Recent zbMATH articles in MSC 54https://zbmath.org/atom/cc/542022-11-17T18:59:28.764376ZWerkzeugA topological approach to infinity in physics and biophysicshttps://zbmath.org/1496.000572022-11-17T18:59:28.764376Z"Tozzi, Arturo"https://zbmath.org/authors/?q=ai:tozzi.arturo"Peters, James F."https://zbmath.org/authors/?q=ai:peters.james-f-iiiSummary: Physical and biological measurements might display range values extending towards infinite. The occurrence of infinity in equations, such as the black hole singularities, is a troublesome issue that causes many theories to break down when assessing extreme events. Different methods, such as re-normalization, have been proposed to avoid detrimental infinity. Here a novel technique is proposed, based on geometrical considerations and the Alexander Horned sphere, that permits to undermine infinity in physical and biophysical equations. In this unconventional approach, a continuous monodimensional line becomes an assembly of countless bidimensional lines that superimpose in quantifiable knots and bifurcations. In other words, we may state that Achilles leaves the straight line and overtakes the turtle.Topologies and ranks for families of theories in various languageshttps://zbmath.org/1496.031492022-11-17T18:59:28.764376Z"Sudoplatov, Sergeĭ Vladimirovich"https://zbmath.org/authors/?q=ai:sudoplatov.sergei-vladimirovichSummary: Topological properties and characteristics of families of theories reflect possibilities of separation of theories and a complexity both for theories and their neighbourhoods. Previously, topologies were studied for families of complete theories, in general case and for a series of natural classes, and for various families of incomplete theories in a fixed language. The ranks were defined and described for complete theories in a given language, for a hierarchy of theories, for families of incomplete theories, for formulae and for a series of natural families of theories, including families of ordered theories, families of theories of permutations and families of theories of abelian groups. In this paper, we study properties and characteristics for topologies and ranks for families of theories in various languages. It is based on special relations connecting formulae in a given language. These relations are used to define and describe kinds of separations with respect to \(T_0\)-topologies, \(T_1\)-topologies and Hausdorff topologies. Besides special relations are used to define and study ranks for families of theories in various languages. Possibilities of values for the rank are described, and these possibilities are characterized in topological terms.The open dihypergraph dichotomy and the second level of the Borel hierarchyhttps://zbmath.org/1496.031832022-11-17T18:59:28.764376Z"Carroy, Raphaël"https://zbmath.org/authors/?q=ai:carroy.raphael"Miller, Benjamin D."https://zbmath.org/authors/?q=ai:miller.benjamin-david"Soukup, Dániel T."https://zbmath.org/authors/?q=ai:soukup.daniel-tamasSummary: We show that several dichotomy theorems concerning the second level of the Borel hierarchy are special cases of the \(\aleph_0\)-dimensional generalization of the open graph dichotomy, which itself follows from the usual proof(s) of the perfect set theorem. Under the axiom of determinacy, we obtain the generalizations of these results from analytic to separable metric spaces. We also consider connections between cardinal invariants and the chromatic numbers of the corresponding dihypergraphs.
For the entire collection see [Zbl 1454.03009].Algorithmic Sahlqvist preservation for modal compact Hausdorff spaceshttps://zbmath.org/1496.032682022-11-17T18:59:28.764376Z"Zhao, Zhiguang"https://zbmath.org/authors/?q=ai:zhao.zhiguangSummary: In this paper, we use the algorithm ALBA to reformulate the proof in [\textit{G. Bezhanishvili} et al., J. Log. Comput. 25, No. 1, 1--35 (2015; Zbl 1382.03089); \textit{N. Bezhanishvili} and \textit{S. Sourabh}, J. Log. Comput. 27, No. 3, 679--703 (2017; Zbl 1444.03044)] that over modal compact Hausdorff spaces, the validity of Sahlqvist sequents are preserved from open assignments to arbitrary assignments. In particular, we prove an adapted version of the topological Ackermann lemma based on the Esakia-type lemmas proved in [Zbl 1382.03089; Zbl 1444.03044].
For the entire collection see [Zbl 1369.03021].Characterization of posets for liminf convergence being topologicalhttps://zbmath.org/1496.060062022-11-17T18:59:28.764376Z"Shen, Ao"https://zbmath.org/authors/?q=ai:shen.ao"Lu, Jing"https://zbmath.org/authors/?q=ai:lu.jing"Li, Qingguo"https://zbmath.org/authors/?q=ai:li.qingguo|li.qingguo.1In this paper, the authors supply an example to illustrate that a dcpo in which the liminf convergence is topological may not be continuous, and proved that a poset is continuous if and only if it is meet continuous and the liminf convergence is topological and the Lawson topology equals the liminf topology. In particular, the authors obtain a complete characterization of posets for the liminf convergence being topological. These results are interesting and extend the framework of domain theory.
Reviewer: Wenfeng Zhang (Nanchang)Insertion theorems for countably paracompact frames and stratifiable frameshttps://zbmath.org/1496.060132022-11-17T18:59:28.764376Z"Yang, Er-Guang"https://zbmath.org/authors/?q=ai:yang.erguangSummary: The notion of localic real functions was introduced by \textit{J. Gutiérrez García} et al. [J. Pure Appl. Algebra 213, No. 6, 1064--1074 (2009; Zbl 1187.06005)] with which pointfree forms of insertion theorems for some classes of topological spaces were obtained. In this paper, we present insertion theorems for countably paracompact frames and stratifiable frames in terms of localic real functions, yielding the insertion theorems for countably paracompact spaces and stratifiable spaces while applied to the frame of open subsets of these spaces.Generated new classes of permutation I/B-algebrashttps://zbmath.org/1496.060192022-11-17T18:59:28.764376Z"Khalil, Shuker Mahmood"https://zbmath.org/authors/?q=ai:khalil.shuker-mahmood"Suleiman, Enoch"https://zbmath.org/authors/?q=ai:suleiman.enoch"Torki, Modhar M."https://zbmath.org/authors/?q=ai:torki.modhar-m(no abstract)On \(S\)-second spectrum of a modulehttps://zbmath.org/1496.130132022-11-17T18:59:28.764376Z"Çeken, Seçil"https://zbmath.org/authors/?q=ai:ceken.secilSummary: Let \(R\) be a commutative ring with identity, \(S\) be a multiplicatively closed subset of \(R\). A submodule \(N\) of an \(R\)-module \(M\) with \(\mathrm{ann}_R(N)\cap S=\emptyset\) is called an \(S\)-second submodule of \(M\) if there exists a fixed \(s\in S\), and whenever \(rN\subseteq K\), where \(r\in R\) and \(K\) is a submodule of \(M\), then either \(rsN=0\) or \(sN\subseteq K\). The set of all \(S\)-second submodules of \(M\) is called \(S\)-second spectrum of \(M\) and denoted by \(S\)-\(Spec^s(M)\). In this paper, we construct and study two topologies on \(S\)-\(Spec^s(M)\). We investigate some connections between algebraic properties of \(M\) and topological properties of \(S\)-\(Spec^s(M)\) such as seperation axioms, compactness, connectedness and irreducibility.Assouad-Nagata dimension of finitely generated \(C^\prime(\frac{1}{6})\) groupshttps://zbmath.org/1496.200712022-11-17T18:59:28.764376Z"Sledd, Levi"https://zbmath.org/authors/?q=ai:sledd.leviSummary: This paper is the first in a two-part series. In this paper, we prove that the Assouad-Nagata dimension of any finitely generated (but not necessarily finitely presented) \( C^\prime(\frac{1}{6})\) group is at most \(2\). In the next paper, we use this result, along with techniques of classical small cancellation theory, to answer two open questions in the study of asymptotic and Assouad-Nagata dimension of finitely generated groups.On the gamma spectrum of multiplication gamma actshttps://zbmath.org/1496.201262022-11-17T18:59:28.764376Z"Abbas, Mehdi S."https://zbmath.org/authors/?q=ai:abbas.mehdi-sadiq"Gubeir, Samer A."https://zbmath.org/authors/?q=ai:gubeir.samer-aSummary: In this paper, we introduce the concept of topological gamma acts as a generalization of Zariski topology. Some topologica lproperties of this topology are studied. Various algebraic properties of topologica lgamma acts have been discussed. We clarify the interplay between this topological space's properties and the algebraic properties of the gamma acts under consideration. Also, the relation between this topological space and (multiplication, cyclic) gamma act was discussed. We also study some separation axioms and the compactness of this topological space.Ohlin type theorem and its applications for strongly convex set-valued mapshttps://zbmath.org/1496.260132022-11-17T18:59:28.764376Z"Bakula, Milica Klaricic"https://zbmath.org/authors/?q=ai:klaricic-bakula.milica"Nikodem, Kazimierz"https://zbmath.org/authors/?q=ai:nikodem.kazimierzSummary: A version of the Ohlin theorem for strongly convex set-valued maps is presented. As an application counterparts of the Jensen, converse Jensen and Hermite-Hadamard inequalities for strongly convex set-valued maps are proved in a simple unified way.When a convergence of filters is measure-theoretichttps://zbmath.org/1496.280012022-11-17T18:59:28.764376Z"Dolecki, Szymon"https://zbmath.org/authors/?q=ai:dolecki.szymonSummary: Convergence almost everywhere cannot be induced by a topology, and if measure is finite, it coincides with almost uniform convergence and is finer than convergence in measure, which is induced by a metrizable topology.
Measures are assumed to be finite. It is proved that convergence in measure is the Urysohn modification of convergence almost everywhere, which is pseudotopological. Extensions of these convergences from sequences to arbitrary filters are discussed, and a concept of measure-theoretic convergence is introduced. A natural extension of convergence almost everywhere is neither measure-theoretic, nor finer than a natural extension of convergence in measure. A straightforward extension of almost uniform convergence is not pseudotopologically induced; it is finer than a natural extension of convergence in measure.Semisolid sets and topological measureshttps://zbmath.org/1496.280152022-11-17T18:59:28.764376Z"Butler, Svetlana V."https://zbmath.org/authors/?q=ai:butler.svetlana-vThe paper investigates topological measures, that is, measures that are defined on open and closed sets of a locally compact space and which are finitely additive on the collection of open and compact sets, inner regular on open sets and outer regular on closed sets. In this context, semisolid sets and solid-set functions are studied and several examples of finite and infinite topological measures are also provided.
Reviewer: Alina Gavrilut (Iasi)Global weak solutions to the Euler-Vlasov equations with finite energyhttps://zbmath.org/1496.352902022-11-17T18:59:28.764376Z"Cao, Wentao"https://zbmath.org/authors/?q=ai:cao.wentaoSummary: This paper concentrates on the global existence of weak solutions in \(L^p\) with finite energy to a type of one-dimensional compressible Euler-Vlasov equations, which models the interaction between the isentropic gas and dispersed particles. Approximate solutions are constructed by adding artificial viscosity. Then the uniform \(L^p\) estimates of the approximate solutions with respect to the artificial viscosity are established through some subtle analysis on level sets of density and relative velocity. The convergence of approximate solutions to the desired weak solutions is guaranteed by the \(L^p\) compensated compactness framework.Lifting the regionally proximal relation and characterizations of distal extensionshttps://zbmath.org/1496.370072022-11-17T18:59:28.764376Z"Cao, Kai"https://zbmath.org/authors/?q=ai:cao.kai"Dai, Xiongping"https://zbmath.org/authors/?q=ai:dai.xiongpingSummary: We consider a commutative diagram (CD) of flows with discrete phase group T and extensions as follows:
\[
\begin{tikzcd}
{(T,X)} \arrow[rr, "\pi"] \arrow[rd, "\phi"] & & {(T,Y)} \arrow[ld, "\psi" '] \arrow[rd, "\psi_K", dotted] & \\
& {(T,Z)} & & {(T,Y_K)} \arrow[ll, "\psi^{\prime}_K" ', dotted]
\end{tikzcd}
\]
where \((T,Y)\) is minimal.
By \(\operatorname{RP}_{\phi}\) and \(\operatorname{RP}_{\psi}\) we denote the relativized regionally proximal relations in \(X\) and \(Y\), respectively. We mainly prove, among other things, the following:
1. If \(X\) is topologically transitive \(\phi\)-distal, then \(X\) is minimal.
2. \((\pi \times \pi)\operatorname{RP}_{\phi} = \operatorname{RP}_{\psi}\).
3. If \(Y\) is locally \(\psi\)-Bronstein, then \(\operatorname{RP}_{\psi} \circ P\psi\) is an equivalence relation, and, \(\bar{y} \in \operatorname{RP}_{\psi}\) whenever \(\bar{y} \in \operatorname{RP}_{\psi} \circ \operatorname{RP}_{\psi}\) is almost periodic.
4. If \(Y_d\) is the maximal distal extension of \(Z\) below \(Y\) and \(Z^d\) is the universal minimal distal extension of \(Z\), then \(Y \bot_{Y_d} Z^d\).
5. (a) \(Y\) is locally \(\psi\)-Bronstein iff \(F^d < F^{\prime} A\) where \(A\), \(F\), \(F^d\) are respectively
the Ellis groups of \(Y\), \(Z\), \(Z^d\). (b) If \(Y\) is locally \(\psi\)-Bronstein and \(K\) a \(\tau\)-
closed group with \(F^{\prime} A < K < F\), then there is a unique \(Y_K\) which is \(\psi^{\prime}_K\)-equicontinuous and has the Ellis group \(K\).
We also prove the above theorems in the case \(T\) is a semigroup. Moreover, we show the following in minimal semiflows:
6. \(Y\) is \(\psi\)-distal iff \(psi\) has a DE-tower iff there is a least group-like extension \(X\) via \(\phi\) (i.e., \(\phi^{-1} \phi x = \operatorname{Aut}_\phi (T, X)x\) for all \(x \in X\)).
7. \(\psi\) is group-like iff \(\psi\) has a G-tower that consists of group extensions and inverse limits.Turnpike properties for discrete-time optimal control problems with a Lyapunov functionhttps://zbmath.org/1496.370112022-11-17T18:59:28.764376Z"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: We study the turnpike phenomenon for discrete disperse dynamical systems introduced in [Sib. Mat. Zh. 21, No. 4, 136--145 (1980; Zbl 0453.90024)] by \textit{A. M. Rubinov}, which have a prototype in mathematical economics.Conglomerated filters and statistical measureshttps://zbmath.org/1496.400102022-11-17T18:59:28.764376Z"Kadets, Vladimir"https://zbmath.org/authors/?q=ai:kadets.vladimir-m"Seliutin, Dmytro"https://zbmath.org/authors/?q=ai:seliutin.dmytro"Tryba, Jacek"https://zbmath.org/authors/?q=ai:tryba.jacekThe paper under review is an interdisciplinary paper which deals with measure-theoretical problems connected to statistical convergence, from the point of view of analysis, and with filters and ultrafilters, from a set-theoretical point of view.
Ultrafilters have been introduced by \textit{F.~Riesz} [``Stetigkeitsbegriff und abstrakte Mengenlehre'', Atti del IV Congresso Intern. Matem., Roma 1908, II, Roma, 18--24 (1909; JFM 40.0098.07)], see [\textit{H. L. Bentley} et al., in: Handbook of the history of general topology. Volume 2. Dordrecht: Kluwer Academic Publishers. 577--629 (1998; Zbl 0936.54028)], but have not received due attention until they were rediscovered by the French school.
In the terminology of the paper under review, a \emph{statistical measure} is a non-negative finitely additive measure \(\mu \) defined on the collection of all subsets of $\mathbb N$ and such that \( \mu (\mathbb N) = 1\) and \( \mu (\{ k \} ) = 0\), for all \(k \in \mathbb N\). The name is motivated by the notion of statistical convergence, which plays an important role in mathematical analysis, measure theory and functional analysis. Notice that a \(\{ 0,1\}\)-valued statistical measure corresponds to a free ultrafilter over \(\mathbb N\). As pointed out by the authors, statistical measures, without using this name, have been considered earlier by other scholars working in axiomatic set theory, model theory and descriptive set theory.
The \emph{filter generated by a statistical measure} \(\mu \) is the collection of all subsets \(A \subseteq \mathbb N\) such that \(\mu (A)=1\). With the aim of characterizing such filters, the authors introduce the notions of a \emph{poor} and of a \emph{conglomerated} filter. They show that every filter generated by a statistical measure is poor, that every conglomerated filter is not poor, and that there is a filter which is neither poor nor conglomerated. A filter with the Baire property is conglomerated, hence it is not generated by a statistical measure.
Many examples of statistical measures are known such that the corresponding filter cannot be represented as a countable intersection of ultrafilters. In Section~3, the authors study intersections of families of ultrafilters. If some filter can be represented as a finite intersection of ultrafilters, the representation is unique; this is not the case for infinite intersections.
In Section 4, among other things, an extract from a letter by Piotr Koszmider is reproduced, showing that there is a poor filter which is not generated by a statistical measure. The argument relies on Boolean algebra techniques. The authors credit valuable comments also to István Juhász.
This well-written paper will surely foster further the collaboration among analysts and set theorists.
Reviewer: Paolo Lipparini (Roma)On ideal convergence of triple sequences in intuitionistic fuzzy normed space defined by compact operatorhttps://zbmath.org/1496.400122022-11-17T18:59:28.764376Z"Khan, Vakeel A."https://zbmath.org/authors/?q=ai:khan.vakeel-a"Idrisi, Mohd. Imran"https://zbmath.org/authors/?q=ai:idrisi.mohd-imran"Tuba, Umme"https://zbmath.org/authors/?q=ai:tuba.ummeSummary: The main purpose of this article is to introduce and study some new spaces of \(I\)-convergence of triple sequences in intuitionistic fuzzy normed space defined by compact operator i.e., \(_3 S^I_{(\mu,\nu)}(T)\) and \(_3 S^I_{0(\mu,\nu)}(T)\) and examine some fundamental properties, fuzzy topology and verify inclusion relations lying under these spaces.\(G\)-connectedness in topological groups with operationshttps://zbmath.org/1496.400202022-11-17T18:59:28.764376Z"Mucuk, Osman"https://zbmath.org/authors/?q=ai:mucuk.osman"Çakallı, Hüseyin"https://zbmath.org/authors/?q=ai:cakalli.huseyinSummary: It is a well-known fact that for a Hausdorff topological group \(X\), the limits of convergent sequences in \(X\) define a function denoted by \(\lim\) from the set of all convergent sequences in \(X\) to \(X\). This notion has been modified by \textit{J. Connor} and \textit{K. G. Grosse-Erdmann} [Rocky Mt. J. Math. 33, No. 1, 93--121 (2003; Zbl 1040.26001)] for real functions by replacing \(\lim\) with an arbitrary linear functional \(G\) defined on a linear subspace of the vector space of all real sequences. Recently, some authors have extended the concept to the topological group setting and introduced the concepts of \(G\)-continuity, \(G\)-compactness and \(G\)-connectedness. In this paper, we present some results about \(G\)-hulls, \(G\)-connectedness and \(G\)-fundamental systems of \(G\)-open neighbourhoods for a wide class of topological algebraic structures called groups with operations, which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.On the algebraic dimension of Riesz spaceshttps://zbmath.org/1496.460012022-11-17T18:59:28.764376Z"Baziv, N. M."https://zbmath.org/authors/?q=ai:baziv.n-m"Hrybel, O. B."https://zbmath.org/authors/?q=ai:hrybel.o-bSummary: We prove that the algebraic dimension of an infinite dimensional \(C\)-\(\sigma \)-complete Riesz space (in particular, of a Dedekind \(\sigma \)-complete and a laterally \(\sigma \)-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known.Abundance of independent sequences in compact spaces and Boolean algebrashttps://zbmath.org/1496.460142022-11-17T18:59:28.764376Z"Avilés, Antonio"https://zbmath.org/authors/?q=ai:aviles.antonio"Martínez-Cervantes, Gonzalo"https://zbmath.org/authors/?q=ai:martinez-cervantes.gonzalo"Plebanek, Grzegorz"https://zbmath.org/authors/?q=ai:plebanek.grzegorzLet \(\mathcal{C}\) be a class of compact spaces. By \(\mathcal{C}^{\perp}\) is denoted the orthogonal class of \(\mathcal{C}\), whose elements are those compact spaces \(K\) such that every continuous image of \(K\) that belongs to \(\mathcal{C}\) is metrizable. The aim of the paper under review is to study the orthogonal class of several well-known classes of compact spaces (denoted by capital letters).
In particular, the article can be split in three main parts. In the first part the authors deal with centeredness. It is shown that the class of compact spaces satisfying the countable chain condition (CCC) is the orthogonal class of the following classes of compact spaces: uniformly Eberlein, Eberlein, Talagrand and Gul'ko (Proposition~9). Moreover, Martin's axiom MA\(_{\omega_1}\) is equivalent to CCC=CORSON\(^{\perp}\) (Proposition~13).
The second part is devoted to the classes of Radon-Nikodým, weakly Radon-Nykodým and the class of weakly Radon-Nikodým Boolean algebras denoted by RN, WRN and WRN(B), respectively. Among others it is proved that dyadic compact spaces are contained in WRN\(^{\perp}\) (Proposition~20). If MA\(_{\omega_1}\) does not hold, then there exists a nonmetrizable zero-dimensional compact space in CORSON \(\cap\) WRN(B)\(^{\perp}\) (Corollary~27). The class WRN(B)\(^{\perp}\setminus\)WRN\(^{\perp}\) contains zero-dimensional compact spaces (Corollary~32).
In the final section, the class of zero-dimensional compact spaces have been investigated. The class ZERO-DIMENSIONAL\(^{\perp}\) is characterized by compact spaces containing at most countably many different clopens (Lemma~33). It is also proved that it is consistent with Martin's axiom and the negation of the continuum hypothesis that there exists a weakly Radon-Nikodým nonmetrizable compact space in ZERO-DIMENSIONAL\(^{(\perp)}\) (Corollary~39).
The symbol \(\mathcal{C}^{(\perp)}\) denotes the class of those compact spaces \(K\) such that every continuous image of any closed subspace of \(K\) that belongs to \(\mathcal{C}\) is metrizable. In Lemma~7 it is shown that \(\mathcal{C}^{(\perp)}\) coincides with the class of hereditarily \(\mathcal{C}^{\perp}\) compact spaces.
Reviewer: Jacopo Somaglia (Pavia)Coupled fixed point theorems on FLM algebrashttps://zbmath.org/1496.470812022-11-17T18:59:28.764376Z"Amini, Kheghat"https://zbmath.org/authors/?q=ai:amini.kheghat"Hosseinzadeh, Hasan"https://zbmath.org/authors/?q=ai:hosseinzadeh.hasan"Vakilabad, Ali Bagheri"https://zbmath.org/authors/?q=ai:vakilabad.ali-bagheri"Abazari, Rasoul"https://zbmath.org/authors/?q=ai:abazari.rasoulThe authors establish some couple fixed point results for fundamental locally multiplicative (FLM) algebras. Some basic results on FLM algebras are established in Section 2. The main result contained in this article is Theorem 3.1, and the coupled fixed point result is proved on a unital complete semi-simple metrizable FLM algebra. Finally, a characterization of couple fixed point
of holomorphic function on FlM algebra is given in Theorem 3.4.
Reviewer: Mewomo Oluwatosin Temitope (Durban)A new approach to the generalization of Darbo's fixed point problem by using simulation functions with application to integral equationshttps://zbmath.org/1496.470822022-11-17T18:59:28.764376Z"Asadi, Mehdi"https://zbmath.org/authors/?q=ai:asadi.mehdi"Gabeleh, Moosa"https://zbmath.org/authors/?q=ai:gabeleh.moosa"Vetro, Calogero"https://zbmath.org/authors/?q=ai:vetro.calogeroSummary: We investigate the existence of fixed points of self-mappings via simulation functions and measure of noncompactness. We use different classes of additional functions to get some general contractive inequalities. As an application of our main conclusions, we survey the existence of a solution for a class of integral equations under some new conditions. An example will be given to support our results.Two abstract approaches in vectorial fixed point theoryhttps://zbmath.org/1496.470832022-11-17T18:59:28.764376Z"Cardinali, Tiziana"https://zbmath.org/authors/?q=ai:cardinali.tiziana"Precup, Radu"https://zbmath.org/authors/?q=ai:precup.radu"Rubbioni, Paola"https://zbmath.org/authors/?q=ai:rubbioni.paolaSummary: In this paper a fixed point theory is established for operators defined on Cartesian product spaces. Two abstract approaches are presented in terms of closure operators and of general functionals called measures of deviations from zero resembling the measures of noncompactness. In particular, we give vectorial versions to Mönch's fixed point theorems. An application is included to illustrate the theory.Near fixed point theorems in hyperspaceshttps://zbmath.org/1496.470842022-11-17T18:59:28.764376Z"Wu, Hsien-Chung"https://zbmath.org/authors/?q=ai:wu.hsien-chungSummary: The hyperspace consists of all the subsets of a vector space. It is well-known that the hyperspace is not a vector space because it lacks the concept of inverse element. This also says that we cannot consider its normed structure, and some kinds of fixed point theorems cannot be established in this space. In this paper, we shall propose the concept of null set that will be used to endow a norm to the hyperspace. This normed hyperspace is clearly not a conventional normed space. Based on this norm, the concept of Cauchy sequence can be similarly defined. In addition, a Banach hyperspace can be defined according to the concept of Cauchy sequence. The main aim of this paper is to study and establish the so-called near fixed point theorems in Banach hyperspace.Viscosity approximation methods for hierarchical optimization problems in \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1496.470982022-11-17T18:59:28.764376Z"Liu, Xin-Dong"https://zbmath.org/authors/?q=ai:liu.xindong"Chang, Shih-Sen"https://zbmath.org/authors/?q=ai:chang.shih-senSummary: This paper aims at investigating viscosity approximation methods for solving a system of variational inequalities in a \(\mathrm{CAT}(0)\) space. Two algorithms are given. Under certain appropriate conditions, we prove that the iterative schemes converge strongly to the unique solution of the hierarchical optimization problem. The result presented in this paper mainly improves and extends the corresponding results of \textit{L. Y. Shi} and \textit{R. D. Chen} [J. Appl. Math. 2012, Article ID 421050, 11 p. (2012; Zbl 1281.47059)], \textit{R. Wangkeeree} and \textit{P. Preechasilp} [J. Inequal. Appl. 2013, Paper No. 93, 15 p. (2013; Zbl 1292.47056)] and others.On the strong and \(\delta\)-convergence of new multi-step and \(S\)-iteration processes in a \(\mathrm{CAT}(0)\) spacehttps://zbmath.org/1496.471122022-11-17T18:59:28.764376Z"Başarır, Metin"https://zbmath.org/authors/?q=ai:basarir.metin"Şahin, Aynur"https://zbmath.org/authors/?q=ai:sahin.aynurSummary: In this paper, we introduce a new class of mappings and prove the demiclosedness
principle for mappings of this type in a \(\mathrm{CAT}(0)\) space. Also, we obtain the strong and \(\Delta\)-convergence theorems of new multi-step and \(S\)-iteration processes in a \(\mathrm{CAT}(0)\) space. Our results extend and improve the corresponding recent results announced by many authors in the literature.Strong and \(\Delta \)-convergence theorems for a countable family of multi-valued demicontractive maps in Hadamard spaceshttps://zbmath.org/1496.471202022-11-17T18:59:28.764376Z"Minjibir, Ma'aruf Shehu"https://zbmath.org/authors/?q=ai:minjibir.maaruf-shehu"Salisu, Sani"https://zbmath.org/authors/?q=ai:salisu.saniSummary: In this paper, iterative algorithms for approximating a common fixed point of a countable family of multi-valued demicontractive maps in the setting of Hadamard spaces are presented. Under different mild conditions, the sequences generated are shown to strongly convergent and \(\Delta \)-convergent to a common fixed point of the considered family, accordingly. Our theorems complement many results in the literature.Some convergence results for monotone nonexpansive mappings in ordered \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1496.471212022-11-17T18:59:28.764376Z"Uddin, Izhar"https://zbmath.org/authors/?q=ai:uddin.izhar"Garodia, Chanchal"https://zbmath.org/authors/?q=ai:garodia.chanchal"Yildirim, İsa"https://zbmath.org/authors/?q=ai:yildirim.isaSummary: In this paper, we study Picard-S iteration scheme for monotone nonexpansive mappings in the setting of \(\mathrm{CAT}(0)\) spaces and establish some convergence results. We prove \(\Delta\) and strong convergence results. Further, we provide a non trivial numerical example to illustrate the convergence of our iteration scheme and its stability with respect to the different parameters and initial values.Strong and \(\triangle\)-convergence theorems for total asymptotically nonexpansive nonself mappings in \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1496.471232022-11-17T18:59:28.764376Z"Yang, Li"https://zbmath.org/authors/?q=ai:yang.li.2"Zhao, Fu Hai"https://zbmath.org/authors/?q=ai:zhao.fuhaiSummary: The purpose of this paper is to study the existence theorems of fixed points, \(\triangle\)-convergence and strong convergence theorems for total asymptotically nonexpansive nonself mappings in the framework of \(\mathrm{CAT}(0)\) spaces. The convexity and closedness of a fixed point set of such mappings are also studied. Our results generalize, unify and extend several comparable results in the existing literature.Convergence rate of implicit iteration process and a data dependence resulthttps://zbmath.org/1496.471242022-11-17T18:59:28.764376Z"Yildirim, Isa"https://zbmath.org/authors/?q=ai:yildirim.isa"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahidSummary: The aim of this paper is to introduce an implicit S-iteration process and study its convergence in the framework of W-hyperbolic spaces. We show that the implicit S-iteration process has higher rate of convergence than implicit Mann type iteration and implicit Ishikawa-type iteration processes. We present a numerical example to support the analytic result proved herein. Finally, we prove a data dependence result for a contractive type mapping using implicit S-iteration process.Convex contractions of order \(n\) in \(\mathrm{CAT}(0)\) spaceshttps://zbmath.org/1496.471252022-11-17T18:59:28.764376Z"Yildirim, Isa"https://zbmath.org/authors/?q=ai:yildirim.isa"Tekmanli, Yücel"https://zbmath.org/authors/?q=ai:tekmanli.yucel"Khan, Safeer Hussain"https://zbmath.org/authors/?q=ai:khan.safeer-hussainSummary: In this paper, we work on convex contraction of order \(n\). Our first
result in general metric spaces shows that each convex contraction of order \(n\) is a Bessaga mapping. We then turn our attention to \(\mathrm{CAT}(0)\) spaces. We prove a demiclosedness principle for such mappings in this setting. Next, we consider modified Mann iteration process and prove some convergence theorems for fixed points of such mappings in \(\mathrm{CAT}(0)\) spaces. Our results are new in \(\mathrm{CAT}(0)\) setting. Our results remain true in linear spaces like Hilbert and Banach spaces. Finally, we give an example in order to support our main results and to demonstrate the efficiency of modified Mann iteration process.A unified approach to collectively maximal elements in abstract convex spaceshttps://zbmath.org/1496.520022022-11-17T18:59:28.764376Z"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieThe author establishes a very KKM type theorem in abstract convex spaces from which he then obtains an abstract collectively maximal element theorem. Finally, the author shows that a large number of previous theorems on the existence of maximal element and of equilibrium can be derived from his result.
Reviewer: Mircea Balaj (Oradea)On cosymplectic dynamics. I.https://zbmath.org/1496.530832022-11-17T18:59:28.764376Z"Tchuiaga, Stephane"https://zbmath.org/authors/?q=ai:tchuiaga.stephane"Houenou, Franck"https://zbmath.org/authors/?q=ai:houenou.franck-djideme"Bikorimana, Pierre"https://zbmath.org/authors/?q=ai:bikorimana.pierreThe paper under review is an introduction to cosymplectic topology. By adapting methods from symplectic topology, the authors characterize and study several subgroups of diffeomophisms of a cosymplectic manifold.
Recall that a cosymplectic structure on a smooth manifold is given by a closed \(2\)-form \(\omega\) and a closed \(1\)-form \(\eta\) such that \(\eta \wedge \omega\) is a nowhere vanishing top-form.
In particular, among many other results, the authors show that the Reeb vector field determines the almost cosymplectic nature of a uniform limit of a sequence of almost cosymplectic diffeomorphisms. They also define and study the cosymplectic setting of Hofer and Hofer-like geometries with respect to the group of all cosymplectic diffeomorphisms isotopic to the identity map.
Reviewer: Daniele Angella (Firenze)Soft \(\beta\)-rough sets and their application to determine COVID-19https://zbmath.org/1496.540012022-11-17T18:59:28.764376Z"El Bably, Mostafa K."https://zbmath.org/authors/?q=ai:el-bably.mostafa-k"El Atik, Abd El Fattah A."https://zbmath.org/authors/?q=ai:el-atik.abd-el-fattah-aSummary: Soft rough set theory has been presented as a basic mathematical model for decision-making for many real-life data. However, soft rough sets are based on a possible fusion of rough sets and soft sets which were proposed by \textit{F. Feng} et al. [Inf. Sci. 181, No. 6, 1125--1137 (2011; Zbl 1211.68436)]. The main contribution of the present article is to introduce a modification and a generalization for Feng's approximations, namely, soft \(\beta\)-rough approximations, and some of their properties will be studied. A comparison between the suggested approximations and the previous one [loc. cit.] will be discussed. Some examples are prepared to display the validness of these proposals. Finally, we put an actual example of the infections of coronavirus (COVID-19) based on soft \(\beta\)-rough sets. This application aims to know the persons most likely to be infected with COVID-19 via soft \(\beta\)-rough approximations and soft \(\beta\)-rough topologies.Set-theoretic properties of generalized topologically open sets in relator spaceshttps://zbmath.org/1496.540022022-11-17T18:59:28.764376Z"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-m"Salih, Muwafaq M."https://zbmath.org/authors/?q=ai:salih.muwafaq-m"Száz, Árpád"https://zbmath.org/authors/?q=ai:szaz.arpadSummary: A family \(\mathscr{R}\) of binary relations on a set \(X\) is called a relator on \(X\), and the ordered pair \(X(\mathscr{R})= (X, \mathscr{R})\) is called a relator space. Sometimes relators on \(X\) to \(Y\) are also considered.
By using an obvious definition of the generated open sets, each generalized topology \(\mathscr{T}\) on \(X\) can be easily derived from the family \(\mathscr{R}_{\mathscr{T}}\) of all Pervin's preorder relations \(R_V = V^2 \cup (V^c \times X)\) with \(V\in \mathscr{T}\), where \(V^2 = V \times V\) and \(V^c = X\backslash V\).
For a subset A of the relator space \(X(\mathscr{R})\), we define
\[
A^{\circ}= \operatorname{int}_{\mathscr{R}}\,(A)= \big \{x\in X: \quad \exists \, R\in \mathscr{R}: \quad R\,(x)\subseteq A\,\big \}
\]
and \(A^-= \operatorname{cl}_{\mathscr{R}}\,(A)= \operatorname{int}_{\mathscr{R}}\,(A^c)^c\). And, for instance, we also define
\[
\mathscr{T}_{\mathscr{R}}=\big \{A\subseteq X: \, A\subseteq A^{\circ}\,\big \} \quad \text{and}\quad \mathcal{F}_{\mathscr{R}}=\big \{A\subseteq X: \, A^c\in \mathscr{T}_{\mathscr{R}}\,\big \}.
\]
Moreover, motivated by some basic definitions in topological spaces, for a subset \(A\) of the relator space \(X(\mathscr{R})\) we shall write
\begin{itemize}
\item[(1)] \(A\in \mathscr{T}_{\mathscr{R}}^r\) if \(A = A^{- \circ}\);
\item[(2)] \(A\in \mathscr{T}_{\mathscr{R}}^p\) if \(A \subseteq A^{- \circ}\);
\item[(3)] \(A\in \mathscr{T}_{\mathscr{R}}^s\) if \(A \subseteq A^{\circ -}\);
\item[(4)] \(A\in \mathscr{T}_{\mathscr{R}}^{\alpha}\) if \(A \subseteq A^{\circ - \circ}\);
\item[(5)] \(A\in \mathscr{T}_{\mathscr{R}}^{\beta}\) if \(A \subseteq A^{- \circ -}\);
\item[(6)] \(A\in \mathscr{T}_{\mathscr{R}}^a\) if \(A \subseteq A^{- \circ} \cap A^{\circ -}\)
\item[(7)] \(A\in \mathscr{T}_{\mathscr{R}}^b\) if \(A \subseteq A^{- \circ} \cup A^{\circ -}\)
\item[(8)] \(A\in \mathscr{T}_{\mathscr{R}}^q\) if there exists \(V\in \mathscr{T}_{\mathscr{R}}\) such that \(V \subseteq A \subseteq V^-\);
\item[(9)] \(A\in \mathscr{T}_{\mathscr{R}}^{ps}\) if there exists \(V\in \mathscr{T}_{\mathscr{R}}\) such that \(A \subseteq V \subseteq A^-\);
\item[(10)] \(A\in \mathscr{T}_{\mathscr{R}}^{\gamma}\) if there exists \(V\in \mathscr{T}_{\mathscr{R}}^s\) such that \(A \subseteq V \subseteq A^-\);
\item[(11)] \(A\in \mathscr{T}_{\mathscr{R}}^{\delta}\) if there exists \(V\in \mathscr{T}_{\mathscr{R}}^p\) such that \(V \subseteq A \subseteq V^-\).
\end{itemize}
And, the members of the above families will be called the topologically regular open, preopen, semi-open, \(\alpha\)-open, \(\beta\)-open, \(a\)-open, \(b\)-open, quasi-open, pseudo-open, \(\gamma\)-open, and \(\delta\)-open subsets of the relator space \(X(\mathscr{R})\), respectively.
In a former paper, we have systematically investigated the various relationships among the families \(\mathscr{T}_{\mathscr{R}}^{\kappa}\). Moreover, we have tried to establish several illuminating characterizations of the families \(\mathscr{T}_{\mathscr{R}}^{\kappa}\).
Here, we shall mainly be interested in the most simple set-theoretic properties of the families \(\mathscr{T}_{\mathscr{R}}^{\kappa}\). First of all, we shall briefly investigate their dual families \(\mathcal{F}_{\mathscr{R}}^{\kappa}=\{A\subseteq X: A^c\in \mathscr{T}_{\mathscr{R}}^{\kappa}\}\).
Then, we shall establish some intrinsic characterizations of the families \(\mathscr{T}_{\mathscr{R}}^{\kappa}\). Moreover, we shall give some necessary and sufficient conditions in order that \(\emptyset\), \(\{x\}\), with \(x \in X\), and \(X\) could be contained in \(\mathscr{T}_{\mathscr{R}}^{\kappa}\).
Finally, we shall show that, with the exception of \(\mathscr{T}_{\mathscr{R}}^r\), the families \(\mathscr{T}_{\mathscr{R}}^{\kappa}\) are closed under arbitrary unions. Moreover, for every \(\mathscr{T}_{\mathscr{R}}^{\kappa}\), we shall try to determine those subsets \(A\) of \(X\) which satisfy \(A\cap B\in \mathscr{T}_{\mathscr{R}}^{\kappa}\) for all \(B\in \mathscr{T}_{\mathscr{R}}^{\kappa}\).
Furthermore, we shall indicate that, analogously to the family \(\mathscr{T}_{\mathscr{R}}\) of all topologically open subsets of the relator spaces \(X(\mathscr{R})\), the families \(\mathscr{T}_{\mathscr{R}}^{\kappa}\) can also be used to introduce some interesting classifications of relators.
For the entire collection see [Zbl 1483.00042].Ordinary, super and hyper relators can be used to treat the various generalized open sets in a unified wayhttps://zbmath.org/1496.540032022-11-17T18:59:28.764376Z"Rassias, Themistocles M."https://zbmath.org/authors/?q=ai:rassias.themistocles-m"Száz, Árpád"https://zbmath.org/authors/?q=ai:szaz.arpadSummary: If \(\mathscr{R}\) is a family of relations on \(X\) to \(Y\), \(\mathscr{U}\) is a family of relations on \(\mathscr{P}(X)\) to \(Y\), and \(\mathscr{V}\) is a family of relations on \(\mathscr{P}(X)\) to \(\mathscr{P}(Y)\), then we say that \(\mathscr{R}\) is an ordinary relator, \(\mathscr{U}\) is a super relator, and \(\mathscr{V}\) is a hyper relator on \(X\) to \(Y\).
We show that the \(X = Y\), \(\mathscr{U}=\{U\}\) and \(\mathscr{V}=\{V\}\) particular case of the non-conventional three relator space \((X, Y)(\mathscr{R}, \mathscr{U}, \mathscr{V})\) can be used to treat, in a unified way, the various generalized open sets studied by a great number of topologists.
For the entire collection see [Zbl 1485.65002].The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaceshttps://zbmath.org/1496.540042022-11-17T18:59:28.764376Z"Chigr, Fadoua"https://zbmath.org/authors/?q=ai:chigr.fadoua"Mynard, Frédéric"https://zbmath.org/authors/?q=ai:mynard.fredericSummary: This article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Fréchet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizable (respectively, weakly first-countable, respectively sequential) space is semi-metrizable (respectively first-countable, respectively Fréchet) that clarifies the situation for non-Hausdorff spaces. Using additionally known results on the commutation of the topologizer with product, we obtain simple algebraic proofs of various results of Y. Tanaka on the stability under product of symmetrizability and weak first-countability, and we obtain the same way a new characterization of spaces whose product with every metrizable topology is weakly first-countable, respectively symmetrizable.Three problems in convergence theoryhttps://zbmath.org/1496.540052022-11-17T18:59:28.764376Z"Wojciechowski, Jerzy"https://zbmath.org/authors/?q=ai:wojciechowski.jerzyIn the present paper the author proves three theorems within the realm of convergence theory. To that aim he considers certain classes of filters. If one takes \(\mathbb{H}\) to be the class of all countably based filters, then one obtains the class of paratopologies. That one enables a unification of various classes of quotient maps. In particular, the class of hereditary quotient maps corresponds to the class of pretopologies and the class of biquotient maps to pseudotopologies. The class of paratopologies is obtained by considering countably biquotient maps.
If \(\mathbb{H}\) is the class of all countably complete filters, those that are closed under countable intersections, then the obtained class of \(\mathbb{H}\)-adherence determined convergences is the class of hypotopologies.
Next the author investigates the concept of an initial convergence, and by supposing a class of convergences \(\Phi\) he defines for a convergence \(\eta\) the property for being initially dense in \(\Phi\). Consequently, a class of convergence is said to be \textit{simple} provided that it includes an initially dense convergence. By using this terminology one can say that the standard topology on the set of real numbers is initially dense in the class of completely regular topologies, and in particular the class of completely regular topologies is simple.
Then the author proves the theorem that the class of paratopologies is simple. If assuming that for each cardinal there exists a larger measurable cardinal then the class of hypotopologies is not simple.
Another property of convergences, studied in this paper, is diagonality. This property is important since a topology can be characterized as a diagonal pretopology. The convergences with idempotent set adherence are called subdiagonal, and in general it is true that each diagonal convergence is subdiagonal. A further weakening of the diagonality leads to the notion of being weakly diagonal; it is known by \textit{E. Lowen-Colebunders} [Proc. Am. Math. Soc. 72, 205--210 (1978; Zbl 0369.54002)] that a convergence is weakly diagonal iff filters have closed adherences. Then the author notes that every weakly diagonal convergence is subdiagonal. At the end it is shown that there exists a Hausdorff subdiagonal convergence which is not weakly diagonal.
Reviewer: Dieter Leseberg (Berlin)Intuitionistic fuzzy N-compactness by intuitionistic fuzzy netshttps://zbmath.org/1496.540062022-11-17T18:59:28.764376Z"Pankajam, N."https://zbmath.org/authors/?q=ai:pankajam.natarajan"Pushpalatha, A."https://zbmath.org/authors/?q=ai:pushpalatha.a-pSummary: In this paper, we introduce the notion of R-neighbourhood of an intuitionistic fuzzy point to investigate the compactness by intuitionistic fuzzy nets for arbitrary intuitionistic fuzzy sets and we discuss a series of properties.Projective points over matrices and their separability propertieshttps://zbmath.org/1496.540072022-11-17T18:59:28.764376Z"Agnew, Alfonso F."https://zbmath.org/authors/?q=ai:agnew.alfonso-f"Rathbun, Matt"https://zbmath.org/authors/?q=ai:rathbun.matt"Terry, William"https://zbmath.org/authors/?q=ai:terry.williamThe authors focus their work on topological quotients of real and complex matrices, by various subgroups, and they study their separation properties, as this finds an immediate application to twistor spaces in mathematical physics. As a main conclusion, the authors find that as the group one quotients by gets smaller, the separability properties of the quotient improve. The authors then pose a number of topological questions. For example, the quotient by the general linear group is compact, but the others are not; is there a ground of a further topological study, apart from the separation properties? What about the homotopy or homology properties?
Reviewer: Kyriakos Papadopoulos (Madīnat al-Kuwait)Set-valued maps and some generalized metric spaceshttps://zbmath.org/1496.540082022-11-17T18:59:28.764376Z"Yang, Er-Guang"https://zbmath.org/authors/?q=ai:yang.erguangA sequence \(\{B_n\}_{n\in\mathbb N}\) of closed subsets of a space \(Y\) is called a strictly increasing closed cover of \(Y\) if \(Y=\bigcup_{n\in\mathbb N}B_n\), \(B_1\neq\varnothing\), and \(B_n\varsubsetneq B_{n+1}\) for each \(n\in\mathbb N\). \textit{K. Yamazaki} in [Topology Appl. 154, No. 15, 2817--2825 (2007; Zbl 1131.54011)] introduced the notion of strictly increasing closed covers of topological spaces with which the boundedness of set-valued maps was defined, and gave characterizations of countably paracompact spaces. Yamazaki asked whether stratifiable spaces can be characterized with set-valued maps along the same lines. \textit{L.-H. Xie} and \textit{P.-F. Yan} [Houston J. Math. 43, No. 2, 611--624 (2017; Zbl 1390.54022)] answered the question by presenting some characterizations of stratifiable spaces with set-valued maps.
In this paper the author shows that some generalized metric spaces can also be characterized with set-valued maps with values in \(\mathcal{F}(Y)\) where \(Y\) is a space with a strictly increasing closed cover, a condition which is fulfilled by e.g. the following generalized metric spaces: \(\gamma\)-spaces, first-countable spaces, strongly first-countable spaces, \(k\)-semi-stratifiable spaces, semi-stratifiable spaces, semi-metrizable spaces, Nagata spaces, strongly quasi-metrizable spaces and so on.
Reviewer: Shou Lin (Ningde)Simultaneous extension of continuous and uniformly continuous functionshttps://zbmath.org/1496.540092022-11-17T18:59:28.764376Z"Gutev, Valentin"https://zbmath.org/authors/?q=ai:gutev.valentin-gThis paper deals with the problem of finding simultaneous uniformly continuous extensions of bounded uniformly continuous functions on metric spaces. A classical results of Tietze states that any bounded continuous function \(\phi\colon A \to \mathbb{R}\), where \(A\subset X\) is a closed subset of a metric space \(X\), can be extended to a continuous function \(f\colon X \to \mathbb{R}\). The function \(f\) can be constructed by means of an explicit formula and in the past century many authors have studied variants of Tietze's construction. For example, Hausdorff showed that
\[
f(p)= \inf_{a\in A} \Big[ \phi(a)+\frac{d(a, p)}{d(p, A)}-1 \Big] \quad \quad \text{ for \(p \in X \setminus A\)},
\]
is a continuous extension of \(\phi\). In the article under review it is shown that Hausdorff's construction also preserves uniform continuity. This answers a question of W. S. Watson which was asked in [\textit{M. Mandelkern}, Arch. Math. 55, No. 4, 387--388 (1990; Zbl 0712.54009)]. Moreover, the author develops a general framework for the construction of extension operators. More concretely, Tietze extenders are introduced, which are functions \(\mathbf{F} \colon \{ (s, t)\in \mathbb{R}^2 : s \geq t >0 \} \to (0,1]\) satisfying three natural properties. Using Tietze extenders one can define
\[
\Omega_{\mathbf{F}}[\phi](p)=\sup_{a\in A} \phi(a) \cdot \mathbf{F}(d(a, p), d(p, A)) \quad \quad \text{ for \(p \in X \setminus A\)}.
\]
It is shown that \(\Omega_{\mathbf{F}} \colon \ell_\infty^+(A) \to \ell_\infty^+(X)\) is a sublinear isotone isometry which preserves both continuity and uniform continuity. This recovers many well-known extension results, most notably Tietze's original construction and constructions of Riesz and Dieudonné. Finally, in the last section an extension construction is considered which is due to H. Bohr and it is shown that it also preserves uniform continuity. The paper is well written and provides a good overview of the existing literature on the topic.
Reviewer: Giuliano Basso (Fribourg)On quasi-small loop groupshttps://zbmath.org/1496.540102022-11-17T18:59:28.764376Z"Mashayekhy, Behrooz"https://zbmath.org/authors/?q=ai:mashayekhy.behrooz"Mirebrahimi, Hanieh"https://zbmath.org/authors/?q=ai:mirebrahimi.hanieh"Torabi, Hamid"https://zbmath.org/authors/?q=ai:torabi.hamid"Babaee, Ameneh"https://zbmath.org/authors/?q=ai:babaee.amenehThis paper introduces a notion of ``closeness'' of path-homotopy classes designed to be related to the ``homotopically path-Hausdorff'' property. The concepts studied become particularly relevant for studying fundamental groups of locally complicated spaces. The ``quasi-small loop group'' and related subgroups of the fundamental group end up being closely related to the natural quotient topology on the fundamental group. Overall, this paper was a pleasure to read.
Given two paths \(f,g:[0,1]\to X\) with the same starting and endpoints, \(f\) is \textit{homotopically close to} \(g\) (and write \(f\xrightarrow{\text{close}}g\)) if for every subdivision \(0=t_0<t_1<t_2<\cdots <t_n=1\) of \([0,1]\) and sequence of open sets \(U_1,U_2,\dots , U_n\) with \(g([t_{i-1},t_i])\subseteq U_i\) for \(1\leq i\leq n\), there exists a path \(\gamma\simeq f\), which satisfies \(\gamma([t_{i-1},t_i])\subseteq U_i\) and \(\gamma(t_i)=g(t_i)\) for \(1\leq i\leq n\). Although the authors do not state it this way, if \(X\) is locally path-connected, then \(f\) is homotopically close to \(g\) if and only if every neighborhood of \(g\) in the compact-open topology on the path-space \(P(X)\) contains a path homotopic to \(f\), that is, if \([f]\) is represented by paths arbitrarily close to \(g\).
The authors then define the quasi-small loop group of \(X\) to be
\[
\pi_{1}^{qs}(X,x)=\{[f]\in \pi_1(X,x)\mid f\xrightarrow{\text{close}}g\text{ where }[g]=1\}.
\]
A main result of the paper under review is to show that \(\pi_{1}^{qs}(X,x)\) is a normal subgroup of \(\pi_1(X,x)\), which is trivial if and only if \(X\) is homotopically path-Hausdorff in the sense of [\textit{H. Fischer} et al., Topology Appl. 158, No. 3, 397--408 (2011; Zbl 1219.54028)].
More generally, for a subset \(H\subseteq \pi_1(X,x)\), the authors define \(\pi_{H}^{qs}(X,x)\subseteq \pi_1(X,x)\) to consist of all \([f]\in \pi_1(X,x)\) such that \(f\xrightarrow{\text{close}}h\) for \([h]\in H\). The authors prove in Theorem 3.1 that \(\pi_{H}^{qs}(X,x)\) is a (normal) subgroup of \(\pi_1(X,x)\) whenever \(H\) is. Moreover, \(\pi_{H}^{qs}(X,x)=\pi_{1}^{qs}(X,x)H\) when \(H\) is a subgroup (Prop. 3.2).
In the last section of the paper, the authors begin to explore the relationship between \(\pi_{1}^{qs}(X,x)\) and the topology of \(\pi_{1}^{qtop}(X,x)\), that is, the fundamental group \(\pi_1(X,x)\) equipped with the natural quotient topology inherited from the loop space with the compact-open topology. Here, the reviewer provides a little extra context, not included in the paper, which may aid readers interested in addressing problems asked by the authors or relating their ideas more closely to the topology of \(\pi_{1}^{qtop}(X,x)\). When \(X\) is not locally path-connected, let \(lpc(X)\) be the locally path-connected coreflection and recall that we can identify the (non-topologized) groups \(\pi_1(lpc(X),x)=\pi_1(X,x)\). Under this identification it is clear that \(\pi_{1}^{qs}(lpc(X),x)=\pi_{1}^{qs}(X,x)\). Hence, there is no information lost if we assume \(X\) is locally path-connected. In the locally path-connected case, one can show that \(\pi_{1}^{qs}(X,x)\) is precisely the closure \(\overline{e}\) of the trivial subgroup \(e\) in \(\pi_{1}^{qtop}(X,x)\), i.e. the fundamental group equipped with the natural quotient topology. Thus, Proposition 3.2 in the current paper may be restated as the equality \(\pi_{H}^{qs}(X,x)=\overline{e}H=\bigcup_{h\in H}\overline{h}\) for a subgroup \(H\leq \pi_1(X,x)\). The authors then ask in Problem 3.8: if \(\pi_{H}^{qs}(X,x)=H\), when can it be concluded that \(X\) is homotopically path-Hausdorff rel. \(H\)? Equipped with these observations, we see that for a subgroup \(H\), this problem is asking precisely when \(H=\bigcup_{h\in H}\overline{h}\) implies \(H=\overline{H}\) in \(\pi_{1}^{qtop}(X,x)\).
Reviewer: Jeremy Thomas Brazas (West Chester)Compositions of porouscontinuous functionshttps://zbmath.org/1496.540112022-11-17T18:59:28.764376Z"Kowalczyk, Stanisław"https://zbmath.org/authors/?q=ai:kowalczyk.stanislaw"Turowska, Małgorzata"https://zbmath.org/authors/?q=ai:turowska.malgorzataSummary: The notion of porouscontinuity was introduced by \textit{J. Borsík} and \textit{J. Holos} [Math. Slovaca 64, No. 3, 741--750 (2014; Zbl 1340.54028)]. We find classes of functions whose compositions with porouscontinuous functions are still porouscontinuous. Next, we study compositions of porouscontinuous functions with homeomorphisms. We investigate connections between \(IC(A) \cap HOM\) and \(IC(B) \cap HOM\) for different classes \(A,B \in PC\), where \(PC\) is the set of all classes of porouscontinuous functions and \(HOM\) is the set of all homeomorphisms \(f: \mathbb R\to\mathbb R\).Separately continuous functions with a given rectangular set of points of discontinuityhttps://zbmath.org/1496.540122022-11-17T18:59:28.764376Z"Kozlovskyi, Mykola"https://zbmath.org/authors/?q=ai:kozlovskyi.mykola"Mykhaylyuk, Volodymyr"https://zbmath.org/authors/?q=ai:mykhaylyuk.volodymyr-vWe say that a function \(f(x,y)\) is separately continuous if at any point the restricted functions \(f_x(y)\) and \(f_y(x)\) are continuous as functions of one variable.
In the manuscript, the authors investigate the problem of existence of a separately continuous function \(f\colon X\times Y\to\mathbb{R}\) defined on a product of topological spaces \(X\) and \(Y\) with a given discontinuity points set of the form \(A\times B\). Using an approach based on the classical Schwartz function they prove that there exists a separately continuous function \(f\colon X\times Y\to\mathbb{R}\) whose discontinuity points set is equal to the product \(A\times B\) of nowhere dense functionally closed sets in a quite general case of spaces \(X\) and \(Y\). Moreover, they show that if \(X=Y=\beta\omega\), where \(\beta\omega\) is the Čech-Stone compactification of the countable discrete space \(\omega\), and \(A=B=\beta\omega\setminus\omega\), then there is no separately continuous function \(f\colon X\times Y\to\mathbb{R}\) whose discontinuity points set is equal to \(A\times B\).
Reviewer: Waldemar Sieg (Bydgoszcz)Games relating to weak covering properties in bitopological spaceshttps://zbmath.org/1496.540132022-11-17T18:59:28.764376Z"Eysen, Ali Emre"https://zbmath.org/authors/?q=ai:eysen.ali-emre"Özçağ, Selma"https://zbmath.org/authors/?q=ai:ozcag.selmaSummary: We study topological games related to weak forms of the Menger property in bitopological spaces. In particular we investigate almost Menger game and its connections to games which are associated with the covering properties consisting of covers containing \(G_\delta\) subsets.Some topological and combinatorial properties preserved by inverse limitshttps://zbmath.org/1496.540142022-11-17T18:59:28.764376Z"Camargo, Javier"https://zbmath.org/authors/?q=ai:camargo.javier"Uzcátegui, Carlos"https://zbmath.org/authors/?q=ai:uzcategui.carlos-enriqueSummary: We show that the following properties are preserved under inverse limits: countable fan-tightness, \(q^+\), discrete generation and selective separability. We also present several examples based on inverse limits of countable spaces.Base and weight of Boolean \(\mathrm{I}\)-contact algebrashttps://zbmath.org/1496.540152022-11-17T18:59:28.764376Z"Estaji, Ali Akbar"https://zbmath.org/authors/?q=ai:estaji.ali-akbar"Farokhi Ostad, Javad"https://zbmath.org/authors/?q=ai:farokhi-ostad.javad"Haghdadi, Toktam"https://zbmath.org/authors/?q=ai:haghdadi.toktamSummary: We define the notions of base and weight for Boolean \(\mathrm{I}\)-contact algebras. Using base concept, we present some interesting properties concerning the dense Boolean \(\mathrm{I}\)-contact subalgebra. In addition, we prove that the weight of a Boolean \(\mathrm{I}\)-contact algebra and that of its compactification are the same.On functions preserving products of certain classes of semimetric spaceshttps://zbmath.org/1496.540162022-11-17T18:59:28.764376Z"Lichman, Mateusz"https://zbmath.org/authors/?q=ai:lichman.mateusz"Nowakowski, Piotr"https://zbmath.org/authors/?q=ai:nowakowski.piotr"Tcroboś, Filip"https://zbmath.org/authors/?q=ai:tcrobos.filip``The investigation of functions which coalesce multiple metric spaces into a single one dates back to the beginning of the 1980s when \textit{J. Borsík} and \textit{J. Doboš} [Math. Slovaca 31, 193--205 (1981; Zbl 0562.54016) and ibid. 32, 97--102 (1982; Zbl 0483.54019)] laid formal foundations to the concept of the product-wise metric preserving functions.''
In the present paper, the authors continue the research on functions preserving products of certain classes of generalized metric spaces, extend their scope to a broader class of spaces which satisfy some weaker form of the triangle inequality, and provide analogues of the results of Borsík and Doboš adjusted to the new broader setting.
Reviewer: Shou Lin (Ningde)Chains in partially ordered spaceshttps://zbmath.org/1496.540172022-11-17T18:59:28.764376Z"Lawson, Jimmie"https://zbmath.org/authors/?q=ai:lawson.jimmie-dAfter a careful introduction and survey of existing results, the author provides some elegant new results, some of which generalize results of \textit{L. W. Anderson} and \textit{L. E. Ward jun.} [Proc. Glasg. Math. Assoc. 5, 1--3 (1961; Zbl 0098.25801)]. For example, the author shows that in a locally compact, semiclosed (= \(T_1\)-ordered) partially ordered topological space \(X\), for any \(a, b\) in a closed connected chain \(C\) with \(a < b\), the interval \([a,b]\) is compact, connected, and has the order topology. If \(S\) is a locally compact connected semilattice with \(0\) and either (a) \(S\) is locally order dense, has a closed order, and is semitopological or (b) \(S\) has small semilattices and every \(x, y \in S\) can be connected with a subcontinuum of \(S\), then for each \(x \in S\), there exists a compact connected chain \(C\) with \(0 = \inf C \in C\) and \(x = \sup C \in C\).
Reviewer: Thomas Richmond (Bowling Green)Semi-Kelley continuahttps://zbmath.org/1496.540182022-11-17T18:59:28.764376Z"Illanes, Alejandro"https://zbmath.org/authors/?q=ai:illanes.alejandroIn this article written by one of the most productive researchers in the field of continuum theory in recent times, semi-Kelly continua are studied widely. The first significant results are characterizations of semi-Kelly continua and Kelly continua, where tools are provided to facilitate the study of such continua. By using one of the characterizations, the result is proved saying that the following are equivalent
\begin{itemize}
\item[1.] \(X \times [0,1]\) is semi-Kelly,
\item[2.] \(\text{cone}(X)\) is semi-Kelly, and
\item[3.] \(\text{suspension}(X)\) is semi-Kelly.
\end{itemize}
Furthermore, this holds even if we change semi-Kelly to Kelly.
Later we find the result saying that a semi-Kelly continuum does not contain \(R^3\)-sets, which generalizes the result saying that a semi-Kelly continuum contains no \(R^2\)-continua (a non-empty proper subset \(K\) of a continuum \(X\) is an \(R^3\)-set if there is an open set \(U\) such that \(K \subset U\) and a sequence \((C_n)_{n=1}^{\infty}\) of components of \(U\) such that \(\liminf C_n = K\)).
In the paper, the author also considers the question of which mappings preserve the property of being semi-Kelly. For this purpose, a family of mappings is given, which contains retractions, open mappings, and monotone mappings.
In the last part of the paper, semi-Kelly fans are studied. Among other things, we find two crucial results. The first result says that semi-Kelly fans are semi-smooth, and the second is that a fan is hereditarily semi-Kelly if and only if it is smooth.
The results in the paper present an important contribution to the theory of semi-Kelly continua and offer a further study of these continua with the obtained tools.
Reviewer: Matevž Črepnjak (Maribor)A universal coregular countable second-countable spacehttps://zbmath.org/1496.540192022-11-17T18:59:28.764376Z"Banakh, Taras"https://zbmath.org/authors/?q=ai:banakh.taras-o"Stelmakh, Yaryna"https://zbmath.org/authors/?q=ai:stelmakh.yarynaIn this interesting paper, the authors present a topological characterization of the infinite rational projective space \({\mathbb Q}P^\infty\). It is topologically, the unique countable, second countable space that possesses a superskeleton. Among its properties are the Hausdorff property, it is coregular and has very strong homogeneity properties. Moreover, it is a universal object for the class of all countable, second countable coregular spaces. The proof of the characterization theorem is quite involved and long. As the paper demonstrates, there are many spaces homeomorphic to \({\mathbb Q}P^\infty\) that surface in several seemingly unrelated situations. It is unknown whether the famous Golomb (or Kirch) space contains a subspace homeomorphic to \({\mathbb Q}P^\infty\). This paper is an absolute must for anybody interested in countable connected Hausdorff spaces.
Reviewer: Jan van Mill (Amsterdam)Some pseudocompact-like properties in certain topological groupshttps://zbmath.org/1496.540202022-11-17T18:59:28.764376Z"Tomita, Artur Hideyuki"https://zbmath.org/authors/?q=ai:tomita.artur-hideyuki"Trianon-Fraga, Juliane"https://zbmath.org/authors/?q=ai:trianon-fraga.julianeLet \(\omega^{\ast}\) be the set of non-principal (free) ultrafilters on \(\omega\). Let \(X\) be a topological space, for each \(x\in X\), \(p\in\omega^{\ast}\) and a sequence \((x_n)_{n\in\omega}\) in \(X\), recall that \(x\) is a \textit{\(p\)-limit point} of \((x_n)_{n\in\omega}\) if, for every neighborhood \(U\) of \(x\), \(\{n\in\omega: x_n\in U\}\in p\). It is easy to show that for a space \(X\), \(X\) is countably compact if and only if every sequence in \(X\) has a \(p\)-limit, for some \(p\in\omega^{\ast}\), \(X\) is countably pracompact if and only if there exists a dense subset \(D\) of \(X\) such that every sequence in \(D\) has a \(p\)-limit in \(X\), for some \(p\in\omega^{\ast}\), and \(X\) is pseudocompact if and only if for every countable family \(\{U_n:n\in\omega\}\) of nonempty open subsets of \(X\), there exist \(x\in X\) and \(p\in\omega^{\ast}\) such that, for each neighborhood \(V\) of \(x\), \(\{n\in\omega: V\cap U_n\neq\emptyset\}\in p\). A topological space \(X\) is called \textit{selectively pseudocompact}, defined in [\textit{S. García-Ferreira} and \textit{Y. F. Ortiz-Castillo}, Commentat. Math. Univ. Carol. 55, No. 1, 101--109 (2014; Zbl 1313.54054)] under the name \textit{strong pseudompactness}, if for each sequence \((U_n)_{n\in\omega}\) of nonempty open subsets of \(X\) there are a sequence \((x_n)_{n\in\omega}\), \(x\in X\) and \(p\in\omega^{\ast}\) such that \(x\) is a \(p\)-limit of \((x_n)_{n\in\omega}\) and, for each \(n\in\omega\), \(x_n\in U_n\). These notions are related as follows:
\[
\text{countable compactness} \Rightarrow \text{countable pracompactness}
\]
\[ \Rightarrow \text{selective pseudocompactness} \Rightarrow \text{pseudocompactness.}
\]
\textit{S. Garcia-Ferreira} and \textit{A. H. Tomita} [Topology Appl. 192, 138--144 (2015; Zbl 1330.54045)] gave examples of a selectively pseudocompact group which is not countably compact, and of a pseudocompact group which is not selectively pseudocompact. In this paper, the authors prove that there exists a topological group which is selectively pseudocompact but is not countably pracompact. They also prove that assuming the existence of a single selective ultrafilter, there exists a topological group which is not countably pracompact and has all powers selectively pseudocompact.
The question whether there exists a countably compact group without non-trivial convergent sequences in ZFC has been left open, however, \textit{M. Hrušák} et al. [Trans. Am. Math. Soc. 374, No. 2, 1277--1296 (2021; Zbl 1482.22002)] finally proved that in ZFC, there exists a Hausdorff countably compact topological Boolean group (of size \(\mathfrak c\)) without non-trivial convergent sequences. In this paper, the authors construct, in ZFC, a Hausdorff countably compact topological Boolean group of size \(2^{\mathfrak c}\) without non-trivial convergent sequences answering a question posed in [\textit{M. K. Bellini} et al., Topology Appl. 296, Article ID 107684, 14 p. (2021; Zbl 1481.54029)].
Reviewer: Kohzo Yamada (Shizuoka)Group topologies making every continuous homomorphic image to a compact group connectedhttps://zbmath.org/1496.540212022-11-17T18:59:28.764376Z"Yañez, Víctor Hugo"https://zbmath.org/authors/?q=ai:yanez.victor-hugoIn this paper, the author proves that if an abelian group \(G\) can be equipped with a group topology making all of its continuous homomorphic images to a compact group connected, then it admits a MinAP (minimally almost periodic) group topology, hence for every positive natural number \(m\) the subgroup \(mG\) of \(G\) is either the trivial group or has infinite cardinality by a result of \textit{D. Dikranjan} and \textit{D. Shakhmatov} [``Final solution of Protasov-Comfort's problem on minimally almost periodic group topologies'', Preprint, \url{arXiv:1410.3313}].
Reviewer: Fucai Lin (Zhangzhou)On orbit spaces of distributive binary \(G\)-spaceshttps://zbmath.org/1496.540222022-11-17T18:59:28.764376Z"Gevorgyan, P. S."https://zbmath.org/authors/?q=ai:gevorgyan.pavel-sSummary: The orbit space of a distributive binary \(G\)-space is studied. A number of its properties in the case of a compact binarily acting group \(G\) are established.Generalized multivalued integral type contraction on weak partial metric spacehttps://zbmath.org/1496.540232022-11-17T18:59:28.764376Z"Acar, Özlem"https://zbmath.org/authors/?q=ai:acar.ozlem"Coşkun, Sümeyye"https://zbmath.org/authors/?q=ai:coskun.sumeyye(no abstract)Some common fixed point theorems in complex valued metric spaceshttps://zbmath.org/1496.540242022-11-17T18:59:28.764376Z"Adegani, Ebrahim Analouei"https://zbmath.org/authors/?q=ai:adegani.ebrahim-analouei"Motamednezad, Ahmad"https://zbmath.org/authors/?q=ai:motamednezad.ahmadSummary: In this work, some common fixed point results for the mappings satisfying rational expressions on a closed ball in complex valued metric spaces will be proposed. Presented theorems can be realized as extensions of some well-known results in the literature. Further, our result is well supported by nontrivial example which shows that the improvement is a actual.Best proximity points of \(F\)-proximal contractions under the influence of an \(\alpha\)-functionhttps://zbmath.org/1496.540252022-11-17T18:59:28.764376Z"Ali, Muhammad Usman"https://zbmath.org/authors/?q=ai:ali.muhammad-usman"Fahimuddin"https://zbmath.org/authors/?q=ai:fahimuddin."Kamran, Tayyab"https://zbmath.org/authors/?q=ai:kamran.tayyab"Houmani, Hassan"https://zbmath.org/authors/?q=ai:houmani.hassanSummary: In this paper, we introduce the notions of \(F\)-\(\alpha\)-proximal contractions for Hardy-Rogers type mappings as well as for Ćirić-type mappings. Then we discuss the existence of best proximity for nonself multivalued mappings satisfying at least one of these notions along with few other conditions. An example is also constructed to support the result.Fixed point theorems in \(b\)-multiplicative metric spaceshttps://zbmath.org/1496.540262022-11-17T18:59:28.764376Z"Ali, Muhammad Usman"https://zbmath.org/authors/?q=ai:ali.muhammad-usman"Kamran, Tayyab"https://zbmath.org/authors/?q=ai:kamran.tayyab"Kurdi, Alia"https://zbmath.org/authors/?q=ai:kurdi.aliaSummary: In this paper, we introduce the new notion of \(b\)-multiplicative metric space. We prove fixed point theorems for single and multivalued mappings on \(b\)-multiplicative metric spaces, endowed with a graph. We construct examples to illustrate our notions and results. As illustrative application, we give an existence theorem for the solution of a class of Fredholm multiplicative integral equations.Fixed point stability for \(\alpha_{\ast}\)-\(\psi\)-contraction mappingshttps://zbmath.org/1496.540272022-11-17T18:59:28.764376Z"Ali, Muhammad Usman"https://zbmath.org/authors/?q=ai:ali.muhammad-usman"Kiran, Quanita"https://zbmath.org/authors/?q=ai:kiran.quanitaSummary: \textit{B. S. Choudhury} and \textit{C. Bandyopadhyay} [J. Egypt. Math. Soc. 23, No. 2, 321--325 (2015; Zbl 1327.54041)] discussed the stability of fixed point sets of mappings satisfying the notion of multivalued \(\alpha\)-\(\psi\)-contraction and raised an open problem: can \(\alpha\)-\(\psi\)-contractions extended to multivalued case in some other way and in those case whether the stability of fixed point sets still holds? As an answer to this problem, in this paper, we study the stability of fixed point sets of mappings satisfying a new multivalued generalization of \(\alpha\)-\(\psi\)-contractive mappings.Fixed point theorems of Wardowski type mappings in \(S_b\)-metric spaceshttps://zbmath.org/1496.540282022-11-17T18:59:28.764376Z"Anwar, Muhammad"https://zbmath.org/authors/?q=ai:anwar.muhammad-zishan|anwar.muhammad-adnan|anwar.muhammad-sabieh|anwar.muhammad-imran|anwar.muhammad-fazeel|anwar.muhammad-shoaib"Sagheer, Dur-e-Shehwar"https://zbmath.org/authors/?q=ai:sagheer.dur-e-shehwar"Ali, Rashid"https://zbmath.org/authors/?q=ai:ali.rashid"Atul, Samina B."https://zbmath.org/authors/?q=ai:atul.samina-bSummary: In this paper, we introduced the concept of \((\alpha,F)\)-contractive pair in the structure of \(S_b\)-metric spaces. And we generalized the Wardowski type mappings in this structure and proved the fixed point and common fixed point theorems for \((\alpha,F)\)-contractive pair with some suitable conditions. Furthermore, we give some examples for the guarantee of the main theorems.Recent thought of \(\alpha_\ast\)-Geraghty $F$-contraction with applicationhttps://zbmath.org/1496.540292022-11-17T18:59:28.764376Z"Arshad, Muhammad"https://zbmath.org/authors/?q=ai:arshad.muhammad-junaid|arshad.muhammad-sarmad"Mudhesh, Mustafa"https://zbmath.org/authors/?q=ai:mudhesh.mustafa"Hussain, Aftab"https://zbmath.org/authors/?q=ai:hussain.aftab"Ameer, Eskandar"https://zbmath.org/authors/?q=ai:ameer.eskandarSummary: In this paper, we use the concept of a generalized \(\alpha \)-Geraghty contraction type mapping to introduce the new notion of \(\alpha_\ast\)-Geraghty type F-contraction multivalued mapping and prove some new common fixed point results for such contraction in b-metric-like spaces. Also, we give some examples to illustrate our main results, we also discuss existence a solution for a system of non-linear integral equation.Fixed point theorem via Meir-Keeler contraction in rectangular \(M_b\)-metric spacehttps://zbmath.org/1496.540302022-11-17T18:59:28.764376Z"Asim, Mohammad"https://zbmath.org/authors/?q=ai:asim.mohammad"Meenu"https://zbmath.org/authors/?q=ai:meenu.Summary: In this paper, we present a fixed point theorem for Meir-Keeler contraction in the framework of Rectangular \(M_b\)-metric Space. Our main result improves some existing results in literature. An example is also adopted to exhibit the utility of our main result.Remarks on monotone contractive type mappings in weighted graphshttps://zbmath.org/1496.540312022-11-17T18:59:28.764376Z"Bin Dehaish, Buthinah A."https://zbmath.org/authors/?q=ai:bin-dehaish.buthinah-abdullatif"Khamsi, Mohamed A."https://zbmath.org/authors/?q=ai:khamsi.mohamed-amineSummary: In this work, we will discuss the recent work of \textit{J. Górnicki} [Fixed Point Theory Appl. 2017, Paper No. 8, 12 p. (2017; Zbl 1458.54037)] in the context of weighted graphs. This extension is valuable since it relaxes any order structure defined on a metric space. This approach finds its origin in the work of \textit{J. Jachymski} [Proc. Am. Math. Soc. 136, No. 4, 1359--1373 (2008; Zbl 1139.47040)]. To be more specific, we prove that continuous Ćirić-Jachymski-Matkowski contraction mappings monotone in the graphical sense have a fixed point.Fixed point results for couplings on metric spaceshttps://zbmath.org/1496.540322022-11-17T18:59:28.764376Z"Choudhury, Binayak S."https://zbmath.org/authors/?q=ai:choudhury.binayak-samadder"Maity, P."https://zbmath.org/authors/?q=ai:maity.pranati"Konar, P."https://zbmath.org/authors/?q=ai:konar.pulakSummary: In this paper we define the concept of coupling between two non-empty subsets in metric space. The definition is motivated by the concept of cyclic mapping of a metric space. We show that these coupling have strong unique coupled fixed point whenever they satisfy Banach type or Chatterjea type contractive inequalities. We illustrate our main results with examples.End point of multivalued cyclic admissible mappingshttps://zbmath.org/1496.540332022-11-17T18:59:28.764376Z"Choudhury, Binayak S."https://zbmath.org/authors/?q=ai:choudhury.binayak-samadder"Metiya, Nikhilesh"https://zbmath.org/authors/?q=ai:metiya.nikhilesh"Kundu, Sunirmal"https://zbmath.org/authors/?q=ai:kundu.sunirmalSummary: In this paper we introduce an admissibility condition for multivalued mappings. Also we introduce two types of multivalued almost contractions with the help of \(\delta \)-distance. We show that these contractions, under the assumption of the admissibility condition defined here, have end point property. The results are obtained without any reference to continuity. There is also an illustrative example.Perov-type contractionshttps://zbmath.org/1496.540342022-11-17T18:59:28.764376Z"Cvetković, Marija"https://zbmath.org/authors/?q=ai:cvetkovic.marija-s"Karapınar, Erdal"https://zbmath.org/authors/?q=ai:karapinar.erdal"Rakočević, Vladimir"https://zbmath.org/authors/?q=ai:rakocevic.vladimir"Yeşilkaya, Seher Sultan"https://zbmath.org/authors/?q=ai:yesilkaya.seher-sultanSummary: Fixed point theory is rapidly growing in various directions, so the goal of this chapter is to collect and underline recent results on Perov-type contractions and talk about various generalizations of this result. Perov contraction is defined on generalized metric space firstly introduced by Russian mathematician \textit{A. I. Perov} [Priblizhen. Metody Reshen. Differ. Uravn. 2, 115--134 (1964; Zbl 0196.34703)]. The main difference and strength of this result is in changed view on contractive constant since, in Perov results, that role is played by a matrix with positive entries. The question is what do we gain in this case? And also can we talk about scientific novelty of this concrete results and all other generalizations published in the last 10 years? We will try to answer at least partially on these questions and gather most important results regarding Perov contractions.
For the entire collection see [Zbl 1485.65002].Fixed point results for \((\alpha\)-\(\beta)\)-admissible almost \(Z\)-contractions in metric-like space via simulation functionhttps://zbmath.org/1496.540352022-11-17T18:59:28.764376Z"Dewangan, Archana"https://zbmath.org/authors/?q=ai:dewangan.archana"Dubey, Anil Kumar"https://zbmath.org/authors/?q=ai:dubey.anil-kumar"Mishra, Urmila"https://zbmath.org/authors/?q=ai:mishra.urmila"Dubey, R. P."https://zbmath.org/authors/?q=ai:dubey.ravi-prakashSummary: In this paper, we establish the existence and uniqueness of a fixed point of \(( \alpha, \beta)\)-admissible almost \(Z\)-contractions via simulation functions in metric-like spaces. Our results generalize and unify several fixed point theorem in literature.Coincidence point theorems in \(S\)-metric spaces using integral type of contractionhttps://zbmath.org/1496.540362022-11-17T18:59:28.764376Z"Došenović, Tatjana"https://zbmath.org/authors/?q=ai:dosenovic.tatjana"Radenović, Stojan"https://zbmath.org/authors/?q=ai:radenovic.stojan"Rezvani, Asieh"https://zbmath.org/authors/?q=ai:rezvani.asieh"Sedghi, Shaban"https://zbmath.org/authors/?q=ai:sedghi.shabanSummary: In the present paper, we consider a coupled coincidence point results in partially ordered \(S\)-metric spaces using integral type of contraction as well as the mixed monotone property of the mappings. We also generalize the famous Branciari fixed point theorem \textit{A. Branciari} [Int. J. Math. Math. Sci. 29, No. 9, 531--536 (2002; Zbl 0993.54040)] from the metric spaces to the framework of partially ordered \(S\)-metric spaces.Contraction-type functions and some applications to GIIFShttps://zbmath.org/1496.540372022-11-17T18:59:28.764376Z"Dumitru, Dan"https://zbmath.org/authors/?q=ai:dumitru.danSummary: The aim of this paper is to study some properties of the contraction-type functions of one or multiple variables such as weak Meir-Keeler, Meir-Keeler, strong Meir-Keeler, contractive and non-expansive functions. We establish some equivalences on compact metric spaces. We also investigate the families of uniformly strong Meir-Keeler functions and their relation to Meir-Keeler-type functions. In the end, we give some applications to generalized infinite iterated function systems.Simultaneous generalizations of known fixed point theorems for a Meir-Keeler type condition with applicationshttps://zbmath.org/1496.540382022-11-17T18:59:28.764376Z"Du, Wei-Shih"https://zbmath.org/authors/?q=ai:du.wei-shih"Rassias, Th. M."https://zbmath.org/authors/?q=ai:rassias.themistocles-mSummary: In this paper, we first establish a new fixed point theorem for a Meir-Keeler type condition. As an application, we derive a simultaneous generalization of Banach contraction principle, Kannan's fixed point theorem, Chatterjea's fixed point theorem and other fixed point theorems. Some new fixed point theorems are also obtained.Fixed-point theorems for \((\phi,\psi,\beta)\)-Geraghty contraction type mappings in partially ordered fuzzy metric spaces with applicationshttps://zbmath.org/1496.540392022-11-17T18:59:28.764376Z"Goswami, Nilakshi"https://zbmath.org/authors/?q=ai:goswami.nilakshi"Patir, Bijoy"https://zbmath.org/authors/?q=ai:patir.bijoySummary: In this paper, we prove some fixed-point theorems in partially ordered fuzzy metric spaces for \((\phi,\psi,\beta)\)-Geraghty contraction type mappings which are generalization of mappings with Geraghty contraction type condition. Application of the derived results are shown in proving the existence of unique solution to some boundary value problems.A new version of coupled fixed point results in ordered metric spaces with applicationshttps://zbmath.org/1496.540402022-11-17T18:59:28.764376Z"Isik, Huseyin"https://zbmath.org/authors/?q=ai:isik.huseyin"Radenović, Stojan"https://zbmath.org/authors/?q=ai:radenovic.stojanSummary: The purpose of the present paper is to give more general results than many results in literature without mixed monotone property and use a new method of reducing coupled common fixed point results in ordered metric spaces to the respective results for mappings with one variable. In addition, an example and an application to integral equations are given to verify the effectiveness of the obtained results.An extension of Cantor's intersection theorem and well posedness of the fixed point problem for discontinuous mappingshttps://zbmath.org/1496.540412022-11-17T18:59:28.764376Z"Jachymski, Jacek"https://zbmath.org/authors/?q=ai:jachymsky.jacek-rSummary: We establish a simple extension of Cantor's intersection theorem in which we weaken the assumption that all sets are closed. This result leads to a characterization of a class of mappings (not necessarily continuous) for which the fixed point problem is well posed. We also present an example of a mapping from that class for which the existence of a fixed point cannot be deduced from the classical Cantor's intersection theorem whereas our result is applicable.Fixed points of Mizoguchi-Takahashi's type contraction on metric spaces with a graphhttps://zbmath.org/1496.540422022-11-17T18:59:28.764376Z"Kamran, Tayyab"https://zbmath.org/authors/?q=ai:kamran.tayyab"Ali, Muhammad Usman"https://zbmath.org/authors/?q=ai:ali.muhammad-usman"Shahzad, Naseer"https://zbmath.org/authors/?q=ai:shahzad.naseerSummary: In this paper, we introduce the notion of Mizoguchi-Takahashi's type \(G\)-contraction and using this new notion we prove a fixed point theorem to generalize a recent fixed point theorem by \textit{A. Sultana} and \textit{V. Vetrivel} [J. Math. Anal. Appl. 417, No. 1, 336--344 (2014; Zbl 1368.54030)]. Some examples are constructed to demonstrate generality of our result. We also obtain Mizoguchi-Takahashi's type, and Hicks and Rhoades type fixed point theorems on \(\varepsilon\)-chainable metric space.Fixed points of multivalued maps via \((G,\varphi)\)-contractionhttps://zbmath.org/1496.540432022-11-17T18:59:28.764376Z"Kamran, Tayyab"https://zbmath.org/authors/?q=ai:kamran.tayyab"Rakocevic, Vladimir"https://zbmath.org/authors/?q=ai:rakocevic.vladimir"Waheed, Mehwish"https://zbmath.org/authors/?q=ai:waheed.mehwish"Ali, Muhammad Usman"https://zbmath.org/authors/?q=ai:ali.muhammad-usmanSummary: In this paper, we extend the notion of \((G,\varphi)\)-contraction, introduced by \textit{M. Öztürk} and \textit{E. Girgin} [J. Inequal. Appl. 2014, Paper No. 39, 10 p. (2014; Zbl 1332.54214)], to multi-valued mappings. By using our new notion we prove a fixed point theorem for multi-valued mappings. Our results imply Nadler's theorem, and generalized version of Nadler's theorem on partial metric spaces.Fixed point theorems for \(F\)-contraction in generalized asymmetric metric spaceshttps://zbmath.org/1496.540442022-11-17T18:59:28.764376Z"Kari, Abdelkarim"https://zbmath.org/authors/?q=ai:kari.abdelkarim"Rossafi, Mohamed"https://zbmath.org/authors/?q=ai:rossafi.mohamed"Saffaj, Hamza"https://zbmath.org/authors/?q=ai:saffaj.hamza"Marhrani, El Miloudi"https://zbmath.org/authors/?q=ai:marhrani.elmiloudi"Aamri, Mohamed"https://zbmath.org/authors/?q=ai:aamri.mohamedSummary: Recently, a new type of mapping called \(F\)-contraction was introduced in the literature as a generalization of the concepts of contractive mappings. This present article extends the new notion in generalized asymmetric metric spaces and establishing the existence and uniqueness of fixed point for them. Non-trivial examples are further provided to support the hypotheses of our results.Fixed point theorems for multivalued mappings by the convex compactness notion and applications to mathematical economicshttps://zbmath.org/1496.540452022-11-17T18:59:28.764376Z"Koukkous, Abellatif"https://zbmath.org/authors/?q=ai:koukkous.abellatif"Chadli, Ouayl"https://zbmath.org/authors/?q=ai:chadli.ouayl"Saidi, Asma"https://zbmath.org/authors/?q=ai:saidi.asmaSummary: In this paper, we give new versions of the Knaster-Kuratowski-Mazurkiewicz theorem under the concept of convex compactness. The results obtained are then used to derive some new fixed points theorems for set-valued mappings. As applications, we give some results on the existence of greatest and maximal elements for preference relations under very weak assumptions.Multivalued and random version of Perov fixed point theorem in generalized gauge spaceshttps://zbmath.org/1496.540462022-11-17T18:59:28.764376Z"Laadjel, A."https://zbmath.org/authors/?q=ai:laadjel.a"Nieto, Juan J."https://zbmath.org/authors/?q=ai:nieto.juan-jose"Ouahab, Abdelghani"https://zbmath.org/authors/?q=ai:ouahab.abdelghani"Rodríguez-López, Rosana"https://zbmath.org/authors/?q=ai:rodriguez-lopez.rosanaSummary: In this paper, we present some random fixed point theorems in complete gauge spaces. We establish then a multivalued version of a Perov-Gheorghiu's fixed point theorem in generalized gauge spaces. Finally, some examples are given to illustrate the results.Fixed point results of generalized Suzuki-Geraghty contractions on \(f\)-orbitally complete b-metric spaceshttps://zbmath.org/1496.540472022-11-17T18:59:28.764376Z"Leyew, Bahru Tsegaye"https://zbmath.org/authors/?q=ai:leyew.bahru-tsegaye"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahidSummary: In this paper, we introduce the concept of a generalized \(\alpha\)-Suzuki-Geraghty type contraction mapping and obtain fixed point results in the framework of an \(f\)-orbitally complete b-metric space. The results proved herein improve, generalize, and unify various comparable results in the existing literature. Some examples are presented to validate the effectiveness and applicability of our main result. It is also shown that the presented results are proper extensions of corresponding results in the existing literature. As an application of our obtained result, we establish the existence of a solution of integral equations in the setup of b-metric spaces.Note on the existence of best proximity points in partially ordered metric spaceshttps://zbmath.org/1496.540482022-11-17T18:59:28.764376Z"Liu, Huahua"https://zbmath.org/authors/?q=ai:liu.huahua"Hong, Shihuang"https://zbmath.org/authors/?q=ai:hong.shihuangSummary: In this note, we adopt a new approach to establish the existence of best proximity point for multivalued mappings satisfying a sort of proximal contractions in a complete ordered metric space. To wit, we consider the best proximity point problems as an application, rather than as a generalization, of fixed point theory and then the existence of best proximity points can be deduced from the corresponding fixed point theorems. In other words, the many existing best proximity point theorems are in fact immediate consequences of well-known fixed point theorems. In addition, as an application of our results, we also show that many coupled best proximity point theorems are in fact immediate consequences of best proximity point theorems and explore the existence of equilibrium pairs for a constrained generalized game.Fixed point theorems of generalized \((\Psi,\Phi)_s\)-rational type contractive mapping in quasi \(b\)-metric spaceshttps://zbmath.org/1496.540492022-11-17T18:59:28.764376Z"Maheswari, J. Uma"https://zbmath.org/authors/?q=ai:maheswari.j-uma"Ravichandran, M."https://zbmath.org/authors/?q=ai:ravichandran.m-k|ravichandran.mohan"Lakshmi, A. Anbarasan"https://zbmath.org/authors/?q=ai:lakshmi.a-anbarasan"Mishra, Narayan"https://zbmath.org/authors/?q=ai:mishra.narayan"Mishra, Vishnu Narayan"https://zbmath.org/authors/?q=ai:mishra.vishnu-narayanSummary: In this paper, we consider the setting of quasi \(b\)-metric spaces to establish results regarding the fixed point theorems with help of new notion \((\Psi,\Phi)_s\)-rational type quasi \(b\)-metric spaces with application. An example is presented to support our results comparing with existing ones.Some classes of Meir-Keeler contractionshttps://zbmath.org/1496.540502022-11-17T18:59:28.764376Z"Manolescu, Laura"https://zbmath.org/authors/?q=ai:manolescu.laura"Găvruţa, Paşc"https://zbmath.org/authors/?q=ai:gavruta.pasc"Khojasteh, Farshid"https://zbmath.org/authors/?q=ai:khojasteh.farshidSummary: In the present paper, we prove that \(\mathcal{Z}\)-contractions, weakly type contractions, and some type of \(F\)-contractions are actually Meir-Keeler contractions.
For the entire collection see [Zbl 1485.65002].Fuzzy fixed point results via simulation functionshttps://zbmath.org/1496.540512022-11-17T18:59:28.764376Z"Mohammed, Shehu Shagari"https://zbmath.org/authors/?q=ai:mohammed.shehu-shagari"Fulatan, Ibrahim Aliyu"https://zbmath.org/authors/?q=ai:fulatan.ibrahim-aliyuSummary: We inaugurate two concepts, admissible hybrid fuzzy \(\mathcal{Z} \)-contractions and hybrid fuzzy \(\mathcal{Z} \)-contractions in the bodywork of \(b\)-metric spaces and establish sufficient criteria for fuzzy fixed points for such contractions. Nontrivial illustrations are constructed to support the hypotheses of our main notions. From application point of view, a handful of fixed point results of \(b\)-metric spaces endowed with partial ordering and graph are deduced. The ideas established herein unify and complement several well-known crisp and fuzzy fixed point theorems in the framework of both single-valued and set-valued mappings involving either linear or nonlinear contractions. A few important consequences of our main theorem are highlighted and analysed by using various forms of simulation functions.Approximate fixed points of operators on \(G\)-metric spaceshttps://zbmath.org/1496.540522022-11-17T18:59:28.764376Z"Mohsenialhosseini, S. A. M."https://zbmath.org/authors/?q=ai:mohsenialhosseini.seyed-ali-mohammadSummary: In this paper, we will first introduce the approximate fixed point property and a new class of operators and contraction mapping for a cyclic map \(T\) on \(G\)-metric spaces. Also, we prove two general lemmas regarding approximate fixed Point of cyclic maps on \(G\)-metric spaces. Using these results we prove several approximate fixed point theorems for a new class of operators and contraction mapping on \(G\)-metric spaces. In addition, some examples are presented to illustrate our results for a new class of operators and contraction mapping on \(G\)-metric spaces.Nadler mappings in cone \(b\)-metric spaces over Banach algebrashttps://zbmath.org/1496.540532022-11-17T18:59:28.764376Z"Özavşar, Muttalip"https://zbmath.org/authors/?q=ai:ozavsar.muttalipSummary: In this study we first define the concept of Nadler type contraction in the setting of \(H\)-cone \(b\)-metric space with respect to cone \(b\)-metric spaces over Banach algebras. Next we prove the Banach contraction principle for such contractions by means of the notion of spectral radius and a solid cone in underlying Banach algebra. Finally we observe that the main result achieved in this work extends and generalizes the well known results associated with contractions of Nadler type.Generalizations of metric spaces: from the fixed-point theory to the fixed-circle theoryhttps://zbmath.org/1496.540542022-11-17T18:59:28.764376Z"Özgür, Nihal Yılmaz"https://zbmath.org/authors/?q=ai:yilmaz-ozgur.nihal"Taş, Nihal"https://zbmath.org/authors/?q=ai:tas.nihal-arabaciogluSummary: This paper is a research survey about the fixed-point (resp. fixed-circle) theory on metric and some generalized metric spaces. We obtain new generalizations of the well-known Rhoades' contractive conditions, Ćirić's fixed-point result and Nemytskii-Edelstein fixed-point theorem using the theory of an \(S_b\)-metric space. We present some fixed-circle theorems on an \(S_b\)-metric space as a generalization of the known fixed-circle (fixed-point) results on a metric and an \(S\)-metric space.
For the entire collection see [Zbl 1402.47001].Some fixed point theorems via asymptotic regularityhttps://zbmath.org/1496.540552022-11-17T18:59:28.764376Z"Panja, Sayantan"https://zbmath.org/authors/?q=ai:panja.sayantan"Roy, Kushal"https://zbmath.org/authors/?q=ai:roy.kushal"Saha, Mantu"https://zbmath.org/authors/?q=ai:saha.mantu"Bisht, Ravindra K."https://zbmath.org/authors/?q=ai:bisht.ravindra-kishorSummary: In this article, we introduce some generalized contractive mappings over a metric space as extensions of various contractive mappings given by Kannan, Ćirić, Proinov and Górnicki. Some fixed point theorems have been proved for such new contractive type mappings via asymptotic regularity and some weaker versions of continuity. Supporting examples have been given in strengthening the hypothesis of our established theorems. As a by-product we explore some new answers to the open question posed by \textit{B. E. Rhoades} [Contemp. Math. 72, 233--245 (1988; Zbl 0649.54024)].Fixed point theory in graph metric spaceshttps://zbmath.org/1496.540562022-11-17T18:59:28.764376Z"Petruşel, A."https://zbmath.org/authors/?q=ai:petrusel.adrian"Petruşel, G."https://zbmath.org/authors/?q=ai:petrusel.gabrielaSummary: Let \((X, d)\) be a metric space, \(G\) be a graph associated with \(X\) and \(f : X \to X\) be an operator which satisfies two main assumptions:
\begin{itemize}
\item[(1)] \(f\) is generalized \(G\)-monotone;
\item[(2)] \(f\) is a \(G\)-contraction with respect to \(d\).
\end{itemize}
In the above framework, we will present sufficient conditions under which:
\begin{itemize}
\item[(i)] \(f\) is a Picard operator;
\item[(ii)] the fixed point problem \(x = f(x)\), \(x \in X\) is well-posed in the sense of Reich and Zaslavski;
\item[(iii)] the fixed point problem \(x = f(x)\), \(x \in X\) has the Ulam-Hyers stability property;
\item[(iv)] \(f\) has the Ostrowski stability property;
\item[(v)] \(f\) satisfies to some Gronwall type inequalities.
\end{itemize}
Some open questions are presented.
For the entire collection see [Zbl 1485.65002].A fixed point theorem for hybrid pairs satisfying a new type of limit range property in symmetric spaces and applicationshttps://zbmath.org/1496.540572022-11-17T18:59:28.764376Z"Popa, Valeriu"https://zbmath.org/authors/?q=ai:popa.valeriu|popa.valeriu.1Summary: A general fixed point theorem for two pairs of hybrid mappings satisfying a new type of limit range property in symmetric spaces is proved, generalizing Theorem 2 and Theorem 3 of \textit{M. Imdad} et al. [Demonstr. Math. 47, No. 4, 949--962 (2014; Zbl 1304.54081)]. As application, we obtain new results for hybrid mappings satisfying contractive conditions of integral type and for mappings satisfying a \(\varphi\)-contractive conditions.Coincidence and common fixed point results in \(G\)-metric spaces using generalized cyclic contractionhttps://zbmath.org/1496.540582022-11-17T18:59:28.764376Z"Puvar, Sejal V."https://zbmath.org/authors/?q=ai:puvar.sejal-v"Vyas, Rajendra G."https://zbmath.org/authors/?q=ai:vyas.rajendra-gSummary: Here, we have established the generalized cyclic contractive condition in \(G\)-metric spaces which can't be reduced to the contractive condition in standard metric spaces. The coincidence and common fixed point results are obtained for the pair of \((A,B)\)-weakly increasing mappings in \(G\)-metric spaces.Fixed point and convergence results for nonexpansive set-valued mappingshttps://zbmath.org/1496.540592022-11-17T18:59:28.764376Z"Reich, Simeon"https://zbmath.org/authors/?q=ai:reich.simeon"Zaslavski, Alexander J."https://zbmath.org/authors/?q=ai:zaslavski.alexander-jSummary: We present several fixed point and convergence results for nonexpansive set-valued mappings which map a closed subset of a complete metric space into the space.Common fixed point theorem for six mappingshttps://zbmath.org/1496.540602022-11-17T18:59:28.764376Z"Roy, Kakali"https://zbmath.org/authors/?q=ai:roy.kakali"Tiwary, Kalishankar"https://zbmath.org/authors/?q=ai:tiwary.kalishankarSummary: In this paper we shall obtain a common fixed point of six mappings in a metric space which extend the results proved in [\textit{G. Jungck}, Proc. Am. Math. Soc. 103, No. 3, 977--983 (1988; Zbl 0661.54043); Far East J. Math. Sci. 4, No. 2, 199--215 (1996; Zbl 0928.54043); \textit{B. E. Rhoades} et al., Indian J. Pure Appl. Math. 26, No. 5, 403--409 (1995; Zbl 0870.54044)].A new kind of \(F\)-contraction and some best proximity point results for such mappings with an applicationhttps://zbmath.org/1496.540612022-11-17T18:59:28.764376Z"Şahin, Hakan"https://zbmath.org/authors/?q=ai:sahin.hakanSummary: In this paper, we aim to present a new and unified way, including the previously mentioned solution methods, to overcome the problem in [\textit{I. Altun} et al., J. Nonlinear Convex Anal. 16, No. 4, 659--666 (2015; Zbl 1315.54032)] for closed and bounded valued \(F\)-contraction mappings. We also want to obtain a real generalization of fixed point results existing in the literature by using best proximity point theory. Further, considering the strong relationship between homotopy theory and various branches of mathematics such as category theory, topological spaces, and Hamiltonian manifolds in quantum mechanics, our objective is to present an application to homotopy theory of our best proximity point results obtained in the paper. In this sense, we first introduce a new family, which is larger than \(\mathcal{F}^\ast\) that has often been used to give a positive answer to the problem. Then, we prove some best proximity point results for the new kind of \(F\)-contractions on quasi metric spaces via the new family. Additionally, we show that the note given by \textit{A. Almeida} et al. [Abstr. Appl. Anal. 2014, Article ID 716825, 4 p. (2014; Zbl 1478.54036)] is not valid for our results. Therefore, our results are real generalizations of fixed point results in the literature. Moreover, we give comparative examples to demonstrate that our results unify and generalize some well-known results in the literature. As an application, we show that each homotopic mapping to \(\varphi\) satisfying all the hypotheses of our best proximity point result has also a best proximity point.Best proximity point results in generalized metric spaceshttps://zbmath.org/1496.540622022-11-17T18:59:28.764376Z"Saleem, Naeem"https://zbmath.org/authors/?q=ai:saleem.naeem"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahid"Farooq, Sadia"https://zbmath.org/authors/?q=ai:farooq.sadiaSummary: In this paper we define a new class of mappings called \((\theta ,\alpha^+)\)-proximal admissible contractions and obtain a unique best proximity point for such mappings in the setting of complete generalized metric space. Our result is an extension of comparable results in the existing literature. Some examples are presented to support the results proved herein.Common fixed points for mappings of cyclic form satisfying linear contractive conditions with Omega-distancehttps://zbmath.org/1496.540632022-11-17T18:59:28.764376Z"Shatanawi, Wasfi"https://zbmath.org/authors/?q=ai:shatanawi.wasfi-a"Maniu, Georgeta"https://zbmath.org/authors/?q=ai:maniu.georgeta"Bataihah, Anwar"https://zbmath.org/authors/?q=ai:bataihah.anwar"Bani Ahmad, Feras"https://zbmath.org/authors/?q=ai:bani-ahmad.feras-aliSummary: In this paper we utilize the concept of cyclic form and \(\Omega\)-distance to derive and prove some common fixed point theorems for self mappings of cyclic form by using the concept of \(\Omega\)-distance. Our results are extensions on some results on \(\Omega\)-distance.Fixed point results for rational contraction in function weighted dislocated quasi-metric spaces with an applicationhttps://zbmath.org/1496.540642022-11-17T18:59:28.764376Z"Shoaib, Abdullah"https://zbmath.org/authors/?q=ai:shoaib.abdullah"Mahmood, Qasim"https://zbmath.org/authors/?q=ai:mahmood.qasim"Shahzad, Aqeel"https://zbmath.org/authors/?q=ai:shahzad.aqeel"Noorani, Mohd Salmi Md"https://zbmath.org/authors/?q=ai:noorani.mohd-salmi-mohd"Radenović, Stojan"https://zbmath.org/authors/?q=ai:radenovic.stojanSummary: The objective of this article is to introduce function weighted \(L\)-\(R\)-complete dislocated quasi-metric spaces and to present fixed point results fulfilling generalized rational type F-contraction for a multivalued mapping in these spaces. A suitable example confirms our results. We also present an application for a generalized class of nonlinear integral equations. Our results generalize and extend the results of \textit{E. Karapınar} et al. [``Function weighted quasi-metric spaces and fixed point results'', IEEE Access 7, 89026--89032 (2019; \url{doi:10.1109/ACCESS.2019.2926798})].Best proximity coincidence point theorem for \(G\)-proximal generalized Geraghty auxiliary function in a metric space with graph \(G\)https://zbmath.org/1496.540652022-11-17T18:59:28.764376Z"Sinsongkham, Khamsanga"https://zbmath.org/authors/?q=ai:sinsongkham.khamsanga"Atiponrat, Watchareepan"https://zbmath.org/authors/?q=ai:atiponrat.watchareepanSummary: In a complete metric space endowed with a directed graph \(G\), we investigate the best proximity coincidence points of a pair of mappings that is \(G\)-proximal generalized auxiliary function. We show that the best proximity coincidence point is unique if any pair of two best proximity coincidence points is an edge of the graph \(G\). In addition, we provide an example as well as corollaries that are pertinent to our main theorem.A remark on the paper ``A best proximity point theorem for Geraghty-contractions''https://zbmath.org/1496.540662022-11-17T18:59:28.764376Z"Som, Sumit"https://zbmath.org/authors/?q=ai:som.sumitSummary: In the year 2012, \textit{J. Caballero} et al. [Fixed Point Theory Appl. 2012, Paper No. 231, 9 p. (2012; Zbl 1281.54021)] introduced the notion of Geraghty-contraction for non-self mappings and studied the existence and uniqueness of best proximity point for this class of mappings to generalize the fixed point result due to Geraghty. In this short note, we show that the existence of best proximity point for Geraghty-contraction follows from fixed point theorem 2.1 of \textit{M. A. Geraghty} [Proc. Am. Math. Soc. 40, 604--608 (1973; Zbl 0245.54027)] \textit{i.e}., the existence of best proximity point for Geraghty-contraction follows from the same conclusion in fixed point theory.Fixed point to fixed circle and activation function in partial metric spacehttps://zbmath.org/1496.540672022-11-17T18:59:28.764376Z"Tomar, Anita"https://zbmath.org/authors/?q=ai:tomar.anita"Joshi, Meena"https://zbmath.org/authors/?q=ai:joshi.meena"Padaliya, S. K."https://zbmath.org/authors/?q=ai:padaliya.sanjay-kumarSummary: We familiarize a notion of a fixed circle in a partial metric space, which is not necessarily the same as a circle in a Euclidean space. Next, we establish novel fixed circle theorems and verify these by illustrative examples with geometric interpretation to demonstrate the authenticity of the postulates. Also, we study the geometric properties of the set of non-unique fixed points of a discontinuous self-map in reference to fixed circle problems and responded to an open problem regarding the existence of a maximum number of points for which there exist circles. This paper is concluded by giving an application to activation function to exhibit the feasibility of results, thereby providing a better insight into the analogous explorations.Krasnoselskii-Mann method for multi-valued non-self mappings in CAT(0) spaceshttps://zbmath.org/1496.540682022-11-17T18:59:28.764376Z"Tufa, Abebe Regassa"https://zbmath.org/authors/?q=ai:tufa.abebe-regassa"Zegeye, Habtu"https://zbmath.org/authors/?q=ai:zegeye.habtuSummary: We define Mann iterative scheme in CAT(0) spaces and obtain \(\vartriangle\)-convergence and strong convergence of the iterative scheme to a fixed point of multi-valued nonexpansive non-self mappings. We also obtain strong convergence of the scheme to a fixed point of multi-valued quasi-nonexpansive non-self mappings under appropriate conditions. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.Perturbed geometric contractions in ordered metric spaceshttps://zbmath.org/1496.540692022-11-17T18:59:28.764376Z"Turinici, Mihai"https://zbmath.org/authors/?q=ai:turinici.mihaiSummary: A geometric extension is given for the perturbed contraction principle in [\textit{H. Aydi} et al., Abstr. Appl. Anal. 2013, Article ID 312479, 10 p. (2013; Zbl 1266.54083)].
For the entire collection see [Zbl 1485.65002].Meir-Keeler sequential contractions and Pata fixed point resultshttps://zbmath.org/1496.540702022-11-17T18:59:28.764376Z"Turinici, Mihai"https://zbmath.org/authors/?q=ai:turinici.mihaiSummary: The (contractive) maps introduced by \textit{V. Pata} [J. Fixed Point Theory Appl. 10, No. 2, 299--305 (2011; Zbl 1264.54065)] are in fact Meir-Keeler sequential maps. This allows us treating in a unitary manner all fixed point results of this type.
For the entire collection see [Zbl 1483.00042].Fixed points of functional Meir-Keeler contractionshttps://zbmath.org/1496.540712022-11-17T18:59:28.764376Z"Turinici, Mihai"https://zbmath.org/authors/?q=ai:turinici.mihaiSummary: A fixed point result involving functional Meir-Keeler maps is
established on ordered metric spaces. Then, it is shown that some related
statements, including the ones due to \textit{B. S. Choudhury} and \textit{A. Kundu} [Demonstr. Math. 46, No. 2, 327--334 (2013; Zbl 1296.54060)] or \textit{W.-S. Du} and \textit{Th. M. Rassias} [Int. J. Nonlinear Anal. Appl. 11, No. 1, 55--66 (2020; Zbl 1496.54038)] are ultimately reducible to such techniques.On extended interpolative single and multivalued \(F\)-contractionshttps://zbmath.org/1496.540722022-11-17T18:59:28.764376Z"Yildirim, İsa"https://zbmath.org/authors/?q=ai:yildirim.isaSummary: The main objective of this paper is to study an extended interpolative single and multivalued Hardy-Rogers type \(F\)-contractions in complete metric spaces. We prove some fixed point theorems for such mappings. Further, we give an application to integral equations to verify our main results. The results presented in this paper improve the recent works of \textit{E. Karapınar} et al. [Symmetry 11, No. 1, Paper No. 8, 7 p. (2019; Zbl 1423.47027)] and \textit{B. Mohammadi} et al. [J. Inequal. Appl. 2019, Paper No. 290, 11 p. (2019; Zbl 07459318)].On the existence problem for a fixed point of a generalized contracting multivalued mappinghttps://zbmath.org/1496.540732022-11-17T18:59:28.764376Z"Zhukovskiĭ, Evgeniĭ Semenovich"https://zbmath.org/authors/?q=ai:zhukovskiy.evgeny-sSummary: We discuss the still unresolved question, posed in [\textit{S. Reich}, Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 57, 194--198 (1975; Zbl 0329.47019)], of existence in a complete metric space \(X\) of a fixed point for a generalized contracting multivalued map \(\Phi: X \rightrightarrows X\) having closed values \(\Phi (x) \subset X\) for all \(x \in X\). Generalized contraction is understood as a natural extension of the Browder-Krasnoselskii definition of this property to multivalued maps:
\[ \forall x, u \in X : h \bigl(\varphi(x), \varphi(u) \bigr) \leq \eta \bigl(\rho(x, u) \bigr),\]
where the function \(\eta: \mathbb{R}_+\to\mathbb{R}_+\) is increasing, right continuous, and for all \(d>0\), \(\eta(d)<d (h(\cdot, \cdot)\) denotes the Hausdorff distance between sets in the space \(X\). We give an outline of the statements obtained in the literature that solve Reich's problem with additional requirements on the generalized contraction \(\Phi\). In the simplest case, when the multivalued generalized contraction map \(\Phi\) acts in \(\mathbb{R}\), without any additional conditions, we prove the existence of a fixed point for this map.Common fixed point theorems for a class of \((s,q)\)-contractive mappings in \(b\)-metric-like spaces and applications to integral equationshttps://zbmath.org/1496.540742022-11-17T18:59:28.764376Z"Zoto, Kastriot"https://zbmath.org/authors/?q=ai:zoto.kastriot"Rhoades, Billy E."https://zbmath.org/authors/?q=ai:rhoades.billy-e"Radenović, Stojan"https://zbmath.org/authors/?q=ai:radenovic.stojanSummary: In this paper, we establish fixed point theorems for one and two selfmaps in \(b\)-metric-like spaces, using \((s,q)\)-contractive and \(F\)-\((\psi,\varphi,s,q)\)-contractive conditions, defined by means of altering distances and \(\mathcal C\)-class functions. Our theorems unify, extend and generalize corresponding results in the literature.Cold and freezing sets in the digital planehttps://zbmath.org/1496.540752022-11-17T18:59:28.764376Z"Boxer, Laurence"https://zbmath.org/authors/?q=ai:boxer.laurenceSummary: Cold sets and freezing sets belong to the theory of (approximate) fixed points for continuous self-maps on digital images. We study some properties of cold sets for digital images in the digital plane, and we examine some relationships between cold sets and freezing sets.Rational spaces and set constraintshttps://zbmath.org/1496.681852022-11-17T18:59:28.764376Z"Kozen, Dexter"https://zbmath.org/authors/?q=ai:kozen.dexter-cSummary: Set constraints are inclusions between expressions denoting sets of ground terms. They have been used extensively in program analysis and type inference. In this paper we investigate the topological structure of the spaces of solutions to systems of set constraints. We identify a family of topological spaces called \textit{rational spaces}, which formalize the notion of a topological space with a regular or self-similar structure, such as the Cantor discontinuum or the space of runs of a finite automaton. We develop the basic theory of rational spaces and derive generalizations and proofs from topological principles of some results in the literature on set constraints.
For the entire collection see [Zbl 0835.68002].Directed homotopy in non-positively curved spaceshttps://zbmath.org/1496.682252022-11-17T18:59:28.764376Z"Goubault, Éric"https://zbmath.org/authors/?q=ai:goubault.eric"Mimram, Samuel"https://zbmath.org/authors/?q=ai:mimram.samuelThe notion of non-positively curved precubical set, which can be thought of as an algebraic analogue of the well-known one for metric spaces, captures the geometric properties of the precubical sets associated with concurrent programs using only mutexes, which are the most widely used synchronization primitives. A precubical set is non-positively curved if it is geometric, satisfies the cube property and satisfies the unique \(n\)-cube property for \(n\geq 3\). Using this, as well as categorical rewriting techniques, the authors are then able to show that directed and non-directed homotopy coincide for directed paths in these precubical sets. Finally, they study the geometric realization of precubical sets in metric spaces, to show that the conditions on precubical sets actually coincide with those for metric spaces. Since the category of metric spaces is not cocomplete, they are led to work with generalized metric spaces and study some of their properties.
Reviewer: Philippe Gaucher (Paris)The Einstein equations and multipole moments at null infinityhttps://zbmath.org/1496.830032022-11-17T18:59:28.764376Z"Tafel, J."https://zbmath.org/authors/?q=ai:tafel.jacek