\). Let \(V\) be a projective variety (not necessarily irreducible) over \(K\) and \(X \subset V\) a closed subscheme. A local height is a function \(\lambda_X \colon V(\bar{K}) \times \mathcal{M}_K \longrightarrow \mathbb{R} \cup \infty\) determined by the following properties:
\begin{enumerate}
\item[(1)] If \(D\) is an effective divisor, we get the usual local height, i.e., \(\lambda_X = \lambda_D\).
\item[(2)] if \(X,X'\) are subschemes \(\lambda_{X \cap X'} = \min{(\lambda_X,\lambda_{X'})}\) (where \(X \cap X'\) denotes the subscheme with ideal sheaf .\(\mathcal{I}_X+\mathcal{I}_{X'}\) ).
\end{enumerate}
and having also many other nice properties:
\begin{enumerate}
\item[(3)] (functoriality) If \(\varphi \colon V \longrightarrow W\) denotes a morphism of varieties and \(X\) is a closed subscheme of \(W\), we have the equality: \[\lambda_{\varphi^{-1} W,V} = \lambda_{X,W}.\]
\item[(4)] Local height functions are bounded below, so up a \(\mathcal{M}_K\)-constant, we can assume that \(\lambda_X \geq 0\).
\end{enumerate}
Remark: Using the local height function associated to the boundary divisor, the local height function machinery can be extended to quasi-projective varieties.
Definition: Let \(\Delta(V)\) denotes the diagonal subvariety in \(V \times V\). The arithmetic distance function on \(V\) is the local height \[\delta_V = \lambda_{\Delta(V)}.\] It is well-defined up to an \(\mathcal{M}_K\)-bounded function and satisfies many nice properties as well, for example:
\begin{enumerate}
\item[(1)] \(\delta(P,R) \geq \min(\delta(P,Q),\delta(Q,R))\)
\item[(2)] \(\lambda_X(Q) \geq \min(\lambda_X(P),\delta(P,Q))\)
\end{enumerate}
Defintion: Let \(\varphi \colon W \longrightarrow V\) be a finite flat morphism between schemes of finite type over a field \(k\). Let \(k'\) be an algebraically closed field containing \(k\). For \(x \in W(k')\), define the multiplicity of \(\varphi\) at \(x\) by the formula \[e_\varphi(x)= \text{length}_{\mathcal{O}_{W_{k'},x}} \mathcal{O}_{W_{k'},x}/\varphi^{-1} m_{\varphi(x)} \mathcal{O}_{W_{k'},x}.\]
For finite flat morphisms \(W \longrightarrow V\), the distribution inequality bounds the arithmetic distance in the target variety in terms of the arithmetic distance of the pre-images in \(W\). In good cases, it will give a distribution relation.
Theorem: (Arithmetic distribution relation/inequality) Let \(\varphi \colon W \longrightarrow V\) be a generically étale finite flat morphism between quasi-projective geometrically integral varieties over \(K\). For all \((P,q,v) \in W(\bar{K}) \times V(\bar{K}) \times v\), we have \[\delta_V(\varphi(P),q,v) \leq \sum_{Q \in W(\bar{K}),\, \varphi(Q)=q} e_\varphi(Q) \delta_W(P,Q,v)+ O(\lambda_{ \partial(V \times W)}(P,q,v))+ O(\mathcal{M}_K).\] We say that we have an arithmetic distribution relation when the above inequality becomes an equality. For example, assuming that \(V,W\) are both smooth, we have an arithmetic relation in the following two situations:
\begin{enumerate}
\item[(1)] The map \(\varphi \colon W \longrightarrow V\) is étale (where there is not ramification).
\item[(2)] The dimensions \(\dim(V)=\dim(W)=1\) (where the ramification divisor is at most zero-dimensional).
\end{enumerate}
Remark: The above inequality does not always became an arithmetic relation. Even when we take a Galois cover \(\varphi \colon W \longrightarrow V\) with Galois group \(\text{Gal}(V/W)=\{\tau_1,\dots,\tau_n\}\), we have an inclusion of associated sheaves of ideals \[\mathcal{I}(\sum_{i=1}^n (1\times \tau_i)^* \Delta(W)) \subset \mathcal{I}((\varphi \times \varphi)^*(\Delta(V))\] that does not always gives an equality of closed schemes. Take for example \(\varphi \colon \mathbb{P}^1 \times \mathbb{P}^1 \longrightarrow \mathbb{P}^1 \times \mathbb{P}^1 \) defined by \(\varphi([x,y],[z,w])=([x^2,y^2],[z,w])\).
Remark: The above inequality can be used to obtain a quantitative inverse theorem, namely, given a finite map, how far apart from the ramification locus and the boundary we need to be, to be able to define a local inverse. Also, the inverse obtained can be shown to behave nicely with respect to the distance functions. In some sense, the distance between points is close to the distance between the inverses.
Theorem: (Inverse function theorem version \(1\)) Suppose that \(V\) and \(W\) are quasi-projective geometrically integral varieties defined over \(K\). Assume that the map \(\varphi \colon W \longrightarrow V\) is a generically étale finite flat surjective morphism of degree \(d\) also defined over K. Let us denote by \(\text{Ann}(\Omega_{W/V})\), the annihilator ideal sheaf of \(\Omega_{W/V}\) and by \(A(\varphi) \subset W\) the closed subscheme defined by \(\text{Ann}(\Omega_{W/V})\).
\begin{enumerate}
\item[(a)] There exist constants \(C_2,C_3\) and \(\mathcal{M}_K\)-constants \(C_4,C_5\) such that the following holds:\\
If the triple \((P,q,v) \in W(K) \times V(K) \times M(K)\) satisfies \[\delta_V (\varphi(P),q;v) \geq d \lambda_{A(\varphi)}(P;v) + C_2 \lambda_{ \partial_{W \times V} }(P,q;v) + C_4(v)\] then there exists a point \(Q \in W(K)\) satisfying \(\varphi(Q) = q\) and \[\delta_W(P,Q;v) \geq \delta_V (\varphi(P),q;v) -(d-1)\lambda_{A(\varphi)}(P;v)- C_3 \lambda_{\partial(W \times V )}(P, q; v) - C_5(v).\]
\item[(b)]If we take \(C_4\) to be an appropriate positive real number, instead of an \(\mathcal{M}_K\) constant, and if we also assume that \(P \notin A(\varphi)\), then the point \(Q\) in (a) is unique.
\end{enumerate}
Remark: The arithmetic distribution relation and the inverse function theorem have been used to study integral points in the following situations:
\begin{enumerate}
\item[(1)] To find uniform height estimates while working with the étale map \([n] \colon A \longrightarrow A\) on an Abelian scheme \(A \longrightarrow T\) over a base variety \(T\).
\item[(2)] To find an analogous of Siegel's theorem while working with Iterates \(f^n\) of a rational map \(f \colon \mathbb{P}^1 \longrightarrow \mathbb{P}^1\) of degree at least two.
\end{enumerate}
The first version (version \(1\)) of the inverse function theorem works simultaneously over several places. In the next version the authors present a stronger result working only over a complete field \(K\). By working over a complete field, the exponents or coefficients of the inverse function theorem are improved from \((d,d-1)\) to \((2,1)\).
Theorem: (Inverse function theorem version \(2\)). Let \( (K, | . |)\) be a complete field. Let \(W, V\) be smooth quasi-projective varieties over \(K\), and let \(\varphi \colon W \longrightarrow V\) be a generically finite generically étale morphism. Let \(E \subset W\) be the closed subscheme defined by the \(0\)-th fitting ideal sheaf of \(\Omega_{W/V}\). Fix arithmetic distance functions \(\delta_W\), \(\delta_V\), a local height function \(\lambda_E\), and a boundary function \(\lambda_{\partial V}\). Let \(B \subset W(K)\) be a bounded subset. Then there are constants \(C_{36}, C_{37}, C_{38}, C_{39} > 0\) and a bounded subset \(\tilde{B} \subset W (K)\) containing \(B\) such that for all \(P \in B\) and \(q \in V(K)\) satisfying \[P \notin E \quad \text{and} \quad \delta_V(\varphi(P),q) \geq 2 \lambda_E(P) + C_{36} \lambda_{\partial V}(q) + C_{37},\] there is a unique \(Q \in \tilde{B}\) satisfying \[\varphi(Q)=q \quad \text{and} \quad \delta_W(P,Q) \geq \delta_V(\varphi(P),q) - \lambda_E(P)-C_{38} \lambda_{\partial V}(q) - C_{39}.\]
Remark: Both versions of the inverse function theorem are suitable to prove results analogous to the continuity of the roots for polynomials. For example from the second version, we could get the following result. Let \( (K, | . |)\) be a complete field. Let \(D \in \mathbb{R}_{>0}\) and \(n \in \mathbb{Z}_{>0}\). Then there are positive constants \(C_{40},C_{41} > 0\) such that the following holds. Suppose that:
\begin{itemize}
\item \(f,g \in K[t]\) are monic polynomials of degree \(n\);
\item The Gauss norms \(|f | \leq D\) and \(|g| \leq D\);
\item There is an \(\alpha \in K\) such that \(f(\alpha) = 0\) and \(|f-g| \leq C_{40} |f''\alpha)|\).
\end{itemize}
Then there is \(\beta \in K\) such that \[g(\beta) = 0 \quad \text{and} \quad |\alpha - \beta||f'(\alpha)| \leq C_{41} |f - g|.\]
Remark: The second version of the inverse function theorem is based on a higher dimensional version of Newton's method. The point \(Q\), will be obtained as limit of a Cauchy sequence of points \(Q_0=P, Q_1, Q_2 \dots\).Spectrum of a linear differential equation with constant coefficientshttps://zbmath.org/1472.120062021-11-25T18:46:10.358925Z"Azzouz, Tinhinane A."https://zbmath.org/authors/?q=ai:azzouz.tinhinane-aThe author computes the spectrum, in the sense of [\textit{V. G. Berkovich}, Spectral theory and analytic geometry over non-Archimedean fields. Providence, RI: American Mathematical Society (1990; Zbl 0715.14013), Chapter 7], of an ultrametric linear differential operator with constant coefficients, defined over an affinoid domain of the analytic affine line. Its spectrum is a finite union of either closed disks or topological closures of open disks. It is shown to satisfy a continuity property.Global properties of eigenvalues of parametric rank one perturbations for unstructured and structured matriceshttps://zbmath.org/1472.150122021-11-25T18:46:10.358925Z"Ran, André C. M."https://zbmath.org/authors/?q=ai:ran.andre-c-m"Wojtylak, Michał"https://zbmath.org/authors/?q=ai:wojtylak.michalThe paper contributes to eigenvalue perturbation theory. The authors previously considered perturbations of the form \(A+tuv^*\), where \(t\in \mathbb R\), in [\textit{A. C. M. Ran} and \textit{M. Wojtylak}, Linear Algebra Appl. 437, No. 2, 589--600 (2012; Zbl 1247.15009)]. Here, they also consider angular perturbations \(A+e^{i\theta}uv^*\) where \(\theta\in [0,2\pi)\). Connecting these two cases, the authors prove new results on the global behaviour of the eigenvalues. The two main problems under consideration are defining the eigenvalues as functions of the parameter \(\tau\) (where either \(\tau\in \mathbb R\) or \(\tau=e^{i\theta}\)), in which case the eigenvalues can be defined as analytic functions, and considering the case of large \(|\tau|\rightarrow \infty\), where the eigenvalues are studied through the roots of the polynomial \(m_A(\lambda)-\tau p_{uv}(\lambda)\), where \(m_A(\lambda)\) is the minimal polynomial of \(A\) and \(p_{uv}(\lambda)=v^*m_A(\lambda)(\lambda I_n-A)^{-1}u\). The results are applied to various families of matrices.An alternative canonical form for quaternionic \(H\)-unitary matriceshttps://zbmath.org/1472.150162021-11-25T18:46:10.358925Z"Groenewald, G. J."https://zbmath.org/authors/?q=ai:groenewald.gilbert-j"Janse van Rensburg, D. B."https://zbmath.org/authors/?q=ai:janse-van-rensburg.dawie-b"Ran, A. C. M."https://zbmath.org/authors/?q=ai:ran.andre-c-mLet $H$ be the quaternion ring. Let $(A,H)$ be a pair of matrices over $H$. The matrix $A$ is \(H\)-unitary if $H=H^*$ is invertible and $A^*HA=H$. The authors find an invertible matrix $S$ such that the transformations from $(A,H)$ to $(S^{-1}AS,S^*HS)$ brings the matrix A in Jordan form and simultaneously brings $H$ into a canonical form. The authors found inspiration for their work in the results in [the authors, Oper. Matrices 10, No. 4, 739--783 (2016; Zbl 1360.15015); Oper. Theory: Adv. Appl. 271, 269--290 (2018; Zbl 07137673)]. Their main goal is the study of a quaternionic pair of matrices, a topic that is still under development (see [\textit{L. Rodman}, Topics in quaternion linear algebra. Princeton, NJ: Princeton University Press (2014; Zbl 1304.15004)]).Rank one perturbations of matrix pencilshttps://zbmath.org/1472.150182021-11-25T18:46:10.358925Z"Dodig, Marija"https://zbmath.org/authors/?q=ai:dodig.marija"Stošić, Marko"https://zbmath.org/authors/?q=ai:stosic.markoUsing some of their earlier results (see for example [the first author, Linear Algebra Appl. 438, No. 8, 3155--3173 (2013; Zbl 1269.15029)]), the authors completely resolve the open problem of describing all possible Kronecker invariants of an arbitrary matrix pencil under rank one perturbations. Their solution is explicit and constructive, and is valid for arbitrary pencils. They combine results on one-row matrix pencil completions with combinatorial results on double generalized majorization, and develop some new techniques.Exact conditions for preservation of the partial indices of a perturbed triangular \(2 \times 2\) matrix functionhttps://zbmath.org/1472.150192021-11-25T18:46:10.358925Z"Adukov, Victor M."https://zbmath.org/authors/?q=ai:adukov.viktor-mikhailovich|adukov.viktor-michailovich"Mishuris, Gennady"https://zbmath.org/authors/?q=ai:mishuris.gennady-s"Rogosin, Sergei V."https://zbmath.org/authors/?q=ai:rogosin.sergei-vSummary: The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial.On the rank and the approximation of symmetric tensorshttps://zbmath.org/1472.150362021-11-25T18:46:10.358925Z"Rodríguez, Jorge Tomás"https://zbmath.org/authors/?q=ai:rodriguez.jorge-tomasSummary: In this work we study different notions of ranks and approximation of tensors. We consider the tensor rank, the nuclear rank and we introduce the notion of symmetric decomposable rank, a notion of rank defined only on symmetric tensors. We show that when approximating symmetric tensors, using the symmetric decomposable rank has some significant advantages over the tensor rank and the nuclear rank.Optimal rank-1 Hankel approximation of matrices: Frobenius norm and spectral norm and Cadzow's algorithmhttps://zbmath.org/1472.150412021-11-25T18:46:10.358925Z"Knirsch, Hanna"https://zbmath.org/authors/?q=ai:knirsch.hanna"Petz, Markus"https://zbmath.org/authors/?q=ai:petz.markus"Plonka, Gerlind"https://zbmath.org/authors/?q=ai:plonka.gerlindSummary: We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm.Strong Novikov conjecture for low degree cohomology and exotic group \(\mathrm{C}^\ast\)-algebrashttps://zbmath.org/1472.190052021-11-25T18:46:10.358925Z"Antonini, Paolo"https://zbmath.org/authors/?q=ai:antonini.paolo"Buss, Alcides"https://zbmath.org/authors/?q=ai:buss.alcides"Engel, Alexander"https://zbmath.org/authors/?q=ai:engel.alexander"Siebenand, Timo"https://zbmath.org/authors/?q=ai:siebenand.timoLet \(G\) be a discrete group, \(\Lambda^*(G) \subset H^*(BG;\mathbb Q)\) the subring generated by the rational cohomology classes of degree at most two, and let \(\mathrm{ch}: K_*(BG) \to H_*(BG;\mathbb Q)\) be the homological Chern character from the \(K\)-homology to the homology of the classifying space \(BG\) of \(G\).
It was shown in [\textit{B. Hanke} and \textit{T. Schick}, Geom. Dedicata 135, 119--127 (2008; Zbl 1149.19006)] that if for \(h\in K_*(BG)\) there exists \(c \in\Lambda^*(G)\) with \(\langle c, \mathrm{ch}(h)\rangle\neq 0\) then \(h\) is not mapped to zero under the assembly map \(K_*(BG) \to K_*(C^*_{\max}(G)) \otimes\mathbb R\).
The paper under review shows that the maximal group \(C^*\)-algebra here can be replaced by a smaller one, namely, by the exotic group \(C^*\)-algebra \(C^*_\epsilon (G)\), which is obtained from the so-called minimal exact and strongly Morita compatible crossed product functor introduced in [\textit{P. Baum} et al., Ann. \(K\)-Theory 1, No. 2, 155--208 (2016; Zbl 1331.46064)].An algebraic approach to the Weyl groupoidhttps://zbmath.org/1472.220012021-11-25T18:46:10.358925Z"Bice, Tristan"https://zbmath.org/authors/?q=ai:bice.tristan-matthewThe Kumjian-Renault Weyl groupoid construction and the Lawson-Lenz version of Exel's tight groupoid construction are unified by utilising only a weak algebraic fragment of the \(C^*\)-algebra structure, that is, its *-semigroup reduct. The author also prove that local compactness is still valid in general classes of *-rings.Contractive iterated function systems enriched with nonexpansive mapshttps://zbmath.org/1472.280122021-11-25T18:46:10.358925Z"Strobin, Filip"https://zbmath.org/authors/?q=ai:strobin.filipSummary: Motivated by a recent paper of Leśniak and Snigireva [\textit{Iterated function systems enriched with symmetry}, preprint], we investigate the properties of the semiattractor \(A_{\mathcal{F}\cup\mathcal{G}}^*\) of an IFS \(\mathcal{F}\) enriched by some other IFS \(\mathcal{G}\). We show that in natural cases, the semiattractor \(A_{\mathcal{F}\cup\mathcal{G}}^*\) is in fact the attractor of certain IFSs related naturally with the IFSs \(\mathcal{F}\) and \(\mathcal{G}\). We also give an example when \(A_{\mathcal{F}\cup\mathcal{G}}^*\) is not compact, yet still being the attractor of considered related IFSs. Finally, we use presented machinery to prove that the so called \textit{lower transition attractors} due to Vince are semiattractors of enriched IFSs.Generalized integration operators on Hardy spaceshttps://zbmath.org/1472.300252021-11-25T18:46:10.358925Z"Chalmoukis, Nikolaos"https://zbmath.org/authors/?q=ai:chalmoukis.nikolaosLet $H^p$, with a positive exponent $p$, be the Hardy space of analytic function in the unit disc, i.e., functions $f$ holomorphic in $\mathbb D$ such that $$\|f\|_p^p:=\frac1{2\pi}\sup_{0<r<1}\int_0^{2\pi}|f(re^{i\theta})|^p\,d\theta<\infty.$$
Let $I$ be the integration operator, i.e., $If(z):=\int_0^zf(t)\,dt$. Fixing now an analytic symbol $g$ and a sequence of coefficients $a=(a_1,\dots,a_{n-1})\in \mathbb C^{n-1}$, define the generalized integration operator $T_{g,a}$ by $$T_{g,a}f(z):=I^n\left(fg^{(n)}+a_1f'g^{(n-1)}+\cdots+a_{n_1}f^{(n-1)}g'\right)(z).$$
The paper gives sufficient and necessary conditions on $g$ for the boundedness and compactness of the operator $T_{g,a}$. Indeed, the author obtains
\begin{enumerate}
\item $T_{g,a}$ is bounded from $H^p$ to itself if and only if $g\in \operatorname{BMOA}$;
\item $T_{g,a}$ is compact from $H^p$ to itself if and only if $g\in \operatorname{VMOA}$.
\end{enumerate}
Moreover, the author also proves that: Let $0<p<q<\infty$ and $a\in\mathbb C^{n-1}$. If $g\in H^s$, where $\frac1s=\frac1q-\frac1p$, then $T_{g,a}$ is bounded from $H^p$ to $H^q$. In the special case that $n=2$ and $a=0$, the boundedness of $T_{g,a}$ from $H^p$ to $H^q$ implies $g\in H^s$.Correction to: ``On self-adjointness of symmetric diffusion operators''https://zbmath.org/1472.310112021-11-25T18:46:10.358925Z"Robinson, Derek W."https://zbmath.org/authors/?q=ai:robinson.derek-wFrom the text: Due to a typesetting error, the sentence just below equation (31) has been published incorrectly in the original publication [the author, ibid. 21, No. 1, 1089--1116 (2021; Zbl 1469.31029)]. The correct sentence is given.A short proof of the symmetric determinantal representation of polynomialshttps://zbmath.org/1472.320012021-11-25T18:46:10.358925Z"Stefan, Anthony"https://zbmath.org/authors/?q=ai:stefan.anthony"Welters, Aaron"https://zbmath.org/authors/?q=ai:welters.aaron-tA recent theorem [\textit{J. W. Helton} et al., J. Funct. Anal. 240, No. 1, 105--191 (2006; Zbl 1135.47005)] asserts that a real, multivariate polynomial can be written as the determinant of an affine pencil of real, symmetric matrices. The proof by Helton et al. was derived from an elaborate construct of non-commutative algebra, in its turn inspired by control system theory. The note by Stefan and Welters offers an elementary proof of the same result. The authors ingeniously exploit Schur complement identities, allowing a generalization of the main result to certain fields of finite characteristic. The strong algorithmic flavor of the proof may appeal to wider groups of scientists touching numerical matrix analysis in their studies.Applications of spectral theory to special functionshttps://zbmath.org/1472.330062021-11-25T18:46:10.358925Z"Koelink, Erik"https://zbmath.org/authors/?q=ai:koelink.erikSummary: Many special functions are eigenfunctions to explicit operators, such as difference and differential operators, which is in particular true for the special functions occurring in the Askey scheme, its q-analogue and extensions. The study of the spectral properties of such operators leads to explicit information for the corresponding special functions. We discuss several instances of this application, involving orthogonal polynomials and their matrix-valued analogues.
For the entire collection see [Zbl 1460.33001].The Heun-Racah and Heun-Bannai-Ito algebrashttps://zbmath.org/1472.330112021-11-25T18:46:10.358925Z"Bergeron, Geoffroy"https://zbmath.org/authors/?q=ai:bergeron.geoffroy"Crampé, Nicolas"https://zbmath.org/authors/?q=ai:crampe.nicolas"Tsujimoto, Satoshi"https://zbmath.org/authors/?q=ai:tsujimoto.satoshi"Vinet, Luc"https://zbmath.org/authors/?q=ai:vinet.luc"Zhedanov, Alexei"https://zbmath.org/authors/?q=ai:zhedanov.alexei-sSummary: The Heun-Racah and Heun-Bannai-Ito algebras are introduced. Specializations of these algebras are seen to be realized by the operators obtained by applying the algebraic Heun construct to the bispectral operators of the Racah and Bannai-Ito polynomials. The study supplements the results on the Heun-Askey-Wilson algebra and completes the description of the Heun algebras associated with the polynomial families at the top of the Askey scheme, its q-analog, and the Bannai-Ito one.
{\copyright 2020 American Institute of Physics}Implicit fractional differential equation involving \(\psi\)-Caputo with boundary conditionshttps://zbmath.org/1472.340082021-11-25T18:46:10.358925Z"Abdellatif, Boutiara"https://zbmath.org/authors/?q=ai:abdellatif.boutiara"Benbachir, Maamar"https://zbmath.org/authors/?q=ai:benbachir.maamarThis paper deals with the existence and uniqueness of solutions for boundary-value problems of the nonlinear \(\psi\)-Caputo fractional differential equations
\[
\begin{aligned} ^CD^{\alpha, \psi}_{a^+}u(t) &= f(t, u(t),^CD^{\alpha, \psi}_{a^+}u(t)), \quad t\in [a, T],\\
u(T) &= \lambda u(\eta). \end{aligned}
\]
where \(^CD^{\alpha, \psi}_{a^+}\) is the \(\psi\)-Caputo fractional derivative of order \(\alpha \in (0, 1]\), \(f : [a, T]\times \mathbb{R} \to \mathbb{R}\) is a given continuous function, \(\lambda\) is a real constant and \(\eta\in (a, T).\) The results are obtained by using standard fixed point theorems. Further some types of fractional Ulam-Hyers stability are established. An example is given to illustrate the existence results.Existence and uniqueness of tripled fixed points for mixed monotone operators with perturbations and applicationhttps://zbmath.org/1472.340092021-11-25T18:46:10.358925Z"Afshari, Hojjat"https://zbmath.org/authors/?q=ai:afshari.hojjat"Kheiryan, Alireza"https://zbmath.org/authors/?q=ai:kheiryan.alireza(no abstract)Existence of solution to fractional differential equation with fractional integral type boundary conditionshttps://zbmath.org/1472.340102021-11-25T18:46:10.358925Z"Ali, Anwar"https://zbmath.org/authors/?q=ai:ali.anwar"Sarwar, Muhammad"https://zbmath.org/authors/?q=ai:sarwar.muhammad"Zada, Mian Bahadur"https://zbmath.org/authors/?q=ai:zada.mian-bahadur"Shah, Kamal"https://zbmath.org/authors/?q=ai:shah.kamalSummary: This paper is devoted by developing sufficient condition required for the existence of solution to a nonlinear fractional order boundary value problem
\[
D^{\gamma} \mathfrak{u} (\ell ) = \psi (\ell , \mathfrak{u} (\lambda \ell )), \ell \in \mathfrak{Z} = [0 , 1],
\]
with fractional integral boundary conditions
\[
\mathfrak{p}_1 \mathfrak{u} (0) + \mathfrak{q}_1 \mathfrak{u} (1) = \frac{1}{\Gamma (\gamma )} \int_0^1 (1 - \rho )^{\gamma - 1} g_1 (\rho , \mathfrak{u} (\rho )) d \rho ,
\]
and
\[
\mathfrak{p}_2 \mathfrak{u}^{\prime} (0) + \mathfrak{q}_2 \mathfrak{u}^{\prime} (1) = \frac{1}{\Gamma (\gamma )} \int_0^1 (1 - \rho )^{\gamma - 1} g_2 (\rho , \mathfrak{u} (\rho )) d \rho ,
\]
where \(\gamma \in (1, 2], 0 < \lambda < 1, D\) denotes the Caputo fractional derivative (in short CFD), \(\psi, g_1, g_2 : \mathfrak{Z} \times \mathfrak{R} \to \mathfrak{R}\) are continuous functions and \(\mathfrak{p}_i, \mathfrak{q}_i (i = 1, 2)\) are positive real numbers. Using topological degree theory sufficient results are constructed for the existence of at least one and unique solution to the concerned problem. For the validity of our result, a concrete example is presented in the end.A coupled system of nonlinear Caputo-Hadamard Langevin equations associated with nonperiodic boundary conditionshttps://zbmath.org/1472.340142021-11-25T18:46:10.358925Z"Matar, Mohammed M."https://zbmath.org/authors/?q=ai:matar.mohammed-m"Alzabut, Jehad"https://zbmath.org/authors/?q=ai:alzabut.jehad-o"Jonnalagadda, Jagan Mohan"https://zbmath.org/authors/?q=ai:jonnalagadda.jaganmohanSummary: In this paper, we study the coupled system of nonlinear Langevin equations involving Caputo-Hadamard fractional derivative and subject to nonperiodic boundary conditions. The existence, uniqueness, and stability in the sense of Ulam are established for the proposed system. Our approach is based on the features of the Hadamard fractional derivative, the implementation of fixed point theorems, and the employment of Urs's stability approach. An example is introduced to facilitate the understanding of the theoretical findings.Existence theory and Ulam's stabilities of fractional Langevin equationhttps://zbmath.org/1472.340152021-11-25T18:46:10.358925Z"Rizwan, Rizwan"https://zbmath.org/authors/?q=ai:rizwan.rizwan"Zada, Akbar"https://zbmath.org/authors/?q=ai:zada.akbarSummary: In this paper, we consider fractional Langevin equation and derive a formula of solutions for fractional Langevin equation involving two Caputo fractional derivatives. Secondly, we implement the concept of Ulam-Hyers as well as Ulam-Hyers-Rassias stability. Then, we choose Generalized Diaz-Margolis's fixed point approach to derive Ulam-Hyers as well as Ulam-Hyers-Rassias stability results for our proposed model, over generalized complete metric space. We give several examples which support our main results.Correction to: ``Solvability and stability of the inverse Sturm-Liouville problem with analytical functions in the boundary condition''https://zbmath.org/1472.340292021-11-25T18:46:10.358925Z"Bondarenko, Natalia P."https://zbmath.org/authors/?q=ai:bondarenko.natalia-pCorrection to the author's paper [ibid. 43, No. 11, 7009--7021 (2020; Zbl 1456.34014)].Inverse problems for Sturm-Liouville operators on a compact equilateral graph by partial nodal datahttps://zbmath.org/1472.340302021-11-25T18:46:10.358925Z"Wang, Yu Ping"https://zbmath.org/authors/?q=ai:wang.yuping.1"Shieh, Chung-Tsun"https://zbmath.org/authors/?q=ai:shieh.chung-tsunSummary: Partial inverse nodal problems for Sturm-Liouville operators on a compact equilateral star graph are investigated in this paper. Uniqueness theorems from partial twin-dense nodal subsets in interior subintervals or arbitrary interior subintervals having the central vertex are proved. In particular, we posed and solved a new type partial inverse nodal problems for the Sturm-Liouville operator on the compact equilateral star graph.Ambarzumyan theorems for Dirac operatorshttps://zbmath.org/1472.340322021-11-25T18:46:10.358925Z"Yang, Chuan-fu"https://zbmath.org/authors/?q=ai:yang.chuanfu"Wang, Feng"https://zbmath.org/authors/?q=ai:wang.feng.3|wang.feng.2|wang.feng.4|wang.feng.1"Huang, Zhen-you"https://zbmath.org/authors/?q=ai:huang.zhenyouSummary: We consider the inverse eigenvalue problems for stationary Dirac systems with differentiable self-adjoint matrix potential. The theorem of Ambarzumyan for a Sturm-Liouville problem is extended to Dirac operators, which are subject to separation boundary conditions or periodic (semi-periodic) boundary conditions, and some analogs of Ambarzumyan's theorem are obtained. The proof is based on the existence and extremal properties of the smallest eigenvalue of corresponding vectorial Sturm-Liouville operators, which are the second power of Dirac operators.Existence of solutions of \(\alpha \in(2,3]\) order fractional three-point boundary value problems with integral conditionshttps://zbmath.org/1472.340382021-11-25T18:46:10.358925Z"Mahmudov, N. I."https://zbmath.org/authors/?q=ai:mahmudov.nazim-idrisoglu"Unul, S."https://zbmath.org/authors/?q=ai:unul.sinemSummary: Existence and uniqueness of solutions for \(\alpha \in(2,3]\) order fractional differential equations with three-point fractional boundary and integral conditions involving the nonlinearity depending on the fractional derivatives of the unknown function are discussed. The results are obtained by using fixed point theorems. Two examples are given to illustrate the results.Multiple solutions for mixed boundary value problems with \(\varphi\)-Laplacian operatorshttps://zbmath.org/1472.340402021-11-25T18:46:10.358925Z"Dallos Santos, Dionicio Pastor"https://zbmath.org/authors/?q=ai:santos.dionicio-pastor-dallosSummary: Using Leray-Schauder degree theory and the method of upper and lower solutions we establish existence and multiplicity of solutions for problems of the form \[\begin{aligned} (\varphi(u'))' = f(t,u,u') \\ u(0)= u(T)=u'(0), \end{aligned}\] where \(\varphi\) is an increasing homeomorphism such that \(\varphi(0)=0\), and \(f\) is a continuous function.Existence of positive solutions for multi-point semi-positive boundary value problemshttps://zbmath.org/1472.340452021-11-25T18:46:10.358925Z"Su, Hua"https://zbmath.org/authors/?q=ai:su.huaThe author considers the existence of positive solutions for the nonlinear multi-point boundary value problem \[ - \big(u''(t) + a(t)u'(t) \big ) = \lambda f(t, u) + \mu g(t, u), \, t \in (0, 1), \] \[ u'(0) = 0, \, \, u(1) = \sum_{i=1}^k \alpha_i u(\eta_i) - \sum_{i = k+1}^{m-2} \alpha_i u(\eta_i). \] Using cone theoretic techniques, the author shows the existence of at least one positive solution when \(\lambda\) and \(\mu\) are small under various suitable conditions on \(f\) and \(g\). They also establish the existence of at least three positive solutions under other conditions on \(f\) and \(g\) when \(\lambda\) and \(\mu\) are small.Two classes of conformable fractional Sturm-Liouville problems: theory and applicationshttps://zbmath.org/1472.340482021-11-25T18:46:10.358925Z"Mortazaasl, Hamid"https://zbmath.org/authors/?q=ai:mortazaasl.hamid"Akbarfam, Ali Asghar Jodayree"https://zbmath.org/authors/?q=ai:akbarfam.aliasghar-jodayreeSummary: In this paper, we investigate in more detail some useful theorems related to conformable fractional derivative (CFD) and integral and introduce two classes of conformable fractional Sturm-Liouville problems (CFSLPs): namely, regular and singular CFSLPs. For both classes, we study some of the basic properties of the Sturm-Liouville theory. In the class of \(r\)-CFSLPs, we discuss two types of CFSLPs which include left- and right-sided CFDs, each of order \(\alpha \in (n,n+1]\), and prove properties of the eigenvalues and the eigenfunctions in a certain Hilbert space. Also, we apply a fixed-point theorem for proving the existence and uniqueness of the eigenfunctions. As an operator for the class of \(s\)-CFSLPs, we first derive two fractional types of the hypergeometric differential equations of order \(\alpha \in (0,1]\) and obtain their analytical eigensolutions as Gauss hypergeometric functions. Afterwards, we define the conformable fractional Legendre polynomial/functions (CFLP/Fs) as Jacobi polynomial and investigate their basic properties. Moreover, the conformable fractional integral Legendre transforms (CFILTs) based on CFLP/Fs-I and -II are introduced, and using these new transforms, an effective procedure for solving explicitly certain ordinary and partial conformable fractional differential equations (CFDEs) are given. Finally, as a theoretical application, some fractional diffusion equations are solved.Fractional boundary value problem on the half-linehttps://zbmath.org/1472.340512021-11-25T18:46:10.358925Z"Khamessi, Bilel"https://zbmath.org/authors/?q=ai:khamessi.bilelThe author investigates a nonlinear boundary value problem of a fractional differential equation. Using results on the functions in Karamata's classes and the estimate on Green's function, the existence and the uniqueness of a positive solution of this equation are studied. Finally, a description of the global behavior of this solution is given.On heteroclinic solutions for BVPs involving \(\phi\)-Laplacian operators without asymptotic or growth assumptionshttps://zbmath.org/1472.340522021-11-25T18:46:10.358925Z"Minhós, Feliz"https://zbmath.org/authors/?q=ai:minhos.feliz-manuelSummary: In this paper we consider the second order discontinuous equation in the real line, \[\begin{aligned}(\phi(a(t)u'(t)))' &=f(t,u(t),u'(t)), \ \mathrm{a}.\mathrm{e} \ t \in \mathbb R \\ u(-\infty) &= A, u(+\infty)=B \end{aligned}\] with \(\phi\) an increasing homeomorphism such that \(\phi (0)=0\) and \(\phi (\mathbb R)=\mathbb R, a \in C(\mathbb R)\) with \(a(t)>0\), for \(t \in \mathbb R, f: \mathbb R^3 \to \mathbb R\), a \(L^1\)-Carathéodory function and \(A, B \in \mathbb R\) verifying an adequate relation. We remark that the existence of heteroclinic solutions is obtained without asymptotic or growth assumptions on the nonlinearities \(\phi\) and \(f\). Moreover, as far as we know, our main result is even new when \(\phi(y)=y\), that is, for the equation \[(a(t)u'(t))'=f(t,u(t),u'(t)),\ \mathrm{a}.\mathrm{e} \ t \in \mathbb R. \]Multiple periodic solutions for one-sided sublinear systems: a refinement of the Poincaré-Birkhoff approachhttps://zbmath.org/1472.340762021-11-25T18:46:10.358925Z"Dondè, Tobia"https://zbmath.org/authors/?q=ai:donde.tobia"Zanolin, Fabio"https://zbmath.org/authors/?q=ai:zanolin.fabioThe paper investigates the existence of periodic solutions, both harmonic and subharmonic, for planar Hamiltonian systems of the type \[ x' = h(y), \qquad y' = - a(t)g(x), \] where \(a(t)\) is a sign-changing periodic function and at least one of \(g\) and \(h\) is bounded on \(\mathbb{R}^-\) or \(\mathbb{R}^+\).
At first, by further assuming the global continuability for the solutions, a multiplicity result is proved via the Poincaré-Birkhoff theorem; as usual, solutions are distinguished via their nodal properties. Then, a refinement of this result, obtained with the theory of topological horseshoses, is presented; here, the assumption of global continuability is replaced by a largeness condition on the weight function \(a(t)\) in its negativity intervals. In this latter case, the existence of chaotic dynamics is also ensured.
Applications of the results are finally described for a Minkowksi-curvature equation like \[ \left( \frac{u'}{\sqrt{1-(u')^2}} \right)' + a(t) g(u) = 0 \] as well as for the equation, with exponential nonlinearity, \[ u'' + k(t)e^u = p(t). \]Exact multiplicity and stability of periodic solutions for Duffing equation with bifurcation methodhttps://zbmath.org/1472.340772021-11-25T18:46:10.358925Z"Liang, Shuqing"https://zbmath.org/authors/?q=ai:liang.shuqingSummary: Under some \(L^p\)-norms \((p\in [1,\infty ])\) assumptions for the derivative of the restoring force, the exact multiplicity and the stability of \(2\pi\)-periodic solutions for Duffing equation are considered. The nontrivial \(2\pi\)-periodic solutions of it are positive or negative, and the bifurcation curve of it is a unique reversed \(S\)-shaped curve. The class of the restoring force is extended, comparing with the class of \(L^{\infty }\)-norm condition. The proof is based on the global bifurcation theorem, topological degree and the estimates for periodic eigenvalues of Hill's equation by \(L^p\)-norms\((p\in [1,\infty ])\).A new fixed point theorem and periodic solutions of nonconservative weakly coupled systemshttps://zbmath.org/1472.340782021-11-25T18:46:10.358925Z"Liu, Chunlian"https://zbmath.org/authors/?q=ai:liu.chunlian"Qian, Dingbian"https://zbmath.org/authors/?q=ai:qian.dingbianSummary: In this paper, we prove a new fixed point theorem for the coupling of twist conditions and Poincaré-Bohl type conditions, as its applications we prove the existence of periodic solutions for various nonconservative mixed type weakly coupled systems.Non-resonance and double resonance for a planar system via rotation numbershttps://zbmath.org/1472.340792021-11-25T18:46:10.358925Z"Liu, Chunlian"https://zbmath.org/authors/?q=ai:liu.chunlian"Qian, Dingbian"https://zbmath.org/authors/?q=ai:qian.dingbian"Torres, Pedro J."https://zbmath.org/authors/?q=ai:torres.pedro-joseThe authors consider a general planar periodic system and propose two existence results.
In the first one they compare the nonlinearity with two positively homogeneous functions with ``rotation numbers'' larger than some \(n\) and smaller than \(n+1\). They thus prove that the system has a periodic solution, by the use of the Poincaré-Bohl fixed point theorem. This is a generalization of some classical ``nonresonance'' results.
In the second one the above two functions have rotation numbers exactly equal to \(n\) and \(n+1\). Then, in order to avoid possible resonance phenomena, they add two Landesman-Lazer conditions, and they prove again the existence of a periodic solution.
The proofs involve delicate analysis in the phase-plane, in order to precisely estimate the rotational properties of the solutions.Periodic solutions for a singular Liénard equation with indefinite weighthttps://zbmath.org/1472.340802021-11-25T18:46:10.358925Z"Lu, Shiping"https://zbmath.org/authors/?q=ai:lu.shiping"Xue, Runyu"https://zbmath.org/authors/?q=ai:xue.runyuIn this paper, the authors study the following singular Liénard equation \[ x''(t)+f(x(t))x'(t)+\frac{\alpha(t)}{x^\mu(t)}= h(t),\tag{1} \] where \(f\in C((0, +\infty), \mathbb{R})\) may have a singularity at \(x=0,\, \mu\in(0, +\infty)\) is a constant, \(\alpha\) and \(h\) are \(T\)-periodic functions with \(\alpha,\, h \in L^1 ([0, T], \mathbb{R}).\) The weight function \(\alpha\) may change sign on \([0, T].\) A new method for estimating a priori bounds of all possible positive \(T\)-periodic solutions is obtained. By using a continuation theorem of Mawhin's coincidence degree theory, some new results on the existence of positive periodic solutions for the equation (1) are established.Existence and uniqueness of globally attractive positive almost periodic solution in a predator-prey dynamic system with Beddington-DeAngelis functional responsehttps://zbmath.org/1472.341002021-11-25T18:46:10.358925Z"Wu, Wenquan"https://zbmath.org/authors/?q=ai:wu.wenquanSummary: This paper is concerned with a predator-prey system with Beddington-DeAngelis functional response on time scales. By using the theory of exponential dichotomy on time scales and fixed point theory based on monotone operator, some simple conditions are obtained for the existence of at least one positive (almost) periodic solution of the above system. Further, by means of Lyapunov functional, the global attractivity of the almost periodic solution for the above continuous system is also investigated. The main results in this paper extend, complement, and improve the previously known result. And some examples are given to illustrate the feasibility and effectiveness of the main results.Corrigendum to: ``Existence and stability results for semilinear systems of impulsive stochastic differential equations with fractional Brownian motion''https://zbmath.org/1472.341122021-11-25T18:46:10.358925Z"Blouhi, T."https://zbmath.org/authors/?q=ai:blouhi.tayeb"Caraballo, T."https://zbmath.org/authors/?q=ai:caraballo.tomas"Ouahab, A."https://zbmath.org/authors/?q=ai:ouahab.abdelghaniSummary: In this paper we correct an error made in our paper [ibid. 34, No. 5, 792--834 (2016; Zbl 1380.34091)]. In fact, in this corrigendum we present the correct hypotheses and results, and highlight that the results can be proved using the same method used in the original work. The main feature is that we used a result which has been proved only when the diffusion term does not depend on the unknown.Boundary value problems for fractional-order differential inclusions in Banach spaces with nondensely defined operatorshttps://zbmath.org/1472.341182021-11-25T18:46:10.358925Z"Obukhovskii, Valeri"https://zbmath.org/authors/?q=ai:obukhovskii.valeri"Zecca, Pietro"https://zbmath.org/authors/?q=ai:zecca.pietro"Afanasova, Maria"https://zbmath.org/authors/?q=ai:afanasova.mariaA general nonlocal boundary value problem for a fractional-order semilinear differential inclusion in a separable Banach space \(E\) of a fractional order \(0<q<1,\)
\[
\begin{cases}
^{C}D^{q}x(t)\in Ax(t)+F(t,x(t)), ~~ t\in [0,T],\\
\mathcal{Q}(x)\in \mathcal{S}(x)
\end{cases}
\]
is considered, where \(A: D(A)\subset E\to E\) is a Hille-Yosida operator generating a locally Lipschitz integrated semigroup, \(F: [0,T]\times E\to Kv(E),\) is a nonlinear multivalued map, \(\mathcal{Q}: C([0,T]; \overline{D(A)})\to \overline{D(A)}\) is a bounded linear operator and \(\mathcal{S}: C([0,T]; \overline{D(A)})\to K(\overline{D(A)})\) is a completely upper semicontinuous quasi-\(R_{\delta}\)-multioperator. Here \(Kv(E), K(E)\) denote the collections of all nonempty compact convex and, respectively, compact subsets of \(E.\) By using the theory of integrated semigroups, fractional calculus and the fixed point theory of condensing multivalued maps, the existence of mild solutions is proved. Some important particular cases including a nonlocal Cauchy problem, periodic and anti-periodic boundary value problems are presented.Partial approximate controllability of fractional systems with Riemann-Liouville derivatives and nonlocal conditionshttps://zbmath.org/1472.341192021-11-25T18:46:10.358925Z"Haq, Abdul"https://zbmath.org/authors/?q=ai:haq.abdul"Sukavanam, N."https://zbmath.org/authors/?q=ai:sukavanam.nagarajanSummary: In this work, we investigate the partial approximate controllability of nonlocal Riemann-Liouville fractional systems with integral initial conditions in Hilbert spaces without assuming the Lipschitz continuity of nonlinear function. We also exclude the conditions of Lipschitz continuity and compactness for the nonlocal function. The existence results are derived using Schauder fixed point theorem, then the partial approximate controllability result is proved by assuming that the associated linear system is partial approximately controllable for \(\varphi =0\), where \(\varphi\) is nonlocal function. Lastly, an example is provided to apply our results.Twin semigroups and delay equationshttps://zbmath.org/1472.341202021-11-25T18:46:10.358925Z"Diekmann, O."https://zbmath.org/authors/?q=ai:diekmann.odo"Verduyn Lunel, S. M."https://zbmath.org/authors/?q=ai:verduyn-lunel.sjoerd-mAuthors' abstract: In the standard theory of delay equations, the fundamental solution does not `live' in the state space. To eliminate this age-old anomaly, we enlarge the state space. As a consequence, we lose the strong continuity of the solution operators and this, in turn, has as a consequence that the Riemann integral no longer suffices for giving meaning to the variation-of-constants formula. To compensate, we develop the Stieltjes-Pettis integral in the setting of a norming dual pair of spaces. Part I provides general theory, Part II deals with ``retarded'' equations, and in Part III we show how the Stieltjes integral enables incorporation of unbounded perturbations corresponding to neutral delay equations.Boundary value problems associated with singular strongly nonlinear equations with functional termshttps://zbmath.org/1472.341242021-11-25T18:46:10.358925Z"Biagi, Stefano"https://zbmath.org/authors/?q=ai:biagi.stefano"Calamai, Alessandro"https://zbmath.org/authors/?q=ai:calamai.alessandro"Marcelli, Cristina"https://zbmath.org/authors/?q=ai:marcelli.cristina"Papalini, Francesca"https://zbmath.org/authors/?q=ai:papalini.francescaThe authors study the existence of a solution to the non-local boundary value problem for the functional differential equation
\begin{gather*}
(\Phi(k(t)x'(t)))'+f(t,G(x)(t))\rho(t,x'(t))=0, \\
x(a)=H_a(x),\ \ x(b)=H_b(x)
\end{gather*}
with the \(\Phi\)-Laplacian operator and the Carathéodory functions \(f\), \(\rho\). Under the assumption of the existence of a well-ordered pair of lower and upper functions, a quite general existence result is proved by means of fixed point arguments.Existence of positive solutions of second-order delayed differential system on infinite intervalhttps://zbmath.org/1472.341252021-11-25T18:46:10.358925Z"Ding, Ran"https://zbmath.org/authors/?q=ai:ding.ran"Wang, Fanglei"https://zbmath.org/authors/?q=ai:wang.fanglei"Yang, Nannan"https://zbmath.org/authors/?q=ai:yang.nannan"Ru, Yuanfang"https://zbmath.org/authors/?q=ai:ru.yuanfangSummary: The present paper is focused on the analysis on the existence of positive solutions of a second-order differential system with two delays
\[
\begin{cases}x_1^{\prime\prime}(t)-a_1(t)x_1(t)+m_1(t)f_1(t,x(t),x_\tau(t))=0,t > 0, \\ x_2^{\prime\prime}(t)-a_2(t)x_2(t)+m_2(t)f_2(t,x(t),x_\tau(t))=0,t > 0,\\ x_1(t)=0,-\tau_1\leq t\leq 0,\text{ and }\lim_{t\to\infty}x_1(t)=0, \\ x_2(t)=0,-\tau_2\leq t\leq 0,\text{ and }\lim_{t\to\infty}x_2(t)=0\end{cases}.
\] by using two well-known fixed point theorems.Green's function for periodic solutions in alternately advanced and delayed differential systemshttps://zbmath.org/1472.341292021-11-25T18:46:10.358925Z"Chiu, Kuo-Shou"https://zbmath.org/authors/?q=ai:chiu.kuo-shouIn the paper, the differential system
\[
x'(t)=A(t)x(t)+f(t,x(t),x(\gamma(t)))+g(t,x(t),x(\gamma(t)))
\]
is considered with a piecewise continuous argument deviation \(\gamma\). The existence of an \(\omega\)-periodic solution is studied under the assumption that the corresponding linear system has only the trivial \(\omega\)-periodic solution. By using Krasnoselskii's fixed point theorem, the author proves the existence (resp. the existence and uniqueness) of an \(\omega\)-periodic solution to the given system. Applications and illustrative examples are discussed as well.New result of existence of periodic solutions for a generalized \(p\)-Laplacian Liénard type differential equation with a variable delayhttps://zbmath.org/1472.341302021-11-25T18:46:10.358925Z"Eswari, R."https://zbmath.org/authors/?q=ai:eswari.rajendiran"Piramanantham, V."https://zbmath.org/authors/?q=ai:piramanantham.veeraraghavanIn this paper, the authors study the following generalized \(p\)-Laplacian Liénard type differential equation with a variable delay:
\[
(\phi_p(u'(t)))' = f(t, u(t), u'(t))u'(t) + \beta(t)g(t, u(t), u(t-\tau(t))) +s(t),
\]
where \(p\) is a constant, \(\phi_p: \mathbb{R}\to\mathbb{R}\) is defined by \(\phi_p(x) = |x|^{p-2}x\) for \(x\not =0\) and \(\phi_p(0) = 0\), \(\tau, \beta, s\) are periodic continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\) with period \(T>0\), and \(f, g\) are continuous and \(T\)-periodic in the first argument. There are many results available for periodic solutions of Duffing type, Rayleigh type and Liénard equations. Motivated by some of the results in literature, the authors consider this generalized equation under the assumptions that \(p\geq q >2\) with \(\frac{1}{p} + \frac{1}{q} = 1\) and \(\beta(t) > 0\) for \(t\in [0, T]\). By applying the Mawhin continuation theorem, a set of sufficient conditions is established for existence of at least one \(T\)-periodic solution. This improves and generalizes some existing results.The existence of periodic solutions for three-order neutral differential equationshttps://zbmath.org/1472.341312021-11-25T18:46:10.358925Z"Huang, Manna"https://zbmath.org/authors/?q=ai:huang.manna"Guo, Chengjun"https://zbmath.org/authors/?q=ai:guo.chengjun"Liu, Junming"https://zbmath.org/authors/?q=ai:liu.junmingIn this paper, the following third-order neutral type delay differential equation is considered:
\begin{eqnarray*}
p(t) &=& x'''(t) +c x'''(t-\tau) +a_2(t) x''(t) + a_1(t)x'(t) + a_0(t)x(t)\\
&& + \sum^n_{i=1}\beta_i(t)g_i(x(t-\tau_i(t))),
\end{eqnarray*}
where \(c, \tau\) are constants satisfying \(|c|<1\) and \(\tau >0\), \(a_0, a_1, a_2, \beta_i, \tau_i (i = 1, \ldots, n)\) and \(p\) are continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\) with period \(T> 0\), and the \(g_i\) are continuous from \(\mathbb{R}\) to \(\mathbb{R}\). Under suitable assumptions, sufficient conditions are provided for existence of nontrivial \(T\)-periodic solutions.Existence of \(P\)-mean almost periodic mild solution for fractional stochastic neutral functional differential equationhttps://zbmath.org/1472.341322021-11-25T18:46:10.358925Z"Sun, Xiao-ke"https://zbmath.org/authors/?q=ai:sun.xiaoke"He, Ping"https://zbmath.org/authors/?q=ai:he.pingSummary: A class of fractional stochastic neutral functional differential equation is analyzed in this paper. With the utilization of the fractional calculations, semigroup theory, fixed point technique and stochastic analysis theory, a sufficient condition of the existence for \(p\)-mean almost periodic solution is obtained, which are supported by two examples.Existence of global mild solutions for a class of fractional partial functional differential equationshttps://zbmath.org/1472.341382021-11-25T18:46:10.358925Z"Xi, Xuan-Xuan"https://zbmath.org/authors/?q=ai:xi.xuanxuan"Hou, Mimi"https://zbmath.org/authors/?q=ai:hou.mimi"Zhou, Xian-Feng"https://zbmath.org/authors/?q=ai:zhou.xianfengIn this paper, the authors establish sufficient conditions for the local and global existence of mild solutions of a class of fractional partial functional differential equations in Banach spaces. The results are obtained by using standard fixed point theorems. A specific nonlinear function is provided to verify the assumptions. It should be noted that the solution representation (22) for the abstract equation (19) is not true in all cases of the operator \(A\). As claimed in the abstract, explicit nonlocal conditions are not discussed.On critical dipoles in dimensions \(n \geq 3\)https://zbmath.org/1472.350122021-11-25T18:46:10.358925Z"Blake Allan, S."https://zbmath.org/authors/?q=ai:blake-allan.s"Gesztesy, Fritz"https://zbmath.org/authors/?q=ai:gesztesy.fritzSummary: We reconsider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials \(V_\gamma(x) = \gamma(u, x) | x |^{- 3}\), \(x \in \mathbb{R}^n\setminus\{0 \}\), \(\gamma \in [0, \infty)\), \(u \in \mathbb{R}^n\), \(| u | = 1\), \(n \in \mathbb{N}\), \(n \geqslant 3\). More precisely, for \(n \geqslant 3\), we provide an alternative proof of the existence of a critical dipole coupling constant \(\gamma_{c , n} > 0\), such that
\begin{align*}
&\text{for all } \gamma \in [ 0 , \gamma_{c , n} ] \text{, and all } u \in \mathbb{R}^n,\ | u | = 1 , \\
&\quad\int_{\mathbb{R}^n} d^n x |({\nabla} f)(x) |^2 \geqslant \pm \gamma \int_{\mathbb{R}^n} d^n x(u, x) | x |^{- 3} | f(x) |^2,\quad f \in D^1( \mathbb{R}^n)
\end{align*}
with \(D^1( \mathbb{R}^n)\) denoting the completion of \(C_0^\infty( \mathbb{R}^n)\) with respect to the norm induced by the gradient. Here \(\gamma_{c , n}\) is sharp, that is, the largest possible such constant. Moreover, we discuss upper and lower bounds for \(\gamma_{c , n} > 0\) and develop a numerical scheme for approximating \(\gamma_{c , n} \).
This quadratic form inequality will be a consequence of the fact
\[\overline{\left[ - {\Delta} + \gamma ( u , x ) | x |^{- 3} \right]\! |_{C_0^\infty ( \mathbb{R}^n \setminus \{ 0 \} )}} \geqslant 0 \text{ if and only if } 0 \leqslant \gamma \leqslant \gamma_{c , n}\]
in \(L^2( \mathbb{R}^n)\) (with \(\overline{T}\) the operator closure of the linear operator \(T\)).
We also consider the case of multicenter dipole interactions with dipoles centered on an infinite discrete set.Upscaling of a system of diffusion-reaction equations coupled with a system of ordinary differential equations originating in the context of crystal dissolution and precipitation of minerals in a porous mediumhttps://zbmath.org/1472.350292021-11-25T18:46:10.358925Z"Mahato, Hari Shankar"https://zbmath.org/authors/?q=ai:mahato.hari-shankar"Kräutle, Serge"https://zbmath.org/authors/?q=ai:krautle.serge"Knabner, Peter"https://zbmath.org/authors/?q=ai:knabner.peter"Böhm, Michael"https://zbmath.org/authors/?q=ai:bohm.michael-j|bohm.michael-cSummary: In this paper, we consider diffusion and reaction of mobile chemical species, and dissolution and precipitation of immobile species present inside a porous medium. The transport of mobile species in the pores is modeled by a system of semilinear parabolic partial differential equations. The reactions amongst the mobile species are assumed to be reversible. i.e. both forward and backward reactions are considered. These reversible reactions lead to highly nonlinear reaction rate terms on the right-hand side of the partial differential equations. This system of equations for the mobile species is complemented by flux boundary conditions at the outer boundary. Furthermore, the dissolution and precipitation of immobile species on the surface of the solid parts are modeled by mass action kinetics which lead to a nonlinear precipitation term and a multivalued dissolution term. The model is posed at the pore (micro) scale. The contribution of this paper is two-fold: first we show the existence of a unique positive global weak solution for the coupled systems and then we upscale (homogenize) the model from the micro scale to the macro scale. For the existence of solution, some regularization techniques, Schaefer's fixed point theorem and Lyapunov type arguments have been used whereas the concepts of two-scale convergence and periodic unfolding are used for the homogenization.Stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problemhttps://zbmath.org/1472.350372021-11-25T18:46:10.358925Z"Carvalho, Alexandre N."https://zbmath.org/authors/?q=ai:nolasco-de-carvalho.alexandre"Moreira, Estefani M."https://zbmath.org/authors/?q=ai:moreira.estefani-mSummary: In this work, we present results on stability and hyperbolicity of equilibria for a scalar nonlocal one-dimensional quasilinear parabolic problem. We show that this nonlocal version of the well-known Chafee-Infante equation bears some resemblance with the local version. However, its nonlocal characteristic requires a fine analysis of the spectrum of the associated linear operators, a lot more elaborated than the local case. The saddle point property of equilibria is shown to hold for this quasilinear model.Large time behavior of solutions to Schrödinger equation with complex-valued potentialhttps://zbmath.org/1472.350432021-11-25T18:46:10.358925Z"Aafarani, Maha"https://zbmath.org/authors/?q=ai:aafarani.mahaSummary: We study the large-time behavior of the solutions to the Schrödinger equation associated with a quickly decaying potential in dimension three. We establish the resolvent expansions at threshold zero and near positive resonances. The large-time expansions of solutions are obtained under different conditions, including the existence of positive resonances and zero resonance or/and zero eigenvalue.Asymptotics for 2-D wave equations with Wentzell boundary conditions in the squarehttps://zbmath.org/1472.350552021-11-25T18:46:10.358925Z"Li, Chan"https://zbmath.org/authors/?q=ai:li.chan"Jin, Kun-Peng"https://zbmath.org/authors/?q=ai:jin.kunpengThis paper deals with the linear wave equations with frictional dampings on Wentzell boundary,
\[
\left\{\!\!\! \begin{array}{lll} &u_{tt}-\Delta u=0, &x\in\Omega,\> t\ge0,\\
&u=0, &x\in\Gamma_0,\> t\ge0,\\
&u_{tt}-\Delta_Tu+\partial_\nu u+u_t=0, &x\in\Gamma_1,\> t\ge0,\\
&u(0,x) = u_0(x),\>u_t(0,x) = u_1(,x),&x\in \Omega, \end{array} \right.
\]
which models the vertical motion of membranes edged with a thin boundary coil of high rigidity. Here,
\[
\begin{array}{ll} &\Omega=(0,1)\times(0,1),\\
&\Gamma_0 = \{(x,1):\, 0 < x < 1\}\cup\{(0,y):\, 0 < y < 1\},\\
&\Gamma_1 = \{(x,0):\, 0 < x < 1\} \cup \{(1, y):\, 0 < y < 1\}, \end{array}
\]
\(\nu(\cdot)\) is the unit outer normal vector at boundary, \(\Delta_T\) is the tangential Laplacian operator on \(\Gamma_1\). The motion at the boundary \(\Gamma_1\) is constrained to vertical motion and is governed by the Newton equation of motion. The boundary condition on \(\Gamma_1\) is a dynamic Wentzell boundary one. The authors study the stabilization of the problem.Correction to: ``Scattering threshold for the focusing nonlinear Klein-Gordon equation''https://zbmath.org/1472.350632021-11-25T18:46:10.358925Z"Ibrahim, Slim"https://zbmath.org/authors/?q=ai:ibrahim.slim"Masmoudi, Nader"https://zbmath.org/authors/?q=ai:masmoudi.nader"Nakanishi, Kenji"https://zbmath.org/authors/?q=ai:nakanishi.kenji.1Summary: This article resolves some errors in the paper [the authors, ibid. 4, No. 3, 405--460 (2011; Zbl 1270.35132)]. The errors are in the energy-critical cases in two and higher dimensions.Corrigendum to: ``Investigation of the analyticity of dissipative-dispersive systems via a semigroup method''https://zbmath.org/1472.350762021-11-25T18:46:10.358925Z"Ioakim, Xenakis"https://zbmath.org/authors/?q=ai:ioakim.xenakis"Smyrlis, Yiorgos-Sokratis"https://zbmath.org/authors/?q=ai:smyrlis.yiorgos-sokratisCorrects [the authors, ibid. 420, No. 2, 1116--1128 (2014; Zbl 1304.35172)].Some global results for a class of homogeneous nonlocal eigenvalue problemshttps://zbmath.org/1472.351632021-11-25T18:46:10.358925Z"Dai, Guowei"https://zbmath.org/authors/?q=ai:dai.guoweiSpectral heat content for Lévy processeshttps://zbmath.org/1472.352092021-11-25T18:46:10.358925Z"Grzywny, Tomasz"https://zbmath.org/authors/?q=ai:grzywny.tomasz"Park, Hyunchul"https://zbmath.org/authors/?q=ai:park.hyunchul"Song, Renming"https://zbmath.org/authors/?q=ai:song.renmingSummary: In this paper we study the spectral heat content for various Lévy processes. We establish the small time asymptotic behavior of the spectral heat content for Lévy processes of bounded variation in \(\mathbb R^d, d \geq 1\). We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in \(\mathbb R\) with respect to symmetric Lévy processes of unbounded variation under certain conditions on their characteristic exponents. Finally, we establish that the small time asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measure.Correction to: ``Control and controllability of PDEs with hysteresis''https://zbmath.org/1472.352242021-11-25T18:46:10.358925Z"Gavioli, Chiara"https://zbmath.org/authors/?q=ai:gavioli.chiara"Krejčí, Pavel"https://zbmath.org/authors/?q=ai:krejci.pavelCorrection of Equation 4.17 in the authors' paper [ibid. 84, No. 1, 829--847 (2021; Zbl 1470.35197)].A direct approach to quasilinear parabolic equations on unbounded domains by Brézis's theory for subdifferential operatorshttps://zbmath.org/1472.352282021-11-25T18:46:10.358925Z"Kurima, Shunsuke"https://zbmath.org/authors/?q=ai:kurima.shunsuke"Yokota, Tomomi"https://zbmath.org/authors/?q=ai:yokota.tomomiSummary: This paper deals with nonlinear diffusion equations and their approximate equations under homogeneous Neumann boundary conditions in unbounded domains with smooth bounded boundary. \textit{P. Colli} and \textit{T. Fukao} [J. Differ. Equations 260, No. 9, 6930--6959 (2016; Zbl 1334.35154)] studied similar equations in bounded domains by applying an abstract theory for doubly nonlinear evolution inclusions; however, the proof is based on compactness methods and hence the case of unbounded domains is excluded from the framework. The present paper asserts that one can solve the original problem and the approximate problem individually and directly in unbounded domains by applying Brézis theory.Widths of resonances above an energy-level crossinghttps://zbmath.org/1472.352532021-11-25T18:46:10.358925Z"Fujiié, S."https://zbmath.org/authors/?q=ai:fujiie.setsuro"Martinez, A."https://zbmath.org/authors/?q=ai:martinez.andre"Watanabe, T."https://zbmath.org/authors/?q=ai:watanabe.takuyaThe authors study a \(2\times2\) Schrödinger operator
\[
Pu = Eu,\qquad P = \left( \begin{matrix} P_1 & hW\\
hW^* & P_2 \end{matrix} \right),
\]
where \(D_x:=-i\frac{d}{d x}\), \(P_j:=h^2D_x^2 + V_j(x)\) (\(j=1,2\)), \(W=W(x,hD_x)\) is a semiclassical differential operator, and \(W^*\) is the formal adjoint of \(W\). The main aim is to study the asymptotic distribution of resonances in the semiclassical limit \(h\to 0_+\) in a neighborhood of a fixed real energy \(E_0\).
A series of conditions is supposed:
\textbf{Assumption (A1)} \(V_1(x)\), \(V_2(x)\) are real-valued analytic functions on \(\mathbb{R}\), and extend to holomorphic functions in the complex domain, \(\mathcal{S}=\{x\in\mathbb{C}\,;\,|\mathrm{IM}\,\, x|<\delta_0\langle\mathrm{Re}\, \,x\rangle\}\), where \(\delta_0>0\) is a constant, and \(\langle t\rangle:=(1+|t|^2)^{1/2}\).
\textbf{Assumption (A2)} For \(j=1,2\), \(V_j\) admits limits as \(\mathrm{Re}\,\, x\to \pm\infty\) in \(\mathcal{S}\), and they satisfy
\[
\begin{aligned}
\lim_{\substack{\mathrm{Re}\,\,x\to -\infty \\ x\in \mathcal{S}}} V_1(x)>E_0\, ;\, \lim_{\substack{\mathrm{Re}\,\,x\to -\infty \\ x\in \mathcal{S}}} V_2(x)>E_0\, ;\\
\lim_{\substack{\mathrm{Re}\,\,x\to +\infty \\ x\in \mathcal{S}}} V_1(x)>E_0\, ;\, \lim_{\substack{\mathrm{Re}\,\,x\to +\infty \\ x\in \mathcal{S}}} V_2(x)<E_0.
\end{aligned}
\]
\textbf{Assumption (A3)} There exist three numbers \(a<b<0<c\) such that
\[
V_1(a)=V_1(c)=V_2(b)=E_0, V_1'(a) < 0, V_1'(c) > 0, V_2'(b) < 0,
\]
and that
\[
\begin{array}{ll}
V_1>E_0\text{ on }(-\infty, a)\cup (c,+\infty), V_1<E_0\text{ on }(a,c), V_2>E_0\text{ on }(-\infty, b), V_2<E_0\text{ on }(b,+\infty).
\end{array}
\]
\textbf{Assumption (A4)} The set \(\{x\in \mathbb{R}; V_1(x)=V_2(x)\, , \, V_1(x)\leq E_0\, ,\, V_2(x)\leq E_0\}\) is reduced to \(\{0\}\), and one has \(V_1(0)=V_2(0)=0\), \(V'_1(0)>0\), \(V_2'(0)<0\).
In particular, in the phase-space, the characteristic sets \(\Gamma_j:=\{\xi^2 +V_j(x)=E_0\}\) (\(j=1,2\)) intersect transversally at \((0,\pm\sqrt{E_0})\).
\textbf{Assumption (A5)} \(W(x,hD_x)\) is a first order differential operator,
\[
W(x,hD_x)=r_0(x)+ir_1(x)hD_x,
\]
where \(r_0\) and \(r_1\) are bounded analytic functions on \(\mathcal{S}\), are real on the real, and such that \(W\) is elliptic at the crossing points \((0,\pm\sqrt{E_0})\), that is, \( (r_0(0),r_1(0)) \not=(0,0). \)
Under the above assumptions, in a neighbourhood of the energy \(E_0\), the spectrum of \(P\) is essential only. Let \(\mathrm{Res}(P)\) be the set of the resonances of the operator \(P\). For \(E\in \mathbb{C}\) close enough to \(E_0\), the authors define the action,
\[
\mathcal{A}(E):= \int_{a(E)}^{c(E)}\sqrt{ E-V_1(t)} \, dt,
\]
where \(a(E)\) (respectively \(c(E)\)) is the unique solution of \(V_1(x)=E\) close to \(a\) (respectively close to \(c\)). In this situation, \(\mathcal{A}(E)\) is an analytic function of \(E\) near \(E_0\) and \(\mathcal{A}'(E)\) is strictly positive for any real \(E\) near \(E_0\). Then fix \(\delta_0>0\) sufficiently small and \(C_0>0\) arbitrarily large and let
\[
\mathcal{D}_h(\delta_0,C_0):= [E_0-\delta_0, E_0+\delta_0]-i[0,C_0h].
\]
For \(h>0\) and \(k\in\mathbb{Z}\) such that \((k+\frac12)\pi h\) belongs to \(\mathcal{A}( [E_0-2\delta_0, E_0+2\delta_0])\), let
\[
e_k(h):=\mathcal{A}^{-1}\left( (k+\frac12)\pi h\right).
\]
The main result is as follows. Under Assumptions (A1)-(A5), there exists \(\delta_0>0\) such that for any \(C_0>0\), one has, for \(h>0\) small enough
\[
\mathrm{Res}\,(P)\cap \mathcal{D}_h(\delta_0, C_0) =\{E_k(h); k\in\mathbb{Z}\}\cap\mathcal{D}_h(\delta_0, C_0)
\]
where the \(E_k(h)\)'s are complex numbers that satisfy \begin{align*}
\mathrm{Re}\, \,E_k(h) &= e_k(h) + \mathcal{O}(h^{2}),\\
\mathrm{Im}\, \,E_k(h) &= - C(e_k(h))h^2+ \mathcal{O}(h^{7/3}), \end{align*}
uniformly as \(h \to 0\). Here
\[
C(E)=\frac {\pi }{\gamma\mathcal{A}'(E)} \left |r_0(0)E^{-\frac 14}\sin \left(\frac{\mathcal{B}(E)}{h} + \frac{\pi}{4}\right) + r_1(0)E^{\frac 14}\cos \left(\frac{\mathcal{B}(E)}{h} + \frac{\pi}{4}\right)\right |^2
\]
with \(\gamma := V_1'(0)-V_2'(0) > 0\) and
\[
\mathcal{B}(E):=\int_{b(E)}^0\!\!\!\!\sqrt{E-V_2(x)}dx+\int_0^{c(E)}\!\!\!\!\!\sqrt{E-V_1(x)}dx,
\]
where \(b(E)\) is the unique root of \(V_2(x)=E\) close to \(b\).Semiclassical resonances generated by crossings of classical trajectorieshttps://zbmath.org/1472.352552021-11-25T18:46:10.358925Z"Higuchi, Kenta"https://zbmath.org/authors/?q=ai:higuchi.kentaSummary: We consider a \(2\times 2\) system of one-dimensional semiclassical Schrödinger operators with small interactions with respect to the semiclassical parameter \(h\). We study the asymptotics in the semiclassical limit of the resonances near a non-trapping energy for both corresponding classical Hamiltonians. We show the existence of resonances of width \(T^{-1}h\log (1/h)\), contrary to the scalar case, under the condition that two classical trajectories cross and compose a periodic trajectory with period \(T\).On the generalization of Moyal equation for an arbitrary linear quantizationhttps://zbmath.org/1472.353162021-11-25T18:46:10.358925Z"Borisov, Leonid A."https://zbmath.org/authors/?q=ai:borisov.leonid-a"Orlov, Yuriy N."https://zbmath.org/authors/?q=ai:orlov.yurii-nEvolution by Schrödinger equation of Aharonov-Berry superoscillations in centrifugal potentialhttps://zbmath.org/1472.353212021-11-25T18:46:10.358925Z"Colombo, F."https://zbmath.org/authors/?q=ai:colombo.fabrizio"Gantner, J."https://zbmath.org/authors/?q=ai:gantner.jonathan"Struppa, D. C."https://zbmath.org/authors/?q=ai:struppa.daniele-carloSummary: In recent years, we have investigated the evolution of superoscillations under Schrödinger equation with non-singular potentials. In all those cases, we have shown that superoscillations persist in time. In this paper, we investigate the centrifugal potential, which is a singular potential, and we show that the techniques developed to study the evolution of superoscillations in the case of the Schrödinger equation with a quadratic Hamiltonian apply to this setting. We also specify, in the case of the centrifugal potential, the notion of super-shift of the limit solution, a fact explained in the last section of this paper. It then becomes apparent that superoscillations are just a particular case of super-shift.Transport equations and perturbations of boundary conditionshttps://zbmath.org/1472.353282021-11-25T18:46:10.358925Z"Tyran-Kamińska, Marta"https://zbmath.org/authors/?q=ai:tyran-kaminska.martaSummary: We provide a new perturbation theorem for substochastic semigroups on abstract AL spaces extending Kato's perturbation theorem to nondensely defined operators. We show how it can be applied to piecewise deterministic Markov processes and transport equations with abstract boundary conditions. We give particular examples to illustrate our results.Solving nonlinear non-local problems using positive square-root operatorshttps://zbmath.org/1472.353382021-11-25T18:46:10.358925Z"Montagu, E. L."https://zbmath.org/authors/?q=ai:montagu.e-l"Norbury, John"https://zbmath.org/authors/?q=ai:norbury.john-wSummary: A non-constructive existence theory for certain operator equations \[L u = D u,\] using the substitution \(u = B^{\frac{1}{2}} \xi\) with \(B = L^{-1} \), is developed, where \(L\) is a linear operator (in a suitable Banach space) and \(D\) is a homogeneous nonlinear operator such that \(D \lambda u = \lambda^{} \alpha D u\) for all \(\lambda \geq 0\) and some \(\alpha \in \mathbb{R}, \alpha \neq \) ~1. This theory is based on the positive-operator approach of Krasnosel'skii. The method has the advantage of being able to tackle the nonlinear right-hand side \(D\) in cases where conventional operator techniques fail. By placing the requirement that the operator \(B\) must have a positive square root, it is possible to avoid the usual regularity condition on either the mapping \(D\) or its Fréchet derivative. The technique can be applied in the case of elliptic PDE problems, and we show the existence of solitary waves for a generalization of Benjamin's fluid dynamics problem.Non-convex \(\ell_p\) regularization for sparse reconstruction of electrical impedance tomographyhttps://zbmath.org/1472.353692021-11-25T18:46:10.358925Z"Wang, Jing"https://zbmath.org/authors/?q=ai:wang.jing|wang.jing.13|wang.jing.1|wang.jing.11|wang.jing.14|wang.jing.2|wang.jing.3|wang.jing.6|wang.jing.17|wang.jing.16|wang.jing.5|wang.jing.15Summary: This work is to investigate the image reconstruction of electrical impedance tomography from the electrical measurements made on an object's surface. An \(\ell_p\)-norm \((0<p<1)\) sparsity-promoting regularization is considered to deal with the fully non-linear electrical impedance tomography problem, and a novel type of smoothing gradient-type iteration scheme is introduced. To avoid the difficulty in calculating its gradient in the optimization process, a smoothing Huber potential function is utilized to approximate the \(\ell_p\)-norm penalty. We then propose the smoothing algorithm in the general frame and establish that any accumulation point of the generated iteration sequence is a first-order stationary point of the original problem. Furthermore, one iteration scheme based on the homotopy perturbation technology is derived to find the minimizers of the Huberized approximated objective function. Numerical experiments show that non-convex \(\ell_p\)-norm sparsity-promoting regularization improves the spatial resolution and is more robust with respect to noise, in comparison with \(\ell_p\)-norm regularization.Singular limits of sign-changing weighted eigenproblemshttps://zbmath.org/1472.354022021-11-25T18:46:10.358925Z"Kielty, Derek"https://zbmath.org/authors/?q=ai:kielty.derekSummary: Consider the eigenvalue problem generated by a fixed differential operator with a sign-changing weight on the eigenvalue term. We prove that as part of the weight is rescaled towards negative infinity on some subregion, the spectrum converges to that of the original problem restricted to the complementary region. On the interface between the regions the limiting problem acquires Dirichlet-type boundary conditions. Our main theorem concerns eigenvalue problems for sign-changing bilinear forms on Hilbert spaces. We apply our results to a wide range of PDEs: second and fourth order equations with both Dirichlet and Neumann-type boundary conditions, and a problem where the eigenvalue appears in both the equation and the boundary condition.Discrete curve flows in two-dimensional Cayley-Klein geometrieshttps://zbmath.org/1472.354242021-11-25T18:46:10.358925Z"Benson, Joseph"https://zbmath.org/authors/?q=ai:benson.joseph"Valiquette, Francis"https://zbmath.org/authors/?q=ai:valiquette.francisSummary: Using the method of equivariant moving frames, we study geometric flows of discrete curves in the nine Cayley-Klein planes. We show that, under a certain arc-length preserving flow, the curvature invariant \(\kappa_ n\) evolves according to the differential-difference equation \(\frac{\partial \kappa_n}{\partial t} = (1+\epsilon \kappa_{n+1}^2)(\kappa_{n+1}-\kappa_{n-1})\), where the value of \(\varepsilon \in\{-1, 0, 1\}\) is linked to the geometry of the Cayley-Klein plane.
For the entire collection see [Zbl 1471.81009].Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacianhttps://zbmath.org/1472.354272021-11-25T18:46:10.358925Z"Bidi, Younes"https://zbmath.org/authors/?q=ai:bidi.younes"Beniani, Abderrahmane"https://zbmath.org/authors/?q=ai:beniani.abderrahmane"Zennir, Khaled"https://zbmath.org/authors/?q=ai:zennir.khaled"Himadan, Ahmed"https://zbmath.org/authors/?q=ai:himadan.ahmedSummary: We consider strong damped wave equation involving the fractional Laplacian with nonlinear source. The results of global solution under necessary conditions on the critical exponent are established. The existence is proved by using the Galerkin approximations combined with the potential well theory. Moreover, we showed new decay estimates of global solution.Existence and non-existence results for fractional Kirchhoff Laplacian problemshttps://zbmath.org/1472.354402021-11-25T18:46:10.358925Z"Nyamoradi, Nemat"https://zbmath.org/authors/?q=ai:nyamoradi.nemat"Ambrosio, Vincenzo"https://zbmath.org/authors/?q=ai:ambrosio.vincenzoSummary: In this paper, we study the following fractional Kirchhoff-type problem:
\[\begin{aligned}
\left[ a+b\Big (\iint_{{\mathbb{R}}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dx dy \Big )^{\theta -1}\right] (-\Delta )^s u= &|u|^{2^*_s- 2} u\\
&\;+\lambda f(x) |u|^{q-2}u, \text{ in }\mathbb{R}^N,
\end{aligned}\]
where \((-\Delta )^s\) is the fractional Laplacian operator with \(0<s<1\), \(\lambda \ge 0\), \(a \ge 0\), \(b> 0\), \(1<q<2\), \(N>2s\), and \(2^*_s= \frac{2 N}{N-2s}\) is fractional critical Sobolev exponent. When \(\lambda =0\), under suitable values of the parameters \(\theta\), \(a\) and \(b\), we obtain a non-existence result and the existence of infinitely many nontrivial solutions for the above problem. Also, for suitable weight function \(f(x)\), using the Nehari manifold technique and the fibbing maps, we prove the existence of at least two positive solutions for a sufficiently small choice of \(\lambda\).Existence of three nontrivial solutions of asymptotically linear second order operator equationshttps://zbmath.org/1472.354472021-11-25T18:46:10.358925Z"Chen, Yingying"https://zbmath.org/authors/?q=ai:chen.yingyingSummary: In this article, we prove the existence of three nontrivial solutions for some second order operator equations, especially the asymptotically linear ones. The main methods are the Leray-Schauder degree theory and mountain pass theorem.Corrigendum to: ``Analysis of regularized inversion of data corrupted by white Gaussian noise''https://zbmath.org/1472.354542021-11-25T18:46:10.358925Z"Kekkonen, Hanne"https://zbmath.org/authors/?q=ai:kekkonen.hanne"Lassas, Matti"https://zbmath.org/authors/?q=ai:lassas.matti-j"Siltanen, Samuli"https://zbmath.org/authors/?q=ai:siltanen.samuliCorrigendum to the authors' paper [ibid. 30, No. 4, Article ID 045009, 18 p. (2014; Zbl 1287.35101)].Infinite order \(\Psi\mathrm{DOs}\): composition with entire functions, new Shubin-Sobolev spaces, and index theoremhttps://zbmath.org/1472.354692021-11-25T18:46:10.358925Z"Pilipović, Stevan"https://zbmath.org/authors/?q=ai:pilipovic.stevan-r"Prangoski, Bojan"https://zbmath.org/authors/?q=ai:prangoski.bojan"Vindas, Jasson"https://zbmath.org/authors/?q=ai:vindas.jassonSummary: We study global regularity and spectral properties of power series of the Weyl quantisation \(a^w\), where \(a(x,\xi)\) is a classical elliptic Shubin polynomial. For a suitable entire function \(P\), we associate two natural infinite order operators to \(a^w\), \(P(a^w)\) and \((P\circ a)^w\), and prove that these operators and their lower order perturbations are globally Gelfand-Shilov regular. They have spectra consisting of real isolated eigenvalues diverging to \(\infty\) for which we find the asymptotic behaviour of their eigenvalue counting function. In the second part of the article, we introduce Shubin-Sobolev type spaces by means of \(f\)-\(\Gamma^{*,\infty}_{A_p,\rho}\)-elliptic symbols, where \(f\) is a function of ultrapolynomial growth and \(\Gamma^{*,\infty}_{A_p,\rho}\) is a class of symbols of infinite order studied in this and our previous papers. We study the regularity properties of these spaces, and show that the pseudo-differential operators under consideration are Fredholm operators on them. Their indices are independent on the order of the Shubin-Sobolev spaces; finally, we show that the index can be expressed via a Fedosov-Hörmander integral formula.Fourier multipliers and transfer operatorshttps://zbmath.org/1472.370262021-11-25T18:46:10.358925Z"Pollicott, Mark"https://zbmath.org/authors/?q=ai:pollicott.markThe author gives a rigorous proof of a conjectured numerical value proposed by \textit{X. Chen} and \textit{H. Volkmer} [J. Fractal Geom. 5, No. 4, 351--386 (2018; Zbl 1400.37026)] which estimates a quantity related to the spectral radius of a transfer operator. The problem is significantly connected to the theory of Fourier multipliers. More specifically, the author takes the bounded linear operator \(\mathcal{L}: C^{0}([0, 1])\rightarrow C^{0}([0, 1])\) defined by \[(\mathcal{L}u)(t) = \frac{1}{3} \sum_{i=0}^{3}\left|\sin\left(\frac{\pi (t+i)}{3}\right)\right|u\left(\frac{t+i}{3}\right).\] For estimating the conjectured numerical value \(c=\lim_{n\rightarrow +\infty}||\mathcal{L}^{n}||^{1/n}\), the following complex function is used: \[d(z)=\exp\bigg(-\sum_{n=1}^{\infty}\frac{z^{n}}{n}\frac{1}{3^{n}-1}\sum_{j=0}^{3^{n}-1}\prod_{k=0}^{n-1}\sin\bigg(\frac{3^{k}j\pi}{3^{n}-1}\bigg)\bigg), \quad z\in \mathbb{C}.\] Note that \(d(z)\) extends analytically to \(\mathbb{C}\). The smallest positive zero \(\alpha>0\) is the reciprocal of the spectral radius \(c\), i.e., \(c=1/\alpha\). He describes a rigorous computation to determine a better estimate of \(c\), namely \[c= 0.648314752798325682324771447 \dots \pm 10^{-27}.\]
The author also considers a more general form of the above bounded linear operator \(\mathcal{L}\) and estimates its spectral radius. He gives two applications to justify the importance of his results.A \(\beta\)-Sturm-Liouville problem associated with the general quantum operatorhttps://zbmath.org/1472.390362021-11-25T18:46:10.358925Z"Cardoso, J. L."https://zbmath.org/authors/?q=ai:cardoso.joao-lopes|cardoso.jose-luis|cardoso-cortes.jose-luisSummary: Let \(I \subseteq \mathbb{R}\) be an interval and \(\beta : I \to I\) a strictly increasing and continuous function with a unique fixed point \(s_0 \in I\) that satisfies \((s_0 - t)(\beta(t)-t)\geq 0\) for all \(t \in I\), where the equality holds only when \(t = s_0\). The general quantum operator defined by \textit{A. E. Hamza} et al. [Adv. Difference Equ. 2015, Paper No. 182, 19 p. (2015; Zbl 1422.39010)], \(D_{\beta}[f](t):=\frac{f(\beta(t))-f(t)}{\beta(t)-t}\) if \(t \neq s_0\) and \(D_{\beta}[f](s_0):=f'(s_0)\) if \(t=s_0\) generalizes the Jackson \(q\)-operator \(D_q\) and also the Hahn (quantum derivative) operator, \(D_{q,\omega}\). Regarding a \(\beta\)-Sturm-Liouville eigenvalue problem associated with the above operator \(D_{\beta}\), we construct the \(\beta\)-Lagrange's identity, show that it is self-adjoint in \(\mathscr{L}_{\beta}^2([a,b])\) and exhibit some properties for the corresponding eigenvalues and eigenfunctions.Positive solution of discrete BVPs involving the mean curvature operator on the set of nonnegative integershttps://zbmath.org/1472.390392021-11-25T18:46:10.358925Z"Wei, Liping"https://zbmath.org/authors/?q=ai:wei.liping"Ma, Ruyun"https://zbmath.org/authors/?q=ai:ma.ruyun"Zhao, Zhongzi"https://zbmath.org/authors/?q=ai:zhao.zhongziSummary: Setting \(\mathbb{N}=\{1,2,\ldots ,\infty\}\), \(\mathbb{N}_0=\{0,1,2,\ldots ,\infty \}\), we show the existence of positive solutions of the quasilinear boundary value problem
\[
\begin{aligned}
&-\nabla \Big (\frac{\Delta u(x)}{\sqrt{1+(\Delta u(x))^2}}\Big)= g(x,u(x),\Delta u(x)), \quad x\in{\mathbb{N}},\\
&\quad u(0)=0, \quad \lim \limits_{x\rightarrow +\infty}\Delta u(x)=0,
\end{aligned}
\]
where \(\Delta\) is the forward difference operator, \(\nabla\) is the backward difference operator, and \(g: \mathbb{N}_0\times [0,+\infty)\times [0,+\infty)\rightarrow [0,+\infty)\) is continuous. Under suitable assumptions on nonlinearity, we obtained the existence of at least one positive solution by a simple application of a Fixed Point Theorem in cones.Existence of solutions of polynomial-like iterative equation with discontinuous known functionshttps://zbmath.org/1472.390422021-11-25T18:46:10.358925Z"Yu, Zhiheng"https://zbmath.org/authors/?q=ai:yu.zhiheng"Liu, Jinghua"https://zbmath.org/authors/?q=ai:liu.jinghuaThe authors study the existence of solutions of polynomial-like iterative equation \(\lambda_{1}f(x)+\lambda_{2}f^{2}(x)+\dots+\lambda_{n}f^{n}(x)=F(x)\), \(x\in I\), with discontinuous known functions. Let \(I=[a, b]\). Then the authors consider the set
\[
\mathcal{F}=\{f \mid f : I \rightarrow I \text{ is strictly increasing,} \; f(a)=a \text{ and } f(b)=b\},
\]
which contains continuous and discontinuous functions. They define the following three classes of ``good'' functions. For a given \(t\in \operatorname{int} (I)\) define:
\begin{gather*}
\mathcal{C}_{t}(I) :=\{f \mid f \in\mathcal{F}(I) \text{ is continuous on } I \text{ and } f(t)=t\}, \\
\tilde{\mathcal{C}}_{t}(I) :=\{f \mid f \in\mathcal{F}(I) \text{ is continuous on } \operatorname{int} I \text{ and } f(t)=t\}, \\
\mathcal{H}_{t}(I) :=\{f \mid f \in \mathcal{F}(I) \text{ is discontinuous exactly at point } t \text{ and } f(t)=t\}.
\end{gather*}
The authors show that if \(f_{1}, f_{2} \in \mathcal{C}_{t}\), then \(f_{1} \circ f_{2} \in\mathcal{C}_{t}\). Similarly, if \(f_{1}, f_{2} \in \mathcal{H}_{t}\), then \(f_{1} \circ f_{2} \in\mathcal{H}_{t}\).
Further, the authors introduce piecewise bi-Lipschitz functions and construct a functional space consisting of such functions. They denote the subclasses of all functions \(f\) from \(\mathcal{C}_{t}(I)\) and from \(\mathcal{H}_{t}(I)\) by \(\mathcal{A}_{t}(I, m, M)\) and \(\mathcal{B}_{t}(I, m, M)\) respectively. Moreover, \(\mathcal{G}_{t}(I, m, M)=\mathcal{A}_{t}(I, m, M) \cup\mathcal{B}_{t}(I, m, M)\). They show that the set \(\mathcal{G}_{t}(I, m, M)\) endowed with the distance \(\mathcal{D}(f_{1}, f_{2})=\sup\{|f_{1}(x)-f_{2}(x)|, x\in I\}\) is a complete metric space.
Then they define the operator \(\mathcal{T}: \mathcal{G}_{t}(I, m, M) \rightarrow\mathcal{F}(I)\) as \[\mathcal{T}f=\frac{1}{\lambda_{1}}\bigg(F-\sum_{i=2}^{n}\lambda_{i}f^{i}\bigg)\] and prove the existence of solutions by means of the Banach fixed point principle. An example to justify their main results
is discussed.Hypercontractivity of the semigroup of the fractional Laplacian on the \(n\)-spherehttps://zbmath.org/1472.390482021-11-25T18:46:10.358925Z"Frank, Rupert L."https://zbmath.org/authors/?q=ai:frank.rupert-l"Ivanisvili, Paata"https://zbmath.org/authors/?q=ai:ivanisvili.paataThe paper presents a further contribution to a problem concerning the hypercontractivity of the Poisson semigroup of \(e^{-t\sqrt{-\Delta}}\) from \(L^p(\mathbb{S}^n)\) to \(L^q(\mathbb{S}^n)\) for \(t>0\) on the sphere \(\mathbb{S}^n\) of dimension \(n\). This question was posed by \textit{C. E. Mueller} and \textit{F. B. Weissler} [J. Funct. Anal. 48, 252--283 (1982; Zbl 0506.46022)].
The main result of the paper states that for \(1<p\leq q\) the condition \(e^{-t\sqrt{n}}\leq \sqrt{\frac{p-1}{q-1}}\), carrying the smallest nonzero eigenvalue \(\sqrt{n}\) of \(\sqrt{-\Delta}\), is necessary and sufficient for dimension \(n\leq 3\). Noteworthy, in case of \(q>\max\{2,p\}\) the aforementioned condition is not sufficient for dimension \(n\geq 4\).
The reason of the exceptionality for \(n=1,2,3\) is explained in detail in Subsection 2.2. In the remaining part of the paper it is proved, by contradiction, why the sufficient condition does not hold in general.
Summing up, the question of finding an hypercontractivity equivalence for the Poisson semigroup on \(t>0\) for \(n\geq 4\) remains open.A variant of Wigner's theorem in normed spaceshttps://zbmath.org/1472.390502021-11-25T18:46:10.358925Z"Ilišević, Dijana"https://zbmath.org/authors/?q=ai:ilisevic.dijana"Turnšek, Aleksej"https://zbmath.org/authors/?q=ai:turnsek.aleksejLet \(X\) and \(Y\) be normed spaces over \(\mathbb{F}\) and let \(U:X \rightarrow Y\) be a linear (or a conjugate linear) isometry. If a function \(f:X \rightarrow Y\) has the property \[ f(x)=\sigma(x) Ux, \quad x\in H, \] where \(\sigma\) is a phase function, i.e., \(\sigma\) takes values in modulus one scalars, then a function \(f\) is called phase equivalent to a linear (or a conjugate linear) isometry.
The main result of this paper is given in the following theorem.
Theorem. Let \(X\) and \(Y\) be normed spaces over \(\mathbb{F}\) and \(f:X \rightarrow Y\) a surjective mapping. Suppose that for all semi-inner products on \(X\) and \(Y\), we have \[ | [f(x),f(y)]| = |[x,y]|, \quad x,y \in X.\]
The following statements hold true:
\begin{itemize}
\item[(i)] If dim \(X=1\), then \(f\) is phase equivalent to a linear surjective isometry;
\item[(ii)] If dim \(X\geq 2\) and \(\mathbb{F} = \mathbb{R}\), then \(f\) is phase equivalent to a linear surjective isometry;
\item[(iii)] If dim \(X\geq 2\) and \(\mathbb{F} =\mathbb{C}\), then \(f\) is phase equivalent to a linear or conjugate linear surjective isometry.
\end{itemize}The uniform convergence of a double sequence of functions at a point and Korovkin-type approximation theoremshttps://zbmath.org/1472.410022021-11-25T18:46:10.358925Z"Dirik, Fadime"https://zbmath.org/authors/?q=ai:dirik.fadime"Demirci, Kamil"https://zbmath.org/authors/?q=ai:demirci.kamil"Yıldız, Sevda"https://zbmath.org/authors/?q=ai:yildiz.sevda"Acu, Ana Maria"https://zbmath.org/authors/?q=ai:acu.ana-mariaIn this paper, the authors considered a new kind of uniform convergence of a double sequence of functions at a point. They have given an example and demonstrated a Korovkin-type approximation theorem for a double sequence of functions using the uniform convergence at a point. Then they have shown that their result is stronger than the Korovkin theorem given by Volkov. In the last section, they studied the rates of convergence via a new kind of convergence.On the compactness of oscillation and variation of commutatorshttps://zbmath.org/1472.420192021-11-25T18:46:10.358925Z"Guo, Weichao"https://zbmath.org/authors/?q=ai:guo.weichao"Wen, Yongming"https://zbmath.org/authors/?q=ai:wen.yongming"Wu, Huoxiong"https://zbmath.org/authors/?q=ai:wu.huoxiong"Yang, Dongyong"https://zbmath.org/authors/?q=ai:yang.dongyongThe singular integral operator with homogeneous kernel is defined by
\[T_{\Omega}f(x):=p.v.\int_{\mathbb{R}^{n}}\frac{\Omega(x-y)}{|x-y|^{n}}f(y)dy,\] where \(\Omega\) is a homogeneous function of degree zero and satisfies the following mean value zero property:
\[\int_{S^{n-1}}\Omega(x^\prime)d_{\sigma}(x^\prime)=0,\] where \(d_{\sigma}\) is the spherical measure on the sphere \(S^{n-1}\). Given a locally integrable function \(b\) and a linear operator \(T\), the commutator \([b,T]\) is defined by:
\[T^{b}(f)(x):=[b,T]f(x):=b(x)T(f)(x)-T(bf)(x)\]
for suitable functions \(f\).
The paper is devoted to the weighted \(L_{p}\)-compactness of the oscillation and variation of the commutator of singular integral operator. The authors first establish the weighted compactness result for oscillation and variation associated with the truncated commutator of singular integral operators. Moreover, they establish a new \(CMO(\mathbb{R}^{n})\) characterization via the compactness of oscillation and variation of commutators on weighted Lebesgue spaces.Invariant means on weakly almost periodic functions and generalized fixed point propertieshttps://zbmath.org/1472.430082021-11-25T18:46:10.358925Z"Soliman, Ahmed H."https://zbmath.org/authors/?q=ai:soliman.ahmed-hussein"Imdad, Mohammad"https://zbmath.org/authors/?q=ai:imdad.mohammad"Ahmadullah, Md"https://zbmath.org/authors/?q=ai:ahmadullah.mdSummary: In this paper, we prove common fixed point theorems for Generalized Suzuki Contractions (abbreviated as GSC) involving two semi-topological semigroups of self-mappings \(S_1\) and \(S_2\), besides establishing the existence of a left invariant mean (abbreviated as LIM) on the space of all weakly almost periodic functions on \(S_1\cap S_2\) (abbreviated as WAP\((S_1\cap S_2)\)).A Schur-Nevanlinna type algorithm for the truncated matricial Hausdorff moment problemhttps://zbmath.org/1472.440052021-11-25T18:46:10.358925Z"Fritzsche, Bernd"https://zbmath.org/authors/?q=ai:fritzsche.bernd"Kirstein, Bernd"https://zbmath.org/authors/?q=ai:kirstein.bernd"Mädler, Conrad"https://zbmath.org/authors/?q=ai:madler.conradSummary: The main goal of this paper is to achieve a parametrization of the solution set of the truncated matricial Hausdorff moment problem in the non-degenerate and degenerate situations. We treat the even and the odd cases simultaneously. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version, worked out in our former paper [Linear Algebra Appl. 590, 133--209 (2020; Zbl 1447.44005)], is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and discussion of the function-theoretic version is a central theme of this paper. This leads us to a complete description via Stieltjes transform of the solution set of the moment problem under consideration. Furthermore, we discuss special solutions in detail.Coupled systems of Hammerstein-type integral equations with sign-changing kernelshttps://zbmath.org/1472.450062021-11-25T18:46:10.358925Z"de Sousa, Robert"https://zbmath.org/authors/?q=ai:de-sousa.robert"Minhós, Feliz"https://zbmath.org/authors/?q=ai:minhos.feliz-manuelUsing the Guo-Krasnoselskii fixed point theorem in the context of expansive and compressive cones theory, the authors obtain an existence result for the solution of a generalized coupled system of integral equations of Hammerstein type. Such integral equations contain sign-changing kernels and nonlinearities depending on several derivatives of both variables in different orders. An application is considered. This refers to a coupled system of beam equations modeling the bending of the road-bed and the cable in suspension bridges.An inverse problem for the integro-differential Dirac system with partial information given on the convolution kernelhttps://zbmath.org/1472.450072021-11-25T18:46:10.358925Z"Bondarenko, Natalia Pavlovna"https://zbmath.org/authors/?q=ai:bondarenko.natalia-pThe author obtains uniqueness results for an inverse problem associated to a Dirac system of linear integro-differential equations with convolution kernel. For this purpose, the system of the direct equations is given and the partial inverse problem is formulated. The uniqueness theorem is proved in terms of the completeness of some system of vector functions associated with the given subspectrum. Moreover, a constructive algorithm for the solution of the specific partial inverse problem is provided. Necessary and sufficient conditions for the unique solvability of the inverse problem in this special case are obtained.Time memory effect in entropy decay of Ornstein-Uhlenbeck operatorshttps://zbmath.org/1472.450092021-11-25T18:46:10.358925Z"Agresti, Antonio"https://zbmath.org/authors/?q=ai:agresti.antonio"Loreti, Paola"https://zbmath.org/authors/?q=ai:loreti.paola"Sforza, Daniela"https://zbmath.org/authors/?q=ai:sforza.danielaSummary: We investigate the effect of memory terms on the entropy decay of the solutions to diffusion equations with Ornstein-Uhlenbeck operators. Our assumptions on the memory kernels include Caputo-Fabrizio operators and, more generally, the stretched exponential functions. We establish a sharp rate decay for the entropy. Examples and numerical simulations are also given to illustrate the results.Some remarks on the weak maximizing propertyhttps://zbmath.org/1472.460052021-11-25T18:46:10.358925Z"Dantas, Sheldon"https://zbmath.org/authors/?q=ai:dantas.sheldon"Jung, Mingu"https://zbmath.org/authors/?q=ai:jung.mingu"Martínez-Cervantes, Gonzalo"https://zbmath.org/authors/?q=ai:martinez-cervantes.gonzaloA pair \((E,F)\) of Banach spaces is said to have the {\em weak maximizing property} (wmp) if for every bounded linear operator \(T:E \to F\), \(T\) attains its norm whenever there is a non-weakly null sequence \((x_n)\) of norm one vectors such that \(\| T(x_n)\| \to \|T\|\). The idea for this property comes from a paper by \textit{D.~Pellegrino} and \textit{E.~V. Teixeira} [Bull. Braz. Math. Soc. (N.S.) 40, No.~3, 417--431 (2009; Zbl 1205.47012)], and it was recently adapted by \textit{R.~M. Aron} et al. [Proc. Am. Math. Soc. 148, No.~2, 741--750 (2020; Zbl 1442.46007)]. The latter paper contained several open questions, a number of which are addressed here.
For instance, although it was known that the pair \((\ell_p,\ell_q)\) has the wmp (\(1 < p < \infty$, $1 \leq q < \infty\)), it was unknown whether a similar property holds for \((L_p[0,1],L_q[0,1])\). The authors show that, in fact, wmp fails for \(p > 2\) or \(q < 2\). Among others, this result is a consequence of the authors' main result.
Theorem. Let \(E, F, X\), and \(Y\) be non-trivial Banach spaces such that not every bounded linear operator \(E \to F\) attains its norm. Then (i) whenever \(1 \leq q < p < \infty\), the pair \((E \oplus_p X, F \oplus_q Y)\) fails the wmp, and (ii) \((E \oplus_\infty X, F)\) fails the wmp. On the other hand, \((\ell_s \oplus_p \ell_p, \ell_s \oplus_q \ell_q)\) has the wmp if and only if \(1 < p \leq s \leq q < \infty\).
A number of interesting questions remain, such as: If \(E\) is a reflexive Banach space such that \((E,F)\) has the wmp for every \(F\), does it follow that \(E\) is finite dimensional? Also, the following is known: if the pair \((E, F)\) has the wmp, then an operator \(T +K\) attains its norm whenever a bounded linear operator \(T:E \to F\) and a compact operator \(K:E \to F\) satisfy \(\|T\| < \|T + K\|\). On the other hand, the authors note that the following converse is open: Let \(E\) and \(F\) be reflexive. Suppose that whenever \(T:E \to F\) (resp., \(K:E \to F\)) is a bounded (resp., compact) linear operator such that if \(\|T\| < \|T + K\|\), then necessarily \(T + K\) attains its norm. Does it follow that the pair \((E,F)\) has the wmp?Tingley's problem for \(p\)-Schatten von Neumann classeshttps://zbmath.org/1472.460062021-11-25T18:46:10.358925Z"Fernández-Polo, Francisco J."https://zbmath.org/authors/?q=ai:fernandez-polo.francisco-j"Jordá, Enrique"https://zbmath.org/authors/?q=ai:jorda.enrique"Peralta, Antonio M."https://zbmath.org/authors/?q=ai:peralta.antonio-mSummary: Let \(H\) and \(H'\) be the complex Hilbert spaces. For \(p\in (1,\infty)\setminus \{2\}\), we consider the Banach space \(C_p(H)\) of all \(p\)-Schatten-von Neumann operators, whose unit sphere is denoted by \(S(C_p(H))\). We prove that every surjective isometry \(\Delta:S(C_p(H))\to S(C_p(H'))\) can be extended to a complex linear or to a conjugate linear surjective isometry \(T:C_p(H)\to C_p(H')\).Approximate local isometries on spaces of absolutely continuous functionshttps://zbmath.org/1472.460082021-11-25T18:46:10.358925Z"Hosseini, Maliheh"https://zbmath.org/authors/?q=ai:hosseini.maliheh"Jiménez-Vargas, A."https://zbmath.org/authors/?q=ai:jimenez-vargas.antonioLet \(X,Y\) be compact sets of real numbers. Let \(AC(X)$ $(AC(Y))\) denote space of complex absolutely continous functions, equipped with the norm \(\|f\|_{\Sigma}= \|f\|_{\infty}+V(f)\), for \(f \in AC(X)\), where \(V(f)\) denotes the variation of~\(f\). A description of the surjective isometries between such spaces has long been known (see [\textit{V. D. Pathak}, Can. J. Math. 34, 298--306 (1982; Zbl 0464.46029)]). The authors first note that the group of isometries is not topologically reflexive and go on to provide a description of the objects in the topological closure of the isometry group (Theorem~2).Some open problems related to fixed point properties and amenabilityhttps://zbmath.org/1472.460142021-11-25T18:46:10.358925Z"Lau, Anthony To-Ming"https://zbmath.org/authors/?q=ai:lau.anthony-to-mingSummary: In this paper, we post some open problems related to fixed point theory and amenability of semigroups mainly based on my joint work with Professor Wataru Takahashi.On relative \(k\)-uniform rotundity, normal structure and fixed point property for nonexpansive mapshttps://zbmath.org/1472.460172021-11-25T18:46:10.358925Z"Veena Sangeetha, M."https://zbmath.org/authors/?q=ai:veena-sangeetha.mSummary: The idea of \(k\)-uniform rotundity relative to a \(k\)-dimensional subspace generalizes the classical notion of uniform rotundity in a direction. A normed space that is \(k\)-uniformly rotund relative to every \(k\)-dimensional subspace is said to be URE\(_k\). In this article, relative \(k\)-uniform rotundity is used to obtain: (1) new conditions sufficient for asymptotic centers to be compact; (2) new equivalent conditions for normal structure, weak normal structure and weak fixed point property for nonexpansive maps (WFPP) in a normed space. These results are then applied to study the inheritance of some geometric and fixed point properties in products of normed spaces. In particular, it is proved that if \(N_1\), \(N_2\) are normed spaces such that \(N_1\) is URE\(_{k_1}\) and \(N_2\) is URE\(_{k_2}\), then for \(1<p<\infty\), the \(p\)-direct such of \(N_1\) and \(N_2\) is URE\(_{(k_1+k_2-1)}\). Also, it is proved that if \(N_1\) has WFPP and \(N_2\) is URE\(_k\) for some positive integer \(k\), then with respect to certain norms (including the standard \(p\)-norms for \(1<p<\infty\)) \(N_1\otimes N_2\) has WFPP. In addition to these, relative \(k\)-uniform rotundity in a class of subspaces of the Banach space of all bounded real valued functions on a nonempty set with supremum norm is also studied.Weighted conformal invariance of Banach spaces of analytic functionshttps://zbmath.org/1472.460232021-11-25T18:46:10.358925Z"Aleman, Alexandru"https://zbmath.org/authors/?q=ai:aleman.alexandru"Mas, Alejandro"https://zbmath.org/authors/?q=ai:mas.alejandroSummary: We consider Banach spaces of analytic functions in the unit disc which satisfy a weighted conformal invariance property, that is, for a fixed \(\alpha > 0\) and every conformal automorphism \(\varphi\) of the disc, \(f \mapsto f \circ \varphi ( \varphi^\prime )^\alpha\) defines a bounded linear operator on the space in question, and the family of all such operators is uniformly bounded in operator norm. Many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces satisfy this condition. The aim of the paper is to develop a general approach to the study of such spaces based on this property alone. We consider polynomial approximation, duality and complex interpolation, we identify the largest and the smallest as well as the ``unique'' Hilbert space satisfying this property for a given \(\alpha > 0\). We investigate the weighted conformal invariance of the space of derivatives, or anti-derivatives with the induced norm, and arrive at the surprising conclusion that they depend entirely on the properties of the (modified) Cesàro operator acting on the original space. Finally, we prove that this last result implies a John-Nirenberg type estimate for analytic functions \(g\) with the property that the integration operator \(f \mapsto \int_0^z f(t) g^\prime(t) d t\) is bounded on a Banach space satisfying the weighted conformal invariance property.Atomic decomposition and Carleson measures for weighted mixed norm spaceshttps://zbmath.org/1472.460252021-11-25T18:46:10.358925Z"Peláez, José Ángel"https://zbmath.org/authors/?q=ai:pelaez.jose-angel"Rättyä, Jouni"https://zbmath.org/authors/?q=ai:rattya.jouni"Sierra, Kian"https://zbmath.org/authors/?q=ai:sierra.kianSummary: The purpose of this paper is to establish an atomic decomposition for functions in the weighted mixed norm space \(A^{p,q}_\omega\) induced by a radial weight \(\omega\) in the unit disc admitting a two-sided doubling condition. The obtained decomposition is further applied to characterize Carleson measures for \(A^{p,q}_\omega\), and bounded differentiation operators \(D^{(n)}(f)=f^{(n)}\) acting from \(A^{p,q}_\omega\) to \(L^s_\mu\), induced by a positive Borel measure \(\mu\), on the full range of parameters \(0<p,q,s<\infty\).Full proof of Kwapień's theorem on representing bounded mean zero functions on \([0,1]\)https://zbmath.org/1472.460272021-11-25T18:46:10.358925Z"Ber, Aleksei"https://zbmath.org/authors/?q=ai:ber.aleksey"Borst, Matthijs"https://zbmath.org/authors/?q=ai:borst.matthijs"Sukochev, Fedor"https://zbmath.org/authors/?q=ai:sukochev.fedor-aThe authors are able to fill a gap in the proof of a theorem in [\textit{S.~Kwapień}, Math. Nachr. 119, 175--179 (1984; Zbl 0575.46003)]. Whereas Kwapień's original proof holds for continuous functions, a gap appears for functions with discontinuities. Indeed, a proof of the following theorem in full generality is given. For every mean zero function \(f \in L_\infty[0, 1]\), there exists \(g \in L_\infty[0,1]\) and a mod \(0\) measure preserving transformation \(T\) of \([0, 1]\) such that \(f=g\circ T - g\). The original gap is discussed and a counterexample is also given.A note on Riemann-Liouville fractional Sobolev spaceshttps://zbmath.org/1472.460352021-11-25T18:46:10.358925Z"Carbotti, Alessandro"https://zbmath.org/authors/?q=ai:carbotti.alessandro"Comi, Giovanni E."https://zbmath.org/authors/?q=ai:comi.giovanni-eSummary: Taking inspiration from a recent paper by \textit{M. Bergounioux} et al. [Fract. Calc. Appl. Anal. 20, No. 4, 936--962 (2017; Zbl 1371.26013)], we study the Riemann-Liouville fractional Sobolev space \(W^{s,p}_{RL,a+}(I)\), for \(I=(a,b)\) for some \(a,b\in\mathbb{R}\), \(a<b\), \(s\in(0,1)\) and \(p\in [1,\infty]\); that is, the space of functions \(u \in L^p(I)\) such that the left Riemann-Liouville \((1-s)\)-fractional integral \(I_{a+}^{1-s}[u]\) belongs to \(W^{1,p}(I)\). We prove that the space of functions of bounded variation \(BV(I)\) and the fractional Sobolev space \(W^{s,1}(I)\) continuously embed into \(W^{s,1}_{RL,a+}(I)\). In addition, we define the space of functions with left Riemann-Liouville \(s\)-fractional bounded variation, \(BV^s_{RL,a+}(I)\), as the set of functions \(u\in L^1(I)\) such that \(I^{1-s}_{a+}[u]\in BV(I)\), and we analyze some fine properties of these functions. Finally, we prove some fractional Sobolev-type embedding results and we analyze the case of higher order Riemann-Liouville fractional derivatives.Characterizations of continuous operators on \(C_b(X)\) with the strict topologyhttps://zbmath.org/1472.460462021-11-25T18:46:10.358925Z"Nowak, Marian"https://zbmath.org/authors/?q=ai:nowak.marian"Stochmal, Juliusz"https://zbmath.org/authors/?q=ai:stochmal.juliuszSummary: Let \(X\) be a completely regular Hausdorff space and \(C_b(X)\) be the space of all bounded continuous functions on \(X\), equipped with the strict topology \(\beta \). We study some important classes of \((\beta,\Vert\cdot\Vert_E)\)-continuous linear operators from \(C_b(X)\) to a Banach space \((E,\Vert\cdot\Vert_E)\): \(\beta\)-absolutely summing operators, compact operators and \(\beta\)-nuclear operators. We characterize compact operators and \(\beta\)-nuclear operators in terms of their representing measures. It is shown that dominated operators and \(\beta\)-absolutely summing operators \(T:C_b(X)\rightarrow E\) coincide and if, in particular, \(E\) has the Radon-Nikodym property, then \(\beta\)-absolutely summing operators and \(\beta\)-nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concerning the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space \(C(X)\), where \(X\) is a compact Hausdorff space.Extensions of uniform algebrashttps://zbmath.org/1472.460502021-11-25T18:46:10.358925Z"Morley, Sam"https://zbmath.org/authors/?q=ai:morley.samCole's counterexample to the peak point conjecture
[\textit{B.~J. Cole}, One-point parts and the peak point conjecture. Ph.D. dissertation, Yale University, New Haven, CT (1968)]
provides a construction of extensions of uniform algebras having various desirable properties, and has been used in the theory of uniform algebras (e.g., [\textit{J.~F. Feinstein}, Stud. Math. 148, No.~1, 67--74 (2001; Zbl 1055.46035); Proc. Am. Math. Soc. 132, No.~8, 2389--2397 (2004; Zbl 1055.46036)]).
The paper under review introduces some new classes of uniform algebra extensions and investigates the properties that are inherited by any uniform algebra extensions such as being nontrivial, natural, regular and normal. The relationship between peak sets in the weak sense and the Shilov boundary of an extension to those of the original uniform algebra is studied as well.
Let \(X\) and \(Y\) be compact Hausdorff spaces, let \(A\) be a uniform algebra on \(X\) and \(B\) be a uniform algebra on \(Y\). If there exists a continuous surjection \(\Pi: Y \to X\) such that \(\Pi^*(A) \subseteq B\), then \(B\) is called a uniform algebra extension of \(A\). The author introduces a class of uniform algebra extensions called generalised Cole extensions, namely, if there exists a continuous surjection \(\Pi: Y \to X\) and a unital linear map \(T: C(X) \to C(Y)\) with \(\|T\|= 1\) such that \(T\circ \Pi^*= \mathrm{ id}_{C(X)}\), \(\Pi^*(A) \subseteq B\) and \(T(B)= A\). In this case, \(\Pi\) and \(T\) are called associated maps to the extension. It is shown in the paper, Section 5, that generalised Cole extensions preserve several properties of the original uniform algebra. The author also studies generalised Cole extensions implemented by a compact group. In general, a generalised Cole extension need not be implemented by a group; however, if the associated maps to the extension \(B\) satisfy \(\|\mathrm{id}_{C(Y)}- \Pi^*\circ T\|= 1\), then there exists a finite group \(G\) such that \(B\) is a generalised Cole extension implemented by \(G\). The paper is concluded with some examples and two open questions.Involutive operator algebrashttps://zbmath.org/1472.460532021-11-25T18:46:10.358925Z"Blecher, David P."https://zbmath.org/authors/?q=ai:blecher.david-p"Wang, Zhenhua"https://zbmath.org/authors/?q=ai:wang.zhenhuaLet \(\mathcal{H}\) be a complex Hilbert space and \(\mathcal{A}\) be an operator algebra on \(\mathcal{H}\), that is, a closed subalgebra of \(\mathcal{B}(\mathcal{H})\). When \(\mathcal{A}\) is equipped with operator space norms inherited from \(\mathcal{B}(\mathcal{H})\) and a completely isometric algebra involution \(\dagger\), it is called an operator \(*\)-algebra. There are four types of involutions (bijections of period two) on an operator algebra, including the above one as the first type. In the paper under review, the authors investigate the structure of operator \(*\)-algebras and remark that their results are applicable to algebras with other types of involution, after appropriate modifications in the arguments.
The paper begins with establishing some characterisations of operator algebras and operator \(*\)-algebras \(\mathcal{A}\), in terms of their biggest \(C^*\)-cover \(C^*_{\max}(\mathcal{A})\) and smallest \(C^*\)-cover \(C^*_e(\mathcal{A})\) of \(\mathcal{A}\).
In the third section of the paper, a wide range of examples of operator \(*\)-algebras are presented, including some uniform algebras, operator algebras generated by a single element, and operator algebras obtained from operator systems by Paulsen's off diagonal technique. Then several characterisations of operator \(*\)-algebras with contractive approximate identities and, in particular, those with countable contractive approximate identities are given. Moreover, analogues of the Arens-Kadison theorem and Cohen's factorisation theorem for operator \(*\)-algebras are proved.
The last section of the paper is devoted to hereditary subalgebras of operator \(*\)-algebras and their relations with Akemann's noncommutative topology. To be more precise, suppose that \(\mathcal{A}\) with the involution \(\dagger\) turns into an operator \(*\)-algebra. An orthogonal projection \(p\in\mathcal{A}^{**}\) is open in \(\mathcal{A}^{**}\) if there is a net \((x_t)\) in \(\mathcal{A}\) such that \[ x_t=px_t=x_tp\overset{w^*}\longrightarrow p. \] If, moreover, \(p=p^\dagger\), then we say that \(p\) is \(\dagger\)-open. In this case, the closed \(\dagger\)-subalgebra \(\mathcal{D}=p\mathcal{A}^{**}p\cap \mathcal{A}\) is called a \(\dagger\)-hereditary subalgebra of \(\mathcal{A}\) and \(p\) is called the support projection of \(\mathcal{D}\). The authors characterise \(\dagger\)-hereditary subalgebras in terms of one-sided ideals, analogous to a well-known characterisation of hereditary \(C^*\)-subalgebras. Besides, for every \(\dagger\)-hereditary subalgebra \(\mathcal{B}\) of \(\mathcal{A}\) a set \(E\) of real positive elements of \(\mathcal{A}\) is identified which generates \(\mathcal{B}\), that is, \(\mathcal{B}=\overline{E\mathcal{A} E}\). In particular separable \(\dagger\)-hereditary subalgebras or \(\dagger\)-hereditary subalgebras with countable contractive approximate identities are of the form \(\overline{x\mathcal{A} x}\) for certain \(x\in\mathcal{A}\).
A projection \(q\in\mathcal{A}^{**}\) is called \(\dagger\)-compact in \(\mathcal{A}^{**}\) if \(1-q\) is open in \(\mathcal{A}^{**}\) and there is an \(x\) in the unit ball of \(\mathcal{A}\) such that \(q=qx\). The paper is concluded with characterisations of compact projections and noncommutative analogues of Urysohn's lemma and Tietze's extension theorem.Unbounded operator algebrashttps://zbmath.org/1472.460592021-11-25T18:46:10.358925Z"Asadi, Mohammad B."https://zbmath.org/authors/?q=ai:asadi.mohammad-b"Hassanpour-Yakhdani, Z."https://zbmath.org/authors/?q=ai:hassanpour-yakhdani.zahra"Shamloo, S."https://zbmath.org/authors/?q=ai:shamloo.saraSummary: In this paper, we introduce the notion of abstract local operator algebras and operator modules, and provide a representation theorem for them which extends the BRS theorem for operator algebras. Furthermore, we give a new proof for the representation theorem of local operator systems. Also, we investigate the Haagerup tensor product of local operator spaces and Morita equivalence of local operator algebras.Maximality and finiteness of type 1 subdiagonal algebrashttps://zbmath.org/1472.460622021-11-25T18:46:10.358925Z"Ji, Guoxing"https://zbmath.org/authors/?q=ai:ji.guoxingSummary: Let \(\mathfrak{A}\) be a type 1 subdiagonal algebra in a \(\sigma\)-finite von Neumann algebra \(\mathcal{M}\) with respect to a faithful normal conditional expectation \(\Phi\). We give necessary and sufficient conditions for which \(\mathfrak{A}\) is maximal among the \(\sigma\)-weakly closed subalgebras of \(\mathcal{M}\). In addition, we show that a type 1 subdiagonal algebra in a finite von Neumann algebra is automatically finite which gives a positive answer of \textit{W. B. Arveson}'s finiteness problem in [Am. J. Math. 89, 578--642 (1967; Zbl 0183.42501)] in type 1 case.A Beurling-Blecher-Labuschagne type theorem for Haagerup noncommutative \(L^p\) spaceshttps://zbmath.org/1472.460662021-11-25T18:46:10.358925Z"Bekjan, Turdebek N."https://zbmath.org/authors/?q=ai:bekjan.turdebek-n"Raikhan, Madi"https://zbmath.org/authors/?q=ai:raikhan.madiSummary: Let \(\mathcal{M}\) be a \(\sigma\)-finite von Neumann algebra, equipped with a normal faithful state \(\varphi\), and let \(\mathcal{A}\) be maximal subdiagonal subalgebra of \(\mathcal{M}\) and \(1\leq p<\infty\). We prove a Beurling-Blecher-Labuschagne type theorem for \(\mathcal{A}\)-invariant subspaces of Haagerup noncommutative \(L^p(\mathcal{A})\) and give a characterization of outer operators in Haagerup noncommutative \(H^p\)-spaces associated with \(\mathcal{A}\).A weak expectation property for operator modules, injectivity and amenable actionshttps://zbmath.org/1472.460762021-11-25T18:46:10.358925Z"Bearden, Alex"https://zbmath.org/authors/?q=ai:bearden.alex"Crann, Jason"https://zbmath.org/authors/?q=ai:crann.jasonNote on the invariance properties of operator products involving generalized inverseshttps://zbmath.org/1472.470012021-11-25T18:46:10.358925Z"Liu, Xiaoji"https://zbmath.org/authors/?q=ai:liu.xiaoji"Zhang, Miao"https://zbmath.org/authors/?q=ai:zhang.miao"Yu, Yaoming"https://zbmath.org/authors/?q=ai:yu.yaomingSummary: We investigate further the invariance properties of the bounded linear operator product \(A C^{\left(1\right)} B^{\left(1\right)} D\) and its range with respect to the choice of the generalized inverses \(X\) and \(Y\) of bounded linear operators. Also, we discuss the range inclusion invariance properties of the operator product involving generalized inverses.2-strict convexity and continuity of set-valued metric generalized inverse in Banach spaceshttps://zbmath.org/1472.470022021-11-25T18:46:10.358925Z"Shang, Shaoqiang"https://zbmath.org/authors/?q=ai:shang.shaoqiang"Cui, Yunan"https://zbmath.org/authors/?q=ai:cui.yunanSummary: Authors investigate the metric generalized inverses of linear operators in Banach spaces. Authors prove by the methods of geometry of Banach spaces that, if \(X\) is approximately compact and \(X\) is 2-strictly convex, then metric generalized inverses of bounded linear operators in \(X\) are upper semicontinuous. Moreover, authors also give criteria for metric generalized inverses of bounded linear operators to be lower semicontinuous. Finally, a sufficient condition for set-valued mapping \(T^\partial\) to be continuous mapping is given.On the characteristic operator of an integral equation with a Nevanlinna measure in the infinite-dimensional casehttps://zbmath.org/1472.470032021-11-25T18:46:10.358925Z"Bruk, V. M."https://zbmath.org/authors/?q=ai:bruk.vladislav-moiseevichSummary: We define the families of maximal and minimal relations generated by an integral equation with a Nevanlinna operator measure in the infinite-dimensional case and prove their holomorphic property. We show that if the restrictions of maximal relations are continuously invertible, then the operators inverse to these restrictions are integral. By using these results, we prove the existence of the characteristic operator and describe the families of linear relations generating the characteristic operator.Paired operators in asymmetric space settinghttps://zbmath.org/1472.470042021-11-25T18:46:10.358925Z"Speck, Frank-Olme"https://zbmath.org/authors/?q=ai:speck.frank-olmeSummary: Relations between paired and truncated operators acting in Banach spaces are generalized to asymmetric space settings, i.e., to matrix operators acting between different spaces. This allows more direct proofs and further results in factorization theory, here in connection with the Cross Factorization Theorem and the Bart-Tsekanovsky Theorem. Concrete examples from mathematical physics are presented: the construction of resolvent operators to problems of diffraction of time-harmonic waves from plane screens which are not convex.
For the entire collection see [Zbl 1367.47005].On subscalarity of some \(2 \times 2\) \(M\)-hyponormal operator matriceshttps://zbmath.org/1472.470052021-11-25T18:46:10.358925Z"Zuo, Fei"https://zbmath.org/authors/?q=ai:zuo.fei"Shen, Junli"https://zbmath.org/authors/?q=ai:shen.junliSummary: We provide some conditions for \(2 \times 2\) operator matrices whose diagonal entries are \(M\)-hyponormal operators to be subscalar. As a consequence, we obtain that a Weyl-type theorem holds for such operator matrices.Uniform convergence and spectra of operators in a class of Fréchet spaceshttps://zbmath.org/1472.470062021-11-25T18:46:10.358925Z"Albanese, Angela A."https://zbmath.org/authors/?q=ai:albanese.angela-anna"Bonet, José"https://zbmath.org/authors/?q=ai:bonet.jose"Ricker, Werner J."https://zbmath.org/authors/?q=ai:ricker.werner-jSummary: Well-known Banach space results (e.g., due to J. Koliha and Y. Katznelson/L. Tzafriri), which relate conditions on the spectrum of a bounded operator \(T\) to the operator norm convergence of certain sequences of operators generated by \(T\), are extended to the class of quojection Fréchet spaces. These results are then applied to establish various mean ergodic theorems for continuous operators acting in such Fréchet spaces and which belong to certain operator ideals, for example, compact, weakly compact, and Montel.Numerical range of a simple compressionhttps://zbmath.org/1472.470072021-11-25T18:46:10.358925Z"Spain, Philip G."https://zbmath.org/authors/?q=ai:spain.philip-gSummary: The numerical range of the contraction \(K:\begin{bmatrix} a & b\\ c & d\end{bmatrix}\mapsto\begin{bmatrix} a & 0\\ 0 & 0\end{bmatrix}\) acting on \(L(\mathbb C^2)\) is identified, so allowing one to exhibit a hermitian projection that is not ultrahermitian.
An explicit formula for the norm of the operator \(\kappa_m:=\begin{bmatrix} a & b\\ c & d\end{bmatrix}\mapsto \begin{bmatrix} ma & b\\ c & d\end{bmatrix}\) \((m\in\mathbb C)\) translates into a family of inequalities in four complex variables.Powers of convex-cyclic operatorshttps://zbmath.org/1472.470082021-11-25T18:46:10.358925Z"León-Saavedra, Fernando"https://zbmath.org/authors/?q=ai:leon-saavedra.fernando"Romero-de la Rosa, María del Pilar"https://zbmath.org/authors/?q=ai:del-pilar-romero-de-la-rosa.mariaSummary: A bounded operator \(T\) on a Banach space \(X\) is convex cyclic if there exists a vector \(x\) such that the convex hull generated by the orbit \(\left\{T^n x\right\}_{n \geq 0}\) is dense in \(X\). In this note we study some questions concerned with convex-cyclic operators. We provide an example of a convex-cyclic operator \(T\) such that the power \(T^n\) fails to be convex cyclic. Using this result we solve three questions posed by \textit{H. Rezaei} [Linear Algebra Appl. 438, No. 11, 4190--4203 (2013; Zbl 1311.47012)].The index theorem for Toeplitz operators as a corollary of Bott periodicityhttps://zbmath.org/1472.470092021-11-25T18:46:10.358925Z"Baum, Paul F."https://zbmath.org/authors/?q=ai:baum.paul-f"van Erp, Erik"https://zbmath.org/authors/?q=ai:van-erp.erikSummary: This is an expository paper about the index of Toeplitz operators, and in particular Boutet de Monvel's theorem [\textit{L. Boutet de Monvel}, Invent. Math. 50, 249--272 (1979; Zbl 0398.47018)]. We prove Boutet de Monvel's theorem as a corollary of Bott periodicity, and independently of the Atiyah-Singer index theorem.The oriented degree of multivalued perturbations of Fredholm mappings of positive indexhttps://zbmath.org/1472.470102021-11-25T18:46:10.358925Z"Zvyagin, V. G."https://zbmath.org/authors/?q=ai:zvyagin.viktor-grigorevich(no abstract)On matrix-valued Stieltjes functions with an emphasis on particular subclasseshttps://zbmath.org/1472.470112021-11-25T18:46:10.358925Z"Fritzsche, Bernd"https://zbmath.org/authors/?q=ai:fritzsche.bernd"Kirstein, Bernd"https://zbmath.org/authors/?q=ai:kirstein.bernd"Mädler, Conrad"https://zbmath.org/authors/?q=ai:madler.conradSummary: The paper deals with particular classes of \(q{\times}q\) matrix-valued functions which are holomorphic in \(\mathbb{C}\setminus [\alpha, +\infty)\), where \(\alpha\) is an arbitrary real number. These classes are generalizations of classes of holomorphic complex-valued functions studied by \textit{I. S. Kac} and \textit{M. G. Krein} [Transl., Ser. 2, Am. Math. Soc. 103, 1--18 (1974; Zbl 0291.34016)] and by \textit{M. G. Krein} and \textit{A. A. Nudel'man} [The Markov moment problem and extremal problems. Providence, RI: American Mathematical Society (AMS) (1977; Zbl 0361.42014)]. The functions are closely related to truncated matricial Stieltjes problems on the interval \([\alpha+\infty)\). Characterizations of these classes via integral representations are presented. Particular emphasis is placed on the discussion of the Moore-Penrose inverse of these matrix-valued functions.
For the entire collection see [Zbl 1367.47005].The Bézout equation on the right half-plane in a Wiener space settinghttps://zbmath.org/1472.470122021-11-25T18:46:10.358925Z"Groenewald, G. J."https://zbmath.org/authors/?q=ai:groenewald.gilbert-j"ter Horst, S."https://zbmath.org/authors/?q=ai:ter-horst.sanne"Kaashoek, M. A."https://zbmath.org/authors/?q=ai:kaashoek.marinus-aSummary: This paper deals with the Bézout equation \({G}(s){X}(s) = {I}_{m}, \mathfrak{R}{s} \leq {0}\), in the Wiener space of analytic matrix-valued functions on the right halfplane. In particular, \(G\) is an \(m \times p\) matrix-valued analytic Wiener function, where \(p\geq m\), and the solution \(X\) is required to be an analytic Wiener function of size \(p\times m\). The set of all solutions is described explicitly in terms of a \(p\times p\) matrix-valued analytic Wiener function \(Y\), which has an inverse in the analytic Wiener space, and an associated inner function \(\Theta\) defined by \(Y\) and the value of \(G\) at infinity. Among the solutions, one is identified that minimizes the \(H^{2}\)-norm. A~Wiener space version of Tolokonnikov's lemma plays an important role in the proofs. The results presented are natural analogues of those obtained for the discrete case in [the authors, Complex Anal. Oper. Theory 10, No. 1, 115--139 (2016; Zbl 1337.47018)].
For the entire collection see [Zbl 1367.47005].Strong unitary equivalence and approximately unitary equivalence of normal compact operators over topological spaceshttps://zbmath.org/1472.470132021-11-25T18:46:10.358925Z"Jingming, Zhu"https://zbmath.org/authors/?q=ai:jingming.zhuA well-known theorem in operator theory states that two compact normal operators on a Hilbert space \(H\) are unitarily equivalent if and only if they have the same eigenvalues, counting multiplicities. In the paper under review, the author considers a ``parametrized'' analog of this theorem. Given a topological space \(X\), let \(\mathbb{K}(C(X))\) denote the set of bounded norm continuous functions from \(X\) to \(\mathbb{K}(H)\). Define \(A\) and \(B\) in \(\mathbb{K}(C(X))\) to be strongly unitarily equivalent if there exists a strongly continuous family \(\{U(x)\}_{x\in X}\) of unitaries over \(H\) such that \(B(x) = U^*(x)A(x)U(x)\) for every \(x\) in \(X\). The author adapts ideas of \textit{G. Friedman} and the reviewer [J. Topol. Anal. 8, No.~2, 313--348 (2016; Zbl 1352.15026)] to construct a fiber bundle whose fibers are countably infinite products of circles. Suppose that \(A\) and \(B\) are compact normal elements of \(\mathbb{K}(C(X))\) whose eigenvalues coincide at each point of \(X\). Then \(A\) and \(B\) jointly determine a continuous map from \(X\) into the base of the author's fiber bundle, and he proves that \(A\) and \(B\) are unitarily equivalent if and only this map continuously lifts to the total space of the fiber bundle.
The author defines \(A\) and \(B\) in \(\mathbb{K}(H)\) to be approximately unitarily equivalent if, for each \(\varepsilon > 0\), there exists a norm continuous unitary \(U\) in \(I + \mathbb{K}(H)\) such that \(\Vert B(x) - U^*(x)A(x)U(x)\Vert < \varepsilon\) for every \(x\) in \(X\), and asks when compact normal elements \(A\) and \(B\) of \(\mathbb{K}(C(X))\) are approximately unitarily equivalent. In the case where \(X = S^1\) and \(A\) and \(B\) have the same pointwise eigenvalues that vary continuously in \(x\), the author constructs a fiber bundle that is a variation on the one he constructed in the first part of the paper, and proves an analogous result concerning lifts from the base to the total space.Some inequalities of Čebyšev type for functions of operators in Hilbert spaceshttps://zbmath.org/1472.470142021-11-25T18:46:10.358925Z"Dragomir, S. S."https://zbmath.org/authors/?q=ai:dragomir.sever-silvestruSummary: Some operator inequalities for synchronous functions that are related to the Čebyšev inequality for sequences of real numbers are given. Natural examples for pairs of functions that have the same monotonicity on an interval are presented as well.Some properties of Furuta type inequalities and applicationshttps://zbmath.org/1472.470152021-11-25T18:46:10.358925Z"Yuan, Jiangtao"https://zbmath.org/authors/?q=ai:yuan.jiangtao"Wang, Caihong"https://zbmath.org/authors/?q=ai:wang.caihongSummary: This work is to consider Furuta type inequalities and their applications. Firstly, some Furuta-type inequalities under \(A \geq B \geq 0\) are obtained via Loewner-Heinz inequality; as an application, a proof of Furuta inequality is given without using the invertibility of operators. Secondly, we show a unified satellite theorem of grand Furuta inequality which is an extension of the results by \textit{M. Fujii} et al. [Linear Algebra Appl. 438, No. 4, 1580--1586 (2013; Zbl 1270.47016)]. At the end, a kind of Riccati-type operator equation is discussed via Furuta type inequalities.Traces on operator ideals and related linear forms on sequence ideals. IVhttps://zbmath.org/1472.470162021-11-25T18:46:10.358925Z"Pietsch, Albrecht"https://zbmath.org/authors/?q=ai:pietsch.albrechtThe author continues in this paper a sequence of papers on the topic. Here, he deals with dyadic representations of a bounded linear operator \(S\in \mathcal L(X,Y)\), meaning that \(S=\sum_{k=0}^\infty S_k\) where \(S_k\in \mathcal L(X,Y)\) are finite rank operators with \(\operatorname{rank} (S_k)\leq 2^k\) and with shift-monotone sequences ideals \(\mathcal \xi(\mathbb N_0)\), that is, \(S_+\)-invariant linear subspaces of \(\ell_\infty(\mathbb N_0)\), where \(S_+\) stands for the forward shift, satisfying that, if \(a\in \mathcal \xi(\mathbb N_0)\), \(b\in \ell_\infty(\mathbb N_0)\) and \(\sup_{n\ge k}|b_n|\le \sup_{n\ge k}|a_n|\), then \(a\in \mathcal \xi(\mathbb N_0)\). The author refers to \(S=\sum_{k=0}^\infty S_k\) as an \((\mathcal U, \mathcal \xi)\)-representation if \((\|S-\sum_{k=0}^{n-1} S_k\|_{\mathcal U})_n\in \mathbb \xi(\mathbb N_0)\).
It is known [the author, Indag. Math., New Ser. 25, No. 2, 341--365 (2014; Zbl 1319.47067)] that there is a one-to-one correspondence between all symmetric sequence ideals and all shift-monotone sequence ideals. The correspondence maps to each shift-monotone sequence ideal \(\mathcal \xi (\mathbb N_0)\) the symmetric sequence ideal \(s(\mathbb N_0)=\{a\in \ell_\infty(\mathbb N_0): (a_{2^k})\in \mathcal \xi(\mathbb N_0)\}\) and the sequence ideals \(s(\mathbb N_0)\) and \(\xi(\mathbb N_0)\) are said to be associated.
The main result establishes that, if \(\mathcal U\) is a quasi Banach operator ideal of trace type \(\rho\), that is, \(|\operatorname{trace}(F)|\le c n^\rho\|F\|_{\mathcal U}\) whenever \(F\) is a finite rank operator with \(\operatorname{rank}(F)\le n\) and \(c\) is a constant independent of \(F\), and \(\mu\) is a \(2^{-\rho}S_{+}\)-invariant linear form on a shift-monotone sequence ideal \(\xi(\mathbb N_0)\), then \(\tau(S):=\mu(2^{-k\rho}\operatorname{trace}(S_k))\) does not depend on the choice of the \((\mathcal U, \xi)\)-representation \(S=\sum_{k=0}^\infty S_k\). This result is then applied to concrete examples of Banach operator ideals, such as those given by absolutely \((q,2)\)-summing operators and shift-monotone sequence ideals such as those associated to Lorentz sequence ideals. Concrete examples of operators such as convolution operators generated by functions in certain Lipschitz and Besov classes are provided.
For Parts I--III, see [the author, Indag. Math., New Ser. 25, No. 2, 341--365 (2014; Zbl 1319.47067); Integral Equations Oper. Theory 79, No. 2, 255--299 (2014; Zbl 1337.47031); J. Math. Anal. Appl. 421, No. 2, 971--981 (2015; Zbl 1328.47021)].
For the entire collection see [Zbl 1367.47005].Quasinormality and subscalarity of class \(p\)-\(wA(s,t)\) operatorshttps://zbmath.org/1472.470172021-11-25T18:46:10.358925Z"Tanahashia, K."https://zbmath.org/authors/?q=ai:tanahashia.k"Prasadb, T."https://zbmath.org/authors/?q=ai:prasadb.t"Uchiyamac, A."https://zbmath.org/authors/?q=ai:uchiyamac.aSummary: Let \(T\) be a bounded linear operator on a complex Hilbert space \(\mathcal H\) and let \(T=U|T|\) be the polar decomposition of \(T\). An operator \(T\) is called a class \(p\)-\(wA(s,t)\) operator if \((|T^*|^t|T|^{2s}|T^*|^t)^{\frac{tp}{s+t}}\geq |T^*|^{2tp}\) and \((|T|^s|T^*|^{2t}|T|^s)^{\frac{sp}{s+t}}\leq |T|^{2sp}\) where \(0<s,t\) and \(0<p \leq 1\). We investigate quasinormality and subscalarity of class \(p\)-\(wA(s,t)\) operators.Spectral enclosures for non-self-adjoint extensions of symmetric operatorshttps://zbmath.org/1472.470182021-11-25T18:46:10.358925Z"Behrndt, Jussi"https://zbmath.org/authors/?q=ai:behrndt.jussi"Langer, Matthias"https://zbmath.org/authors/?q=ai:langer.matthias"Lotoreichik, Vladimir"https://zbmath.org/authors/?q=ai:lotoreichik.vladimir"Rohleder, Jonathan"https://zbmath.org/authors/?q=ai:rohleder.jonathanIn the description of many quantum mechanical systems, operators appear as a consequence of heuristic arguments which suggest in a first step a formal expression for the Hamiltonian or Schrödinger operator describing the model. These operators \(S\) are typically unbounded and symmetric on a domain \(\operatorname{dom}{S}\) which is a dense subspace of a Hilbert space \(\mathcal{H}\). In a second crucial step for the description of the quantum mechanical system, one has to choose a closed (in many cases self-adjoint) extension of \(S\) in order to start the analysis of the model. Typically, fixing an extension means to specify the relevant boundary conditions for the system. The paper under review focuses on the description of closed non-selfadjoint extensions of \(S\) which appear as restrictions of the adjoint operator \(S^*\) and on the analysis of some of their spectral properties. The article also presents in its final part several applications of their results to elliptic operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with \(\delta\)-interactions, and to quantum graphs with non-self-adjoint vertex couplings.
In the first part of the article, the authors use an abstract and systematic approach to the description of extensions in terms of so-called boundary triples \((\Gamma_0,\Gamma_1,\mathcal{G})\), where \(\Gamma_{0,1}: \operatorname{dom}S^*\to \mathcal{G}\) satisfy a Green identity on the auxiliary Hilbert space \(\mathcal{G}\). Boundary triples (or their generalizations called quasi boundary triples) provide a useful technique to describe extensions encoding abstractly the boundary data of the problem. To formulate their main results in this part (e.g., Theorem 3.1), the authors use the Weyl function, which is an operator-valued function on the auxiliary Hilbert space defined in terms of the boundary triples. In addition, they introduce also a boundary operator \(B\) (in general non-symmetric) which serves to label the different extensions.
The article has an informative and well-written introduction to the topic describing their methods and results, but also introducing the reader to the literature and alternative approaches in this very active field. The bibliography list contains more than 130 references.Composition operators in hyperbolic Bloch-type and \(F \left(p, q, s\right)\) spaceshttps://zbmath.org/1472.470192021-11-25T18:46:10.358925Z"Kotilainen, Marko"https://zbmath.org/authors/?q=ai:kotilainen.marko"Pérez-González, Fernando"https://zbmath.org/authors/?q=ai:perez-gonzalez.fernandoSummary: Composition operators \(C_\varphi\) from Bloch-type \(\mathcal{B}_\alpha\) spaces to \(F \left(p, q, s\right)\) classes, from \(F \left(p, q, s\right)\) to \(\mathcal{B}_\alpha\), and from \(F \left(p_1, q_1, 0\right)\) to \(F \left(p_2, q_2, s_2\right)\) are considered. The criteria for these operators to be bounded or compact are given. Our study also includes the corresponding hyperbolic spaces.Composition operators from \(p\)-Bloch space to \(q\)-Bloch space on the fourth Cartan-Hartogs domainshttps://zbmath.org/1472.470202021-11-25T18:46:10.358925Z"Su, Jianbing"https://zbmath.org/authors/?q=ai:su.jianbing"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.2Summary: We obtain new generalized Hua's inequality corresponding to \(Y_{\text{IV}}(N, n; K)\), where \(Y_{\text{IV}}(N, n; K)\) denotes the fourth Cartan-Hartogs domain in \(\mathbb{C}^{N + n}\). Furthermore, we introduce the weighted Bloch spaces on \(Y_{\text{IV}}(N, n; K)\) and apply our inequality to study the boundedness and compactness of composition operator \(C_\phi\) from \(\beta^p(Y_{\text{IV}}(N, n; K))\) to \(\beta^q(Y_{\text{IV}}(N, n; K))\) for \(p \geq 0\) and \(q \geq 0\).Asymptotic formulas for determinants of a special class of Toeplitz + Hankel matriceshttps://zbmath.org/1472.470212021-11-25T18:46:10.358925Z"Basor, Estelle"https://zbmath.org/authors/?q=ai:basor.estelle-l"Ehrhardt, Torsten"https://zbmath.org/authors/?q=ai:ehrhardt.torstenSummary: We compute the asymptotics of the determinants of certain \(n \times n\) Toeplitz + Hankel matrices \( T_{n}(a)+H_n(b)\) as \( n\rightarrow \infty \) with symbols of Fisher-Hartwig type. More specifically, we consider the case where \(a\) has zeros and poles and where \(b\) is related to \(a\) in specific ways. Previous results of \textit{P. Deift} et al. [Ann. Math. (2) 174, No. 2, 1243--1299 (2011; Zbl 1232.15006)] dealt with the case where \(a\) is even. We are generalizing this in a mild way to certain non-even symbols.
For the entire collection see [Zbl 1367.47005].The theory of generalized locally Toeplitz sequences: a review, an extension, and a few representative applicationshttps://zbmath.org/1472.470222021-11-25T18:46:10.358925Z"Garoni, Carlo"https://zbmath.org/authors/?q=ai:garoni.carlo"Serra-Capizzano, Stefano"https://zbmath.org/authors/?q=ai:serra-capizzano.stefanoSummary: We review and extend the theory of generalized locally Toeplitz (GLT) sequences, which goes back to \textit{P. Tilli}'s work on locally Toeplitz sequences [Linear Algebra Appl. 278, No. 1--3, 91--120 (1998; Zbl 0934.15009)] and was developed by the second author during the last decade. Informally speaking, a GLT sequence \(\{A_{n}\}_n\) is a sequence of matrices with increasing size equipped with a function \(\kappa\) (the so-called symbol). We write \(\{A_{n}\}_n \sim_{\text{\textsc{glt}}}\kappa\) to indicate that \(\{A_{n}\}_n\) is a GLT sequence with symbol \(\kappa\). This symbol characterizes the asymptotic singular value distribution of \(\{A_{n}\}_n\); if the matrices \(A_n\) are Hermitian, it also characterizes the asymptotic eigenvalue distribution of \(\{A_{n}\}_n\). Three fundamental examples of GLT sequences are: (i) the sequence of Toeplitz matrices generated by a function \(f\) in \(L^{1}\); (ii) the sequence of diagonal sampling matrices containing the samples of a Riemann-integrable function \(a\) over equispaced grids; (iii) any zero-distributed sequence, i.e., any sequence of matrices with an asymptotic singular value distribution characterized by 0. The symbol of the GLT sequence (i) is \(f\), the symbol of the GLT sequence (ii) is \(a\), and the symbol of the GLT sequences (iii) is \(0\). The set of GLT sequences is a \(^\ast\)-algebra. More precisely, suppose that \(\{A_{n}^{(i)}\}_{n} \sim_{\text{\textsc{glt}}}\kappa_i\) for \(i = 1,\dots ,r\), and let \(A_n = \mathrm{ops}(A_{n}^{(1)},\dots , A_{n}^{(r)})\) be a matrix obtained from \(A_{n}^{(1)},\dots , A_{n}^{(r)}\) by means of certain algebraic operations ``ops'', such as linear combinations, products, inversions and conjugate transpositions; then \(\{A_{n}\}_{n} \sim_{\text{\textsc{glt}}} \kappa = \mathrm{ops}(\kappa_{1},\dots , k_r)\).
The theory of GLT sequences is a powerful apparatus for computing the asymptotic singular value and eigenvalue distribution of the discretization matrices \(A_n\) arising from the numerical approximation of continuous problems, such as integral equations and, especially, partial differential equations. Indeed, when the discretization parameter \(n\) tends to infinity, the matrices \(A_n\) give rise to a sequence \(\{A_n\}_n\), which often turns out to be a GLT sequence.
Nevertheless, this work is not primarily concerned with the applicative interest of the theory of GLT sequences. Although we will provide some illustrative applications at the end, the attention is focused on the mathematical foundations of the theory. We first propose a modification of the original definition of GLT sequences. With the new definition, we are able to enlarge the applicability of the theory, by generalizing/simplifying a lot of key results. In particular, we remove the Riemann-integrability assumption from the main spectral distribution and algebraic results for GLT sequences. As a final step, we extend the theory. We first prove an approximation result, which is useful to show that a given sequence of matrices is a GLT sequence. By using this result, we provide a new and easier proof of the fact that \(\{A_n^{-1}\}_n \sim _{\text{\textsc{glt}}}\kappa^{-1}\) whenever \(\{A_n\}_n \sim {\text{\textsc{glt}}}\, \kappa\), the matrices \(A_n\) are invertible, and \(\kappa\neq 0\) almost everywhere. Finally, using again the approximation result, we prove that \(\{f(A_{n})\}_n \sim {\text{\textsc{glt}}}f(\kappa)\) whenever \(\{A_n\}_n \sim_{\text{\textsc{glt}}}\kappa\), the matrices \(A_n\) are Hermitian, and \(f : \mathbb{R}\to \mathbb{R}\) is continuous.
For the entire collection see [Zbl 1367.47005].Natural boundary for a sum involving Toeplitz determinantshttps://zbmath.org/1472.470232021-11-25T18:46:10.358925Z"Tracy, Craig A."https://zbmath.org/authors/?q=ai:tracy.craig-a"Widom, Harold"https://zbmath.org/authors/?q=ai:widom.haroldSummary: In the theory of the two-dimensional Ising model, the \textit{diagonal susceptibility} is equal to a sum involving Toeplitz determinants. In terms of a parameter \(k\), the diagonal susceptibility is analytic for \(| k | < 1\), and the authors proved in [J. Math. Phys. 54, No. 12, 123302, 9 p. (2013; Zbl 1288.82018)] the conjecture that this function has the unit circle as a natural boundary. The symbol of the Toeplitz determinants is a \(k\)-deformation of one with a single singularity on the unit circle. Here we extend the result, first, to deformations of a larger class of symbols with a single singularity on the unit circle, and then to deformations of (almost) general Fisher-Hartwig symbols.
For the entire collection see [Zbl 1367.47005].Spectral properties of discrete weighted shift operatorshttps://zbmath.org/1472.470242021-11-25T18:46:10.358925Z"Antonevich, A. B."https://zbmath.org/authors/?q=ai:antonevich.anatolij-borisovich"Akhmatova, A. A."https://zbmath.org/authors/?q=ai:akhmatova.a-aSummary: Discrete weighted shift operators \(B\) in the space \(\ell_2(\mathbb{Z})\) of two-way sequences are considered. Properties of \(B-\lambda I\) for spectral values of \(\lambda\) are described. In particular, conditions for operator \(B-\lambda I\) to be one-sided invertible are obtained.New characterizations for the products of differentiation and composition operators between Bloch-type spaceshttps://zbmath.org/1472.470252021-11-25T18:46:10.358925Z"Liang, Yu-Xia"https://zbmath.org/authors/?q=ai:liang.yuxia"Dong, Xing-Tang"https://zbmath.org/authors/?q=ai:dong.xingtangSummary: We use a brief way to give various equivalent characterizations for the boundedness and the essential norm of the operator \(C_\varphi D^m\) acting on Bloch-type spaces. At the same time, we use this method to easily get a known characterization for the operator \(DC_\varphi\) on Bloch-type spaces.Preservers for the \(p\)-norm of linear combinations of positive operatorshttps://zbmath.org/1472.470262021-11-25T18:46:10.358925Z"Nagy, Gergo"https://zbmath.org/authors/?q=ai:nagy.gergoSummary: We describe the structure of those transformations on certain sets of positive operators which preserve the \(p\)-norm of linear combinations with given nonzero real coefficients. These sets are the collection of all positive \(p\)th Schatten-class operators and the set of its normalized elements. The results of the work generalize, extend, and unify several former theorems.Maps preserving peripheral spectrum of generalized Jordan products of self-adjoint operatorshttps://zbmath.org/1472.470272021-11-25T18:46:10.358925Z"Zhang, Wen"https://zbmath.org/authors/?q=ai:zhang.wen|zhang.wen.3|zhang.wen.2|zhang.wen.1"Hou, Jinchuan"https://zbmath.org/authors/?q=ai:hou.jinchuanSummary: Let \(\mathcal{A}_1\) and \(\mathcal{A}_2\) be standard real Jordan algebras of self-adjoint operators on complex Hilbert spaces \(H_1\) and \(H_2\), respectively. For \(k \geq 2\), let \((i_1, \dots, i_m)\) be a fixed sequence with \(i_1, \dots, i_m \in\{1, \dots, k \}\) and assume that at least one of the terms in \((i_1, \dots, i_m)\) appears exactly once. Define the generalized Jordan product \(T_1 \circ T_2 \circ \cdots \circ T_k = T_{i_1} T_{i_2} \cdots T_{i_m} + T_{i_m} \cdots T_{i_2} T_{i_1}\) on elements in \(\mathcal{A}_i\). Let \(\Phi : \mathcal{A}_1 \rightarrow \mathcal{A}_2\) be a map with the range containing all rank-one projections and trace zero-rank two self-adjoint operators. We show that \(\Phi\) satisfies that \(\sigma_\pi(\Phi(A_1) \circ \cdots \circ \Phi(A_k)) = \sigma_\pi(A_1 \circ \cdots \circ A_k)\) for all \(A_1, \ldots, A_k\), where \(\sigma_\pi(A)\) stands for the peripheral spectrum of \(A\), if and only if
there exist a scalar \(c \in \{- 1,1 \}\) and a unitary operator \(U : H_1 \rightarrow H_2\) such that \(\Phi(A) = c U A U^*\) for all \(A \in \mathcal{A}_1\), or \(\Phi(A) = c U A^t U^*\) for all \(A \in \mathcal{A}_1\), where \(A^t\) is the transpose of \(A\) for an arbitrarily fixed orthonormal basis of \(H_1\). Moreover, \(c = 1\) whenever \(m\) is odd.\(L^0\)-linear modulus of a random linear operatorhttps://zbmath.org/1472.470282021-11-25T18:46:10.358925Z"Liu, Ming"https://zbmath.org/authors/?q=ai:liu.ming.1"Zhang, Xia"https://zbmath.org/authors/?q=ai:zhang.xiaSummary: We prove that there exists a unique \(L^0\)-linear modulus for an a.s.\ bounded random linear operator on a specifical random normed module, which generalizes the classical case.On the norm of linear combinations of projections and some characterizations of Hilbert spaceshttps://zbmath.org/1472.470292021-11-25T18:46:10.358925Z"Krupnik, Nahum"https://zbmath.org/authors/?q=ai:krupnik.naum-yakovlevich"Markus, Alexander"https://zbmath.org/authors/?q=ai:markus.alexander-sSummary: Let \( \mathcal{B} \) be a Banach space and let \(P, Q\) (\(P, Q \neq 0\)) be two complementary projections in \( \mathcal{B}\) (i.e., \(P +Q=I\)). For dim \( \mathcal{B} > 2 \), we show that formulas of the kind \( \| aP \, +\, bQ \| = f(a, b, \| P\|) \) hold if and only if the norm in \( \mathcal{B} \) can be induced by an inner product. The two-dimensional case needs special consideration, which is done in the last two sections.
For the entire collection see [Zbl 1367.47005].On perturbations of generators of \(C_0\)-semigroupshttps://zbmath.org/1472.470302021-11-25T18:46:10.358925Z"Adler, Martin"https://zbmath.org/authors/?q=ai:adler.martin"Bombieri, Miriam"https://zbmath.org/authors/?q=ai:bombieri.miriam"Engel, Klaus-Jochen"https://zbmath.org/authors/?q=ai:engel.klaus-jochenSummary: We present a perturbation result for generators of \(C_0\)-semigroups which can be considered as an operator theoretic version of the Weiss-Staffans perturbation theorem for abstract linear systems. The results are illustrated by applications to the Desch-Schappacher and the Miyadera-Voigt perturbation theorems and to unbounded perturbations of the boundary conditions of a generator.The Kalton-Lancien theorem revisited: maximal regularity does not extrapolatehttps://zbmath.org/1472.470312021-11-25T18:46:10.358925Z"Fackler, Stephan"https://zbmath.org/authors/?q=ai:fackler.stephanSummary: We give a new more explicit proof of a result by \textit{N. J. Kalton} and \textit{G. Lancien} [Math. Z. 235, No. 3, 559--568 (2000; Zbl 1010.47024)] stating that on each Banach space with an unconditional basis not isomorphic to a Hilbert space there exists a generator $A$ of a holomorphic semigroup which does not have maximal regularity. In particular, we show that there always exists a Schauder basis \((f_m)\) such that $A$ can be chosen of the form \(A(\sum_{m = 1}^\infty a_m f_m) = \sum_{m = 1}^\infty 2^m a_m f_m\). Moreover, we show that maximal regularity does not extrapolate: we construct consistent holomorphic semigroups \((T_p(t))_{t \geqslant 0}\) on \(L^p(\mathbb{R})\) for \(p \in(1, \infty)\) which have maximal regularity if and only if \(p = 2\). These assertions were both open problems. Our approach is completely different than the one of Kalton and Lancien [loc.\,cit.]. We use the characterization of maximal regularity by \(\mathcal{R}\)-sectoriality for our construction.Convoluted fractional \(C\)-semigroups and fractional abstract Cauchy problemshttps://zbmath.org/1472.470322021-11-25T18:46:10.358925Z"Mei, Zhan-Dong"https://zbmath.org/authors/?q=ai:mei.zhandong"Peng, Ji-Gen"https://zbmath.org/authors/?q=ai:peng.jigen"Gao, Jing-Huai"https://zbmath.org/authors/?q=ai:gao.jinghuaiSummary: We present the notion of convoluted fractional \(C\)-semigroup, which is the generalization of convoluted \(C\)-semigroup in the Banach space setting. We present two equivalent functional equations associated with convoluted fractional \(C\)-semigroup. Moreover, the well-posedness of the corresponding fractional abstract Cauchy problems is studied.\(n\)-times integrated \(C\)-semigroups and exponential stability of an abstract Cauchy problemhttps://zbmath.org/1472.470332021-11-25T18:46:10.358925Z"Yue, Tian"https://zbmath.org/authors/?q=ai:yue.tian"Song, Xiaoqiu"https://zbmath.org/authors/?q=ai:song.xiaoqiu"Li, Zhigang"https://zbmath.org/authors/?q=ai:li.zhigang(no abstract)Spectrums of solvable pantograph differential-operators for first orderhttps://zbmath.org/1472.470342021-11-25T18:46:10.358925Z"Ismailov, Z. I."https://zbmath.org/authors/?q=ai:ismailov.zameddin-ismailovich"Ipek, P."https://zbmath.org/authors/?q=ai:ipek.pembeSummary: By using the methods of operator theory, all solvable extensions of minimal operator generated by first order pantograph-type delay differential-operator expression in the Hilbert space of vector-functions on finite interval have been considered. As a result, the exact formula for the spectrums of these extensions is presented. Applications of obtained results to the concrete models are illustrated.Wave front sets with respect to the iterates of an operator with constant coefficientshttps://zbmath.org/1472.470352021-11-25T18:46:10.358925Z"Boiti, C."https://zbmath.org/authors/?q=ai:boiti.chiara"Jornet, D."https://zbmath.org/authors/?q=ai:jornet.david"Juan-Huguet, J."https://zbmath.org/authors/?q=ai:juan-huguet.jordiSummary: We introduce the wave front set \(\text{WF}_*^P(u)\) with respect to the iterates of a hypoelliptic linear partial differential operator with constant coefficients of a classical distribution \(u \in \mathcal{D}'(\Omega)\) in an open set \(\Omega\) in the setting of ultradifferentiable classes of \textit{R. W. Braun} et al. [Result. Math. 17, No. 3--4, 206--237 (1990; Zbl 0735.46022)]. We state a version of the microlocal regularity theorem of Hörmander for this new type of wave front set and give some examples and applications of the former result.Wegner estimate for random divergence-type operators monotone in the randomnesshttps://zbmath.org/1472.470362021-11-25T18:46:10.358925Z"Dicke, Alexander"https://zbmath.org/authors/?q=ai:dicke.alexanderSummary: In this note, a Wegner estimate for random divergence-type operators that are monotone in the randomness is proven. The proof is based on a recently shown unique continuation estimate for the gradient and the ensuing eigenvalue liftings. The random model which is studied here contains quite general random perturbations, among others, some that have a non-linear dependence on the random parameters.Pseudodifferential operators in weighted Hölder-Zygmund spaces of variable smoothnesshttps://zbmath.org/1472.470372021-11-25T18:46:10.358925Z"Kryakvin, Vadim"https://zbmath.org/authors/?q=ai:kryakvin.v-d"Rabinovich, Vladimir"https://zbmath.org/authors/?q=ai:rabinovich.vladimir-sSummary: We consider pseudodifferential operators of variable orders acting in Hölder-Zygmund spaces of variable smoothness. We prove the boundedness and compactness of the operators under consideration and study the Fredholm property of pseudodifferential operators with slowly oscillating at infinity symbols in the weighted Hölder-Zygmund spaces of variable smoothness.
For the entire collection see [Zbl 1367.47005].Potential operators associated with Hankel and Hankel-Dunkl transformshttps://zbmath.org/1472.470382021-11-25T18:46:10.358925Z"Nowak, Adam"https://zbmath.org/authors/?q=ai:nowak.adam"Stempak, Krzysztof"https://zbmath.org/authors/?q=ai:stempak.krzysztofSummary: We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those \(1\leq p,q\leq\infty\), for which the potential operators satisfy \(L^p\)-\(L^q\) estimates. In case of the Riesz potentials, we also characterize those \(1\leq p,q\leq\infty\) for which two-weight \(L^p\)-\(L^q\) estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight \(L^p\)-\(L^q\) bounds for the classical Riesz potentials in the radial case. This complements an old result of \textit{B. S. Rubin} [Math. Notes 34, 751--757 (1983; Zbl 0535.45008); translation from Mat. Zametki 34, 521--533 (1983)] and its recent reinvestigations by \textit{P. L. De Nápoli} et al. [Ill. J. Math. 55, No. 2, 575--587 (2011; Zbl 1277.26026)], and \textit{J. Duoandikoetxea} [Ann. Mat. Pura Appl. (4) 192, No. 4, 553--568 (2013; Zbl 1279.47073)].Class \(\mathfrak A\)-\(\text{KKM}(X, Y, Z)\), general KKM type theorems, and their applications in topological vector spacehttps://zbmath.org/1472.470392021-11-25T18:46:10.358925Z"Tang, Gusheng"https://zbmath.org/authors/?q=ai:tang.gusheng"Zhang, Qingbang"https://zbmath.org/authors/?q=ai:zhang.qingbangSummary: The class \(\mathfrak A\)-KKM\((X, Y, Z)\) and generalized KKM mapping are introduced, and some generalized KKM theorems are proved. As applications, Ky Fan's matching theorem and Fan-Browder fixed-point theorem are extended, and some existence theorems of solutions for the generalized vector equilibrium problems are established under noncompact setting, which improve and generalize some known results.On monotone mappings in modular function spaceshttps://zbmath.org/1472.470402021-11-25T18:46:10.358925Z"Alfuraidan, M. R."https://zbmath.org/authors/?q=ai:alfuraidan.monther-rashed"Khamsi, M. A."https://zbmath.org/authors/?q=ai:khamsi.mohamed-amine"Kozlowski, W. M."https://zbmath.org/authors/?q=ai:kozlowski.wojciech-mSummary: Because of its many diverse applications, fixed point theory has been a flourishing area of mathematical research for decades. Banach's formulation of the contraction mapping principle in the early twentieth century signaled the advent of an intense interest in the metric related aspects of the theory. The metric fixed point theory in modular function spaces is closely related to the metric theory, in that it provides modular equivalents of norm and metric concepts. Modular spaces are extensions of the classical Lebesgue and Orlicz spaces, and in many instances, conditions cast in this framework are more natural and more easily verified than their metric analogs. In this chapter, we study the existence and construction of fixed points for monotone nonexpansive mappings acting in modular functions spaces equipped with a partial order or a graph structure.
For the entire collection see [Zbl 1470.47001].Generalized contraction and invariant approximation results on nonconvex subsets of normed spaceshttps://zbmath.org/1472.470412021-11-25T18:46:10.358925Z"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahid"Ali, Basit"https://zbmath.org/authors/?q=ai:ali.basit"Romaguera, Salvador"https://zbmath.org/authors/?q=ai:romaguera.salvadorSummary: \textit{D. Wardowski} [Fixed Point Theory Appl. 2012, Paper No. 94, 6 p. (2012; Zbl 1310.54074)] introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle. In this paper, we introduce a notion of generalized \(F\)-contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spaces. Some theorems on invariant approximations in normed linear spaces are also deduced. Our results extend, unify, and generalize comparable results in the literature.On a fixed point theorem in some nonlocally convex spaceshttps://zbmath.org/1472.470422021-11-25T18:46:10.358925Z"Bayoumi, Aboubakr"https://zbmath.org/authors/?q=ai:bayoumi.aboubakr"Ezat, Ibrahim"https://zbmath.org/authors/?q=ai:ezat.ibrahim(no abstract)Existence of coexistence states for systems of equations in ordered Banach spaceshttps://zbmath.org/1472.470432021-11-25T18:46:10.358925Z"El Khannoussi, Mohammed Said"https://zbmath.org/authors/?q=ai:khannoussi.mohammed-said-el"Zertiti, Abderrahim"https://zbmath.org/authors/?q=ai:zertiti.abderrahimSummary: In this paper we give some sufficient conditions for the existence of coexistence states to systems of the form
\[
\begin{aligned} x&=F_1(x,y), \\
y&=F_2(x,y), \end{aligned}
\]
where \(F_1\) and \(F_2\) satisfy some conditions.Leray-Schauder-type fixed point theorems in Banach algebras and application to quadratic integral equationshttps://zbmath.org/1472.470442021-11-25T18:46:10.358925Z"Khchine, Abdelmjid"https://zbmath.org/authors/?q=ai:khchine.abdelmjid"Maniar, Lahcen"https://zbmath.org/authors/?q=ai:maniar.lahcen"Taoudi, Mohamed-Aziz"https://zbmath.org/authors/?q=ai:taoudi.mohamed-azizThe authors of the paper under review study some fixed point results in Banach algebras relative to the weak topology under Leray-Schauder-type boundary conditions. They establish existence of fixed points results for nonlinear operators in Banach algebras relative to the weak topology. To illustrate the results obtained, the authors study the existence of continuous solutions of nonlinear quadratic integral equations.On sufficient conditions for the existence of past-present-future dependent fixed point in the Razumikhin class and applicationhttps://zbmath.org/1472.470452021-11-25T18:46:10.358925Z"Kutbi, Marwan Amin"https://zbmath.org/authors/?q=ai:kutbi.marwan-amin"Sintunavarat, Wutiphol"https://zbmath.org/authors/?q=ai:sintunavarat.wutipholSummary: We introduce the new type of nonself mapping and study sufficient conditions for the existence of past-present-future (for short PPF) dependent fixed point for such mapping in the Razumikhin class. Also, we apply our result to prove the PPF dependent coincidence point theorems. Finally, we use PPF dependence techniques to obtain solution for a nonlinear integral problem with delay.Fixed point theorems for inward mappings in \(\mathbb{R}\)-treeshttps://zbmath.org/1472.470462021-11-25T18:46:10.358925Z"Markin, Jack"https://zbmath.org/authors/?q=ai:markin.jack-t"Shahzad, Naseer"https://zbmath.org/authors/?q=ai:shahzad.naseerSummary: In an \(\mathbb R\)-tree setting, we develop fixed point theorems for multivalued mappings that are stricly contractiove, nonexpansive or upper continuous and satisfy an inward condition. As applications, we obtain common fixed point theorems for a point-valued and a multivalued mapping that commute.\(\mathcal{J} \mathcal{H}\)-operator pairs with application to functional equations arising in dynamic programminghttps://zbmath.org/1472.470472021-11-25T18:46:10.358925Z"Razani, A."https://zbmath.org/authors/?q=ai:razani.abdolrahman"Moeini, B."https://zbmath.org/authors/?q=ai:moeini.bahmanSummary: Some common fixed point theorems for \(\mathcal{J} \mathcal{H}\)-operator pairs are proved. As an application, the existence and uniqueness of the common solution for systems of functional equations arising in dynamic programming are discussed. Also, an example to validate all the conditions of the main result is presented.A fixed point theorem for nonautonomous type superposition operators and integrable solutions of a general nonlinear functional integral equationhttps://zbmath.org/1472.470482021-11-25T18:46:10.358925Z"Wang, Fuli"https://zbmath.org/authors/?q=ai:wang.fuliSummary: We first establish a new fixed point theorem for nonautonomous type superposition operators. After that, we prove the existence of integrable solutions for a general nonlinear functional integral equation in an \(L^1\) space on an unbounded interval by using our theorem. Our main tool is the measure of weak noncompactness.Existence and approximations for order-preserving nonexpansive semigroups over \(\mathrm{CAT}(\kappa)\) spaceshttps://zbmath.org/1472.470492021-11-25T18:46:10.358925Z"Chaipunya, Parin"https://zbmath.org/authors/?q=ai:chaipunya.parinSummary: In this paper, we discuss the fixed point property for an infinite family of order-preserving mappings which satisfy the Lipschitz condition on comparable pairs. The underlying framework of our main results is a metric space of any global upper curvature bound \(\kappa\in\mathbb{R}\), i.e., a \(\mathrm{CAT}(\kappa)\) space. In particular, we prove the existence of a fixed point for a nonexpansive semigroup on comparable pairs. Then, we propose and analyze two algorithms to approximate such a fixed point.
For the entire collection see [Zbl 1470.47001].On the continuity of superposition operators in the space of functions of bounded variationhttps://zbmath.org/1472.470502021-11-25T18:46:10.358925Z"Maćkowiak, Piotr"https://zbmath.org/authors/?q=ai:mackowiak.piotrGiven a function \(f:\mathbb{R}\to\mathbb{R}\), the (autonomous) superposition operator \(F\) generated by \(f\) is defined by \(F(x)= f\circ x\), where \(x: [0,1]\to\mathbb{R}\) runs over some function space \(X\). A basic problem is to find a condition on \(f\), both necessary and sufficient, under which \(F\) maps \(X\) into itself. For example, for \(X= C[0,1]\) this condition is \(f\in C(\mathbb{R})\), while for \(X= \text{Lip}[0,1]\) or \(X= \text{BV}[0,1]\) it is \(f\in\text{Lip}_{\text{loc}}(\mathbb{R})\). Of course, it is also important to have criteria for the boundedness or continuity of \(F\) in \(X\). Boundedness is guaranteed often ``for free'', while continuity may be a delicate issue.
Thus, for \(X= C[0,1]\) the operator is always continuous whenever \(f\) is continuous. In \(X= \text{Lip}[0,1]\), the operator \(F\) is always bounded, but for \(f(u):=\min\{|u|,1\}\) it is not continuous, as was shown by \textit{M. Z. Berkolajko} in a local Russian journal in 1969. Interestingly, \textit{M. Goebel} and \textit{F. Sachweh} [Z. Anal. Anwend. 18, No. 2, 205--229 (1999; Zbl 0941.47053)] proved that \(f\in C^1(\mathbb{R})\) is both necessary and sufficient for the continuity of \(F\) in \(\text{Lip}[0,1]\).
The continuity problem for \(F\) in \(\text{BV}[0,1]\) is surprisingly hard, and it remained open for many decades. \textit{A. P. Morse} [Trans. Am. Math. Soc. 41, 48--83 (1937; Zbl 0016.10502)] claimed that the condition \(F(\text{BV})\subseteq\text{BV}\) implies the continuity of \(F\). However, the proof is 30 pages long and extremely technical, and its correctness is more than doubtful. It is the author's merit to prove this in the paper under review with a much shorter, though still quite technical, proof. We remark that meanwhile a very short and elegant alternative proof has been given by \textit{S. Reinwand} [Real Anal. Exch. 45, No. 1, 173--204 (2020; Zbl 1440.26004)].On the convergence of Ulm's method for equations with regularly smooth operatorshttps://zbmath.org/1472.470512021-11-25T18:46:10.358925Z"Tanygina, A. N."https://zbmath.org/authors/?q=ai:tanygina.a-nSummary: The article deals with Ulm's method for solving nonlinear operator equations in Banach spaces under the regular smoothness assumption of the operator involved. The convergence theorem is proved and the error bounds for the method are obtained.Existence theorems of generalized quasi-variational-like inequalities for pseudo-monotone type II operatorshttps://zbmath.org/1472.470522021-11-25T18:46:10.358925Z"Chowdhury, Mohammad S. R."https://zbmath.org/authors/?q=ai:chowdhury.mohammad-showkat-rahim"Abdou, Afrah A. N."https://zbmath.org/authors/?q=ai:abdou.afrah-ahmad-noan"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: In this paper, we prove the existence results of solutions for a new class of generalized quasi-variational-like inequalities (GQVLI) for pseudo-monotone type II operators defined on compact sets in locally convex Hausdorff topological vector spaces. In obtaining our results on GQVLI for pseudo-monotone type II operators, we use the generalized version of the first and the third author [J. Inequal. Appl. 2012, Paper No. 79, 12 p. (2012; Zbl 1446.47056)] of \textit{K. Fan's} minimax inequality [in: Proceedings of the 3rd symposium on inequalities, Inequalities III. New York, NY , London: Academic Press. 103--113 (1972; Zbl 0302.49019)] as the main tool.Levitin-Polyak well-posedness of an equilibrium-like problem in Banach spaceshttps://zbmath.org/1472.470532021-11-25T18:46:10.358925Z"Deng, Ru-liang"https://zbmath.org/authors/?q=ai:deng.ruliangSummary: The concept of Levitin-Polyak well-posedness of an equilibrium-like problem in Banach spaces is introduced. Under suitable conditions, some characterizations of its Levitin-Polyak well-posedness are established. Some conditions under which an equilibrium-like problem in Banach spaces is Levitin-Polyak well-posed are also derived.On the existence and essential components of solution sets for systems of generalized quasi-variational relation problemshttps://zbmath.org/1472.470542021-11-25T18:46:10.358925Z"Nguyen Van Hung"https://zbmath.org/authors/?q=ai:nguyen-van-hung."Phan Thanh Kieu"https://zbmath.org/authors/?q=ai:phan-thanh-kieu.Summary: In this paper, we study the existence of a solution for a system of quasi-variational relation problems (in short, (SQVR)). Moreover, we discuss the existence of essentially connected components of the solution set for (SQVR). Then the obtained results are applied to systems of quasi-variational inclusions and to systems of weak vector quasi-equilibrium problems. The results presented in the paper improve and extend many results from the literature. Some examples are given to illustrate our results.Extended mixed vector equilibrium problemshttps://zbmath.org/1472.470552021-11-25T18:46:10.358925Z"Rahaman, Mijanur"https://zbmath.org/authors/?q=ai:rahaman.mijanur"Kılıçman, Adem"https://zbmath.org/authors/?q=ai:kilicman.adem"Ahmad, Rais"https://zbmath.org/authors/?q=ai:ahmad.raisSummary: We study extended mixed vector equilibrium problems, namely, extended weak mixed vector equilibrium problem and extended strong mixed vector equilibrium problem in Hausdorff topological vector spaces. Using generalized KKM-Fan theorem [\textit{H. Ben-El-Mechaiekh} et al., J. Math. Anal. Appl. 309, No. 2, 583--590 (2005; Zbl 1080.54028)], some existence results for both problems are proved in noncompact domain.General variational inclusions involving difference of operatorshttps://zbmath.org/1472.470562021-11-25T18:46:10.358925Z"Noor, Muhammad Aslam"https://zbmath.org/authors/?q=ai:noor.muhammad-aslam"Noor, Khalida Inayat"https://zbmath.org/authors/?q=ai:noor.khalida-inayat"Kamal, Rabia"https://zbmath.org/authors/?q=ai:kamal.rabiaSummary: In this paper, we introduce and consider a new class of general variational inclusions involving the difference of operators in a Hilbert space. We establish the equivalence between the general variational inclusions and the fixed point problems as well as with a new class of resolvent equations using the resolvent operator technique. We use this alternative formulation to discuss the existence of a solution of the general variational inclusions. We again use this alternative equivalent formulation to suggest and analyze a number of iterative methods for finding a zero of the difference of operators. We also discuss the convergence of the iterative method under suitable conditions. Our methods of proofs are very simple as compared with other techniques. Several special cases of these problems are also considered. The results proved in this paper may be viewed as a refinement and an improvement of the known results in this area.New general systems of set-valued variational inclusions involving relative \((A, \eta)\)-maximal monotone operators in Hilbert spaceshttps://zbmath.org/1472.470572021-11-25T18:46:10.358925Z"Xiong, Ting-jian"https://zbmath.org/authors/?q=ai:xiong.tingjian"Lan, Heng-you"https://zbmath.org/authors/?q=ai:lan.hengyouSummary: The purpose of this paper is to introduce and study a class of new general systems of set-valued variational inclusions involving relative \((A, \eta)\)-maximal monotone operators in Hilbert spaces. By using the generalized resolvent operator technique associated with relative \((A, \eta)\)-maximal monotone operators, we also construct some new iterative algorithms for finding approximation solutions to the general systems of set-valued variational inclusions and prove the convergence of the sequences generated by the algorithms. The results presented in this paper improve and extend some known results in the literature.On solutions of variational inequality problems via iterative methodshttps://zbmath.org/1472.470582021-11-25T18:46:10.358925Z"Alghamdi, Mohammed Ali"https://zbmath.org/authors/?q=ai:alghamdi.mohammed-ali"Shahzad, Naseer"https://zbmath.org/authors/?q=ai:shahzad.naseer"Zegeye, Habtu"https://zbmath.org/authors/?q=ai:zegeye.habtuSummary: We investigate an algorithm for a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of \(\gamma\)-inverse strongly accretive mappings. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings.Common solution to generalized general variational-like inequality and hierarchical fixed point problemshttps://zbmath.org/1472.470592021-11-25T18:46:10.358925Z"Ali, Rehan"https://zbmath.org/authors/?q=ai:ali.rehan"Shahzad, Mohammad"https://zbmath.org/authors/?q=ai:shahzad.mohammadSummary: This paper deals with a strong convergence theorem for a hybrid iterative algorithm to approximate the common solution of generalized general variational-like inequality problem for generalized relaxed \(\alpha\)-monotone mapping and hierarchical fixed point problem for nonexpansive mapping in real Hilbert spaces. Some consequences of the strong convergence theorem are also derived. Finally, we give a numerical example to justify the main result. The method and results presented in this paper generalize and unify previously known corresponding results of this area.
For the entire collection see [Zbl 1455.53022].Generalized mixed equilibria, variational inclusions, and fixed point problemshttps://zbmath.org/1472.470602021-11-25T18:46:10.358925Z"Al-Mazrooei, A. E."https://zbmath.org/authors/?q=ai:al-mazrooei.abdullah-eqal"Alofi, A. S. M."https://zbmath.org/authors/?q=ai:alofi.abdulaziz-saleem-moslem"Latif, A."https://zbmath.org/authors/?q=ai:latif.abdul"Yao, J.-C."https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: We propose two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and an asymptotically \(\kappa\)-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions.Iterative scheme for solving a class of set-valued variational inequality problemshttps://zbmath.org/1472.470612021-11-25T18:46:10.358925Z"Al-Shemas, Eman"https://zbmath.org/authors/?q=ai:al-shemas.eman-hamad(no abstract)Relaxed iterative algorithms for generalized mixed equilibrium problems with constraints of variational inequalities and variational inclusionshttps://zbmath.org/1472.470622021-11-25T18:46:10.358925Z"Ceng, Lu-Chuan"https://zbmath.org/authors/?q=ai:ceng.lu-chuan"Chen, Chi-Ming"https://zbmath.org/authors/?q=ai:chen.chiming|chen.chi-ming"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfeng"Pang, Chin-Tzong"https://zbmath.org/authors/?q=ai:pang.chin-tzongSummary: We introduce and analyze a relaxed extragradient-like viscosity iterative algorithm for finding a solution of a generalized mixed equilibrium problem with constraints of several problems: a finite family of variational inequalities for inverse strongly monotone mappings, a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings, and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we derive the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems which also solves a variational inequality problem.Hybrid extragradient method with regularization for convex minimization, generalized mixed equilibrium, variational inequality and fixed point problemshttps://zbmath.org/1472.470632021-11-25T18:46:10.358925Z"Ceng, Lu-Chuan"https://zbmath.org/authors/?q=ai:ceng.lu-chuan"Ho, Juei-Ling"https://zbmath.org/authors/?q=ai:ho.juei-lingSummary: We introduce two iterative algorithms by the hybrid extragradient method with regularization for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings and the set of fixed points of an asymptotically \(\kappa\)-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions.Iterative algorithms for systems of generalized equilibrium problems with the constraints of variational inclusion and fixed point problemshttps://zbmath.org/1472.470642021-11-25T18:46:10.358925Z"Ceng, Lu-Chuan"https://zbmath.org/authors/?q=ai:ceng.lu-chuan"Latif, Abdul"https://zbmath.org/authors/?q=ai:latif.abdul"Al-Mazrooei, Abdullah E."https://zbmath.org/authors/?q=ai:al-mazrooei.abdullah-eqalSummary: We introduce and analyze a hybrid extragradient-like viscosity iterative algorithm for finding a common solution of a systems of generalized equilibrium problems and a generalized mixed equilibrium problem with the constraints of two problems: a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we prove the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems.Iterative schemes for convex minimization problems with constraintshttps://zbmath.org/1472.470652021-11-25T18:46:10.358925Z"Ceng, Lu-Chuan"https://zbmath.org/authors/?q=ai:ceng.lu-chuan"Liao, Cheng-Wen"https://zbmath.org/authors/?q=ai:liao.cheng-wen"Pang, Chin-Tzong"https://zbmath.org/authors/?q=ai:pang.chin-tzong"Wen, Ching-Feng"https://zbmath.org/authors/?q=ai:wen.chingfengSummary: We first introduce and analyze one implicit iterative algorithm for finding a solution of the minimization problem for a convex and continuously Fréchet differentiable functional, with constraints of several problems: the generalized mixed equilibrium problem, the system of generalized equilibrium problems, and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.Hybrid algorithms for solving variational inequalities, variational inclusions, mixed equilibria, and fixed point problemshttps://zbmath.org/1472.470662021-11-25T18:46:10.358925Z"Ceng, Lu-Chuan"https://zbmath.org/authors/?q=ai:ceng.lu-chuan"Petrusel, Adrian"https://zbmath.org/authors/?q=ai:petrusel.adrian"Wong, Mu-Ming"https://zbmath.org/authors/?q=ai:wong.mu-ming"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: We present a hybrid iterative algorithm for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed hybrid iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters. Here, our hybrid algorithm is based on Korpelevič's extragradient method, hybrid steepest-descent method, and viscosity approximation method.Strong convergence theorems of general split equality problems for quasi-nonexpansive mappingshttps://zbmath.org/1472.470672021-11-25T18:46:10.358925Z"Chang, Shih-sen"https://zbmath.org/authors/?q=ai:chang.shih-sen"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pSummary: The purpose of this paper is to introduce and study the general split equality problem and general split equality fixed point problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequences generated by the proposed new algorithm converges strongly to a solution of the general split equality fixed point problem and the general split equality problem for quasi-nonexpansive mappings in Hilbert spaces. As an application, we shall utilize our results to study the null point problem of maximal monotone operators, the split feasibility problem, and the equality equilibrium problem. The results presented in the paper extend and improve the corresponding results announced by \textit{A. Moudafi} [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 79, 117--121 (2013; Zbl 1256.49044)], \textit{M. Eslamian} and \textit{A. Latif} [Abstr. Appl. Anal. 2013, Article ID 805104, 6 p. (2013; Zbl 1266.65091)], \textit{R.-D. Chen} et al. [Fixed Point Theory Appl. 2014, Paper No. 35, 8 p. (2014; Zbl 1345.90106)], \textit{Y. Censor}, \textit{T. Elfving} [Numer. Algorithms 8, No. 2--4, 221--239 (1994; Zbl 0828.65065)], \textit{Y. Censor} and \textit{A. Segal} [J. Convex Anal. 16, No. 2, 587--600 (2009; Zbl 1189.65111)], and some others.Convergence theorems for modified generalized \(f\)-projections and generalized nonexpansive mappingshttps://zbmath.org/1472.470682021-11-25T18:46:10.358925Z"Cheng, Qingqing"https://zbmath.org/authors/?q=ai:cheng.qingqing"Su, Yongfu"https://zbmath.org/authors/?q=ai:su.yongfu"Zhang, Jingling"https://zbmath.org/authors/?q=ai:zhang.jinglingSummary: The purpose of this paper is to study a sequence of modified generalized \(f\)-projections in a reflexive, smooth, and strictly convex Banach space and show that Mosco convergence of their ranges implies their pointwise convergence to the generalized \(f\)-projection onto the limit set. Furthermore, we prove a strong convergence theorem for a countable family of \(\alpha\)-nonexpansive mappings in a uniformly convex and smooth Banach space using the properties of a modified generalized \(f\)-projection operator. Our main results generalize the results of \textit{Z.-M. Wang} et al. [J. Nonlinear Sci. Appl. 5, No. 1, 56--63 (2012; Zbl 1439.47058)] and enrich the research contents of \(\alpha\)-nonexpansive mappings.A new iterative scheme of modified Mann iteration in Banach spacehttps://zbmath.org/1472.470692021-11-25T18:46:10.358925Z"Chen, Jinzuo"https://zbmath.org/authors/?q=ai:chen.jinzuo"Wu, Dingping"https://zbmath.org/authors/?q=ai:wu.dingping"Zhang, Caifen"https://zbmath.org/authors/?q=ai:zhang.caifenSummary: We introduce the modified iterations of Mann's type for nonexpansive mappings and asymptotically nonexpansive mappings to have the strong convergence in a uniformly convex Banach space. We study approximation of common fixed point of asymptotically nonexpansive mappings in Banach space by using a new iterative scheme. Applications to the accretive operators are also included.The hybrid steepest descent method for split variational inclusion and constrained convex minimization problemshttps://zbmath.org/1472.470702021-11-25T18:46:10.358925Z"Deepho, Jitsupa"https://zbmath.org/authors/?q=ai:deepho.jitsupa"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poomSummary: We introduced an implicit and an explicit iteration method based on the hybrid steepest descent method for finding a common element of the set of solutions of a constrained convex minimization problem and the set of solutions of a split variational inclusion problem.A hybrid iteration for asymptotically strictly pseudocontractive mappingshttps://zbmath.org/1472.470712021-11-25T18:46:10.358925Z"Dewangan, Rajshree"https://zbmath.org/authors/?q=ai:dewangan.rajshree"Thakur, Balwant Singh"https://zbmath.org/authors/?q=ai:singh.thakur-balwant"Postolache, Mihai"https://zbmath.org/authors/?q=ai:postolache.mihaiSummary: In this paper, we propose a new hybrid iteration for a finite family of asymptotically strictly pseudocontractive mappings. We also prove that such a sequence converges strongly to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings. Results in the paper extend and improve recent results in the literature.Algorithmic approach to the equilibrium points and fixed pointshttps://zbmath.org/1472.470722021-11-25T18:46:10.358925Z"Guo, Lijin"https://zbmath.org/authors/?q=ai:guo.lijin"Kang, Shin Min"https://zbmath.org/authors/?q=ai:kang.shin-min"Kwun, Young Chel"https://zbmath.org/authors/?q=ai:kwun.young-chelSummary: The equilibrium and fixed point problems are considered. An iterative algorithm is presented. Convergence analysis of the algorithm is provided.Strong convergence for hybrid implicit \(S\)-iteration scheme of nonexpansive and strongly pseudocontractive mappingshttps://zbmath.org/1472.470732021-11-25T18:46:10.358925Z"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: Let \(K\) be a nonempty closed convex subset of a real Banach space \(E\), let \(S : K \rightarrow K\) be nonexpansive, and let \(T : K \rightarrow K\) be Lipschitz strongly pseudocontractive mappings such that \(p \in F \left(S\right) \cap F \left(T\right) = \left\{x \in K : S x = T x = x\right\}\) and \(\| x-Sy\| \leq \| x-Sy\|\) and \(\|x-Ty\| \leq \|Tx-Ty\|\) for all \(x, y \in K\). Let \(\left\{\beta_n\right\}\) be a sequence in \(\left[0, 1\right]\) satisfying (i) \(\sum_{n = 1}^\infty \beta_n = \infty\); (ii) \(\lim_{n \rightarrow \infty} \beta_n = 0 \). For arbitrary \(x_0 \in K\), let \(\left\{x_n\right\}\) be a sequence iteratively defined by \(x_n = S y_n, y_n = \left(1 - \beta_n\right) x_{n - 1} + \beta_n T x_n, n \geq 1 \). Then the sequence \(\left\{x_n\right\}\) converges strongly to a common fixed point \(p\) of \(S\) and \(T\).Fixed point approximation of nonexpansive mappings on a nonlinear domainhttps://zbmath.org/1472.470742021-11-25T18:46:10.358925Z"Khan, Safeer Hussain"https://zbmath.org/authors/?q=ai:khan.safeer-hussainSummary: We use a three-step iterative process to prove some strong and \(\Delta\)-convergence results for nonexpansive mappings in a uniformly convex hyperbolic space, a nonlinear domain. Three-step iterative processes have numerous applications and hyperbolic spaces contain Banach spaces (linear domains) as well as CAT(0) spaces. Thus our results can be viewed as extension and generalization of several known results in uniformly convex Banach spaces as well as CAT(0) spaces.An implicitly defined iterative sequence for monotone operators in Banach spaceshttps://zbmath.org/1472.470752021-11-25T18:46:10.358925Z"Kohsaka, Fumiaki"https://zbmath.org/authors/?q=ai:kohsaka.fumiakiSummary: Given a monotone operator in a Banach space, we show that an iterative sequence, which is implicitly defined by a fixed point theorem for mappings of firmly nonexpansive type, converges strongly to a minimum norm zero point of the given operator. Applications to a convex minimization problem and a variational inequality problem are also included.Iterative algorithms for mixed equilibrium problems, system of quasi-variational inclusion, and fixed point problem in Hilbert spaceshttps://zbmath.org/1472.470762021-11-25T18:46:10.358925Z"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poom"Jitpeera, Thanyarat"https://zbmath.org/authors/?q=ai:jitpeera.thanyaratSummary: We introduce a new iterative algorithm for approximating a common element of the set of solutions for mixed equilibrium problems, the set of solutions of a system of quasi-variational inclusion, and the set of fixed points of an infinite family of nonexpansive mappings in a real Hilbert space. Strong convergence of the proposed iterative algorithm is obtained. Our results generalize, extend, and improve the results of \textit{J.-W. Peng} and \textit{J.-C. Yao} [Math. Comput. Modelling 49, No. 9--10, 1816--1828 (2009; Zbl 1171.90542)], \textit{X.-L. Qin} et al. [Comput. Appl. Math. 29, No. 3, 393--421 (2010; Zbl 1216.47099)] and others.On variational inequality, fixed point and generalized mixed equilibrium problemshttps://zbmath.org/1472.470772021-11-25T18:46:10.358925Z"Li, Dong Feng"https://zbmath.org/authors/?q=ai:li.dongfeng"Zhao, Juan"https://zbmath.org/authors/?q=ai:zhao.juanSummary: In this article, variational inequality, fixed point, and generalized mixed equilibrium problems are investigated based on an extragradient iterative algorithm. Weak convergence of the extragradient iterative algorithm is obtained in Hilbert spaces.Convergence theorems on total asymptotically demicontractive and hemicontractive mappings in CAT(0) spaceshttps://zbmath.org/1472.470782021-11-25T18:46:10.358925Z"Liu, Xin-dong"https://zbmath.org/authors/?q=ai:liu.xindong"Chang, Shih-sen"https://zbmath.org/authors/?q=ai:chang.shih-senSummary: The purpose of this paper is to introduce the concepts of \textit{total asymptotically demicontractive mappings} and \textit{total asymptotically hemicontractive mappings}. Under suitable conditions some strong convergence theorems for these two kinds of mappings to converge to their fixed points in \textit{CAT(0) space} are proved. The results presented in the paper extend and improve some recent results announced in the current literature.Viscosity projection algorithms for pseudocontractive mappings in Hilbert spaceshttps://zbmath.org/1472.470792021-11-25T18:46:10.358925Z"Pan, Xiujuan"https://zbmath.org/authors/?q=ai:pan.xiujuan"Kang, Shin Min"https://zbmath.org/authors/?q=ai:kang.shin-min"Kwun, Young Chel"https://zbmath.org/authors/?q=ai:kwun.young-chelSummary: An explicit projection algorithm with viscosity technique is constructed for finding the fixed points of the pseudocontractive mapping in Hilbert spaces. Strong convergence theorem is demonstrated. Consequently, as an application, we can approximate to the minimum-norm fixed point of the pseudocontractive mapping.Weak and strong convergence theorems for equilibrium problems and countable strict pseudocontraction mappings in Hilbert spacehttps://zbmath.org/1472.470802021-11-25T18:46:10.358925Z"Ram, Tirth"https://zbmath.org/authors/?q=ai:ram.tirthSummary: In this paper, we intend to introduce two iterative sequence for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of strict pseudocontractions in Hilbert space. Then we study the weak and strong convergence of the sequences.Strong convergence of iterative algorithms for the split equality problemhttps://zbmath.org/1472.470812021-11-25T18:46:10.358925Z"Shi, Luo Yi"https://zbmath.org/authors/?q=ai:shi.luoyi"Chen, Rudong"https://zbmath.org/authors/?q=ai:chen.rudong"Wu, Yujing"https://zbmath.org/authors/?q=ai:wu.yujingSummary: Let \(H_1, H_2, H_3\) be real Hilbert spaces, \(C \subseteq H_1\), \(Q \subseteq H_2\) be two nonempty closed convex sets, and let \(A : H_1 \to H_3\), \(B : H_2 \to H_3\) be two bounded linear operators. The split equality problem (SEP) is finding \(x \in C\), \(y \in Q\) such that \(Ax = By\). Recently, Moudafi has presented the ACQA algorithm and the RACQA algorithm to solve SEP [\textit{A. Moudafi}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 79, 117--121 (2013; Zbl 1256.49044)]. However, the two algorithms are weakly convergent. It is therefore the aim of this paper to construct new algorithms for SEP so that strong convergence is guaranteed. Firstly, we define the concept of the minimal norm solution of SEP. Using Tychonov regularization, we introduce two methods to get such a minimal norm solution. And then, we introduce two algorithms which are viewed as modifications of Moudafi's ACQA, RACQA algorithms and KM-CQ algorithm, respectively, and converge strongly to a solution of SEP. More importantly, the modifications of Moudafi's ACQA, RACQA algorithms converge strongly to the minimal norm solution of SEP. At last, we introduce some other algorithms which converge strongly to a solution of SEP.On strong convergence of an iterative algorithm for common fixed point and generalized equilibrium problemshttps://zbmath.org/1472.470822021-11-25T18:46:10.358925Z"Song, Jian-Min"https://zbmath.org/authors/?q=ai:song.jianminSummary: In this article, an iterative algorithm for finding a common element in the solution set of generalized equilibrium problems and in the common fixed point set of a family of nonexpansive mappings. Strong convergence of the algorithm is established in the framework of Hilbert spaces.A modified extragradient method for variational inclusion and fixed point problems in Banach spaceshttps://zbmath.org/1472.470832021-11-25T18:46:10.358925Z"Sunthrayuth, Pongsakorn"https://zbmath.org/authors/?q=ai:sunthrayuth.pongsakorn"Cholamjiak, Prasit"https://zbmath.org/authors/?q=ai:cholamjiak.prasitSummary: In this work, we introduce a modified extragradient method for solving the fixed point problem of a nonexpansive mapping and the variational inclusion problem for two accretive operators in the framework of Banach spaces. We then prove its strong convergence under certain assumptions imposed on the parameters. As applications, we apply our main result to the variational inequality problem, split feasibility problem and the LASSO problem.Viscosity approximation methods for a family of nonexpansive mappings in CAT(0) spaceshttps://zbmath.org/1472.470842021-11-25T18:46:10.358925Z"Tang, Jinfang"https://zbmath.org/authors/?q=ai:tang.jinfangSummary: The purpose of this paper is using the viscosity approximation method to study the strong convergence problem for a family of nonexpansive mappings in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the proposed implicit and explicit iterative schemes to converge to a common fixed point of the family of nonexpansive mappings are proved which is also a unique solution of some kind of variational inequalities. The results presented in this paper extend and improve the corresponding results of some others.An algorithm with general errors for the zero point of monotone mappings in Banach spaceshttps://zbmath.org/1472.470852021-11-25T18:46:10.358925Z"Tang, Yan"https://zbmath.org/authors/?q=ai:tang.yan"Bao, Zhiqing"https://zbmath.org/authors/?q=ai:bao.zhiqing"Wen, Daojun"https://zbmath.org/authors/?q=ai:wen.daojunSummary: In this paper, we introduce a proximal point iterative algorithm with general errors for monotone mappings in Banach spaces. We prove that the proposed algorithm converges strongly to a proximal point for monotone mappings. Our theorems in this paper improve and unify most of the results that have been proposed for this important class of nonlinear mappings.Projection methods for a system of nonlinear mixed variational inequalities in Banach spaceshttps://zbmath.org/1472.470862021-11-25T18:46:10.358925Z"Wang, Zhong-Bao"https://zbmath.org/authors/?q=ai:wang.zhongbao"Tang, Guo-Ji"https://zbmath.org/authors/?q=ai:tang.guoji"Zhang, Hong-Ling"https://zbmath.org/authors/?q=ai:zhang.honglingSummary: The existence and uniqueness of solution for a system of nonlinear mixed variational inequality in Banach spaces is given firstly. A Mann iterative sequences with errors for this system of nonlinear mixed variational inequalities in Banach spaces is studied, by using the generalized \(f\)-projection operator \(\pi_K^f\). Our results extend the main results in [\textit{R. U. Verma}, Appl. Math. Lett. 18, No. 11, 1286--1292 (2005; Zbl 1099.47054)]; Comput. Math. Appl. 41, No. 7--8, 1025--1031 (2001; Zbl 0995.47042)] from Hilbert spaces to Banach spaces.Iterative schemes for finite families of maximal monotone operators based on resolventshttps://zbmath.org/1472.470872021-11-25T18:46:10.358925Z"Wei, Li"https://zbmath.org/authors/?q=ai:wei.li"Tan, Ruilin"https://zbmath.org/authors/?q=ai:tan.ruilinSummary: The purpose of this paper is to present two iterative schemes based on the relative resolvent and the generalized resolvent, respectively. And, it is shown that the iterative schemes converge weakly to common solutions for two finite families of maximal monotone operators in a real smooth and uniformly convex Banach space and one example is demonstrated to explain that some assumptions in the main results are meaningful, which extend the corresponding works by some authors.Approximations for equilibrium problems and nonexpansive semigroupshttps://zbmath.org/1472.470882021-11-25T18:46:10.358925Z"Wu, Huan-chun"https://zbmath.org/authors/?q=ai:wu.huanchun"Cheng, Cao-zong"https://zbmath.org/authors/?q=ai:cheng.cao-zongSummary: We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of all common fixed points of a nonexpansive semigroup and prove the strong convergence theorem in Hilbert spaces. Our result extends the recent result of \textit{H. Zegeye} and \textit{N. Shahzad} [Fixed Point Theory Appl. 2013, Paper No. 1, 12 p. (2013; Zbl 1423.47055)]. In the last part of the paper, by the way, we point out that there is a slight flaw in the proof of the main result in [\textit{Y. Shehu}, J. Glob. Optim. 52, No. 1, 57--77 (2012; Zbl 1247.47073)] and perfect the proof.\(\Delta\)-convergence analysis of improved Kuhfittig iterative for asymptotically nonexpansive nonself-mappings in \(W\)-hyperbolic spaceshttps://zbmath.org/1472.470892021-11-25T18:46:10.358925Z"Yi, Li"https://zbmath.org/authors/?q=ai:yi.li"Bo, Liu Hong"https://zbmath.org/authors/?q=ai:bo.liu-hongSummary: Throughout this paper, we introduce a class of asymptotically nonexpansive nonself-mapping and modify the classical Kuhfittig iteration algorithm in the general setup of hyperbolic space. Also, convergence results are obtained under a limit condition. The results presented in the paper extend various results in the existing literature.Strong convergence on iterative methods of Cesàro means for nonexpansive mapping in Banach spacehttps://zbmath.org/1472.470902021-11-25T18:46:10.358925Z"Zhu, Zhichuan"https://zbmath.org/authors/?q=ai:zhu.zhichuan"Chen, Rudong"https://zbmath.org/authors/?q=ai:chen.rudongSummary: Two new iterations with Cesàro's means for nonexpansive mappings are proposed and the strong convergence is obtained as \(n \rightarrow \infty\). Our main results extend and improve the corresponding results of \textit{H.-K. Xu} [J. Math. Anal. Appl. 298, No. 1, 279--291 (2004; Zbl 1061.47060)], \textit{Y.-S. Song} and the second author [Appl. Math. Comput. 186, No. 2, 1120--1128 (2007; Zbl 1121.65063)], and \textit{Y.-H. Yao} et al. [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 6, 2332--2336 (2009; Zbl 1223.47107)].A general iterative algorithm for monotone operators with \(\lambda\)-hybrid mappings in Hilbert spaceshttps://zbmath.org/1472.470912021-11-25T18:46:10.358925Z"Hong, Chung-Chien"https://zbmath.org/authors/?q=ai:hong.chung-chienSummary: Let \(C\) be a nonempty closed convex subset of a Hilbert space \(\mathcal H\), let \(B\), \(G\) be two set-valued maximal monotone operators on \(C\) into \(\mathcal H\), and let \(g:\mathcal H \to\mathcal H\) be a \(k\)-contraction with \(0<k<1\). \(A:C\to\mathcal H\) is an \(\alpha\)-inverse strongly monotone mapping, \(V:\mathcal H\to\mathcal H\) is a \(\overline\gamma\)-strongly monotone and \(L\)-Lipschitzian mapping with \(\overline\gamma>0\) and \(L>0\), \(T:C\to C\) is a \(\lambda\)-hybrid mapping. In this paper, a general iterative scheme for approximating a point of \(F(T)\cap(A+B)^{-1}0\cap G^{-1}0\neq \emptyset\) is introduced, where \(F(T)\) is the set of fixed points of \(T\), and a strong convergence theorem of the sequence generated by the iterative scheme is proved under suitable conditions. As applications of our strong convergence theorem, the related equilibrium and variational problems are also studied.A strong convergence algorithm for the two-operator split common fixed point problem in Hilbert spaceshttps://zbmath.org/1472.470922021-11-25T18:46:10.358925Z"Hong, Chung-Chien"https://zbmath.org/authors/?q=ai:hong.chung-chien"Huang, Young-Ye"https://zbmath.org/authors/?q=ai:huang.young-yeSummary: The two-operator split common fixed point problem (two-operator SCFP) with firmly nonexpansive mappings is investigated in this paper. This problem covers the problems of split feasibility, convex feasibility, and equilibrium and can especially be used to model significant image recovery problems such as the intensity-modulated radiation therapy, computed tomography, and the sensor network. An iterative scheme is presented to approximate the minimum norm solution of the two-operator SCFP problem. The performance of the presented algorithm is compared with that of the last algorithm for the two-operator SCFP and the advantage of the presented algorithm is shown through the numerical result.Approximating fixed points of Suzuki \((\alpha,\beta)\)-nonexpansive mappings in ordered hyperbolic metric spaceshttps://zbmath.org/1472.470932021-11-25T18:46:10.358925Z"Martínez-Moreno, Juan"https://zbmath.org/authors/?q=ai:martinez-moreno.juan"Calderón, Kenyi"https://zbmath.org/authors/?q=ai:calderon.kenyi"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poom"Rojas, Edixon"https://zbmath.org/authors/?q=ai:rojas.edixon-mSummary: In this chapter, we define the class of monotone \((\alpha,\beta)\)-nonexpansive mappings and prove that they have an approximate fixed point sequence in partially ordered hyperbolic metric spaces. We prove the \(\Delta\) and strong convergence of the CR-iteration scheme.
For the entire collection see [Zbl 1470.47001].Approximating fixed points for generalized nonexpansive mapping in CAT(0) spaceshttps://zbmath.org/1472.470942021-11-25T18:46:10.358925Z"Uddin, Izhar"https://zbmath.org/authors/?q=ai:uddin.izhar"Dalal, Sumitra"https://zbmath.org/authors/?q=ai:dalal.sumitra"Imdad, Mohammad"https://zbmath.org/authors/?q=ai:imdad.mohammadSummary: \textit{W. Takahashi} and \textit{G.-E. Kim} [Math. Japon. 48, No. 1, 1--9 (1998; Zbl 0913.47056)] used the Ishikawa iteration process to prove some convergence theorems for nonexpansive mappings in Banach spaces. The aim of this paper is to prove similar results in CAT(0) spaces for generalized nonexpansive mappings, which, in turn, generalize the corresponding results of Takahashi and Kim [loc.\,cit.], \textit{T. Laokul} and \textit{B. Panyanak} [Int. J. Math. Anal., Ruse 3, No. 25--28, 1305--1315 (2009; Zbl 1196.54077)], \textit{A. Razani} and \textit{H. Salahifard} [Bull. Iran. Math. Soc. 37, No. 1, 235--246 (2011; Zbl 1302.47096)], and some others.Some convergence theorems for a hybrid pair of generalized nonexpansive mappings in CAT(0) spaceshttps://zbmath.org/1472.470952021-11-25T18:46:10.358925Z"Uddin, Izhar"https://zbmath.org/authors/?q=ai:uddin.izhar"Imdad, Mohammad"https://zbmath.org/authors/?q=ai:imdad.mohammadSummary: In [Fixed Point Theory Appl. 2010, Article ID 618767, 9 p. (2010; Zbl 1208.47070)], \textit{K. Sokhuma} and \textit{A. Kaewkhao} introduced a modified Ishikawa iteration scheme for a pair of hybrid mappings in Banach spaces and utilized the same to prove psome convergence theorems. In this paper, we study the convergence of a modified Ishikawa iteration process involving a hybrid pair of generalised nonexpansive mappings in CAT(0) spaces. In process, the result of Sokhuma and Kaewkhao [loc. cit.], \textit{K. Sokhuma} et al. [Int. J. Math. Anal., Ruse 6, No. 17--20, 923--932 (2012; Zbl 1296.47091)] and \textit{I. Uddin} et al. [Bull. Malays. Math. Sci. Soc. (2) 38, No. 2, 695--705 (2015; Zbl 1312.54035)] are generalized and improved.Strong convergence algorithms of the split common fixed point problem for total quasi-asymptotically pseudocontractive operatorshttps://zbmath.org/1472.470962021-11-25T18:46:10.358925Z"Wang, Peiyuan"https://zbmath.org/authors/?q=ai:wang.peiyuan"Zhou, Hy"https://zbmath.org/authors/?q=ai:zhou.hySummary: We present a new algorithm for solving the two-set split common fixed point problem with total quasi-asymptotically pseudocontractive operators and consider the case of quasi-pseudocontractive operators. Under some appropriate conditions, we prove that the proposed algorithms have strong convergence. The results presented in this paper improve and extend the previous algorithms and results in [\textit{Y. Censor} and \textit{A. Segal}, J. Convex Anal. 16, No. 2, 587--600 (2009; Zbl 1189.65111); \textit{A. Moudafi}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 12, 4083--4087 (2011; Zbl 1232.49017); Inverse Probl. 26, No. 5, Article ID 055007, 6 p. (2010; Zbl 1219.90185); \textit{L. B. Mohammed} et al., ``Strong convergence of an algorithm about quasi-nonexpansive mappings for the split common fixed-pint problem in Hilbert space'', J. Nat. Sci. Res. 3, No. 7, 215--220 (2013), \url{https://www.iiste.org/Journals/index.php/JNSR/article/view/6434}; \textit{L. Yang} et al., Fixed Point Theory Appl. 2011, Paper No. 77, 11 p. (2011; Zbl 1311.47101); \textit{S.-S. Chang} et al., ibid. 2012, Article ID 491760, 12 p. (2012; Zbl 1234.47047)] and others.Strong convergence of a unified general iteration for \(k\)-strictly pseudononspreading mapping in Hilbert spaceshttps://zbmath.org/1472.470972021-11-25T18:46:10.358925Z"Wen, Dao-Jun"https://zbmath.org/authors/?q=ai:wen.daojun"Chen, Yi-An"https://zbmath.org/authors/?q=ai:chen.yian"Tang, Yan"https://zbmath.org/authors/?q=ai:tang.yanSummary: We introduce a unified general iterative method to approximate a fixed point of \(k\)-strictly pseudononspreading mapping. Under some suitable conditions, we prove that the iterative sequence generated by the proposed method converges strongly to a fixed point of a \(k\)-strictly pseudononspreading mapping with an idea of mean convergence, which also solves a class of variational inequalities as an optimality condition for a minimization problem. The results presented in this paper may be viewed as a refinement and as important generalizations of the previously known results announced by many other authors.Solving split common fixed-point problem of firmly quasi-nonexpansive mappings without prior knowledge of operators normshttps://zbmath.org/1472.470982021-11-25T18:46:10.358925Z"Zhao, Jing"https://zbmath.org/authors/?q=ai:zhao.jing.2"Zhang, Hang"https://zbmath.org/authors/?q=ai:zhang.hangSummary: Very recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms for the split common fixed-point problem concerned two bounded linear operators [\textit{A. Moudafi}, J. Nonlinear Convex Anal. 15, No. 4, 809--818 (2014; Zbl 1393.47034)]. However, to employ Moudafi's algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. It is the purpose of this paper to introduce a viscosity iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. We prove the strong convergence of the proposed algorithms for split common fixed-point problem governed by the firmly quasi-nonexpansive operators. As a consequence, we obtain strong convergence theorems for split feasibility problem and split common null point problems of maximal monotone operators. Our results improve and extend the corresponding results announced by many others.A solution of the system of integral equations in product spaces via concept of measures of noncompactnesshttps://zbmath.org/1472.470992021-11-25T18:46:10.358925Z"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumar"Arab, Reza"https://zbmath.org/authors/?q=ai:arab.reza"Ibrahim, Rabha W."https://zbmath.org/authors/?q=ai:ibrahim.rabha-waellSummary: In this chapter, we present the role of measures of noncompactness and related fixed point results to study the existence of solutions for the system of integral equations of the form
\begin{multline*}
x_i(t) = a_i(t)+f_i(t,x_1(t),x_2(t),\dots ,x_n(t))\\
+g_i(t,x_1(t),x_2(t),\dots ,x_n(t))\int_0^{\alpha (t)} k_i(t,s,x_1(s),x_2(s),\dots ,x_n(s)))\, ds,
\end{multline*}
for all \(t\in\mathbb{R}_+\), \(x_1,x_2,\dots,x_n\in E=BC(\mathbb{R}_+)\) and \(1\leq i\leq n\). We mainly focus on introducing new notion of \(\mu-(F,\varphi,\psi)\)-set contractive operator and establishing some new generalization of Darbo fixed point theorem and Krasnoselskii fixed point result associated with measures of noncompactness. Moreover, we deal with a system of fractional integral equations when \(k_i\) is defined in a fractal space.
For the entire collection see [Zbl 1470.47001].Berinde-Borcut tripled fixed point theorem in partially ordered (intuitionistic) fuzzy normed spaceshttps://zbmath.org/1472.471002021-11-25T18:46:10.358925Z"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poom"Martínez-Moreno, Juan"https://zbmath.org/authors/?q=ai:martinez-moreno.juan"Roldán-López-de-Hierro, Antonio-Francisco"https://zbmath.org/authors/?q=ai:roldan-lopez-de-hierro.antonio-francisco"Roldán-López-de-Hierro, Concepción"https://zbmath.org/authors/?q=ai:roldan-lopez-de-hierro.concepcionSummary: In this paper, we prove some tripled fixed point theorems in fuzzy normed spaces. Our results improve and restate the proof lines of the main results given in the paper [\textit{M. Abbas} et al., Fixed Point Theory Appl. 2012, Paper No. 187, 16 p. (2012; Zbl 1307.47096)].Singular optimal control problems for doubly nonlinear and quasi-variational evolution equationshttps://zbmath.org/1472.490092021-11-25T18:46:10.358925Z"Kenmochi, Nobuyuki"https://zbmath.org/authors/?q=ai:kenmochi.nobuyuki"Shirakawa, Ken"https://zbmath.org/authors/?q=ai:shirakawa.ken"Yamazaki, Noriaki"https://zbmath.org/authors/?q=ai:yamazaki.noriakiSummary: Doubly nonlinear and quasi-variational evolution equations governed by double time-dependent subdifferentials are treated in uniformly convex Banach spaces. We establish some abstract results on the existence-uniqueness of solutions together with related optimal control problems in cases when, in general, the state equations have multiple solutions. In this paper, we propose a general class of singular optimal control problems that are set up for non-well-posed state systems. Moreover, we establish an approximation procedure for such singular optimal control problems and discuss some applications.The property of the set of equilibria of the equilibrium problem with lower and upper bounds on Hadamard manifoldshttps://zbmath.org/1472.490142021-11-25T18:46:10.358925Z"Zhang, Qing-Bang"https://zbmath.org/authors/?q=ai:zhang.qingbang"Tang, Gusheng"https://zbmath.org/authors/?q=ai:tang.gushengSummary: The existence of equilibrium points, and the essential stability of the set of equilibrium points of the equilibrium problem with lower and upper bounds are studied on Hadamard manifolds.Split generalized vector variational inequalities for set-valued mappings and applications to social utility optimizations with uncertaintyhttps://zbmath.org/1472.490172021-11-25T18:46:10.358925Z"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu"Stone, Glenn"https://zbmath.org/authors/?q=ai:stone.glenn-davisSummary: In this paper, we use the power, upward power and the downward power preorders on the power sets of topological vector spaces to define the split generalized vector variational inequality problems for set-valued mappings on topological vector spaces. By using the Fan-KKM theorem, we prove an existence theorem for solutions to some split generalized vector variational inequality problems for set-valued mappings on topological vector spaces. Consequently, we prove the solvability of some generalized vector variational inequality problems for set-valued mappings. As applications, we study the existence of efficient pair of initial social activity profiles for some split generalized social utility optimization problems.Regularity of implicit solution mapping to parametric generalized equationhttps://zbmath.org/1472.490372021-11-25T18:46:10.358925Z"Ouyang, Wei"https://zbmath.org/authors/?q=ai:ouyang.wei"Zhang, Binbin"https://zbmath.org/authors/?q=ai:zhang.binbinSummary: This paper concerns the study of both local and global metric regularity/Lipschitz-like properties concerning the behavior of the implicit solution mapping associated to parametric generalized equation in metric space. We extend some implicit multifunction results to the addition of two multifunctions both depending on parameters. Through the approach of inverse mapping iteration, several results are established regarding the relations between the (partial) metric regularity/Lipschitz-like moduli of multifunctions used as the defining form of the generalized equation and the corresponding implicit solution mapping, the proof of which is completely self-contained. Finally, a local Lyusternik-Graves Theorem is obtained as an application.Applying twice a minimax theoremhttps://zbmath.org/1472.490382021-11-25T18:46:10.358925Z"Ricceri, Biagio"https://zbmath.org/authors/?q=ai:ricceri.biagioSummary: Here is one of the results obtained in this paper: Let \(X\), \(Y\) be two convex sets each in a real vector space, let \(J:X\times Y\to\mathbb{R}\) be convex and without global minima in \(X\) and concave in \(Y\), and let \(\Phi:X\to\mathbb{R}\) be strictly convex. Also, assume that, for some topology on \(X\), \(\Phi\) is lower semicontinuous and, for each \(y\in Y\) and \(\lambda>0\), \(J(\cdot,y)\) is lower semicontinuous and \(J(\cdot,y)+\lambda\Phi(\cdot)\) is inf-compact.
Then, for each \(r\in]\inf_X\Phi,\sup_X\Phi[\) and for each closed set \(S\subseteq X\) satisfying
\[
\Phi^{-1}(r)\subseteq S\subseteq\Phi^{-1}(]-\infty,r]),
\]
one has
\[
\sup\limits_Y\inf\limits_S J=\inf\limits_S\sup\limits_Y J.
\]On sensitivity of vector equilibria by means of the diagonal subdifferential operatorhttps://zbmath.org/1472.490512021-11-25T18:46:10.358925Z"Al-Homidan, Suliman"https://zbmath.org/authors/?q=ai:al-homidan.suliman-s"Ansari, Qamrul Hasan"https://zbmath.org/authors/?q=ai:ansari.qamrul-hasan"Kassay, Gabor"https://zbmath.org/authors/?q=ai:kassay.gaborSummary: Based on the concept of subdifferential of a convex vector function, we define the so-called diagonal subdifferential operator for vector-valued bifunctions depending on a parameter and show its sensitivity with respect to the parameter. As a byproduct, we obtain Lipschitz continuity results of the solution map for parametric strong vector equilibrium problems.Optimization results for the higher eigenvalues of the \(p\)-Laplacian associated with sign-changing capacitary measureshttps://zbmath.org/1472.490672021-11-25T18:46:10.358925Z"Degiovanni, Marco"https://zbmath.org/authors/?q=ai:degiovanni.marco"Mazzoleni, Dario"https://zbmath.org/authors/?q=ai:mazzoleni.darioLet \(\Omega\) be an open subset of \(\mathbb R^n\), \(|\Omega|\) be its Lebesgue measure, and \(\partial\Omega\) be its boundary. The authors consider the following boundary value problem \[ (*):\ -\text{div}(|\nabla u|^{p-2}\nabla u)=\lambda|u|^{p-2}u\text{ in }\Omega\text{ such that }u_{\restriction{_{\partial\Omega}}}=0.\] The associated eigenvalues to \((*)\) are given by: \[\lambda_{m,p}(\Omega)=\inf_{K\subset \mathcal{K}_m}\left(\sup_{u\in K}\displaystyle\int_\Omega|\nabla u(x)|^{p}dx\right)\text{ such that }m\in\mathbb N,\] \(\mathcal{K}_m=\{K\subset \mathcal{W}^{1,p}(\Omega):K\text{ is a compact and symmetric with }i(K)\ge m\},\) where \(i\) represents the Krasnosel'skii genus and \(\mathcal{W}^{1,p}(\Omega)\) is the space of functions belonging to the standard Sobolev space \(W^{1,p}(\Omega)\) and with unit \(L_p(\Omega)\)-norms. Then, the authors consider \(F\) a real-valued increasing lower semicontinuous function on \(\mathbb R^k\) and state that the following problem: \[\min\{F(\lambda_{1,p}(A),\ldots,\lambda_{k,p}(A)):A\text{ is a }p\text{-quasi open subset of }\Omega\text{ with }|A|=c\},\] has a solution where \(c\in (0,|\Omega|]\) and \(A\) is a \(p\)-quasi open set means that \(A\cup \omega_\varepsilon\) is an open set of \(\mathbb R^n\) where \(\omega_\varepsilon\) is an open subset of \(\mathbb R^n\) such that its \(p\)-capacity is less than \(\varepsilon>0\).A generalization of Caristi's fixed point theorem in the variable exponent weighted formal power series spacehttps://zbmath.org/1472.540222021-11-25T18:46:10.358925Z"Bakery, Awad A."https://zbmath.org/authors/?q=ai:bakery.awad-a"El Dewaik, M. H."https://zbmath.org/authors/?q=ai:el-dewaik.m-hSummary: Suppose \((p_n)\) be sequence of positive reals. By \(\mathscr{H}_w((p_n))\), we represent the space of all formal power series \(\sum_{n = 0}^\infty a_n z^n\) equipped with \(\sum_{n = 0}^\infty | \lambda a_n /(n+1)|^{p_n}<\infty \), for some \(\lambda>0\). Various topological and geometric behavior of \(\mathscr{H}_w((p_n))\) and the prequasi ideal constructs by \(s\)-numbers and \(\mathscr{H}_w((p_n))\) have been considered. The upper bounds for \(s\)-numbers of infinite series of the weighted \(n\)-th power forward shift operator on \(\mathscr{H}_w((p_n))\) with applications to some entire functions are granted. Moreover, we investigate an extrapolation of Caristi's fixed point theorem in \(\mathscr{H}_w((p_n))\).Fixed point of generalized weak contraction in \(b\)-metric spaceshttps://zbmath.org/1472.540252021-11-25T18:46:10.358925Z"Iqbal, Maryam"https://zbmath.org/authors/?q=ai:iqbal.maryam"Batool, Afshan"https://zbmath.org/authors/?q=ai:batool.afshan"Ege, Ozgur"https://zbmath.org/authors/?q=ai:ege.ozgur"de la Sen, Manuel"https://zbmath.org/authors/?q=ai:de-la-sen.manuelSummary: In this manuscript, a class of generalized \((\psi,\alpha,\beta)\)-weak contraction is introduced and some fixed point theorems in the framework of \(b\)-metric space are proved. The result presented in this paper generalizes some of the earlier results in the existing literature. Further, some examples and an application are provided to illustrate our main result.An implicit relation approach in metric spaces under \(w\)-distance and application to fractional differential equationhttps://zbmath.org/1472.540262021-11-25T18:46:10.358925Z"Jain, Reena"https://zbmath.org/authors/?q=ai:jain.reena"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumar"Kumar, Santosh"https://zbmath.org/authors/?q=ai:kumar.santosh|kumar.santosh.4|kumar.santosh.1|kumar.santosh.3|kumar.santosh.2Summary: The purpose of this work is to introduce a new class of implicit relation and implicit type contractive condition in metric spaces under \(w\)-distance functional. Further, we derive fixed point results under a new class of contractive condition followed by three suitable examples. Next, we discuss results about weak well-posed property, weak limit shadowing property, and generalized \(w\)-Ulam-Hyers stability of the mappings of a given type. Finally, we obtain sufficient conditions for the existence of solutions for fractional differential equations as an application of the main result.Fixed point theorems in fuzzy metric spaces for mappings with some contractive type conditionshttps://zbmath.org/1472.540282021-11-25T18:46:10.358925Z"Patir, Bijoy"https://zbmath.org/authors/?q=ai:patir.bijoy"Goswami, Nilakshi"https://zbmath.org/authors/?q=ai:goswami.nilakshi"Mishra, Lakshmi Narayan"https://zbmath.org/authors/?q=ai:mishra.lakshmi-narayanIn this paper, different fixed point theorems on fuzzy metric spaces with several contractive type mappings and by using an altering distance function are proved. Examples are given to validate the results. The obtained results generalize the results of \textit{N. Wairojjana} et al. [Fixed Point Theory Appl. 2015, Paper No. 69, 19 p. (2015; Zbl 1338.54231)].Periodic and fixed points for Caristi-type \(G\)-contractions in extended \(b\)-gauge spaceshttps://zbmath.org/1472.540312021-11-25T18:46:10.358925Z"Zikria, Nosheen"https://zbmath.org/authors/?q=ai:zikria.nosheen"Samreen, Maria"https://zbmath.org/authors/?q=ai:samreen.maria"Kamran, Tayyab"https://zbmath.org/authors/?q=ai:kamran.tayyab"Yeşilkaya, Seher Sultan"https://zbmath.org/authors/?q=ai:yesilkaya.seher-sultanSummary: In this paper, we introduce extended \(b\)-gauge spaces and the extended family of generalized extended pseudo-\(b\)-distances. Moreover, we define the sequential completeness and construct the Caristi-type \(G\)-contractions in the framework of extended \(b\)-gauge spaces. Furthermore, we develop periodic and fixed point results in this new setting endowed with a graph. The obtained results of this paper not only generalize but also unify and improve the existing results in the corresponding literature.Index of equivariant Callias-type operators and invariant metrics of positive scalar curvaturehttps://zbmath.org/1472.580142021-11-25T18:46:10.358925Z"Guo, Hao"https://zbmath.org/authors/?q=ai:guo.haoFor any Lie group \(G\) acting isometrically on a manifold \(M\), the author formulates the general notion of a \(G\)-equivariant elliptic operator that is invertible outside of a \(G\)-cocompact subset of \(M\). To establish the analogue of the Rellich lemma in this setting, the author defines \(G\)-Sobolev modules from the \(G\)-action on the space of compactly supported smooth section. It is shown that \(G\)-Callias-type operators are self-adjoint, regular in the sense of Hilbert modules and hence equivariantly invertible at infinity. The paper also gives an explicit construction of \(G\)-Callias-type operator using \(K\)-theory of an equivariant Higson corona of \(M\). Finally the paper obtains an obstruction to positive scalar curvature metrics on non-cocompact manifolds as an application of the theory.Spectral flow for skew-adjoint Fredholm operatorshttps://zbmath.org/1472.580172021-11-25T18:46:10.358925Z"Carey, Alan L."https://zbmath.org/authors/?q=ai:carey.alan-l"Phillips, John"https://zbmath.org/authors/?q=ai:phillips.john"Schulz-Baldes, Hermann"https://zbmath.org/authors/?q=ai:schulz-baldes.hermannIn the present paper the authors constructed a \(\mathbb{Z}_{2}\)-valued spectral flow for paths of skew-adjoint Fredholms on a real Hilbert space. First the authors defined the \(\mathbb{Z}_{2}\)-valued spectral flow associated to a straight line path in finite dimensions. The definition simply counts the number of orientation changes of the eigenfunctions at eigenvalue crossings through 0 along the path. Then the authors gave the analytic approach to the complex spectral flow for paths of self-adjoint Fredholm operators on a complex Hilbert space, this allows to show relatively directly that the \(\mathbb{Z}_{2}\)-valued spectral flow can be calculated, similarly to the complex spectral flow, as a sum of index type contributions, provided the appropriate notion of index is used. Finally an index formula is proved which connects the \(\mathbb{Z}_{2}\)-valued spectral flow of certain paths in the skew-adjoint operators on a real Hilbert space to the \(\mathbb{Z}_{2}\)-index of an associated Toeplitz operator on the complexification. At the end the authors illustrated the theory by an explicit example given by a matrix-valued shift operator which can be considered to be the analogue in real Hilbert space of the standard Toeplitz operator in the complex case. This example is the canonical non-trivial example of \(\mathbb{Z}_{2}\)-valued spectral flow.Weak Poincaré inequalities for convergence rate of degenerate diffusion processeshttps://zbmath.org/1472.600962021-11-25T18:46:10.358925Z"Grothaus, Martin"https://zbmath.org/authors/?q=ai:grothaus.martin"Wang, Feng-Yu"https://zbmath.org/authors/?q=ai:wang.fengyu|wang.feng-yuThe authors introduce weak Poincaré inequalities for the symmetric and antisymmetric part of the generator to estimate the convergence rate for general degenerate diffusion semigroups. They also present a general result on the weak hypocoercivity for \(C_0\)-semigroups on Hilbert spaces. The main result of the paper applies to a large class of degenerate SDEs, and the state space of the Markov process associated to the semigroup can be very general. In particular, the result applies to degenerate spherical velocity Langevin equations.
The results are important from the viewpoint of applications because solutions to SDEs studied in this paper arise, for instance, in industrial mathematics as so-called fiber laydown processes. They are used as surrogate models for the production process of nonwovens. Here, the rate of convergence to equilibrium is related to the quality of the nonwovens, and so is of practical interest. So, cases in which empirical measurements indicate slow growing potentials are also subsumed as special cases of the model considered in this paper.Infinite delay fractional stochastic integro-differential equations with Poisson jumps of neutral typehttps://zbmath.org/1472.601012021-11-25T18:46:10.358925Z"Hussain, R. Jahir"https://zbmath.org/authors/?q=ai:hussain.r-jahir"Hussain, S. Satham"https://zbmath.org/authors/?q=ai:hussain.s-sathamThe well-posedness and continuous dependence are presented for the mild solution to a class of stochastic neutral integral-differential equations with infinite delay driven by Poisson jumps on a separable Hilbert space \(\mathbb H\):
\[
\begin{aligned}
d P(t,x_t)= &\,\int_0^t \frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)} A P(s, x_s) d s d t\\
&+ f(t,x_t)d t+ \sigma(t,x_t) d W(t)\\
&+ \int_{\mathbb Z} h(t,x_t,y) \tilde N(d t, dy),\\
&\ x_0\in \mathcal B:=C((-\infty,0];\mathbb H), t\in [0,T],
\end{aligned}
\]
where \(T>0\) is a fixed constant, \((A,\mathcal{A})\) is a densely defined linear operator on \(\mathbb H\) of sectorial type, \(W(t)\) is a Wiener process on \(\mathbb H\) with finite trace nuclear covariance, \(\tilde N\) is a compensated Poisson martingale measure over a reference space \(\mathbb Z\), \(x_t\in \mathcal B\) with \(x_t(\theta):= x(t+\theta)\) for \(\theta\in (-\infty,0]\) is the segment of \(x(\cdot)\) up to time \(t\), and \(P, f, \sigma, h\) are proper defined functionals. The main results are illustrated with specific examples.The law of the iterated logarithm for a piecewise deterministic Markov process assured by the properties of the Markov chain given by its post-jump locationshttps://zbmath.org/1472.601252021-11-25T18:46:10.358925Z"Czapla, Dawid"https://zbmath.org/authors/?q=ai:czapla.dawid"Hille, Sander C."https://zbmath.org/authors/?q=ai:hille.sander-cornelis"Horbacz, Katarzyna"https://zbmath.org/authors/?q=ai:horbacz.katarzyna"Wojewódka-Ściążko, Hanna"https://zbmath.org/authors/?q=ai:wojewodka-sciazko.hannaSummary: In the paper, we consider some piecewise deterministic Markov process, whose continuous component evolves according to semiflows, which are switched at the jump times of a Poisson process. The associated Markov chain describes the states of this process directly after the jumps. Certain ergodic properties of these two dynamical systems have been already investigated in our recent papers. We now aim to establish the law of the iterated logarithm for the aforementioned continuous-time process. Moreover, we intend to do this using the already proven properties of the discrete-time system. The abstract model under consideration has interesting interpretations in real-life sciences, such as biology. Among others, it can be used to describe the stochastic dynamics of gene expression.Verified computation for the geometric mean of two matriceshttps://zbmath.org/1472.650552021-11-25T18:46:10.358925Z"Miyajima, Shinya"https://zbmath.org/authors/?q=ai:miyajima.shinyaSummary: An algorithm for numerically computing an interval matrix containing the geometric mean of two Hermitian positive definite (HPD) matrices is proposed. We consider a special continuous-time algebraic Riccati equation (CARE) where the geometric mean is the unique HPD solution, and compute an interval matrix containing a solution to the equation. We invent a change of variables designed specifically for the special CARE. By the aid of this special change of variables, the proposed algorithm gives smaller radii, and is more successful than previous approaches. Solutions to the equation are not necessarily Hermitian. We thus establish a theory for verifying that the contained solution is Hermitian. Finally, the positive definiteness of the solution is verified. Numerical results show effectiveness, efficiency, and robustness of the algorithm.A derivative-free \textit{RMIL} conjugate gradient projection method for convex constrained nonlinear monotone equations with applications in compressive sensinghttps://zbmath.org/1472.650582021-11-25T18:46:10.358925Z"Koorapetse, M."https://zbmath.org/authors/?q=ai:koorapetse.mompati-s"Kaelo, P."https://zbmath.org/authors/?q=ai:kaelo.pro"Lekoko, S."https://zbmath.org/authors/?q=ai:lekoko.s"Diphofu, T."https://zbmath.org/authors/?q=ai:diphofu.tSummary: In this paper, a derivative-free \textit{RMIL} conjugate gradient projection method for solving large-scale nonlinear monotone equations with convex constraints is proposed. The proposed method is a modification of an \textit{RMIL} conjugate gradient method combined with the projection techniques. We establish its global convergence under appropriate conditions. The method is then compared with other existing methods in the literature and the numerical results indicate that the method is efficient. Furthermore, the proposed method is used to recover a sparse signal from an incomplete and contaminated sampling measurements and the results are promising.Convergence theorem for system of pseudomonotone equilibrium and split common fixed point problems in Hilbert spaceshttps://zbmath.org/1472.650632021-11-25T18:46:10.358925Z"Jolaoso, Lateef Olakunle"https://zbmath.org/authors/?q=ai:jolaoso.lateef-olakunle"Lukumon, Gafari Abiodun"https://zbmath.org/authors/?q=ai:lukumon.gafari-abiodun"Aphane, Maggie"https://zbmath.org/authors/?q=ai:aphane.maggieSummary: We consider a system of pseudomonotone equilibrium problem and split common fixed point problem in the framework of real Hilbert spaces. We propose a modified extragradient method with line searching technique for approximating a common element in the sets of solutions of the two nonlinear problems. The convergence result is proved without prior knowledge of the Lipschitz-like constants of the equilibrium bifunctions and the norm of the bounded linear operator of the split common fixed point problem. We further provide some application and numerical example to show the importance of the obtained results in the paper.Strong convergence of self-adaptive inertial algorithms for solving split variational inclusion problems with applicationshttps://zbmath.org/1472.650642021-11-25T18:46:10.358925Z"Tan, Bing"https://zbmath.org/authors/?q=ai:tan.bing.1"Qin, Xiaolong"https://zbmath.org/authors/?q=ai:qin.xiaolong"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: In this paper, four self-adaptive iterative algorithms with inertial effects are introduced to solve a split variational inclusion problem in real Hilbert spaces. One of the advantages of the suggested algorithms is that they can work without knowing the prior information of the operator norm. Strong convergence theorems of these algorithms are established under mild and standard assumptions. As applications, the split feasibility problem and the split minimization problem in real Hilbert spaces are studied. Finally, several preliminary numerical experiments as well as an example in the field of compressed sensing are proposed to support the advantages and efficiency of the suggested methods over some existing ones.Modified forward-backward splitting method for variational inclusionshttps://zbmath.org/1472.650652021-11-25T18:46:10.358925Z"Van Hieu, Dang"https://zbmath.org/authors/?q=ai:dang-van-hieu."Anh, Pham Ky"https://zbmath.org/authors/?q=ai:pham-ky-anh."Muu, Le Dung"https://zbmath.org/authors/?q=ai:muu.le-dungSummary: In this paper we propose an explicit algorithm for solving a variational inclusion problem of the sum of two operators, the one is maximally monotone and the other is monotone and Lipschitz continuous. The algorithm uses the variable stepsizes which are updated over each iteration by some cheap comptutations. These stepsizes are found without the prior knowledge of the Lipschitz constant of operator as well as without using lineseach procedure. The algorithm thus can be implemented easily. The convergence and the convergence rate of the algorithm are established under mild conditions. Several preliminary numerical results are provided to demonstrate the theoretical results and also to compare the new algorithm with some existing ones.Convergence of the forward-backward method for split null-point problems beyond cocoercivenesshttps://zbmath.org/1472.650752021-11-25T18:46:10.358925Z"Moudafi, Abdellatif"https://zbmath.org/authors/?q=ai:moudafi.abdellatif"Shehu, Yekini"https://zbmath.org/authors/?q=ai:shehu.yekiniSummary: The forward-backward algorithm is one of the most attractive algorithm for finding zeroes of the sum of two maximal monotone operators, with one being single-valued. However, it requires the single-valued part to be co-coercive, thus precluding its use in many applications. The aim of this paper is to present and investigate the asymptotic behavior of a forward-backward algorithm with Bregman distances for solving constrained split null-point problems beyond co-coerciveness of its single-valued part. Special attention is given to constrained composite minimization and we illustrate the potential of this approach by answering a question by \textit{H.-K. Xu} [Linear Nonlinear Anal. 4, No. 1, 135--144 (2018; Zbl 1458.94147)] on the non applicability of the proximal gradient algorithm when the \(l_p\)-norm is used to measure the errors in signal processing.An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappingshttps://zbmath.org/1472.650772021-11-25T18:46:10.358925Z"Alakoya, Timilehin Opeyemi"https://zbmath.org/authors/?q=ai:alakoya.timilehin-opeyemi"Taiwo, Adeolu"https://zbmath.org/authors/?q=ai:taiwo.adeolu"Mewomo, Oluwatosin Temitope"https://zbmath.org/authors/?q=ai:mewomo.oluwatosin-temitope"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the framework of Hilbert spaces. First, we introduce a new iterative scheme which combines the inertial subgradient extragradient method with viscosity technique and with self-adaptive stepsize. Unlike in many existing subgradient extragradient techniques in literature, the two projections of our proposed algorithm are made onto some half-spaces. Furthermore, we prove a strong convergence theorem for approximating a common solution of the variational inequality and fixed point of an infinite family of nonexpansive mappings under some mild conditions. The main advantages of our method are: the self-adaptive stepsize which avoids the need to know a priori the Lipschitz constant of the associated monotone operator, the two projections made onto some half-spaces, the strong convergence and the inertial technique employed which accelerates convergence rate of the algorithm. Second, we apply our theorem to solve generalised mixed equilibrium problem, zero point problems and convex minimization problem. Finally, we present some numerical examples to demonstrate the efficiency of our algorithm in comparison with other existing methods in literature. Our results improve and extend several existing works in the current literature in this direction.The appearance of particle tracks in detectorshttps://zbmath.org/1472.810112021-11-25T18:46:10.358925Z"Ballesteros, Miguel"https://zbmath.org/authors/?q=ai:ballesteros.miguel"Benoist, Tristan"https://zbmath.org/authors/?q=ai:benoist.tristan"Fraas, Martin"https://zbmath.org/authors/?q=ai:fraas.martin"Fröhlich, Jürg"https://zbmath.org/authors/?q=ai:frohlich.jurg-martinEver since its observation the seemingly innocuous appearance of the atomic particle tracks in detectors has puzzled the physicists as one of the most striking manifestations of the weird properties of the quantum world. To make a long story short, in the quoted words of M.\ Born at the 1927 Solvay conference: ``If one associates a spherical wave with each emission process, how can one understand that the track of each \(\alpha\)-particle appears as a (very nearly) straight line? In other words: how can the corpuscular character of the phenomenon be reconciled here with the representation by waves?'' In particular, the \(S\)-wave of the outgoing \(\alpha\)-particle is spherically symmetric, but the particle tracks are not. How is, then, that this initial symmetry is broken? In a sense the typical answer -- given in a celebrated W.\ Heisenberg 1927 thought experiment -- is well known: it is an effect of the collapse of the wave packet due to the (repeated) measurements of the particle position. But this is famously an answer that begs a lot of explanation and that, ever since its formulation, has elicited heated discussions not to be summarized here.
Admittedly however the present paper deals neither with a physical model for the atom ionization and the subsequent drop formation in a cloud chamber (along the lines, for example, of the quoted N.F.\ Mott 1929 paper), nor with an explanation of the strange nature of the quantum measurements, so that in particular no new insight is to be found about the wave packet collapse, or the \textit{Heisenberg cut}, notions that are simply accepted and used along the paper. For instance, in the words of the authors (page 438): ``To describe the effect of an instantaneous measurement of the approximate position of the particle on its state we follow the conventional wisdom of quantum mechanics: In the course of such a measurement whose result is given by some vector \(\mathbf q\in\mathbb{R}^d\), the state \(\rho\) of the particle changes according to'' the usual quantum rule of the wave packet collapse summarized in the subsequent equation (29). The focus of the discussion instead is to ``present a mathematically rigorous analysis of the appearance of particle tracks'' within the framework of the said ``conventional wisdom of quantum mechanics,'' namely to show that the track appearance is well accounted for with a scrupulous application of the quantum formalism. This is in any case a commendable task, and not a very easy one to carry out as the complexity of the subsequent discussion shows.
More precisely the authors want to ``present a theoretical analysis of a gedanken experiment of the sort Heisenberg had in mind in 1927,'' with repeated position measurements every \(\tau\) seconds. In their discussion however they do not deal with \textit{idealized} quantum measurements, but they take instead the considerable trouble of discussing the case of \textit{approximate} measurements. The general states of the charged particle of mass \(M\) are here density operators \(\rho\) in a Hilbert space \(\mathcal H\) where \(\mathbf X, \mathbf P\) are the position and momentum operators, while the state vectors \(\Omega\) of the electromagnetic (EM) field plus photomultipliers live in another Hilbert space \(\mathfrak H\). The values \(\mathbf q\in\mathbb{R}^d\) (representing the approximate position measurements) of suitable operators \(\mathbf Q\) in \(\mathfrak H\) are then ``supposed to be tightly correlated with the positions, \(\mathbf x\in\mathbb{R}^d\), of the charged particle.''
Before each measurement the EM field and photomultipliers always are in the state \(\Omega_{in}\); then during the light-scattering the state changes according to a propagator \(U_t(\mathbf x)\) (\(\mathbf x\) being the position of the charged particle during the scattering process, and \(t\ll\tau\) the scattering time span) and quickly relaxes back to \(\Omega_{in}\) long before the subsequent measurement is performed. It is therefore possible to define the transition amplitude \(V_{\mathbf q}(\mathbf x)\) of equation (7) representing the probability density amplitude of finding \(\mathbf q\) when the particle is in \(\mathbf x\). The operators \(V_{\mathbf q}(\mathbf X)\) will then constitute a positive-operator-valued measure (POVM) that turns out to be instrumental to implement the wave packet reduction of every approximate position measurement. On the other hand the evolution between two measurements of the position-momentum pair \(\mathbf X,\mathbf P\) of the freely moving particle is described by a propagator \(U_S\) associated to a symplectic matrix \(S\) in the phase space \(\Gamma\)
With the Gaussian \textit{ansatz} of equation (9) for the transition amplitude, a sequence of approximate measurements \(\mathbf q_0,\dots,\mathbf q_n\) falling in the subsets \(\Delta_0,\dots,\Delta_n\) entails a change in the initial density matrix \(\rho\) produced by a total operator \(W_n(\mathbf q_0,\dots,\mathbf q_n)= U_{S^{n+1}}V_{\mathbf q_n} (\mathbf X_{n\tau}) \ldots V_{\mathbf q_0} (\mathbf X)\) given as a repeated combination of \(V_{\mathbf q}\) and \(S\). Finally this gives rise to a probability measure \(\mathbb P_\rho\) on the process of the sequences \(\mathbf q_n\) that can be used to calculate the probability that the position of the particle at the times \(n\tau\) is within \(\Delta_n\). The aim of the paper now is to show that (page 435) ``with high probability, the cells \(\Delta_0,\dots,\Delta_n\) which indicate the positions of the particle at times \(0, \tau,\dots , n\tau\) , are centered in points ``close'' to \(\mathbf x(0), \mathbf x(\tau), \dots , \mathbf x(n\tau)\), respectively, where \(\mathbf x(t) =\mathbf x+t\mathbf v, t \in [0, n\tau]\), is the trajectory of a freely moving classical particle.'' Here \(\mathbf v\) is a value of \(\mathbf V=\mathbf P/M\)
Without retracing in this short review all the details of their exhaustive discussion we will only recall next that the authors on the one hand study (by means of a suitable family of coherent states \(|W,\zeta\rangle \) centered around phase space points \(\zeta\in\Gamma\)) ``the stochastic dynamics of a (quasi-) freely moving quantum particle subjected to repeated measurements of its approximate position;'' and on the other they ``introduce a stochastic process [equation (39)] with values in the classical phase space of the particle that indexes the trajectory of coherent states occupied by the particle under the forward dynamics.'' Their first main result is then summarized in the Theorem 2.2 that ``relates the sequence of measurement data of approximate particle positions to the sequence of phase space points determined by the stochastic process in Eq. (39)'' by establishing an equality in law between the classical positions \(\xi_n\) of the particle -- plus an independent Gaussian noise \(\eta_n\) -- and the measurement results \(\mathbf Q_n\). In the Theorem 2.4 they next ``determine the best guess of the initial condition of a phase space trajectory of the stochastic process introduced in (39) from its tail,'' and finally the Theorem 2.8 ``relates the positive operator-valued measure (POVM) induced by sequences of approximate particle position measurements to a POVM taking values in the space of coherent states.'' The bulk of the paper is thereafter devoted to a long and rather convoluted sequence of technical arguments peppered with a great deal of lemmas and propositions needed to prove the said results, and ends finally with a few examples of free particles, harmonic oscillators and particles in a constant magnetic field to show the efficacity of the method.Extremal states of qubit-qutrit system with maximally mixed marginalshttps://zbmath.org/1472.810292021-11-25T18:46:10.358925Z"Kanmani, S."https://zbmath.org/authors/?q=ai:kanmani.s-s"Satyanarayana, S. V. M."https://zbmath.org/authors/?q=ai:satyanarayana.s-v-mSummary: We study extremal elements of the convex set of qubit-qutrit states whose marginals are maximally mixed. In the two qubit case, it is known that every extreme state of such a convex set is a maximally entangled pure state. In qubit-qutrit case, pure states do not exist in the convex set. We construct mixed extreme states of ranks 2 and 3. Second rank extremal state is entangled whereas third rank extreme element is separable. Parthasarathy obtained an upper bound on the rank of extreme states of such a convex set of a bipartite system of \(n\) and \(m\) dimensions as \(\sqrt{ n^2+m^2-1}\). Thus for a qubit-qutrit system, the rank of an extreme element should be less than \(\sqrt{12} \). Since Parthasarathy's bound for two qubit system is \(\sqrt{7}\) and all extreme elements are of rank one, Rudolph posed a question about its tightness. We establish that Parthasarathy's upper bound is tight for qubit-qutrit system.A variational formulation for Dirac operators in bounded domains. Applications to spectral geometric inequalitieshttps://zbmath.org/1472.810902021-11-25T18:46:10.358925Z"Antunes, Pedro R. S."https://zbmath.org/authors/?q=ai:antunes.pedro-ricardo-simao"Benguria, Rafael D."https://zbmath.org/authors/?q=ai:benguria.rafael-d"Lotoreichik, Vladimir"https://zbmath.org/authors/?q=ai:lotoreichik.vladimir"Ourmières-Bonafos, Thomas"https://zbmath.org/authors/?q=ai:ourmieres-bonafos.thomaslet \(\Omega \subset {\mathbb R}^2\) be a \(C^\infty\) simply connected domain and let \(n = (n_1,n_2)^\top\) be the outward pointing normal field on \(\partial\Omega\). The Dirac operator with infinite mass boundary conditions in \(L^2(\Omega,{\mathbb C}^2)\) is defined as \[D^\Omega := \begin{pmatrix} 0 & -2\mathrm{i}\partial_z\\
-2\mathrm{i}\partial_{\bar z} & 0 \end{pmatrix}, \] with domain \(\{ u = (u_1,u_2)^\top \in H^1(\Omega,{\mathbb C}^2) : u_2 = \mathrm{i} \mathbf{n}u_1 \text{ on }\partial\Omega \},\) where \(\mathbf{n} := n_1 + \mathrm{i} n_2\) and \(\partial_z, \partial_{\bar{z}}\) are the Wirtinger operators. The spectrum of \(D^\Omega\) is symmetric with respect to the origin and constituted of eigenvalues of finite multiplicity \[ \cdots \leq -E_k(\Omega) \leq\cdots \leq-E_{1}(\Omega) < 0 < E_{1}(\Omega) \leq \cdots \leq E_k(\Omega) \leq \cdots.\] The authors prove the following estimate \[E_1(\Omega) \leq \frac{|\partial\Omega|}{(\pi r_i^2 + |\Omega|)}E_1({\mathbb D}) \] with equality if and only if \(\Omega\) is a disk, where \(r_i\) is the inradius of \(\Omega\) and \(\mathbb D\) is the unit disk. \par The second main result of this paper is the following non-linear variational characterization of \(E_1(\Omega)\). \(E > 0\) is the first non-negative eigenvalue of \(D^\Omega\) if and only if \(\mu^\Omega(E) = 0\), where \[\mu^\Omega(E) := \inf\limits_{u} \frac{4 \int_\Omega |\partial_{\bar z} u|^2 dx - E^2 \int_{\Omega}|u|^2dx + E \int_{\partial\Omega} |u|^2 ds}{\int_\Omega |u|^2 dx}.\] \par The authors propose the following conjecture \[\mu^\Omega(E) \geq \frac{\pi}{|\Omega|}\mu^{\mathbb D}\Big(\sqrt{\frac{|\Omega|}{\pi}}E\Big), \forall E>0\] and provide numerical evidences supporting it. This conjecture implies the validity of the Faber-Krahn-type inequality \(E_1(\Omega) \geq \sqrt{\frac{\pi}{|\Omega|}} E_1({\mathbb D})\) (it is still an open question).Tosio Kato's work on non-relativistic quantum mechanics. IIhttps://zbmath.org/1472.810942021-11-25T18:46:10.358925Z"Simon, Barry"https://zbmath.org/authors/?q=ai:simon.barry.1The work is the second part of a review to Kato's work on nonrelativistic quantum mechanics. It focuses on bounds on the number of eigenvalues of the helium atom, on the absence of embedded bound states, on scattering theory under a trace class condition, Kato smoothness, the adiabatic theorem, and the Trotter product formula.
The author is known for the clarity of his presentation which is reflected in this work as well. The review can also serve as an introduction of the subject, since the results are not merely reviewed but put in a current perspective of the field. An example of this is the appendix where the inequality \(|p|>2/(\pi |x|)\) in \(d=3\), known as Kato's inequality or Herbst inequality, is treated. It is put in the context of the groundstate transform which [\textit{R. L. Frank} et al., J. Am. Math. Soc. 21, No. 4, 925--950 (2008; Zbl 1202.35146)] used to prove a generalization for fractional powers of \(p:=-i\nabla\).
For Part I see the author [Bull. Math. Sci. 8, No. 1, 121--232 (2018; Zbl 1416.81063)]Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random modelshttps://zbmath.org/1472.811042021-11-25T18:46:10.358925Z"Hislop, Peter D."https://zbmath.org/authors/?q=ai:hislop.peter-d"Marx, Christoph A."https://zbmath.org/authors/?q=ai:marx.christoph-aSummary: We prove quantitative bounds on the dependence of the density of states on the potential function for discrete, deterministic Schrödinger operators on infinite graphs. While previous results were limited to random Schrödinger operators with independent, identically distributed potentials, this paper develops a deterministic framework, which is applicable to Schrödinger operators independent of the specific nature of the potential. Following ideas by Bourgain and Klein, we consider the density of states outer measure (DOSoM), which is well defined for all (deterministic) Schrödinger operators. We explicitly quantify the dependence of the DOSoM on the potential by proving a modulus of continuity in the \(\ell^\infty \)-norm. The specific modulus of continuity so obtained reflects the geometry of the underlying graph at infinity. For the special case of Schrödinger operators on \(\mathbb{Z}^d\), this implies the Lipschitz continuity of the DOSoM with respect to the potential. For Schrödinger operators on the Bethe lattice, we obtain log-Hölder dependence of the DOSoM on the potential. As an important consequence of our deterministic framework, we obtain a modulus of continuity for the density of states measure (DOSm) of ergodic Schrödinger operators in the underlying potential sampling function. Finally, we recover previous results for random Schrödinger operators on the dependence of the DOSm on the single-site probability measure by formulating this problem in the ergodic framework using the quantile function associated with the random potential.Stability in the higher derivative abelian gauge field theorieshttps://zbmath.org/1472.811642021-11-25T18:46:10.358925Z"Dai, Jialiang"https://zbmath.org/authors/?q=ai:dai.jialiangSummary: We present an exact derivation of conserved tensors associated to the higher-order symmetries in the higher derivative Abelian gauge field theories. In our model, the wave operator of the derived theory is a \(n\)-th order polynomial expressed in terms of the usual Maxwell operator. Relying on this formalism and utilizing the extension of Noether's theorem, we acquire a series of conserved second-rank tensors which includes the standard canonical energy-momentum tensors. Moreover, with the aid of auxiliary fields, we succeed in obtaining the relations between the root decomposition of characteristic polynomial of the wave operator and the conserved energy-momentum tensors in the context of another equivalent lower-order representation. Under the certain conditions, although the canonical energy of the higher derivative dynamics is unbounded from below, the 00-component of the linear combination of these conserved quantities is bounded. By this reason, the original derived theory is considered stable. Finally, as an instructive example, we elaborate the third-order derived system and analyze the stabilities in different cases of root decomposition of the characteristic polynomial extensively.A quantum information theoretic quantity sensitive to the neutrino mass-hierarchyhttps://zbmath.org/1472.812942021-11-25T18:46:10.358925Z"Naikoo, Javid"https://zbmath.org/authors/?q=ai:naikoo.javid"Alok, Ashutosh Kumar"https://zbmath.org/authors/?q=ai:alok.ashutosh-kumar"Banerjee, Subhashish"https://zbmath.org/authors/?q=ai:banerjee.subhashish|banerjee.subhashish.1"Uma Sankar, S."https://zbmath.org/authors/?q=ai:sankar.s-uma"Guarnieri, Giacomo"https://zbmath.org/authors/?q=ai:guarnieri.giacomo"Schultze, Christiane"https://zbmath.org/authors/?q=ai:schultze.christiane"Hiesmayr, Beatrix C."https://zbmath.org/authors/?q=ai:hiesmayr.beatrix-cSummary: In this work, we derive a quantum information theoretic quantity similar to the Leggett-Garg inequality, which can be defined in terms of neutrino transition probabilities. For the case of \(\nu_\mu \to \nu_e / \bar{\nu}_\mu \to \bar{\nu}_e\) transitions, this quantity is sensitive to CP violating effects as well as the neutrino mass-hierarchy, namely which neutrino mass eigenstate is heavier than the other ones. The violation of the inequality for this quantity shows an interesting dependence on mass-hierarchy. For normal (inverted) mass-hierarchy, it is significant for \(\nu_\mu \to \nu_e (\bar{\nu}_\mu \to \bar{\nu}_e)\) transitions. This is applied to the two ongoing accelerator experiments T2K and NO\(\nu\)A as well as the future experiment DUNE.Spectral representation of lattice gluon and ghost propagators at zero temperaturehttps://zbmath.org/1472.813092021-11-25T18:46:10.358925Z"Dudal, David"https://zbmath.org/authors/?q=ai:dudal.david"Oliveira, Orlando"https://zbmath.org/authors/?q=ai:oliveira.orlando-anibal"Roelfs, Martin"https://zbmath.org/authors/?q=ai:roelfs.martin"Silva, Paulo"https://zbmath.org/authors/?q=ai:silva.paulo-roberto|silva.paulo-m-p|silva.paulo-h-d|silva.paulo-j-s|silva.paulo-f|silva.paulo-a-sSummary: We consider the analytic continuation of Euclidean propagator data obtained from 4D simulations to Minkowski space. In order to perform this continuation, the common approach is to first extract the Källén-Lehmann spectral density of the field. Once this is known, it can be extended to Minkowski space to yield the Minkowski propagator. However, obtaining the Källén-Lehmann spectral density from propagator data is a well known ill-posed numerical problem. To regularise this problem we implement an appropriate version of Tikhonov regularisation supplemented with the Morozov discrepancy principle. We will then apply this to various toy model data to demonstrate the conditions of validity for this method, and finally to zero temperature gluon and ghost lattice QCD data. We carefully explain how to deal with the IR singularity of the massless ghost propagator. We also uncover the numerically different performance when using two -- mathematically equivalent -- versions of the Källén-Lehmann spectral integral.An exponential regulator for rapidity divergenceshttps://zbmath.org/1472.813152021-11-25T18:46:10.358925Z"Li, Ye"https://zbmath.org/authors/?q=ai:li.ye.4|li.ye|li.ye.1|li.ye.3"Neill, Duff"https://zbmath.org/authors/?q=ai:neill.duff"Zhu, Hua Xing"https://zbmath.org/authors/?q=ai:zhu.hua-xingSummary: Finding an efficient and compelling regularization of soft and collinear degrees of freedom at the same invariant mass scale, but separated in rapidity is a persistent problem in high-energy factorization. In the course of a calculation, one encounters divergences unregulated by dimensional regularization, often called rapidity divergences. Once regulated, a general framework exists for their renormalization, the rapidity renormalization group (RRG), leading to fully resummed calculations of transverse momentum (to the jet axis) sensitive quantities. We examine how this regularization can be implemented via a multi-differential factorization of the soft-collinear phase-space, leading to an (in principle) alternative non-perturbative regularization of rapidity divergences. As an example, we examine the fully-differential factorization of a color singlet's momentum spectrum in a hadron-hadron collision at threshold. We show how this factorization acts as a mother theory to both traditional threshold and transverse momentum resummation, recovering the classical results for both resummations. Examining the refactorization of the transverse momentum beam functions in the threshold region, we show that one can directly calculate the rapidity renormalized function, while shedding light on the structure of joint resummation. Finally, we show how using modern bootstrap techniques, the transverse momentum spectrum is determined by an expansion about the threshold factorization, leading to a viable higher loop scheme for calculating the relevant anomalous dimensions for the transverse momentum spectrum.From short-range to contact interactions in the 1d Bose gashttps://zbmath.org/1472.813282021-11-25T18:46:10.358925Z"Griesemer, Marcel"https://zbmath.org/authors/?q=ai:griesemer.marcel"Hofacker, Michael"https://zbmath.org/authors/?q=ai:hofacker.michael"Linden, Ulrich"https://zbmath.org/authors/?q=ai:linden.ulrichIn the paper under review, the authors consider a system of finite bosons in dimension \(1\) with point interactions described by a Hamiltonian \(H\). They introduce a family of operators \(H_\epsilon\) (\(\epsilon>0\)) that are obtained from \(H\) by a regularization procedure of the point interactions. The main result of the paper is the norm resolvent convergence of \(H_\epsilon\) to \(H\), as \(\epsilon\) goes to \(0\). Under a stronger requirement on the regularized potentials, they also get a bound on the convergence rate. As a consequence, the time-evolution group associated to \(H_\epsilon\) provides a good approximation of the one associated to \(H\).
As explained in the paper, the restriction to the dimension \(1\) avoids complicated difficulties. The proof is based on an appropriate representation of the Hamiltonian \(H_\epsilon\) and on the use of the Konno-Kuroda formula. The later provides a good expression of the difference between the resolvent of \(H_\epsilon\) and the one of the free Hamiltonian. Further important ingredients of the proof are the notion of \(\Gamma\)-convergence and an appropriate use of the Green's function of the multidimensional Laplace operator. This yields, together with the above convergence, a formula for the difference between the resolvent of \(H\) and the one of the free Hamiltonian and it can be shown that it only depends on the \(\mathrm{L}^1\)-norm of the regularized potentials.
We refer to the Introduction of the paper for more details. We point out, that, in the definition of the regularized potentials, it seems that they are implicitly considered as real functions.Merging decision-making units with interval datahttps://zbmath.org/1472.900622021-11-25T18:46:10.358925Z"Ghobadi, Saeid"https://zbmath.org/authors/?q=ai:ghobadi.saeidSummary: This paper deals with the problem of merging units with interval data. There are two important problems in the merging units. Estimation of the inherited inputs/outputs of the merged unit from merging units is the first problem while the identification of the least and most achievable efficiency targets from the merged unit is the second one. In the imprecise or ambiguous data framework, the inverse DEA concept and linear programming models could be employed to solve the first and second problem, respectively. To identify the required inputs/outputs from merging units, the merged entity is enabled by the proposed method. This provides a predefined efficiency goal. The best and worst attainable efficiency could be determined through the presented models. The developed merging theory is evaluated through a banking sector application.Metastability of the proximal point algorithm with multi-parametershttps://zbmath.org/1472.900862021-11-25T18:46:10.358925Z"Dinis, Bruno"https://zbmath.org/authors/?q=ai:dinis.bruno"Pinto, Pedro"https://zbmath.org/authors/?q=ai:pinto.pedro-cSummary: In this article we use techniques of proof mining to analyse a result, due to \textit{Y. Yao} and \textit{M. A. Noor} [J. Comput. Appl. Math. 217, No. 1, 46--55 (2008; Zbl 1147.65049)], concerning the strong convergence of a generalized proximal point algorithm which involves multiple parameters. Yao and Noor's result [loc. cit.] ensures the strong convergence of the algorithm to the nearest projection point onto the set of zeros of the operator. Our quantitative analysis, guided by \textit{F. Ferreira} and \textit{P. Oliva}'s [Ann. Pure Appl. Logic 135, No. 1--3, 73--112 (2005; Zbl 1095.03060)] bounded functional interpretation, provides a primitive recursive bound on the metastability for the convergence of the algorithm, in the sense of Terence Tao. Furthermore, we obtain quantitative information on the asymptotic regularity of the iteration. The results of this paper are made possible by an arithmetization of the lim sup.A modified self-adaptive extragradient method for pseudomonotone equilibrium problem in a real Hilbert space with applicationshttps://zbmath.org/1472.901432021-11-25T18:46:10.358925Z"Rehman, Habib Ur"https://zbmath.org/authors/?q=ai:rehman.habib-ur"Kumam, Poom"https://zbmath.org/authors/?q=ai:kumam.poom"Dong, Qiao-Li"https://zbmath.org/authors/?q=ai:dong.qiaoli"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: In this paper, we consider an improvement of the extragradient method to figure out the numerical solution for pseudomonotone equilibrium problems in arbitrary real Hilbert space. A new method is proposed with an inertial scheme and a self adaptive step size rule that is revised on each iteration based on the previous three iterations. The weak convergence of the method is proved by assuming standard cost bifunction assumptions. We also consider the application of our results to solve different kinds of variational inequality problems and a particular class of fixed point problems. For a numerical part, we study the well-known Nash-Cournot equilibrium model and other test problems to support our well-established convergence results and to ensure that our proposed method has a competitive edge over CPU time and a number of iterations.On the existence of solutions and Tikhonov regularization of hemivariational inequality problemshttps://zbmath.org/1472.901442021-11-25T18:46:10.358925Z"Tang, Guo-ji"https://zbmath.org/authors/?q=ai:tang.guoji"Wan, Zhongping"https://zbmath.org/authors/?q=ai:wan.zhongping"Wang, Xianfu"https://zbmath.org/authors/?q=ai:wang.xianfu.1|wang.xianfuIn this paper, the authors establish a Tikhonov regularization theory for a class of hemivariational inequalities. To this end, they prove a very general existence result for the class of hemeivariational inequalities provided that the mapping has the so-called hemivariational inequality property and satisfies a rather weak coercivity condition. Based on the existence result, they derive the Tikhonov regularization result.On controllability for a system governed by a fractional-order semilinear functional differential inclusion in a Banach spacehttps://zbmath.org/1472.930112021-11-25T18:46:10.358925Z"Afanasova, Maria"https://zbmath.org/authors/?q=ai:afanasova.maria"Liou, Yeong-Cheng"https://zbmath.org/authors/?q=ai:liou.yeongcheng"Obukhovskii, Valeri"https://zbmath.org/authors/?q=ai:obukhovskii.valeri"Petrosyan, Garik"https://zbmath.org/authors/?q=ai:petrosyan.garik-garikovich|petrosyan.garikSummary: We obtain some controllability results for a system governed by a semilinear functional differential inclusion of a fractional order in a Banach space assuming that the linear part of inclusion generates a noncompact \(C_0\)-semigroup. We define the multivalued operator whose fixed points are generating solutions of the problem. By applying the methods of fractional analysis and the fixed point theory for condensing multivalued maps we study the properties of this operator, in particular, we prove that under certain conditions it is condensing w.r.t. an appropriate measure of noncompactness. This allows to present the general controllability principle in terms of the topological degree theory and to consider certain important particular cases.Topological structure of solution sets for control problems governed by semilinear fractional impulsive evolution equations with nonlocal conditionshttps://zbmath.org/1472.930252021-11-25T18:46:10.358925Z"Jiang, Yi-rong"https://zbmath.org/authors/?q=ai:jiang.yirong"Zhang, Qiong-fen"https://zbmath.org/authors/?q=ai:zhang.qiongfen"Song, Qi-qing"https://zbmath.org/authors/?q=ai:song.qiqingSummary: This article investigates the topological structural of the mild solution set for a control problem monitored by semilinear fractional impulsive evolution equations with nonlocal conditions. The \(R_\delta\)-property of the mild solution set is obtained by applying the measure of noncompactness and a fixed point theorem of condensing maps and a fixed point theorem of nonconvex valued maps. Then this result is applied to prove that the presented control problem has a reachable invariant set under nonlinear perturbations. The obtained results are also applied to characterize the approximate controllability of the presented control problem.Strong stabilization of non-dissipative operators in Hilbert spaces with input saturationhttps://zbmath.org/1472.931442021-11-25T18:46:10.358925Z"Laabissi, M."https://zbmath.org/authors/?q=ai:laabissi.mohamed"Taboye, A. M."https://zbmath.org/authors/?q=ai:taboye.a-mSummary: This paper investigates the question of strong stabilizability of non-dissipative linear systems in Hilbert spaces with input saturation. It is proved under some verifiable conditions that the origin is asymptotically stable for the closed-loop semilinear systems. The contribution is then applied to the Schrödinger equation.