Recent zbMATH articles in MSC 54https://www.zbmath.org/atom/cc/542022-05-16T20:40:13.078697ZWerkzeugOn densely complete metric spaces and extensions of uniformly continuous functions in ZFhttps://www.zbmath.org/1483.030302022-05-16T20:40:13.078697Z"Keremedis, Kyriakos"https://www.zbmath.org/authors/?q=ai:keremedis.kyriakos"Wajch, Eliza"https://www.zbmath.org/authors/?q=ai:wajch.elizaA metric space \(X\) is called densely complete if there exists a dense set \(D\) in \(X\) such that every Cauchy sequence of points of \(D\) converges in \(X\).
In this work, the authors prove that the countable axiom of choice, CAC for abbreviation, is equivalent to the following statements:
(i) Every densely complete (connected) metric space \(X\) is complete.\
(ii) For every pair of metric spaces \(X\) and \(Y\), if \(Y\) is complete and \(S\) is a dense subspace of \(X\), while \(f : S \rightarrow Y\) is a uniformly continuous function, then there exists a uniformly continuous extension \(F : X\rightarrow Y\) of \(f\).\
(iii) Complete subspaces of metric spaces have complete closures.\
(iv) Complete subspaces of metric spaces are closed.
Also they prove that, for every positive integer \(n\), the space \(\mathbb R^n\) is sequential if and only if \(\mathbb R\) is sequential. Finally, it is shown that \(\mathbb R \times \mathbb Q\) is not densely complete if and only if \(\mathrm{CAC}(\mathbb R)\) holds.
Reviewer: Cenap Özel (İzmir)Subbasic open sets in graphs generated by monophonic eccentric neighborhoodshttps://www.zbmath.org/1483.050472022-05-16T20:40:13.078697Z"Gamorez, Anabel E."https://www.zbmath.org/authors/?q=ai:gamorez.anabel-enriquez"Canoy, Sergio R. jun."https://www.zbmath.org/authors/?q=ai:canoy.sergio-rosales-junSummary: In this paper, we use the monophonic eccentric neighborhoods of a graph to generate a base for some topology on the vertex set of the graph. Using this construction, we describe the subbasic open sets in the join, corona and lexicographic product of two graphs.The Scott topology on posets and continuous posetshttps://www.zbmath.org/1483.060012022-05-16T20:40:13.078697Z"Fan, Lihong"https://www.zbmath.org/authors/?q=ai:fan.lihong"He, Wei"https://www.zbmath.org/authors/?q=ai:he.wei|he.wei.3|he.wei.1|he.wei.2(no abstract)Local directed complete sets and properties of their categorieshttps://www.zbmath.org/1483.060022022-05-16T20:40:13.078697Z"Guan, Xue Chong"https://www.zbmath.org/authors/?q=ai:guan.xuechong"Wang, Ge Ping"https://www.zbmath.org/authors/?q=ai:wang.geping(no abstract)Meet continuity of posets via lim-inf-convergencehttps://www.zbmath.org/1483.060032022-05-16T20:40:13.078697Z"Li, Qingguo"https://www.zbmath.org/authors/?q=ai:li.qingguo"Li, Jibo"https://www.zbmath.org/authors/?q=ai:li.jibo(no abstract)Cartesian closeness of the categories of algebraic local complete posets and FS-local directed complete posetshttps://www.zbmath.org/1483.060052022-05-16T20:40:13.078697Z"Xu, Ai Jun"https://www.zbmath.org/authors/?q=ai:xu.ai-jun"Wang, Ge Ping"https://www.zbmath.org/authors/?q=ai:wang.geping(no abstract)Quasicontinuous domains and generalized completely distributive latticeshttps://www.zbmath.org/1483.060112022-05-16T20:40:13.078697Z"Yang, Jinbo"https://www.zbmath.org/authors/?q=ai:yang.jinbo"Luo, Maokang"https://www.zbmath.org/authors/?q=ai:luo.maokang(no abstract)Frames of continuous functionshttps://www.zbmath.org/1483.060132022-05-16T20:40:13.078697Z"Lowen, Wendy"https://www.zbmath.org/authors/?q=ai:lowen.wendy"Sioen, Mark"https://www.zbmath.org/authors/?q=ai:sioen.mark"Van Den Haute, Wouter"https://www.zbmath.org/authors/?q=ai:van-den-haute.wouterIn this paper, the authors propose a new approach to representing a topological space via a frame of continuous functions with values in what they call a topological frame. A topological frame is a frame \(\mathbb{F}\) equipped with a topology such that the operations \[\wedge\colon \mathbb{F}\times \mathbb{F} \to \mathbb{F}\colon (a,b)\mapsto a\wedge b\] and \[\sup_{i\in I}\colon \mathbb{F}^I\to \mathbb{F}\colon (a_i)_{i\in I}\mapsto \sup_{i\in I} a_i\] are continuous. The idea extends that of pointfree topology of investigating topological spaces via their open-set lattices (which are frames of continuous functions to the Sierpinski space).
The authors investigate properties of a topological space \(X\) via the frame of continuous functions from \(X\) to a topological frame \(\mathbb{F}\), namely the associated notion of sobriety (\(\mathbb{F}\)-sobriety). One of the interesting results provides conditions on \(\mathbb{F}\) ensuring that a Hausdorff topological space is \(\mathbb{F}\)-sober. These conditions are fulfilled as soon as \(\mathbb{F}\) is a chain with \(0\neq 1\) equipped with the Scott topology. Further, \(\mathbb{F}\)-spectra of \(\mathbb{F}\)-function frames are computed for various spaces \(X\) and frames \(\mathbb{F}\) and a number of spaces that are not \(\mathbb{F}\)-sober are exhibited, showing in particular that the Hausdorff condition in the aforementioned result cannot be relaxed to classical sobriety. A final section discusses the relation between \(\mathbb{F}\)-sobriety and the notion of \(\mathbb{F}\)-fuzzy sobriety as considered in [\textit{D. Zhang} and \textit{Y. Liu}, Fuzzy Sets Syst. 76, No. 2, 259--270 (1995; Zbl 0852.54008)].
The paper ends with a brief outline of some open problems.
Reviewer: Jorge Picado (Coimbra)Weakly spatial localeshttps://www.zbmath.org/1483.060142022-05-16T20:40:13.078697Z"Sun, Xiang Rong"https://www.zbmath.org/authors/?q=ai:sun.xiangrong"He, Wei"https://www.zbmath.org/authors/?q=ai:he.wei.2|he.wei|he.wei.3|he.wei.1(no abstract)Closed filters of quantaleshttps://www.zbmath.org/1483.060212022-05-16T20:40:13.078697Z"Liu, Zhi Bin"https://www.zbmath.org/authors/?q=ai:liu.zhibin(no abstract)The Zariski covering number for vector spaces and moduleshttps://www.zbmath.org/1483.130342022-05-16T20:40:13.078697Z"Ghosh, Soham"https://www.zbmath.org/authors/?q=ai:ghosh.sohamSummary: Given a module \(M\) over a commutative unital ring \(R\), let \(\sigma (M,R)\) denote the covering number, i.e. the smallest (cardinal) number of proper submodules whose union covers \(M\); this includes the covering numbers of Abelian groups, which are extensively studied in the literature. Recently, \textit{A. Khare} and \textit{A. Tikaradze} [``Covering modules by proper submodules'', Commun. Algebra 50, No. 2, 498--507 (2021; \url{doi:10.1080/00927872.2021.1959922})] showed in several cases that \(\sigma (M,R) = \min_{\mathfrak{m}\in S_M} | R / \mathfrak{m}| +1\), where \(S_M\) is the set of maximal ideals \(\mathfrak{m}\) with \({\dim_{R /\mathfrak{m}}} (M / \mathfrak{m}M) \geq 2\). Our first main result extends this equality to all \(R\)-modules with small Jacobson radical and finite dual Goldie dimension. We next introduce and study a topological counterpart for finitely generated \(R\)-modules \(M\) over rings \(R\), whose `some' residue fields are infinite, which we call the Zariski covering number \(\sigma_\tau (M,R)\). To do so, we first define the ``induced Zariski topology'' \(\tau\) on \(M\), and now define \(\sigma_\tau (M,R)\) to be the smallest (cardinal) number of proper \(\tau\)-closed subsets of \(M\) whose union covers \(M\). We then show our next main result: \(\sigma_\tau (M,R) = \min_{\mathfrak{m}\in S_M} | R / \mathfrak{m}| +1\), for all finitely generated \(R\)-modules \(M\) for which (a) the dual Goldie dimension is finite, and (b) \(\mathfrak{m} \notin S_M\) whenever \(R /\mathfrak{m}\) is finite. As a corollary, this alternately recovers the aforementioned formula for the covering number \(\sigma(M,R)\) of the aforementioned finitely generated modules. Finally, we discuss the notion of \(\kappa\)-Baire spaces, and show that the inequalities \(\sigma_\tau(M,R) \leq \sigma(M,R) \leq \kappa_M := \min_{\mathfrak{m}\in S_M} | R / \mathfrak{m}| +1\) again become equalities when the image of \(M\) under the continuous map \(q:M \to \prod_{\mathfrak{m} \in m \mathrm{Spec}(R)} M/ \mathfrak{m}M\) (with appropriate Zariski-type topologies) is a \(\kappa_M\)-Baire subspace of the product space.Are Banach spaces monadic?https://www.zbmath.org/1483.180062022-05-16T20:40:13.078697Z"Rosický, J."https://www.zbmath.org/authors/?q=ai:rosicky.jiri\par The paper shows that certain concrete categories from functional analysis are monadic, when taking appropriate ground categories and forgetful functors. The respective proofs involve some elements of the theory of locally presentable categories of, e.g.,~[\textit{J. Adámek} and \textit{J. Rosický}, Locally presentable and accessible categories. Cambridge: Cambridge University Press (1994; Zbl 0795.18007)].
\par First, the author considers the category \textbf{Ban} of (complex) Banach spaces and linear maps of norm \(\leqslant 1\), and the category \textbf{Met} of metric spaces and non-expanding maps. Allowing norms and metrics to take infinite values (while keeping all the other requirements), he arrives at the categories \(\textbf{Ban}_{\infty}\) and \(\textbf{Met}_{\infty}\) of \textit{generalized Banach} and \textit{metric spaces}, respectively, and shows that the forgetful functor \(V_{\infty}:\textbf{Ban}_{\infty}\rightarrow\textbf{Met}_{\infty}\) is monadic (Theorem~2.2).
\par Second, replacing \(\textbf{Met}\) with the category \(\textbf{CMet}^{\bullet}\) of \textit{pointed} complete metric spaces, where pointed metric spaces mean metric spaces \(X\) with a chosen element \(0\in X\) (the morphisms of the category \(\textbf{CMet}^{\bullet}\) then preserve the chosen element \(0\)), the author shows that the forgetful functors \(V^{\bullet}:\textbf{Ban}\rightarrow\textbf{CMet}^{\bullet}\) and \(V^{\bullet}_{\infty}:\textbf{Ban}_{\infty}\rightarrow\textbf{CMet}^{\bullet}_{\infty}\) are monadic (Theorem~3.2 and Remark~3.3~(4)).
\par Third, the author shows that the \textit{unit ball functors} \(U:\textbf{Ban}\rightarrow\textbf{CMet}\) and \(U:\textbf{Ban}\rightarrow\textbf{Met}\) are monadic (Theorem~4.1 and Remark~4.2~(3)), where \(UX=\{x\in X\,|\,\|x\|\leqslant 1\}\) and \(Uf\) is the respective restriction.
\par Fourth, the author considers the category \textbf{BanAlg} of unital Banach algebras and the category \textbf{IBanAlg} of unital involutive Banach algebras~[\textit{T. Bühler} and \textit{D. A. Salamon}, Functional analysis. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1408.46001)], and proves that the respective forgetful functors \(G_0:\textbf{BanAlg}\rightarrow\textbf{Ban}\) and \(G_1:\textbf{IBanAlg}\rightarrow\textbf{Ban}\) are monadic (Theorem~5.1).
\par The author finally asks, whether the forgetful functors \(G:\textbf{CAlg}\rightarrow\textbf{Ban}\) and \(G_c:\textbf{CCAlg}\rightarrow\textbf{Ban}\) are monadic, where \textbf{CAlg} (resp. \textbf{CCAlg}) is the category of (resp. commutative) unital \(C^{\ast}\)-algebras (Remark~5.2~(3)). One should observe that the unit ball functors \(U:\textbf{CAlg}\rightarrow\textbf{Set}\) and \(U:\textbf{CCAlg}\rightarrow\textbf{Set}\) are known to be monadic (Remark~5.2~(2)), where \textbf{Set} is the category of sets and maps.
\par The paper is well written, gives some of its required preliminaries, and will be of interest to the researchers applying categorical methods in functional analysis.
Reviewer: Sergejs Solovjovs (Praha)Norms on categories and analogs of the Schröder-Bernstein theoremhttps://www.zbmath.org/1483.180082022-05-16T20:40:13.078697Z"Insall, Matt"https://www.zbmath.org/authors/?q=ai:insall.matt"Luckhardt, Daniel"https://www.zbmath.org/authors/?q=ai:luckhardt.danielSummary: We generalize the concept of a norm on a vector space to one of
a norm on a category. This provides a unified perspective on many specific matters in many different areas of mathematics like set theory, functional analysis, measure theory, topology, and metric space theory. We will especially address the two last areas in which the monotone-light factorization and, respectively, the Gromov-Hausdorff distance will naturally appear. In our formalization a Schröder-Bernstein property becomes an axiom of a norm which
constitutes interesting properties of the categories in question. The proposed concept provides a convenient framework for metrizations.
For the entire collection see [Zbl 1481.26002].Metrical universality for groups (erratum)https://www.zbmath.org/1483.220022022-05-16T20:40:13.078697Z"Doucha, Michal"https://www.zbmath.org/authors/?q=ai:doucha.michalSummary: The aim of this note is to correct the proof of Proposition 2.15 in the original article [Forum Math. 29, No. 4, 847--872 (2017; Zbl 1375.22001)].Tukey order and diversity of free abelian topological groupshttps://www.zbmath.org/1483.220032022-05-16T20:40:13.078697Z"Gartside, Paul"https://www.zbmath.org/authors/?q=ai:gartside.paul-mSummary: For a Tychonoff space \(X\) the \textit{free abelian topological group} over \(X\), denoted \(A(X)\), is the free abelian group on the set \(X\) with the coarsest topology so that for any continuous map of \(X\) into an abelian topological group its canonical extension to a homomorphism on \(A(X)\) is continuous.
We show there is a family \(\mathcal{A}\) of maximal size, \(2^{\mathfrak{c}}\), consisting of separable metrizable spaces, such that if \(M\) and \(N\) are distinct members of \(\mathcal{A}\) then \(A(M)\) and \(A(N)\) are not topologically isomorphic (moreover, \(A(M)\) neither embeds topologically in \(A(N)\) nor is an open image of \(A(N))\). We show there is a chain \(\mathcal{C}=\{M_\alpha:\alpha<\mathfrak{c}^+\}\), of maximal size, of separable metrizable spaces such that if \(\beta < \alpha\) then \(A( M_\beta)\) embeds as a closed subgroup of \(A( M_\alpha)\) but no subspace of \(A( M_\beta)\) is homeomorphic to \(A( M_\alpha)\).
We show that the character (minimal size of a local base at 0) of \(A(M)\) is \(\mathfrak{d}\) (minimal size of a cofinal set in \(\mathbb{N}^{\mathbb{N}})\) for every non-discrete, analytic \(M\), but consistently there is a co-analytic \(M\) such that the character of \(A(M)\) is strictly above \(\mathfrak{d}\).
The main tool used for these results is the Tukey order on the neighborhood filter at 0 in an \(A(X)\), and a connection with the family of compact subsets of an auxiliary space.Finitely continuous Darboux functionshttps://www.zbmath.org/1483.260062022-05-16T20:40:13.078697Z"Kowalewski, Marcin"https://www.zbmath.org/authors/?q=ai:kowalewski.marcin"Marciniak, Mariola"https://www.zbmath.org/authors/?q=ai:marciniak.mariolaA function \(\varphi\colon\mathbb R\to\mathbb R\) is said to be finitely continuous if the domain can be split into finitely many sets \(E_1,\dots,E_n\) so that \(\varphi\restriction E_i\) is continuous for every \(i=1,\dots,n\). If all sets \(E_i\) in \(\mathbb R=\bigcup_{i=1}^nE_i\) can be chosen closed, then \(\varphi\) is \textit{strongly} finitely continuous. Both properties, finite continuity and strong finite continuity, are considered within the class \(\mathscr D\) of Darboux functions, giving rise to subclasses \(\mathscr D_f\) and~\(\mathscr D_{sf}\), correspondingly. It is relatively clear that finitely continuous functions, even within the Darboux class, can be quite irregular (e.g. if \(n>2\), they can be nonmeasurable; an example with \(n=3\) is demonstrated in the article as Example 1.3). On the other hand, strongly finitely continuous functions are all in the first Baire class. The authors investigate some properties of members of \(\mathscr D_f\) and describe how thin is \(\mathscr D_{sf}\) as a subspace of~\(\mathscr D_f\), under the metric of uniform convergence. The main result (Theorem 3.2) says that \(\mathscr D_{sf}\) is superporous in~\(\mathscr D_f\).
Reviewer: Piotr Sworowski (Bydgoszcz)Remarks on \(\mathbf b\)-metrics, ultrametrics, and metric-preserving functionshttps://www.zbmath.org/1483.260072022-05-16T20:40:13.078697Z"Samphavat, Suchat"https://www.zbmath.org/authors/?q=ai:samphavat.suchat"Khemaratchatakumthorn, Tammatada"https://www.zbmath.org/authors/?q=ai:khemaratchatakumthorn.tammatada"Pongsriiam, Prapanpong"https://www.zbmath.org/authors/?q=ai:pongsriiam.prapanpongSummary: We introduce new classes of functions related to metric-preserving functions, \(\mathbf b\)-metrics, and ultrametrics. We investigate their properties and compare them to those of metric-preserving functions.Weak quasicircles have Lipschitz dimension 1https://www.zbmath.org/1483.301012022-05-16T20:40:13.078697Z"Freeman, David M."https://www.zbmath.org/authors/?q=ai:freeman.david-mandellSummary: We prove that the Lipschitz dimension of any bounded turning Jordan circle or arc is equal to 1. Equivalently, the Lipschitz dimension of any weak quasicircle or arc is equal to 1.Solvability of inclusions of Hammerstein typehttps://www.zbmath.org/1483.340342022-05-16T20:40:13.078697Z"Pietkun, Radosław"https://www.zbmath.org/authors/?q=ai:pietkun.radoslawSummary: We establish a universal rule for solving operator inclusions of Hammerstein type in Lebesgue-Bochner spaces with the aid of some recently proven continuation theorem of Leray-Schauder type for the class of so-called admissible multimaps. Examples illustrating the legitimacy of this approach include the initial value problem for perturbation of \(m\)-accretive multivalued differential equation, the nonlocal Cauchy problem for semilinear differential inclusion, abstract integral inclusion of Fredholm and Volterra type and the two-point boundary value problem for nonlinear evolution inclusion.On totally periodic \(\omega \)-limit sets for monotone maps on regular curveshttps://www.zbmath.org/1483.370252022-05-16T20:40:13.078697Z"Mchaalia, Amira"https://www.zbmath.org/authors/?q=ai:mchaalia.amiraSummary: An \(\omega \)-limit set of a continuous self-mapping of a compact metric space \(X\) is said to be totally periodic if all of its points are periodic. In [Chaos Solitons Fractals 75, 91--95 (2015; Zbl 1352.37048)] \textit{G. Askri} and \textit{I. Naghmouchi} proved that if \(f\) is a one-to-one continuous self mapping of a regular curve, then every totally periodic \(\omega \)-limit set of \(f\) is finite. This also holds whenever \(f\) is a monotone map of a local dendrite by \textit{H. Abdelli} in [Chaos Solitons Fractals 71, 66--72 (2015; Zbl 1352.37123)]. In this paper we generalize these results to monotone maps on regular curves. On the other hand, we give some remarks related to expansivity and totally periodic \(\omega \)-limit sets for every continuous map on compact metric space.Localization of the chain recurrent set using shape theory and symbolical dynamicshttps://www.zbmath.org/1483.370262022-05-16T20:40:13.078697Z"Shoptrajanov, M."https://www.zbmath.org/authors/?q=ai:shoptrajanov.martinSummary: The main aim of this paper is localization of the chain recurrent set in shape theoretical framework. Namely, using the intrinsic approach to shape from [\textit{N. Shekutkovski}, Topol. Proc. 39, 27--39 (2012; Zbl 1215.54008)] we present a result which claims that under certain conditions the chain recurrent set preserves local shape properties. We proved this result in [\textit{N. Shekutkovski} and \textit{M. Shoptrajanov}, Topology Appl. 202, 117--126 (2016; Zbl 1341.54021)] using the notion of a proper covering. Here we give a new proof using the Lebesque number for a covering and verify this result by investigating the symbolical image of a couple of systems of differential equations following [\textit{G. Osipenko}, Differ. Uravn. Protsessy Upr. 1998, No. 4, 59--74 (1998; Zbl 07039068)].Entropy on noncompact setshttps://www.zbmath.org/1483.370272022-05-16T20:40:13.078697Z"Cánovas, Jose S."https://www.zbmath.org/authors/?q=ai:canovas.jose-sSummary: In this paper we review and explore the notion of topological entropy for continuous maps defined on non compact topological spaces which need not be metrizable. We survey the different notions, analyze their relationship and study their properties. Some questions remain open along the paper.Continuous solutions to two iterative functional equationshttps://www.zbmath.org/1483.390102022-05-16T20:40:13.078697Z"Baron, Karol"https://www.zbmath.org/authors/?q=ai:baron.karolLet \((\Omega,\mathcal{A},P)\) be a probability space and \((X,\rho)\) a separable metric space with the \(\sigma\)-algebra \(\mathcal{B}\) of all its Borel subsets. Let \(f:X\times \Omega \to X\) be a \(\mathcal{B}\otimes \mathcal{A}\) measurable function. The author's aim is to look for continuous solutions \(\varphi: X \to \mathbb{R}\) of the equations
\begin{align*}
\varphi(x)&=F(x)-\int_{\Omega} \varphi(f(x,\omega))P(d\omega), \tag{1} \\
\varphi(x)&=F(x)+\int_{\Omega} \varphi(f(x,\omega))P(d\omega). \tag{2}
\end{align*}
Define
\[
f^0(x,\omega_1,\omega_2,\dots)=x, \qquad f^n(x,\omega_1,\omega_2,\dots)=f(f^{n-1}(x,\omega_1,\omega_2,\dots),\omega_n),
\]
and
\[
\pi_n^f(x,B)=P^{\infty}(f^n(x,\cdot)\in B), \quad n\in \mathbb{N}\cup \{0\}, \ B\in \mathcal{B}.
\]
Under the following conditions
\[
\int_{\Omega} \rho(f(x,\omega),f(z,\omega))P(d\omega)\le \lambda \rho(x,z), \quad x,z \in X, \ \lambda \in (0,1), \tag{3}
\]
and
\[
\int_{\Omega} \rho(f(x,\omega),x)P(d\omega)< \infty,
\]
there exists a probability Borel measure \(\pi^f\) on \(X\) such that for every \(x\in X\) the sequence \((\pi_n^f(x,\cdot))\) converges weakly to \(\pi^f\). Assuming these conditions with a fixed \(\lambda \in (0,1)\), let \(\mathcal{F}(X)\) be defined as the set of all continuous functions \(F:X \to \mathbb{R}\) such that there are a sequence \((F_n)\) of real functions on \(X\) and constants \(\theta \in (0,1)\), \(L\in (0,1/\lambda)\) and \(\alpha, \beta \in (0, \infty)\) such that
\[
|F(x)-F_n(x)|\le \alpha \theta^n, \quad x\in X,\ n\in \mathbb{N},
\]
and
\[
|F_n(x)-F_n(z)|\le \beta L^n\rho(x,z), \quad x,z \in X,\ n\in \mathbb{N}.
\]
The first result can now be stated.
Theorem. Assume the previous conditions. If \(F \in \mathcal{F}(X)\) then
\[
\varphi(x)=F(x)-\frac{1}{2}\int_X F(z)\pi^f(dz)+\sum_{n=1}^{\infty} (-1)^n\Big(\int_X F(z)\pi_n^f(x,dz)-\int_X F(z)\pi^f(dz)\Big), \quad x\in X,
\]
defines a continuous solution of (1). If additionally the condition \(\int_X F(x)\pi^f(dx)=0\) holds true, then the formula
\[
\varphi_0(x)=F(x)+\sum_{n=1}^{\infty} \int_X F(z)\pi_n^f(x,dz), \quad x\in X,
\]
defines a continuous solution \(\varphi_0:X \to \mathbb{R}\) of (2).
Concerning the problem of uniqueness of solution the following is proved.
Theorem. Assume the previous conditions. Let \(F \in \mathcal{F}(X)\).
\begin{itemize}
\item[(i)] If \(\varphi_1, \varphi_2\in \mathcal{F}(X)\) are solutions of (1), then \(\varphi_1=\varphi_2\).
\item[(ii)] If \(\varphi_1, \varphi_2\in \mathcal{F}(X)\) are solutions of (2), then \(\varphi_1-\varphi_2\) is a constant function.
\end{itemize}
The last problem investigated in the paper is about the number of functions \(F\) for which (1) and (2) have continuous solutions.
Assume that \((X,\rho)\) is a compact metric space and that condition (3) holds true. Define
\begin{align*}
\mathcal{F}_1&=\{F\in C(X): \text{Eq. (1) has a continuous solution} \}, \\
\mathcal{F}_2&=\{F\in C_f: \text{Eq. (2) has a continuous solution} \},
\end{align*}
where
\[
C_f=\{F\in C(X): \int_X F(x)\pi^f(dx)=0 \}.
\]
Theorem. Under the previous assumptions, the following holds:
\begin{itemize}
\item[(i)] \(\mathcal{F}_1\) is a Borel and dense subset of \(C(X)\), and if \(\mathcal{F}_1\neq C(X)\), then \(\mathcal{F}_1\) is of first category in \(C(X)\) and a Haar zero subset of \(C(X)\).
\item[(ii)] \(\mathcal{F}_2\) is a Borel and dense subset of \(C_f\), and if \(\mathcal{F}_2\neq C_f\), then \(\mathcal{F}_2\) is of first category in \(C_f\) and a Haar zero subset of \(C_f\).
\end{itemize}
Reviewer: Gian Luigi Forti (Milano)When is an invariant mean the limit of a Følner net?https://www.zbmath.org/1483.430042022-05-16T20:40:13.078697Z"Hopfensperger, John"https://www.zbmath.org/authors/?q=ai:hopfensperger.johnThe author deals with the topological invariant mean on \(G\), say \(TLIM(G)\), and the limit points of Følner-nets, say \(TLIM_{0}(G)\), where \(G\) is an amenable locally compact group. The author also discusses whether \(TLIM(G)=TLIM_{0}(G)\). Using these facts some previous results like [\textit{C. Chou}, Trans. Am. Math. Soc. 151, 443--456 (1970; Zbl 0202.14001)] and [\textit{N. Hindman} and \textit{D. Strauss}, Topology Appl. 156, No. 16, 2614--2628 (2009; Zbl 1189.22003)] are improved.
Reviewer: Amir Sahami (Tehran)Higher-order tangent epiderivatives and applications to duality in set-valued optimizationhttps://www.zbmath.org/1483.460432022-05-16T20:40:13.078697Z"Khai, Tran Thien"https://www.zbmath.org/authors/?q=ai:khai.tran-thien"Anh, Nguyen Le Hoang"https://www.zbmath.org/authors/?q=ai:anh.nguyen-le-hoang"Giang, Nguyen Manh Truong"https://www.zbmath.org/authors/?q=ai:giang.nguyen-manh-truongLet \(X,Y\) be real Banach spaces and \(C\) a pointed closed convex cone in \(Y\). For a set-valued mapping \(F:X\to 2^Y\) one defines the graph and the epigraph of \(F\) by \[\mathrm{gr}\, F:=\{(x,y)\in X\times Y:y\in F(x)\}\] and \[\mathrm{epi}_C F=\{(x,y)\in X\times Y: y\in F(x)+C\}\,,\] respectively. Two approaches are known to define contingent derivatives of \(F\) -- using linear approximation of its graph or of its epigraph. In the present paper the authors adopt the second approach -- they consider higher-order tangent epiderivatives, study their properties and give applications to set-valued optimization, namely duality for a mixed constrained optimization problem.
Reviewer: Stefan Cobzaş (Cluj-Napoca)A new class of contractive mappingshttps://www.zbmath.org/1483.470862022-05-16T20:40:13.078697Z"Popescu, O."https://www.zbmath.org/authors/?q=ai:popescu.otilia|popescu.ovidiu|popescu.octavThe author introduces a new class of mappings including enriched contractions, enriched Kannan mappings and enriched Chatterjea mappings, and then he proves some fixed point theorems for these mappings. Some examples illustrating the obtained results are finally presented.
Reviewer: Rodica Luca (Iaşi)The implicit midpoint rule for nonexpansive mappings In 2-uniformly convex hyperbolic spaceshttps://www.zbmath.org/1483.471112022-05-16T20:40:13.078697Z"Fukhar-ud-din, H."https://www.zbmath.org/authors/?q=ai:fukharuddin.h|fukhar-ud-din.hafiz"Khan, A. R."https://www.zbmath.org/authors/?q=ai:khan.abdur-rauf|khan.abdul-rauf-khan|khan.atikur-rahman|khan.amjad-rehman|khan.abdul-rahim|khan.abdul-rahmi|khan.asif-r|khan.abdul-raim|khan.abdul-rahmanSummary: The purpose of this paper is to introduce the implicit midpoint rule (IMR) of nonexpansive mappings in 2- uniformly convex hyperbolic spaces and study its convergence. Strong and \(\Delta\)-convergence theorems based on this algorithm are proved in this new setting. The results obtained hold concurrently in uniformly convex Banach spaces, CAT(0) spaces and Hilbert spaces as special cases.Approximation of endpoints for multivalued nonexpansive mappings in geodesic spaceshttps://www.zbmath.org/1483.471192022-05-16T20:40:13.078697Z"Ullah, Kifayat"https://www.zbmath.org/authors/?q=ai:ullah.kifayat"Khan, Muhammad Safi Ullah"https://www.zbmath.org/authors/?q=ai:khan.muhammad-safi-ullah"Muhammad, Naseer"https://www.zbmath.org/authors/?q=ai:muhammad.naseer"Ahmad, Junaid"https://www.zbmath.org/authors/?q=ai:ahmad.junaidGromov-Hausdorff distance between interval and circlehttps://www.zbmath.org/1483.510042022-05-16T20:40:13.078697Z"Ji, Yibo"https://www.zbmath.org/authors/?q=ai:ji.yibo"Tuzhilin, Alexey A."https://www.zbmath.org/authors/?q=ai:tuzhilin.alexey-aThe authors introduce the new notions of round metric spaces and nonlinearity degree of a metric space.
A metric space \((X, d)\) is called round if, for every \(b \in (0, \operatorname{diam} X)\) and each \(x \in X\), there exists \(y \in X\) such that \(d(x, y) \geqslant b\). The nonlinearity degree \(c(X)\) of \((X, d)\) is defined as
\[
c(X) := \inf_{f \in \operatorname{Lip}_1(X)} \sup \{d(x, y) - |f(x) - f(y)| : x, y \in X\},
\]
where \(\operatorname{Lip}_1(X)\) is the set of all function \(f \colon X \to \mathbb{R}\) which satisfy the inequality \(|f(x) - f(y)| \leqslant d(x, y)\) for all \(x\), \(y \in X\).
Using these concepts, the authors develop an original technique enabled to obtain the exact values of the Gromov-Hausdorff distance \(d_{GH}(I_{\lambda}, S^1)\) between the interval \(I_{\lambda} = [0, \lambda] \subseteq \mathbb{R}\) and the one-dimensional sphere \(S^1 = \{z \in \mathbb{C} : |z| = 1\}\) each of which is equipped with the standard Euclidean metric,
\[
d_{GH}(I_{\lambda}, S^1) = \begin{cases} \frac{\pi}{2} - \frac{\lambda}{4} & \text{if } 0 \leqslant \lambda < \frac{2}{3} \pi,\\
\frac{\pi}{3} & \text{if } \frac{2}{3} \pi \leqslant \lambda \leqslant \frac{5}{3} \pi,\\
\frac{\lambda - \pi}{2} & \text{otherwise}. \end{cases}
\]
The last formula is one of the few exact values of the Gromov-Hausdorff distance between given metric spaces.
Reviewer: Aleksey A. Dovgoshey (Slovyansk)On continuous functions on \(LG\)-topologyhttps://www.zbmath.org/1483.540012022-05-16T20:40:13.078697Z"Badie, Mehdi"https://www.zbmath.org/authors/?q=ai:badie.mehdi"Shahidikia, Ali"https://www.zbmath.org/authors/?q=ai:shahidikia.ali"Kasiri, Hossein"https://www.zbmath.org/authors/?q=ai:kasiri.hosseinSummary: In this article, we introduce \(OLG\), \(CLG\) and \(LG\) maps in the context of \(LGT\)-spaces (\(l\)-generalized topological spaces, see [\textit{A. R. Aliabad} and \textit{A. Sheykhmiri}, Bull. Iran. Math. Soc. 41, No. 1, 239--258 (2015; Zbl 1345.06007)]), show that they are generalizations of continuous function on \(LGT\)-spaces and some properties of them studied. Also, some generalized notions related to continuous functions as weak topology induced, quotient topology and decomposition topology are introduced and studied and is shown that each decomposition space is an \(LG\)-quotient space.On generalized \(\gamma_\micro \)-closed sets and related continuityhttps://www.zbmath.org/1483.540022022-05-16T20:40:13.078697Z"Sen, Ritu"https://www.zbmath.org/authors/?q=ai:sen.rituSummary: In this paper our main interest is to introduce a new type of generalized open sets defined in terms of an operation on a generalized topological space. We have studied some properties of this newly defined sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have studied some preservation theorems in terms of some irresolute functions.A common extension of Lindelöf, H-closed and ccchttps://www.zbmath.org/1483.540032022-05-16T20:40:13.078697Z"Bella, Angelo"https://www.zbmath.org/authors/?q=ai:bella.angeloThis paper defines a new property, called \(\chi\)-net-Lindelöf, that is shared by each of the properties H-closed, Lindelöf, and ccc, and for any space \(X\) with \(\chi\)-net-Lindelöf, \(|X| \leq 2^{\chi(X)}\).
For a subset \(A\) of a space \(X\) and a net \(\xi = \{x_F: F \in [\kappa]^{<\omega}\} \subseteq A\), \(x \in A\) is a \(\theta\)-cluster point of \(\xi\) relative to \(A\) if, given any open set \(U\) in \(X\) with \(x \in U\) and any \(\alpha < \kappa\), there exists \(F \in [\kappa]^{<\omega}\) such that \(\alpha \in F\) and \(x_F \in \overline{U \cap A}\). If \(A = X\), we say \(x\) is a \(\theta\)-cluster point of \(\xi\).
A subset \(A\) of a space \(X\) is \(\chi\)-net-closed if for any net \(\xi = \{ x_F:F \in [\chi(X)]^{<\omega}\} \subseteq A\), if \(\xi\) has a \(\theta\)-cluster point in \(X\), then \(\xi\) has a \(\theta\)-cluster point in \(A\) relative to \(A\).
A space \(X\) is \(\chi\)-net-Lindelöf if for any \(\chi\)-net-closed set \(A \subseteq X\) and any collection \(\mathcal{U} = \{\mathcal{U}_{\alpha} : \alpha < \chi(X)\}\) of subfamilies of open sets in \(X\) whose union covers \(A\) there are countable subfamilies \(\mathcal{V}_{\alpha} \subseteq \mathcal{U}_{\alpha}\) for each \(\alpha < \chi(X)\) such that \(A\subseteq \bigcup \{\overline{\bigcup \mathcal{V}_{\alpha}} :\alpha < \chi(X)\}\).
Reviewer: Jack R. Porter (Lawrence)On star-cellular-Lindelöf spaceshttps://www.zbmath.org/1483.540042022-05-16T20:40:13.078697Z"Singh, Sumit"https://www.zbmath.org/authors/?q=ai:singh.sumit.1|singh.sumitIn the paper under review, the author defines that a topological space \(X\) is said to be star-cellular-Lindelöf if for each open cover \(\mathcal U\) of \(X\) and every cellular family \(\mathcal V\), there exists a Lindelöf subset \(A\) of \(X\) such that \(V\cap A\ne\emptyset\) for every \(V\in\mathcal V\) and \(\mathrm{St}(A,\mathcal U)=X\). It is obvious that a Lindelöf space is necessarily a star-cellular-Lindelöf space. By a classical result proved by \textit{A. V. Arkhangel'skij} in [Sov. Math., Dokl. 10, 951--955 (1969; Zbl 0191.20903); translation from Dokl. Akad. Nauk SSSR 187, 967--970 (1969)], it is well-known that a first countable Lindelöf Hausdorff space has cardinality at most \(\mathfrak{c}\). The author proves in the present paper that a first countable star-cellular-Lindelöf perfect space has cardinality at most \(\mathfrak{c}\). Some other related matters are also discussed.
Reviewer: Wei-Xue Shi (Nanjing)The fundamental group of quotients of products of some topological spaces by a finite group -- a generalization of a theorem of Bauer-Catanese-Grunewald-Pignatellihttps://www.zbmath.org/1483.540052022-05-16T20:40:13.078697Z"Aguilar, Rodolfo"https://www.zbmath.org/authors/?q=ai:aguilar.rodolfoSummary: We provide a description of the fundamental group of the quotient of a product of topological spaces \(X_i\), each admitting a universal cover, by a finite group \(G\), provided that there is only a finite number of path-connected components in \(X^g_i\) for every \(g\in G\). This generalizes previous work of \textit{I. Bauer} et al. [Am. J. Math. 134, No. 4, 993--1049 (2012; Zbl 1258.14043)] and \textit{T. Dedieu} and \textit{F. Perroni} [J. Group Theory 15, No. 3, 439--453 (2012; Zbl 1257.14017)].Inverse systems with simplicial bonding maps and cell structureshttps://www.zbmath.org/1483.540062022-05-16T20:40:13.078697Z"Dębski, Wojciech"https://www.zbmath.org/authors/?q=ai:debski.wojciech"Kawamura, Kazuhiro"https://www.zbmath.org/authors/?q=ai:kawamura.kazuhiro"Tuncali, Murat"https://www.zbmath.org/authors/?q=ai:tuncali.murat"Tymchatyn, Edward D."https://www.zbmath.org/authors/?q=ai:tymchatyn.edward-dThe main result of the paper is: given a topologically complete space \(X\) and a ``defining family of closed covers'' \(\mathcal{A}\) of \(X\) (i.e., it satisfies some local refinement condition and completeness condition), to construct an inverse system \({\mathbf F}_{\mathcal A} = (F_{\lambda}, \pi_{\lambda}^{\mu}, \Lambda)\) of simplicial complexes and simplicial bonding maps, and an inverse system \({\mathbf N}_{\mathcal A} = (N_{\lambda}, \pi_{\lambda}^{\mu}, \Lambda)\) such that the limit spaces \(F_{\infty} = N_{\infty}\), where \(N_{\lambda}\) is a subcomplex of \(F_{\lambda}\) which is almost the same as a cover, and to obtain a proper map \(\pi: N_{\infty} \to X\), and a continuous map \(p: X\to N_{\infty}\) so that \(\pi\circ p = {\mathrm {id}}_X\) and the two maps \(p\circ \pi\) and \({\mathrm {id}}_{N_{\infty}}\) are homotopic (so \(N_{\infty}\) is homotopy equivalent to \(X\)). The construction of the inverse system is based on a modification of the theorem by \textit{S. Mardešić} [Fundam. Math. 114, 53--78 (1981; Zbl 0411.54019)] that every topological space \(X\) admits a polyhedral resolution.
The authors then show that if \(X\) is a compact Hausdorff space and \(\mathcal{A}\) is a family of locally finite, normal, closed covers of \(X\) satisfying the local refinement condition, then the inverse systems \({\mathbf F}_{\mathcal A}\) and \({\mathbf N}_{\mathcal A}\) are HPol-expansions of \(N_{\infty}\) and hence of \(X\), where an expansion is in the shape theoretical sense.
They also show that the restricted inverse system \({\mathbf F}^{(0)} = (F_{\lambda}^{(0)}, \pi_{\lambda}^{\mu}, \Lambda)\) is a cell structure in the sense of [\textit{W. Dębski} and \textit{E. D. Tymchatyn}, Topology Appl. 239, 293--307 (2018; Zbl 1390.54014)] representing a space canonically homeomorphic to \(X\).
Reviewer: Takahisa Miyata (Kobe)Real functions, covers and bornologieshttps://www.zbmath.org/1483.540072022-05-16T20:40:13.078697Z"Bukovský, Lev"https://www.zbmath.org/authors/?q=ai:bukovsky.levIf \(X\) is a topological space, then \(C(X)\) is the set of all real-valued continuous functions on \(X\) while \(USC(X) \subset \mathbb R^X\) consists of upper semicontinuous functions. Given a set \(X\), a proper ideal \(\mathcal B\) of subsets of \(X\) is called a bornology on \(X\) if \(\bigcup\mathcal B=X\). If \(\mathcal B\) is a bornology on \(X\) and \(\mathcal C \subset \mathbb R^X\), then \(\tau_{\mathcal B}\) is the topology on \(\mathcal C\) of uniform convergence on the elements of \(\mathcal B\).
The paper is a survey of recent results about relationships between covering properties of a topological space \(X\) and selection properties in a space of functions on \(X\). The paper considers both pointwise convergence topology and the topology \(\tau_{\mathcal B}\) on the sets \(C(X)\) and \(USC(X)\). The results for the above-mentioned topologies on \(C(X)\) and \(USC(X)\) are also extended to the sets of \(\mathcal A\)-measurable and \(\mathcal A\)-semimeasurable real-valued functions on \(X\) where \(\mathcal A\) is a given family of subsets of the space \(X\).
Reviewer: Vladimir Tkachuk (Ciudad de México)On O'Malley porouscontinuous functionshttps://www.zbmath.org/1483.540082022-05-16T20:40:13.078697Z"Domnik, Irena"https://www.zbmath.org/authors/?q=ai:domnik.irena"Kowalczyk, Stanisław"https://www.zbmath.org/authors/?q=ai:kowalczyk.stanislaw"Turowska, Małgorzata"https://www.zbmath.org/authors/?q=ai:turowska.malgorzataThe reviewed article considers the following modification of the concept of ``porouscontinuity'', see [\textit{J. Borsík} and \textit{J. Holos}, Math. Slovaca 64, No. 3, 741--750 (2014; Zbl 1340.54028)].
\(\bullet\) Let \(r\in [0,1)\). \(x\in\mathbb{R}\) is called a \textit{point of \(\pi O_r\)-density} of a set \(A\subset\mathbb{R}\) if \[\forall_{\eta>0}\; \exists_{\delta>0}\; \exists_{(a,b)\subset A\cap (x-\delta,x+\delta) \setminus\{ x\}}\; \frac{b-a}{\delta}>r.\]
We say that a function \(f\colon\mathbb{R}\to\mathbb{R}\) is \textit{\(\mathcal{SO}_r\) continuous} at a point \(x\) if for each \(\varepsilon>0\), \(x\) is a point of \(\pi O_r\)-density of the set \(f^{-1}(f(x)-\varepsilon,f(x)+\varepsilon)\). Let \(\mathcal{SO}_r\) denote the class of all functions \(f\colon\mathbb{R}\to\mathbb{R}\) which are \(\mathcal{SO}_r\) continuous at each \(x\in\mathbb{R}\).
\(\bullet\) Let \(r\in (0,1]\). \(x\in\mathbb{R}\) is called a \textit{point of \(\mu O_r\)-density} of a set \(A\subset\mathbb{R}\) if \[\forall_{\eta>0}\; \exists_{\delta>0}\; \exists_{(a,b)\subset A\cap (x-\delta,x+\delta) \setminus\{ x\}}\; \frac{b-a}{\delta}\ge r.\]
We say that a function \(f\colon\mathbb{R}\to\mathbb{R}\) is \textit{\(\mathcal{MO}_r\) continuous} at a point \(x\) if for each \(\varepsilon>0\), \(x\) is a point of \(\mu O_r\)-density of the set \(f^{-1}(f(x)-\varepsilon,f(x)+\varepsilon)\). Let \(\mathcal{MO}_r\) denote the class of all functions \(f\colon\mathbb{R}\to\mathbb{R}\) which are \(\mathcal{MO}_r\) continuous at each \(x\in\mathbb{R}\).
The authors study the properties of the families \(\mathcal{SO}_r\) and \(\mathcal{MO}_r\). In particular, they describe the size of these families in terms of porosity.
Reviewer: Tomasz Natkaniec (Gdańsk)Functionally separable subalgebra of \(C(X)\)https://www.zbmath.org/1483.540092022-05-16T20:40:13.078697Z"Soltanpour, Somayeh"https://www.zbmath.org/authors/?q=ai:soltanpour.sSummary: The useful role of \(C_c(X)\) in studying \(C(X)\) motivated us to introduce and study the functionally separable subalgebra \(C_{cd}(Y)\) of \(C(X)\). Let \(Y\) be a dense subset of \(X, C_{cd}(Y)=\{f\in C(X): |f(Y)|\leq \aleph_0\}\). Clearly, \(C_c(X)\subseteq C_{cd}(Y)\subseteq C(X)\) and \(C_{cd}(Y)\) behaves like \(C(X)\) and \(C_c(X)\) in more properties. If \(X\) is a functionally countable or separable space then \(C_{cd}(Y)=C(X)\), in this case \(X\) is called functionally separable space. Whenever \(X\) is pseudocompact and \(\beta X\) is separable, then each \(f\in C(X)\) is countable on a dense subset of \(X\). Conversely, if each \(f\in C(X)\) is countable on a dense subset of \(X\) and each \(G_\delta\)-set has nonempty interior, then \(C(X)=C_c(X)\). Locally functionally separable subalgebra of \(C(X)\) is denoted by \(C_{cod}(X)\) where \(C_{cod}(X)=\{f\in C(X) : |f(Y)|\leq \aleph_0\), for some open dense subset \(Y\) of \(X \}\), clearly \(C_{cod}(X)\subseteq L_c(X)\). For a locally compact and pseudocompact space \(X, C_{cod}(X)=C(X)\) if and only if \(C_{cod}(\beta X)=C(\beta X)\). We introduce \(z_{cod}\)-ideals in \(C_{cod}(X)\) and trivially observe that most of the facts related to \(z\)-ideals are extendable to \(z_{cod}\)-ideals.Regularity and normality in ideal bitopological spaceshttps://www.zbmath.org/1483.540102022-05-16T20:40:13.078697Z"Rubiano, Néstor Raúl Pachón"https://www.zbmath.org/authors/?q=ai:pachon-rubiano.nestor-raulIn this paper, the concepts of pairwise \(I\)-regular ideal bitopological spaces and pairwise \(I\)-normal ideal bitopological spaces are introduced and studied. Also, the concepts of strongly pairwise \(I\)-regular ideal bitopological spaces and strongly pairwise \(I\)-normal ideal bitopological spaces are introduced and studied. Basic properties, examples and the relationships of pairwise \(I\)-regular ideal bitopological spaces, pairwise \(I\)-normal ideal bitopological spaces, strongly pairwise \(I\)-regular ideal bitopological spaces and strongly pairwise \(I\)-normal ideal bitopological spaces are investigated.
Reviewer: Erdal Ekici (Çanakkale)Generalizations of Lindelöf spaces via hereditary classeshttps://www.zbmath.org/1483.540112022-05-16T20:40:13.078697Z"Al-omari, Ahmad"https://www.zbmath.org/authors/?q=ai:al-omari.ahmad-abdullah"Noiri, Takashi"https://www.zbmath.org/authors/?q=ai:noiri.takashiSummary: In this paper by using hereditary classes [\textit{Á. Császár}, Acta Math. Hung. 115, No. 1--2, 29--36 (2007; Zbl 1135.54300)], we define the notion of \(\gamma \)-Lindelöf modulo hereditary classes called \(\gamma \mathcal{H}\)-Lindelöf and obtain several properties of \(\gamma \mathcal{H}\)-Lindelöf spaces.On monotonically normal and transitively (linearly) dually discrete spaceshttps://www.zbmath.org/1483.540122022-05-16T20:40:13.078697Z"Peng, Liang-Xue"https://www.zbmath.org/authors/?q=ai:peng.liangxue"Yang, Zhen"https://www.zbmath.org/authors/?q=ai:yang.zhen"Dong, Hai-Hong"https://www.zbmath.org/authors/?q=ai:dong.hai-hongGiven a topological space \(X\), let \(\tau(X)\) be the topology of \(X\). A \emph{neighborhood assignment} in a space \(X\) is a map \(\phi:X\to \tau(X)\) such that \(x\in \phi(x)\) for every \(x\in X\). The space \(X\) is called \emph{dually discrete} if, for any neighborhood assignment \(\phi\) on \(X\), there exists a discrete set \(D\subset X\) such that \(\bigcup\{\phi(x): x\in D\}= X\). The authors call a space \(X\) \emph{weakly dually discrete} if, for any neighborhood assignment \(\phi\) in \(X\), there exists a discrete subset \(D\subset X\) such that \(X=\bigcup\{\phi(x): x\in D\} \cup \overline D\) and \(\overline D \setminus(\{\phi(x): x\in D\})\) is a closed discrete subspace of \(X\). The notions of linear dual discreteness and transitive dual discreteness are also introduced and studied in the paper.
It is established, among other things, that if \(X\) is a monotonically normal space, then for any neighborhood assignment \(\phi\) in \(X\), there exists a discrete subset \(D \subset X\) such that \(X=\bigcup\{\phi(x): x\in D\} \cup \overline D\) and \(\overline D \setminus(\{\phi(x): x\in D\})\) is left-separated. The authors show that closed subspaces and direct sums of weakly dually discrete spaces are weakly dually discrete while weak dual discreteness implies transitive dual discreteness and the latter implies linear dual discreteness. It is deduced from the above-mentioned results that every monotonically normal space is transitively dually discrete.
Reviewer: Vladimir Tkachuk (Ciudad de México)On almost sober spaceshttps://www.zbmath.org/1483.540132022-05-16T20:40:13.078697Z"Shan, Qidong"https://www.zbmath.org/authors/?q=ai:shan.qidong"Bao, Meng"https://www.zbmath.org/authors/?q=ai:bao.meng"Wen, Xinpeng"https://www.zbmath.org/authors/?q=ai:wen.xinpeng"Xu, Xiaoquan"https://www.zbmath.org/authors/?q=ai:xu.xiaoquanA topological \(T_0\) space \(X\) is called \textit{almost sober} iff each irreducible closed set has a join in the specialization order of \(X\). It is shown that almost sober spaces are closed under retracts, products and saturated subspaces. Other basic properties known from sober spaces are shown not to hold for almost sobriety, for example closed-hereditariness or the stability with respect to forming function spaces with the topology of pointwise convergence. Examples illustrate that almost sobriety of \(X\) is neither necessary nor sufficient for almost sobriety of its Smyth power space (the nonempty saturated compacta of \(X\) equipped with the upper Vietoris topology). The final section proves that the category of almost sober spaces with continuous mappings is not reflective in the category of \(T_0\) spaces and continuous mappings. The famous example of a non-sober Scott space [\textit{P. T. Johnstone}, Lect. Notes Math. 871, 282--283 (1981; Zbl 0469.06002)] functions prominently in this deduction.
Reviewer: Alexander Vauth (Lübbecke)Metrizability of partial metric spaceshttps://www.zbmath.org/1483.540142022-05-16T20:40:13.078697Z"Mykhaylyuk, Volodymyr"https://www.zbmath.org/authors/?q=ai:mykhaylyuk.volodymyr-v"Myronyk, Vadym"https://www.zbmath.org/authors/?q=ai:myronyk.vadymThe concept of \textit{partial metric space}, which is a certain extension of the concept of metric space, was introduced by \textit{S. G. Matthews} [``Partial metric space'', in: 8th British Colloquium for Theoretical Computer Science. University of Warwick (1992)], see also [\textit{S. G. Matthews}, in: Papers on general topology and applications. Papers from the 8th summer conference at Queens College, New York, NY, USA, June 18--20, 1992. New York, NY: The New York Academy of Sciences. 183--197 (1994; Zbl 0911.54025)] as follows:
A function \(p\colon X^2 \to [0,\infty)\) is called a partial metric on \(X\) if
(i) \(x=y\) \(\Leftrightarrow\) \(p(x,x)=p(x,y)=p(y,y)\);
(ii) \(p(x,x)\leqslant p(x,y)\);
(iii) \(p(x,y)=p(y,x)\);
(iv) \(p(x,z)\leqslant p(x,y)+p(y,z)-p(y,y)\)
for all \(x,y,z\in X\).
Abstract: ``We analyze relationship between partial metric spaces and several generalized metric spaces. We first establish that the perfectness of an arbitrary partially metric space \(X\) is equivalent to each of the following properties: developability, semi-stratifiability, \(X\) has a \(\sigma\)-discrete closed network, \(X\) is a \(\beta\)-space with \(T_1\). Using this fact we obtain that the metrizability of a partial metric space \(X\) is equivalent to the stratifiability of \(X\) or to the fact that \(X\) is a perfect regular paracompact space. Moreover, we give examples that indicate the essentiality of certain conditions in the previous two results. Finally, we show that the statement ``every perfectly normal separable partial metric space is metrizable'' is independent of ZFC, similarly as for the Normal Moore Space Problem.''
Reviewer: Evgeniy Petrov (Slovyansk)Growth rate of Lipschitz constants for retractions between finite subset spaceshttps://www.zbmath.org/1483.540152022-05-16T20:40:13.078697Z"Akofor, Earnest"https://www.zbmath.org/authors/?q=ai:akofor.earnest"Kovalev, Leonid V."https://www.zbmath.org/authors/?q=ai:kovalev.leonid-vFor a metric space \(X\) and a positive integer \(n\), \(X(n)\) denotes the family of all nonempty subsets of \(X\) of cardinality at most \(n\). This space \(X(n)\), called the \(nth\) \textit{finite subset space} of \(X\), becomes a metric space with respect to the Hausdorff distance. Thus, the natural embeddings \(X=X(1)\subset X(2)\subset \dots\) are in fact into isometries with respect to this distance.
It is known that some spaces \(X\) present topological obstructions to the existence of Lipschitz retraction maps \(r:X(n)\to X(k)\), for \(k<n\). Nevertheless, \textit{L. V. Kovalev} [Bull. Aust. Math. Soc. 93, No. 1, 146--151 (2016; Zbl 1356.54034)] has shown that for \(X\) a Hilbert space of any dimension (finite or infinite) and for every \(n\), there exists a Lipschitz retraction \(r_n:X(n)\to X(n-1)\) such that the Lipschitz constant \(\mathrm{Lip}(r_n)\leq \max (n^{3/2}, 2n-1)\). The question is whether these Lipschitz constants \(\mathrm{Lip}(r_n)\) can be bounded independently of \(n\).
The paper under review answers this question in the negative. In order to do so, two nice theorems are obtained as main results. More precisely, the following is proved:
Theorem 1.1. Let \(X\) be a normed space over \(\mathbb R\) with \(\dim X \geq 2\). Suppose that \(r:X(n)\to X(k)\) is a Lipschitz retraction, where \(1\leq k \leq n-1\). Then \[\mathrm{Lip}(r) \geq \frac {kn}{2\pi(n-1)} - \frac{1}{2}.\]
Moreover, if \(X\) is a Hilbert space, then \[\mathrm{Lip}(r) \geq \frac {kn}{\pi(n-1)} - 1.\]
On the other hand, for \(X=\mathbb R\), or more generally for the special class of complete geodesic metric spaces called Hadamard spaces, the following is stated:
Theorem 1.3. Let \(X\) be either a normed space over \(\mathbb R\) with \(\dim X \geq 1\), or an Hadamard space with more that one point. If \(r:X(n)\to X(n-1)\) is a Lipschitz retraction, then \(\mathrm{Lip}(r) \geq n - 3\).
Reviewer: Isabel Garrido (Madrid)The McKinsey-Tarski theorem for locally compact ordered spaceshttps://www.zbmath.org/1483.540162022-05-16T20:40:13.078697Z"Bezhanishvili, Guram"https://www.zbmath.org/authors/?q=ai:bezhanishvili.guram"Bezhanishvili, Nick"https://www.zbmath.org/authors/?q=ai:bezhanishvili.nick"Lucero-Bryan, Joel"https://www.zbmath.org/authors/?q=ai:lucero-bryan.joel-gregory"van Mill, Jan"https://www.zbmath.org/authors/?q=ai:van-mill.janThe modal logic \(\mathsf{L}(X)\) of a topological space \(X\) is the set of valid modal formulas, where box is interior and diamond is closure. The McKinsey-Tarski theorem says that the modal logic of any metrizable space without isolated points is \(\mathsf{S}4\). Various analogs of the theorem are known. The main result of the paper, Theorem 3.12, extends the McKinsey-Tarski Theorem to any nonempty locally compact space without isolated points which is a GO-space, that is, a subspace of a topological space coming from a linear order. It is open whether local compactness can be removed. Then Theorem 5.10 computes \(\mathsf{L}(X)\) for each nonempty locally compact GO-space.
Reviewer: Giacomo Lenzi (Fisciano)Convexity in topological betweenness structureshttps://www.zbmath.org/1483.540172022-05-16T20:40:13.078697Z"Anderson, Daron"https://www.zbmath.org/authors/?q=ai:anderson.daron"Bankston, Paul"https://www.zbmath.org/authors/?q=ai:bankston.paul"McCluskey, Aisling"https://www.zbmath.org/authors/?q=ai:mccluskey.aisling-eA betweenness structure is a pair \(\langle X,[\cdot,\cdot,\cdot] \rangle\), where \(X\) is a set and \([\cdot,\cdot,\cdot]\subset X^{3}\) is a ternary relation satisfying that
\begin{itemize}
\item[(B1)] Inclusivity: \((\forall\ xy )\) \(([x,y,y] \wedge [x,x,y])\)
\item[(B2)] Symmetry: \((\forall\ xzy )\) \(([x,z,y] \rightarrow [y,z,x])\)
\item[(B3)] Uniqueness: \((\forall\ xz)\) \(([x,z,x]\rightarrow x=z)\)
\end{itemize}
Given a betweenness structure \(\langle X,[\cdot,\cdot,\cdot] \rangle\), an interval is defined as \([a,b]=\{c\in X:[a,c,b]\}\), a convex subset of \(X\) is a subset \(C\) of \(X\) such that \(a,b\in C\), implies \([a,b]\subset C\). The span of a subset \(A\) of \(X\) is defined as \([A]=\bigcup\{[a,b]:a,b\in A\}\). The convex hull of \(A\) is defined as \([A]^{\omega}=\bigcup\{[A]^{n}:n\in\omega\}\), where \([A]^{0}=A\) and \([A]^{n+1}=[[A]^{n}]\).
With a detailed analysis of examples, in this paper the authors show how the notion of betweenness is related to several important concepts in mathematics. In particular if besides a betweenness structure, \(X\) has a topology, it is possible to define interesting relations between the two structures, starting by asking that intervals are closed. In this sense, the authors define local convexity, upper (and lower) semi-continuity of betweenness, and a type of internal continuity of the betweenness. They obtain results connecting the convexity and the topology in compact connected Hausdorff spaces which are aposyndetic or hereditary unicoherent. In particular, they study how the span and the convex hull interact with the topological closure and interior operators.
Reviewer: Alejandro Illanes (Ciudad de México)Iterated Brownian motion ad libitum is not the pseudo-archttps://www.zbmath.org/1483.540182022-05-16T20:40:13.078697Z"Casse, Jérôme"https://www.zbmath.org/authors/?q=ai:casse.jerome"Curien, Nicolas"https://www.zbmath.org/authors/?q=ai:curien.nicolasSummary: The construction of a random continuum \(\mathcal{C}\) from independent two-sided Brownian motions as considered in [\textit{V. Kiss} and \textit{S. Solecki}, Bull. Lond. Math. Soc. 53, No. 5, 1376--1389 (2021; Zbl 07456946)] almost surely yields a non-degenerate indecomposable continuum. We show that \(\mathcal{C}\) is not-hereditarily indecomposable and, in particular, it is (unfortunately) not the pseudo-arc.On weakly continuum-chainable continuahttps://www.zbmath.org/1483.540192022-05-16T20:40:13.078697Z"López, Rosario A."https://www.zbmath.org/authors/?q=ai:lopez.rosario-a"Macías, Sergio"https://www.zbmath.org/authors/?q=ai:macias.sergioA \textsl{continuum} is a compact connected metric space. The aim of this paper is to introduce and to study two topological properties of continua that generalize the notion of being continuum-chainable.
Both definitions employ elementary notions. The closed connected subsets of a continuum are called \textsl{subcontinua}. Let \(X\) be a continuum, let \(x,y \in X\) and let \(\varepsilon > 0\). A finite set \(\{ K_1, \ldots, K_n \}\) of subcontinua of \(X\) is an \(\varepsilon\)\textsl{-chain of continua from} \(x \) \textsl{to} \(y\) provided that \(K_j \cap K_i \neq \emptyset\) if and only if \(|i-j| \leq 1\), the diameter of each \(K_j\) is less than \(\varepsilon\), \(x \in K_1\) and \(y \in K_n\).
A continuum \(X\) is \textsl{semiweakly continuum-chainable from} \(x\in X\) \textsl{to} \(y \in X\) provided that, for each \(\varepsilon>0\), there exists \(z \in X\) such that \(z\) is \(\varepsilon\)-near to \(y\) and there exists an \(\varepsilon\)-chain from \(x\) to \(z\). A continuum \(X\) is \textsl{semiweakly continuum-chainable} if for each \(x,y \in X\), either \(X\) is semiweakly continuum-chainable from \(x\) to \(y\) or \(X\) is semiweakly continuum-chainable from \(y\) to \(x\). A continuum \(X\) is \textsl{weakly continuum-chainable} if, for each \(x,y \in X\), \(X\) is semiweakly continuum-chainable from \(x\) to \(y\), and \(X\) is semiweakly continuum-chainable from \(y\) to \(x\).
The first part is dedicated to a study of both new notions. Some of the main results are: Continua whose \(n\)-fold hyperspace suspension is \(\frac 1 2\)-homogeneous are semiweakly continuum-chainable, being semiweakly continuum-chainable and being weakly continuum chainable are topological properties carried by continuous images, all homogeneous semiweakly continuum-chainable continua are weakly continuum-chainable, each continuum whose proper subcontinua are arcs is weakly continuum-chainable, the notion of weakly continuum-chainable is preserved under topological products, the conditions that a continuum \(X\) is weakly continuum-chainable and each one of its symmetric products is weakly continuum-chainable are equivalent, an inverse limit whose factor spaces are weakly continuum-chainable continua and the bonding mappings are confluent and surjective is weakly continuum-chainable, and all weakly continuum-chainable continua whose \(n\)-fold hyperspace is finite dimensional are hereditarily decomposable.
The chainable-continuum composant of a point \(p\) of a continuum \(X\) is defined in the last part of the paper by the set of all points \(y \in X\) such that for each \(\varepsilon >0\), there exists an \(\varepsilon\)-chain of continua from \(x\) to \(y\). The authors prove that the collection of chainable-continuum composants is a decomposition of the continuum. They show that the chainable-continuum composants are singletons provided the continuum is hereditarily indecomposable and the converse of this statement fails. Sergio and Rosario characterize the weakly continuum-chainable continua by the density of all of their chainable-continuum composants. A series of conditions on a continuum that imply the arcwise connectedness of each one of its chainable-continuum composants is presented. Finally, the last result gives a relation between the chainable-continuum composants of a Whitney level of a continuum \(X\) and the chainable-continuum composants of \(X\).
Reviewer: David Maya (Toluca)The space of persistence diagrams fails to have Yu's property Ahttps://www.zbmath.org/1483.540202022-05-16T20:40:13.078697Z"Bell, Greg"https://www.zbmath.org/authors/?q=ai:bell.gregory-c"Lawson, Austin"https://www.zbmath.org/authors/?q=ai:lawson.austin"Pritchard, Neil"https://www.zbmath.org/authors/?q=ai:pritchard.neil"Yasaki, Dan"https://www.zbmath.org/authors/?q=ai:yasaki.danIn the first half of the paper, the authors introduce the notion of \(k\)-prisms, and prove that a metric space with \(k\)-prisms for some \(k\geq 1\) fails to have property A of G. Yu. In the second half of the paper, they show that the space of persistence diagrams over \(\mathbb R\) in the Wasserstein \(q\)-metric (\(0<q<\infty\)) has \(k\)-prisms, so it fails to have property A. It is not discussed in the paper whether the space of persistence diagrams admits a uniform embedding into Hilbert space.
Reviewer: Takahisa Miyata (Kobe)An overlooked reality in logic requiring additionshttps://www.zbmath.org/1483.540212022-05-16T20:40:13.078697Z"Dorsett, Charles"https://www.zbmath.org/authors/?q=ai:dorsett.charlesSummary: In this paper, we continue study of foundational topology that revealed a long overlooked reality in logic requiring additions and changes in the current calculus of logic.Minimum topological group topologieshttps://www.zbmath.org/1483.540222022-05-16T20:40:13.078697Z"Chang, Xiao"https://www.zbmath.org/authors/?q=ai:chang.xiao"Gartside, Paul"https://www.zbmath.org/authors/?q=ai:gartside.paul-mSummary: A Hausdorff topological group topology on a group \(G\) is the minimum (Hausdorff) group topology if it is contained in every Hausdorff group topology on \(G\). For every compact metrizable space \(X\) containing an open \(n\)-cell, \(n\geq 2\), the homeomorphism group \(H(X)\) has no minimum group topology. The homeomorphism groups of the Cantor set and the Hilbert cube have no minimum group topology. For every compact metrizable space \(X\) containing a dense open one-manifold, \(H(X)\) has the minimum group topology. Some, but not all, oligomorphic groups have the minimum group topology.Applications of some fixed point theorems for fractional differential equations with Mittag-Leffler kernelhttps://www.zbmath.org/1483.540232022-05-16T20:40:13.078697Z"Afshari, Hojjat"https://www.zbmath.org/authors/?q=ai:afshari.hojjat"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: Using some fixed point theorems for contractive mappings, including \(\alpha\)-\(\gamma\)-Geraghty-type contraction, \(\alpha\)-type \(F\)-contraction, and some other contractions in \(\mathcal{F}\)-metric space, this research intends to investigate the existence of solutions for some Atangana-Baleanu fractional differential equations in the Caputo sense [\textit{A. Atangana} and \textit{D. Baleanu}, ``New fractional derivative with non-local and non-singular kernel. Theory and application to heat transfer model'', Therm. Sci. 20, No. 2, 763--769 (2016; \url{doi:10.2298/TSCI160111018A})].Common fixed point theorems for multivalued mappings in \(b\)-metric spaces with an application to integral inclusionshttps://www.zbmath.org/1483.540242022-05-16T20:40:13.078697Z"Aliouche, Abdelkrim"https://www.zbmath.org/authors/?q=ai:aliouche.abdelkrim"Hamaizia, Taieb"https://www.zbmath.org/authors/?q=ai:hamaizia.taiebSummary: In this paper, we prove common fixed point theorems for two multivalued mappings in complete \(b\)-metric spaces. Our Theorem 5 generalizes Theorem 1 of \textit{F. Khojasteh} et al. [Abstr. Appl. Anal. 2014, Article ID 325840, 5 p. (2014; Zbl 1436.54035)], Theorem 2 of \textit{B. E. Rhoades} [Gen. Math. Notes 27, No. 2, 123--132 (2015), \url{https://www.emis.de/journals/GMN/yahoo_site_admin/assets/docs/12_GMN-7082-V27N2.154194604.pdf}] and Theorem 3 of \textit{M. Demma} and \textit{P. Vetro} [J. Funct. Spaces 2015, Article ID 189861, 6 p. (2015; Zbl 1321.54079)]. Examples are provided to illustrate the validity of our results. Finally, we apply Theorem 5 to establish the existence of common solutions of Fredholm integral inclusions.Nonlinear \(F\)-contractions on \(b\)-metric spaces and differential equations in the frame of fractional derivatives with Mittag-Leffler kernelhttps://www.zbmath.org/1483.540252022-05-16T20:40:13.078697Z"Alqahtani, Badr"https://www.zbmath.org/authors/?q=ai:alqahtani.badr"Fulga, Andreea"https://www.zbmath.org/authors/?q=ai:fulga.andreea"Jarad, Fahd"https://www.zbmath.org/authors/?q=ai:jarad.fahd"Karapınar, Erdal"https://www.zbmath.org/authors/?q=ai:karapinar.erdalSummary: In this manuscript, we aim to refine and characterize nonlinear \(F\)-contractions in a more general framework of \(b\)-metric spaces. We investigate the existence and uniqueness of such contractions in this setting. We discuss the solutions to differential equations in the setting of fractional derivatives involving Mittag-Leffler kernels (Atangana-Baleanu fractional derivative) by using nonlinear \(F\)-contractions that indicate the genuineness of the presented result.Approximate fixed point property and unions of convex disks in the digital planehttps://www.zbmath.org/1483.540262022-05-16T20:40:13.078697Z"Boxer, Laurence"https://www.zbmath.org/authors/?q=ai:boxer.laurenceThis paper studies the approximate fixed point property (AFPP) for digital images in \(\mathbb{Z}^2\). In particular, conditions under which two kinds of union of two convex digital disks have the AFPP and a retraction property have been obtained.
Reviewer: Wei Yao (Nanjing)Existence, data-dependence and stability of coupled fixed point sets of some multivalued operatorshttps://www.zbmath.org/1483.540272022-05-16T20:40:13.078697Z"Choudhury, Binayak S."https://www.zbmath.org/authors/?q=ai:choudhury.binayak-samadder"Metiya, Nikhilesh"https://www.zbmath.org/authors/?q=ai:metiya.nikhilesh"Kundu, Sunirmal"https://www.zbmath.org/authors/?q=ai:kundu.sunirmalSummary: In this work we consider a fixed point problem related to a multivalued coupled mapping which we define here. The function is also assumed to satisfy \(\alpha\)-dominating condition which is a conceptual extension of admissibility condition. We show that the fixed point problem is solvable for the case of the mapping we consider here. Further we examine the data-dependence and stability of fixed point sets associated with the coupled multivalued mapping. The main result is deduced in metric spaces. Its consequences are discussed in partially ordered metric spaces as well as in metric spaces with a graph. There are some illustrative examples. The work is in the domain of set-valued analysis.Fixed point results for generalized nonexpansive and Suzuki mappings with application in \(L^1 (\Omega, \Sigma, \mu)\)https://www.zbmath.org/1483.540282022-05-16T20:40:13.078697Z"Dehici, Abdelkader"https://www.zbmath.org/authors/?q=ai:dehici.abdelkader"Redjel, Najeh"https://www.zbmath.org/authors/?q=ai:redjel.najeh"Atailia, Sami"https://www.zbmath.org/authors/?q=ai:atailia.samiSummary: It is natural to ask whether the weak fixed point property for nonexpansive mappings in Banach spaces is inherited by other generalized nonexpansive mappings without using weak normal structure or close-to normal structure (also called quasi-normal structure) (see [\textit{C. S. Wong}, J. Funct. Anal. 16, 353--358 (1974; Zbl 0281.46015)]). In this paper, we give an affirmative answer to this question for Suzuki mappings and other generalized nonexpansive mappings in the setting of various Banach spaces. In addition, we prove the existence of common fixed points for commuting affine \((c)\)-mappings and Suzuki mappings acting on convex bounded \(L^0\)-closed subsets in the Banach space \(L^1 (\Omega, \Sigma, \mu)\).Existence of coincidence and common fixed points for a sequence of mappings in quasi partial metric spaceshttps://www.zbmath.org/1483.540292022-05-16T20:40:13.078697Z"Dhawan, Pooja"https://www.zbmath.org/authors/?q=ai:dhawan.pooja"Gupta, Vishal"https://www.zbmath.org/authors/?q=ai:gupta.vishal"Kaur, Jatinderdeep"https://www.zbmath.org/authors/?q=ai:kaur.jatinderdeepSummary: Till now, there exists enormous literature showing existence of fixed points using expansive mappings. But the existence of common and coincidence fixed points for a sequence of functions using expansive mappings is still uncharted. In the present article, some coincidence and common fixed point results for a sequence of mappings satisfying generalized weakly expansive conditions in the setting of quasi partial metric spaces have been investigated. The effectiveness of obtained results have been verified with the help of some comparative examples.Common fixed point results for class of set-contraction mappings endowed with a directed graphhttps://www.zbmath.org/1483.540302022-05-16T20:40:13.078697Z"Latif, Abdul"https://www.zbmath.org/authors/?q=ai:latif.abdul"Nazir, Talat"https://www.zbmath.org/authors/?q=ai:nazir.talat"Kutbi, Marwan Amin"https://www.zbmath.org/authors/?q=ai:kutbi.marwan-aminSummary: We present common fixed point results of finite family of set-valued mappings satisfying generalized graphic contractions on set-valued domain endowed with a directed graph. Some examples are presented to support the results proved therein. Consequently, our results unify, improve and generalize various comparable known results.Fixed point theorem for F-contraction mappings, in partial metric spaceshttps://www.zbmath.org/1483.540312022-05-16T20:40:13.078697Z"Luambano, S."https://www.zbmath.org/authors/?q=ai:luambano.sholastica"Kumar, S."https://www.zbmath.org/authors/?q=ai:kumar.santosh"Kakiko, G."https://www.zbmath.org/authors/?q=ai:kakiko.graysonSummary: The purpose of this paper is to establish a fixed point theorem for F-contraction mappings in partial metric spaces. Also, as a consequence, a fixed point theorem for a pair of F-contraction mappings having a unique common fixed point is obtained. In particular, the main results in this paper generalize and extend a fixed point theorem due to [\textit{D. Wardowski}, Fixed Point Theory Appl. 2012, Paper No. 94, 6 p. (2012; Zbl 1310.54074)] in which F-contraction was introduced as a generalization of Banach Contraction Principle. An illustrative example is provided to validate the results.On the KKM theory on ordered spaceshttps://www.zbmath.org/1483.540322022-05-16T20:40:13.078697Z"Park, Sehie"https://www.zbmath.org/authors/?q=ai:park.sehieSummary: Since \textit{C. D. Horvath} and \textit{J. V. Llinares Cisar} [J. Math. Econ. 25, No. 3, 291--306 (1996; Zbl 0852.90006)] began to study maximal elements and fixed points for binary relations on topological ordered spaces, there have appeared many works related to the KKM theory on such spaces by several authors. Independently to these works, we began to study the KKM theory on abstract convex spaces [the author, Nonlinear Anal. Forum 11, No. 1, 67--77 (2006; Zbl 1120.47038)]. Our aim in the present paper is to extend the known results on topological ordered spaces to the corresponding ones on abstract convex spaces.Fixed points and coupled fixed points for multi-valued \((\varphi,\psi)\)-contractions in \(b\)-metric spaceshttps://www.zbmath.org/1483.540332022-05-16T20:40:13.078697Z"Petruşel, Gabriela"https://www.zbmath.org/authors/?q=ai:petrusel.gabriela"Lazăr, Tania"https://www.zbmath.org/authors/?q=ai:lazar.tania-angelica"Lazăr, Vasile L."https://www.zbmath.org/authors/?q=ai:lazar.vasile-lucianSummary: We will study the coupled fixed point problem for multi-valued operators satisfying a nonlinear contraction condition. The approach is based on a fixed point theorem for multi-valued operators in a complete \(b\)-metric space.JS-Prešić contractive mappings in extended modular \(S\)-metric spaces and extended fuzzy \(S\)-metric spaces with an applicationhttps://www.zbmath.org/1483.540342022-05-16T20:40:13.078697Z"Rezaee, M. M."https://www.zbmath.org/authors/?q=ai:rezaee.mohammad-mahdi"Sedghi, S."https://www.zbmath.org/authors/?q=ai:sedghi.shaban"Parvaneh, V."https://www.zbmath.org/authors/?q=ai:parvaneh.vahidSummary: In this paper, we introduce the concept of extended modular \(S\)-metric spaces which induce the notion of extended fuzzy \(S\)-metric spaces and is a generalization of some classes of metric type spaces. We obtain some results for JS-Prešić contractive mappings in this new setting and in the related fuzzy setting. In fact, we obtain the Prešić fixed points via an easier way than the previous methods. An application in integral equations will support our results.An abstract metric space and some fixed point theorems with an application to Markov processhttps://www.zbmath.org/1483.540352022-05-16T20:40:13.078697Z"Roy, Kushal"https://www.zbmath.org/authors/?q=ai:roy.kushal"Saha, Mantu"https://www.zbmath.org/authors/?q=ai:saha.mantuSummary: In this paper we introduce the concept of a new metric-type space named as extended JS-generalized metric space and establish some Banach-type fixed point theorems on such space. We also discuss about the topology of such spaces and prove a theorem like Cantor's intersection theorem therein. Some examples are given in strengthening the hypothesis of our theorems. Moreover our fixed point result is applied to Markov process for finding a stationary distribution.Interpolative Caristi type contractive mapping in an extended \(b\)-metric spacehttps://www.zbmath.org/1483.540362022-05-16T20:40:13.078697Z"Roy, Kushal"https://www.zbmath.org/authors/?q=ai:roy.kushal"Saha, Mantu"https://www.zbmath.org/authors/?q=ai:saha.mantuSummary: In this paper we introduce some interpolative-Caristi-type contractive mappings and prove some fixed point theorems over an extended \(b\)-metric space. Examples are cited in strengthening the hypothesis of our established theorems. Moreover we give an application of our fixed point theorem to obtain solutions of nonlinear integral equations.Saturated fibre contraction principlehttps://www.zbmath.org/1483.540372022-05-16T20:40:13.078697Z"Şerban, Marcel-Adrian"https://www.zbmath.org/authors/?q=ai:serban.marcel-adrianSummary: For a triangular operator \(A:X \times Y \rightarrow X \times Y\), \(A=(B,C)\), where \(B:X \rightarrow X\) and \(C:X \times Y \rightarrow Y\) we study in which conditions on operators \(B:X \rightarrow X\) and \(C:X \times Y \rightarrow Y\) we have that:
\begin{itemize}
\item[(1)] the fixed point problem for triangular operator \(A=(B,C)\) is well posed,
\item[(2)] the operator \(A=(B,C)\) has the Ostrowski property,
\item[(3)] the fixed point equation \((x,y)=A(x,y)\) is generalized Ulam-Hyers stable.
\end{itemize}Common fixed point theorems for T-Hardy-Rodgers contraction mappings in complete cone b-metric spaces with an applicationhttps://www.zbmath.org/1483.540382022-05-16T20:40:13.078697Z"Shashi, Pauline"https://www.zbmath.org/authors/?q=ai:shashi.pauline"Kumar, Santosh"https://www.zbmath.org/authors/?q=ai:kumar.santosh|kumar.santosh.3|kumar.santosh.2|kumar.santosh.1|kumar.santosh.4Summary: This paper presents some fixed point theorems for T-Hardy-Rodgers contraction mappings in complete cone b-metric spaces without the assumption of normality conditions. We prove the existence of the common fixed point in complete cone b-metric spaces for the continuous self mappings. Our results generalize many recent known results in metric spaces, b-metric spaces and cone metric spaces in the literature.Stability results and qualitative properties for Mann's algorithm via admissible perturbations techniquehttps://www.zbmath.org/1483.650812022-05-16T20:40:13.078697Z"Alecsa, Cristian Daniel"https://www.zbmath.org/authors/?q=ai:alecsa.cristian-danielSummary: We will present data dependence results for Mann iteration scheme related to the fixed point inclusion. The approach is based on the admissible perturbation method introduced by A. Petruşel and I. A. Rus. Then we exemplify this approach for the case of multi-valued contractions defined on a metric space endowed with a convexity structure in the sense of Takahashi. Moreover, we will present some qualitative properties of the fixed point problem for multi-valued contractions involving Mann iteration, such as: Ulam-Hyers stability, T-stability and well-posedness of the fixed point problem.A geometrical representation of the quantum information metric in the gauge/gravity correspondencehttps://www.zbmath.org/1483.810362022-05-16T20:40:13.078697Z"Tsuchiya, Asato"https://www.zbmath.org/authors/?q=ai:tsuchiya.asato"Yamashiro, Kazushi"https://www.zbmath.org/authors/?q=ai:yamashiro.kazushiSummary: We study a geometrical representation of the quantum information metric in the gauge/gravity correspondence. We consider the quantum information metric that measures the distance between the ground states of two theories on the field theory side, one of which is obtained by perturbing the other. We show that the information metric is represented by a back reaction to the volume of a codimension-2 surface on the gravity side if the unperturbed field theory possesses the Poincare symmetry.A new scalarizing functional in set optimization with respect to variable domination structureshttps://www.zbmath.org/1483.901152022-05-16T20:40:13.078697Z"Köbis, Elisabeth"https://www.zbmath.org/authors/?q=ai:kobis.elisabeth"Le, Thanh Tam"https://www.zbmath.org/authors/?q=ai:le.thanh-tam"Tammer, Christiane"https://www.zbmath.org/authors/?q=ai:tammer.christiane"Yao, Jen-Chih"https://www.zbmath.org/authors/?q=ai:yao.jen-chihSummary: We introduce a new nonlinear scalarizing functional in set optimization with respect to variable domination structures. By means of this functional, we characterize solutions of set optimization problems, where the solution concept is given by the set approach. We also investigate the relationship between the well-posedness property of a set-valued problem and the Tykhonov well-posedness property of the scalarized problem by means of the proposed scalarizing functional. Also, two classes of well-posed set optimization problems with respect to variable domination structures are identified. Finally, we apply our results to uncertain vector optimization problems.On periodic solutions of random differential inclusionshttps://www.zbmath.org/1483.901172022-05-16T20:40:13.078697Z"Kornev, Sergey"https://www.zbmath.org/authors/?q=ai:kornev.sergei-viktorovich"Liou, Yeong-Cheng"https://www.zbmath.org/authors/?q=ai:liou.yeongcheng"Loi, Nguyen Van"https://www.zbmath.org/authors/?q=ai:nguyen-van-loi."Obukhovskii, Valeri"https://www.zbmath.org/authors/?q=ai:obukhovskii.valeriSummary: By applying the random coincidence degree we develop the methods of random generalized smooth and nonsmooth integral guiding functions and use them for the study of periodic solutions for random differential inclusions in finite dimensional spaces.Extension of monotonic functions and representation of preferenceshttps://www.zbmath.org/1483.910812022-05-16T20:40:13.078697Z"Evren, Özgür"https://www.zbmath.org/authors/?q=ai:evren.ozgur"Hüsseinov, Farhad"https://www.zbmath.org/authors/?q=ai:husseinov.farhadSummary: Consider a dominance relation (a preorder) \( \succsim\) on a topological space \(X\), such as the \textit{greater than or equal to} relation on a function space or a stochastic dominance relation on a space of probability measures. Given a compact set \(K \subseteq X\), we study when a continuous real function on \(K\) that is strictly monotonic with respect to \(\succsim\) can be extended to \(X\) without violating the continuity and monotonicity conditions. We show that such extensions exist for translation invariant dominance relations on a large class of topological vector spaces. Translation invariance or a vector structure are no longer needed when \(X\) is locally compact and second countable. In decision theoretic exercises, our extension theorems help construct monotonic utility functions on the universal space \(X\) starting from compact subsets. To illustrate, we prove several representation theorems for revealed or exogenously given preferences that are monotonic with respect to a dominance relation.