Recent zbMATH articles in MSC 54Hhttps://zbmath.org/atom/cc/54H2021-06-15T18:09:00+00:00WerkzeugFixed point theorems for occasionally weakly compatible mappings. II.https://zbmath.org/1460.540582021-06-15T18:09:00+00:00"Rhoades, B. E."https://zbmath.org/authors/?q=ai:rhoades.billy-eSummary: In this paper we point out that a number of fixed point papers, involving several maps, are special cases of a general result proved several years ago by \textit{G. Jungck} and the second author [Fixed Point Theory 7, No. 2, 287--296 (2006; Zbl 1118.47045)] , and one proved by \textit{A. Aliouche} and \textit{V. Popa} [Filomat 22, No. 2, 99--107 (2008; Zbl 1199.54200)].Common fixed points for generalized \((\alpha{-}\psi)\)-Meir-Keeler-Khan mappings in metric spaces.https://zbmath.org/1460.540332021-06-15T18:09:00+00:00"Arshad, Muhammad"https://zbmath.org/authors/?q=ai:arshad.muhammad-junaid|arshad.muhammad-sarmad"Alshoraify, Shaif"https://zbmath.org/authors/?q=ai:alshoraify.shaif-s"Shoaib, Abdullah"https://zbmath.org/authors/?q=ai:shoaib.abdullah"Ameer, Eskandar"https://zbmath.org/authors/?q=ai:ameer.eskandarSummary: In this article, we prove a common fixed point results for two pairs of weakly compatible self-mappings in a complete metric space satisfying \((\alpha,\psi)\)-Meir-Keeler-Khan-type contractive condition. We present an example to illustrate main result. Some other results and consequences are also given. These results generalize some classical results in the current literature.A Caristi fixed point theorem for complete quasi-metric spaces by using \(mw\)-distances.https://zbmath.org/1460.540292021-06-15T18:09:00+00:00"Alegre, Carmen"https://zbmath.org/authors/?q=ai:alegre.carmen"Marín, Josefa"https://zbmath.org/authors/?q=ai:marin.josefaSummary: In this paper we give a quasi-metric version of Caristi's fixed point theorem by using mw-distances. Our theorem generalizes a recent result obtained by \textit{E. Karapınar} and \textit{S. Romaguera} [Topology Appl. 184, 54--60 (2015; Zbl 1309.54012)].Fuzzy fixed point theorems in ordered cone metric spaces.https://zbmath.org/1460.540352021-06-15T18:09:00+00:00"Azam, Akbar"https://zbmath.org/authors/?q=ai:azam.akbar"Mehmood, Nayyar"https://zbmath.org/authors/?q=ai:mehmood.nayyar"Rashid, Maliha"https://zbmath.org/authors/?q=ai:rashid.maliha"Radenović, Stojan"https://zbmath.org/authors/?q=ai:radenovic.stojanSummary: In this article, we introduce the notion of the multivalued fuzzy mappings satisfying w.l.b property and l.b properties and prove some results for multivalued generalized contractive fuzzy mappings in ordered-cone metric spaces without the assumption of normality on cones. We generalize many results in the literature.Common fixed point results for four mappings on ordered vector metric spaces.https://zbmath.org/1460.540572021-06-15T18:09:00+00:00"Rahimi, Hamidreza"https://zbmath.org/authors/?q=ai:rahimi.hamidreza"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahid"Rad, Ghasem Soleimani"https://zbmath.org/authors/?q=ai:soleimani-rad.ghasemSummary: A vector metric space is a generalization of a metric space, where the metric is Riesz space valued. We prove some common fixed point theorems for four mappings in ordered vector metric spaces. Obtained results extend and generalize well-known comparable results in the literature.Fixed point theorems for various types of \(F\)-contractions in complete \(b\)-metric spaces.https://zbmath.org/1460.540472021-06-15T18:09:00+00:00"Lukács, A."https://zbmath.org/authors/?q=ai:lukacs.andor"Kajántó, S."https://zbmath.org/authors/?q=ai:kajanto.sandorIn this paper, the authors consider a special type of contractions (the so-called \(F\)-contractions) in complete \(b\)-metric spaces (generalizing ordinary Banach contractions) and fixed points of such contractions. The obtained results are more general than the existing ones. In particular, they obtain some improvement of Wardowski's theorems as well as Kannan-type results in \(b\)-metric spaces. The methods of proof are similar to the Banach space case.
Reviewer: Stefan Czerwik (Gliwice)Nonlinear contractions in metric spaces under locally \(T\)-transitive binary relations.https://zbmath.org/1460.540282021-06-15T18:09:00+00:00"Alam, Aftab"https://zbmath.org/authors/?q=ai:alam.aftab"Imdad, Mohammad"https://zbmath.org/authors/?q=ai:imdad.mohammadSummary: In this paper, we present a variant of Boyd-Wong fixed point theorem in a metric space equipped with a locally \(T\)-transitive binary relation, which under universal relation reduces to
[\textit{D. W. Boyd} and \textit{J. S. W. Wong}, Proc. Am. Math. Soc. 20, 458--464 (1969; Zbl 0175.44903); \textit{N. Jotic}, Indian J. Pure Appl. Math. 26, No. 10, 947--952 (1995; Zbl 0841.54034)] fixed point theorems. Also, our results extend several other well-known fixed point theorems such as:
[\textit{A. Alam} and \textit{M. Imdad}, J. Fixed Point Theory Appl. 17, No. 4, 693--702 (2015; Zbl 1335.54040); \textit{E. Karapınar} and \textit{A.-F. Roldán-López-de-Hierro}, J. Inequal. Appl. 2014, Paper No. 522, 12 p. (2014; Zbl 1338.54177)] besides some others.Interval-expressed tree-like continua with the fixed point property.https://zbmath.org/1460.540482021-06-15T18:09:00+00:00"Marsh, M. M."https://zbmath.org/authors/?q=ai:marsh.m-mA continuum is a connected compact metric space. If \(\epsilon>0\), a mapping \(f:X\longrightarrow Y\) is an \(\epsilon\)-mapping if \(\mathrm{diam}(f^{-1}(y))<\epsilon\) for each \(y\in Y\). A continuum \(X\) is tree-like if for each \(\epsilon>0\), there exists an \(\epsilon\)-mapping of \(X\) onto a tree. Let \(\mathcal{T}\) be the class of tree-like continua that admit representations as inverse limits on \([0,1]\) with surjective upper semi-continuous set-valued functions. The subclass of \(\mathcal{T}\) with interval-valued bonding functions has been characterized by properties of the bonding functions in [\textit{M. M. Marsh}, Topol. Proc. 50, 101--109 (2017; Zbl 1370.54018)]. In this paper, the author characterizes the subclass of \(\mathcal{T}\) with set-valued bonding functions \(f_i\), where each \(f^{-1}_i\) is interval-valued. More precisely, he shows that if \(X\in\mathcal{T}\) with interval-valued bonding functions \(f_i\), where there exists \(m\geq 1\) such that, for each \(i\geq m\), the graph of \(f^{-1}_i\) contains the graph of an interval-valued function, then \(X\) has the fixed point property. Also, he shows that if \(X\in\mathcal{T}\) with set-valued bonding functions \(f_i\), where for each \(i\geq 1\), \(f^{-1}_i\) is an interval-valued function, then \(X\) is a \(\lambda\)-dendroid. Finally, he provides an example of an indecomposable, non-arclike continuum in \(\mathcal{T}\) that has the fixed point property.
Reviewer: Tayyebe Nasri (Bojnord)Existence of best proximity points for set-valued cyclic Meir-Keeler contractions.https://zbmath.org/1460.540402021-06-15T18:09:00+00:00"Fakhar, Majid"https://zbmath.org/authors/?q=ai:fakhar.majid"Soltani, Zeinab"https://zbmath.org/authors/?q=ai:soltani.zeinab"Zafarani, Jafar"https://zbmath.org/authors/?q=ai:zafarani.jafarSummary: In this paper the concept of set-valued cyclic Meir-Keeler contraction map is introduced. The existence of best proximity point for such maps on a metric space with the UC property is presented.Notes on multidimensional fixed-point theorems.https://zbmath.org/1460.540272021-06-15T18:09:00+00:00"Akhadkulov, Habibulla"https://zbmath.org/authors/?q=ai:akhadkulov.habibulla"Noorani, Salmi M."https://zbmath.org/authors/?q=ai:noorani.salmi-m"Saaban, Azizan B."https://zbmath.org/authors/?q=ai:saaban.azizan-b"Alipiah, Fathilah M."https://zbmath.org/authors/?q=ai:alipiah.fathilah-m"Alsamir, Habes"https://zbmath.org/authors/?q=ai:alsamir.habesSummary: In this paper, we prove the existence and uniqueness of coincident (fixed) points for nonlinear mappings of any number of arguments under a \((\psi, \theta, \varphi)\)-weak contraction condition without \(O\)-compatibility. The obtained results extend, improve and generalize some well-known results in the literature. Moreover, we present an example to show the efficiency of our results.Iterative approximation of fixed points of generalized weak Prešić type \(k\)-step iterative method for a class of operators.https://zbmath.org/1460.540262021-06-15T18:09:00+00:00"Abbas, Mujahid"https://zbmath.org/authors/?q=ai:abbas.mujahid"Ilić, Dejan"https://zbmath.org/authors/?q=ai:ilic.dejan-b"Nazir, Talat"https://zbmath.org/authors/?q=ai:nazir.talatSummary: In this paper, we study the convergence of the generalized weak Presic type $k$-step iterative method for a class of operators $f:X^k \to X$ satisfying Presic type contractive conditions in a complete metric space $X$. We also obtain the global attractivity results for a class of matrix difference equations.On generalized metric spaces and generalized convex contractions.https://zbmath.org/1460.540462021-06-15T18:09:00+00:00"Li, Cece"https://zbmath.org/authors/?q=ai:li.cece"Zhang, Dong"https://zbmath.org/authors/?q=ai:zhang.dongSummary: In this paper, we study the generalized metric introduced by \textit{A. Branciari} [Publ. Math. 57, No. 1--2, 31--37 (2000; Zbl 0963.54031)]. We find an induced metric of the generalized metric, by which some new properties of the generalized metric are presented. As a main result, we generalize several generalized, unified and extended fixed point theorems on generalized convex contractions.Various examples of the KKM spaces.https://zbmath.org/1460.490162021-06-15T18:09:00+00:00"Park, Sehie"https://zbmath.org/authors/?q=ai:park.sehieSummary: Recall that a partial KKM space is a topological space satisfying an abstract form of the well-known KKM theorem, and a KKM space is a partial KKM space satisfying the `open-valued' version of the form. Recently, we have found several new examples of such spaces. As a continuation of the preceding two works, we are going to introduce old and new examples of the KKM spaces, which play a major role in applications of the KKM theory. Finally, we state why we should use triples \((X,D;\Gamma)\) for abstract convex spaces by giving further examples, and introduce an example of a partial KKM space which is not a KKM space.Remarks on \(\mathcal{P}-\mathcal{D}\) operator.https://zbmath.org/1460.540422021-06-15T18:09:00+00:00"Gopal, Dhananjay"https://zbmath.org/authors/?q=ai:gopal.dhananjay"Karapinar, Erdal"https://zbmath.org/authors/?q=ai:karapinar.erdalThe authors prove that the \(\mathcal{P}-\mathcal{D}\) introduced in [\textit{H. K. Pathak} and \textit{Deepmala}, J. Comput. Appl. Math. 239, 103--113 (2013; Zbl 1296.54082)] operators are weakly compatible maps if they have an unique common fixed point. This holds true also for occasionally weakly compatible in the case of multiple common fixed points.
Reviewer: Salvatore Sessa (Napoli)Topological aspects of epistemology and metaphysics.https://zbmath.org/1460.000272021-06-15T18:09:00+00:00"Mormann, Thomas"https://zbmath.org/authors/?q=ai:mormann.thomasSummary: The aim of this paper is to show that (elementary) topology may be useful for dealing with problems of epistemology and metaphysics. More precisely, I want to show that the introduction of topological structures may elucidate the role of the spatial structures (in a broad sense) that underly logic and cognition. In some detail I'll deal with ``Cassirer's problem'' that may be characterized as an early for runner of Goodman's ``grue-bleen'' problem. On a larger scale, topology turns out to be useful in elaborating the approach of conceptual spaces that in the last twenty years or so has found quite a few applications in cognitive science, psychology, and linguistics. In particular, topology may help distinguish ``natural'' from ``not-so-natural'' concepts. This classical problem that up to now has withstood all efforts to solve (or dissolve) it by purely logical methods. Finally, in order to show that a topological perspective may also offer a fresh look on classical metaphysical problems, it is shown that Leibniz's famous principle of the identity of indiscernibles is closely related to some well-known topological separation axioms. More precisely, the topological perspective gives rise in a natural way to some novel variations of Leibniz's principle.
For the entire collection see [Zbl 1460.00003].Approximation fixed theorems for \(\alpha\)-partial weakly Zamfirescu mappings with application to homotopy invariance.https://zbmath.org/1460.540502021-06-15T18:09:00+00:00"Ninsri, Aphinat"https://zbmath.org/authors/?q=ai:ninsri.aphinat"Sintunavarat, Wutiphol"https://zbmath.org/authors/?q=ai:sintunavarat.wutipholSummary: In this paper, we introduce the concept of $\alpha$-partial weakly Zamfirescu mappings and give some approximate fixed point results for this mapping in $\alpha$-complete metric spaces. We also give some approximate fixed point results in $\alpha$-complete metric space endowed with an arbitrary binary relation and approximate fixed point results in $\alpha$-complete metric space endowed with graph. As application, we give homotopy results for $\alpha$-partial weakly Zamfirescu mapping.A note on \(\mathcal{F}\)-metric spaces.https://zbmath.org/1460.540202021-06-15T18:09:00+00:00"Jahangir, Farhang"https://zbmath.org/authors/?q=ai:jahangir.farhang"Haghmaram, Pouya"https://zbmath.org/authors/?q=ai:haghmaram.pouya"Nourouzi, Kourosh"https://zbmath.org/authors/?q=ai:nourouzi.kouroshSummary: A new generalization of the metric space notion, named \(\mathcal{F}\)-metric space, was given in [\textit{M. Jleli} and \textit{B. Samet}, J. Fixed Point Theory Appl. 20, No. 3, Paper No. 128, 20 p. (2018; Zbl 1401.54015)]. In this paper, we investigate some properties of \(\mathcal{F}\)-metric spaces. A simple proof is given to show that the natural topology induced by an \(\mathcal{F}\)-metric is metrizable. We present a method to construct \(s\)-relaxed\(_p\) spaces and, therefore, \(\mathcal{F}\)-metric spaces from bounded metric spaces. We give some results that reveal differences between metric and \(\mathcal{F}\)-metric spaces. In particular, we show that the ordinary open and closed balls in \(\mathcal{F}\)-metric spaces are not necessarily topological open and closed, respectively. This answers a question posed implicitly in the quoted paper. We also show that \(\mathcal{F}\)-metrics are not necessarily jointly continuous functions. Despite some topological differences between metrics and \(\mathcal{F}\)-metrics, we show that the Nadler fixed point theorem and, therefore, the Banach contraction principle in the frame of \(\mathcal{F}\)-metric spaces can be reduced to their original metric versions. This reduction even happens when the Schauder fixed point theorem is investigated in \(\mathcal{F}\)-normed spaces structure. By applying the given technique in this paper, it turns out that some nonlinear \(\mathcal{F}\)-metric contractions and, therefore, the related \(\mathcal{F}\)-metric fixed point results can naturally be reduced to their metric versions. In addition, the same happens for some topological fixed point results.Fixed point results for admissible \(\mathcal{Z}\)-contractions.https://zbmath.org/1460.540392021-06-15T18:09:00+00:00"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.vladimirSummary: In this paper, we present some fixed point results in the setting of a complete metric spaces by defining a new contractive condition via admissible mapping embedded in simulation function.{\(\alpha\)}-type fuzzy \(H\)-contractive mappings in fuzzy metric spaces.https://zbmath.org/1460.540372021-06-15T18:09:00+00:00"Beg, I."https://zbmath.org/authors/?q=ai:beg.ismat"Gopal, D."https://zbmath.org/authors/?q=ai:gopal.dhananjay"Došenović, T."https://zbmath.org/authors/?q=ai:dosenovic.tatjana"Rakić, D."https://zbmath.org/authors/?q=ai:rakic.dusanSummary: We introduce a new concept of {\(\alpha\)}-fuzzy \(\mathcal{H}\)-contractive mapping which is essentially weaker than the class of fuzzy contractive mapping and stronger than the concept of {\(\alpha\)}-{\(\phi\)}-fuzzy contractive mapping. For this type of contractions, the existence and uniqueness of fixed point in fuzzy \(M\)-complete metric spaces is also established.Existence theorems of a new set-valued MT-contraction in \(b\)-metric spaces endowed with graphs and applications.https://zbmath.org/1460.540602021-06-15T18:09:00+00:00"Tiammee, Jukrapong"https://zbmath.org/authors/?q=ai:tiammee.jukrapong"Suantai, Suthep"https://zbmath.org/authors/?q=ai:suantai.suthep"Cho, Yeol Je"https://zbmath.org/authors/?q=ai:cho.yeol-jeSummary: A new concept of set-valued Mizoguchi-Takahashi G-contractions is introduced in this paper and some fixed point theorems for such mappings in b-metric spaces endowed with directed graphs are established under some sufficient conditions. Our results improve and extend those of [\textit{N. Mizoguchi} and \textit{W. Takahashi}, J. Math. Anal. Appl. 141, No. 1, 177--188 (1989; Zbl 0688.54028); \textit{A. Sultana} and \textit{V. Vetrivel}, J. Math. Anal. Appl. 417, No. 1, 336--344 (2014; Zbl 1368.54030)]. We also give some examples supporting our main results. As an applications, we prove the existence of fixed points for multivalued mappings satisfying generalized MT-contractive condition in \(\epsilon\)-chainable \(b\)-metric spaces and the existence of a solution for some integral equations.Borel complexity up to the equivalence.https://zbmath.org/1460.540242021-06-15T18:09:00+00:00"Bartoš, Adam"https://zbmath.org/authors/?q=ai:bartos.adamThe families \(\mathcal F\) of subsets of the hyperspace \(\mathcal K([0,1]^\omega)\) of compact subsets of \([0,1]^\omega\) are the main objects of the investigation. Two such families are equivalent (up to homeomorphism) if each element of one of them has its homeomorphic copy in the other one. Among the elements of the sets \([\mathcal F]\subset\mathcal K\) of families equivalent with \(\mathcal F\), families \(\mathcal B\) of the possibly lowest complexity are examined.
It is recalled that for every analytic family \(\mathcal A\subset\mathcal K\) there is a Polish subspace \(\mathcal P\) of the hyperspace \(\mathcal K\) which is equivalent to \(\mathcal A\), see also [\textit{A. Bartoš} et al., Topology Appl. 266, Article ID 106836, 25 p. (2019; Zbl 1429.54042)]. Using compactifiability studied in the mentioned paper, it is proved that for every \(F_{\sigma}\) family in \(\mathcal K\) there is a closed equivalent family. Some more constructions are needed to get open families \(\mathcal O_n\subset\mathcal K\), \(n\in\omega\), such that the equivalence classes \([\mathcal O_n]\) are distinct and they are the unique classes of equivalence up to homeomorphism which contain an open family.
Corresponding results on families of continua are also derived.
Reviewer: Petr Holický (Praha)Best proximity point theorems for non-self proximal Reich type contractions in complete metric spaces.https://zbmath.org/1460.540322021-06-15T18:09:00+00:00"Ampadu, Clement Boateng"https://zbmath.org/authors/?q=ai:ampadu.clement-boatengSummary: Recall from [\textit{S. Reich}, Can. Math. Bull. 14, 121--124 (1971; Zbl 0211.26002)], that a mapping \(T : X \mapsto\) is called a Reich mapping if it satisfies for all \(x, y \in X, d(T_x, T_y)\leq ad(x, T_x) + bd(y, T_y) + cd(x, y)\), where \(a,b,c\) are nonnegative and satisfy \(a + b + c < 1\). Alternatively, one could define a Reich mapping as follows: \(T : X \mapsto X\) is called a Reich mapping if there exists a nonnegative constant \(k\) with \(k < \frac {1}{3}\) such that \(d(T_x, T_y)\leq k[d(x, T_x)+d(y, T_y)+d(x, y)]\). In the present paper, we address the following: How do we characterize Theorem 3 [loc. cit.], when \(T\) is a non-self map? We show such a characterization is given by Theorem 3.1 or Corollary 3.2 in this paper.Fixed point theorems in modular spaces with simulation functions and altering distance functions with applications.https://zbmath.org/1460.540632021-06-15T18:09:00+00:00"Zhu, Chuan Xi"https://zbmath.org/authors/?q=ai:zhu.chuanxi"Chen, Jing"https://zbmath.org/authors/?q=ai:chen.jing.5|chen.jing.1|chen.jing.3|chen.jing.2|chen.jing.4"Huang, Xian Jiu"https://zbmath.org/authors/?q=ai:huang.xianjiu"Chen, Jian Hua"https://zbmath.org/authors/?q=ai:chen.jianhuaSummary: In this paper, we introduce three different contractive conditions and prove some new fixed point theorems in \(\omega\)-complete modular spaces via altering distance functions and stimulation functions. Moreover, we give an application on solving integral equation. Our results generalize and extend some results in [\textit{H. Aydi} and \textit{A. Felhi}, J. Nonlinear Sci. Appl. 9, No. 6, 3686--3701 (2016; Zbl 1348.54042); \textit{J. Martínez-Moreno} et al., Bull. Malays. Math. Sci. Soc. (2) 40, No. 1, 335--344 (2017; Zbl 06679135); \textit{C. Vetro} et al., Bull. Malays. Math. Sci. Soc. (2) 38, No. 3, 1085--1105 (2015; Zbl 1317.54021)].\(\Delta\)-convergence theorems for inverse-strongly monotone mappings in CAT(0) spaces.https://zbmath.org/1460.540312021-06-15T18:09:00+00:00"Alizadeh, Sattar"https://zbmath.org/authors/?q=ai:alizadeh.sattar"Dehghan, Hossein"https://zbmath.org/authors/?q=ai:dehghan.hossein"Moradlou, Fridoun"https://zbmath.org/authors/?q=ai:moradlou.fridounSummary: In this paper, we first define and study inverse-strongly monotone mappings in general metric spaces. Then, we prove the existence theorem of solutions for variational inequalities involving such mappings. Finally, we introduce an iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for inverse-strongly monotone mappings in CAT(0) metric spaces.Stationary points of set-valued contractive and nonexpansive mappings on ultrametric spaces.https://zbmath.org/1460.540442021-06-15T18:09:00+00:00"Hosseini, Meraj"https://zbmath.org/authors/?q=ai:hosseini.meraj"Nourouzi, Kourosh"https://zbmath.org/authors/?q=ai:nourouzi.kourosh"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper, we show that contractive set-valued mappings on spherically complete ultrametric spaces have stationary (or end) points if they have the approximate stationary point property. We also extend some known fixed point results to nonexpansive set-valued mappings.Ribbon complexes with approximate descriptive proximities. Ribbon \& Vortex nerves, Betti numbers and planar divisions.https://zbmath.org/1460.540172021-06-15T18:09:00+00:00"Peters, James F."https://zbmath.org/authors/?q=ai:peters.james-f-iiiThis paper deals with so called planar ribbons, Vergili ribbon complexes and ribbon nerves in a cell complex \(K\). ``A planar ribbon (briefly, ribbon) in a CW space is the closure of a pair of nesting, non-concentric filled cycles that includes boundary but does not include the interior of the inner cycle. Each planar ribbon has its own distinctive shape determined by its outer and inner boundaries and the interior within its boundaries. A Vergili ribbon complex (briefly, ribbon complex) in a CW space is a non-void collection of countable planar ribbons. A ribbon nerve is a nonvoid collection of planar ribbons (members of a ribbon complex) that have nonempty intersection.'' Here, a cell complex is a nonempty collection of cells, where a cell in the Euclidean plane is either a 0-cell (vertex) or 1-cell (edge) or 2-cell (filled triangle). Then $K$ has the Closure Finite Weak (CW) topology, provided $K$ is Hausdorff, meaning that every pair of distinct cells is contained in disjoint neighbourhoods, and the collection of cell complexes in $K$ satisfy the Alexandroff-Hopf-Whitehead conditions, namely, containement (the closure of each cell complex is in $K$) and intersection (the nonempty intersection of all complexes is in $K$). In this context suitable pictures in the paper illustrate the notions of planar ribbons, Vergili ribbon complexes and approximate proximities. So, examples of ribbon nerves derived from a ribbon complex on a finite bounded planar region in a CW space are presented by using the latter mentioned pictures. Some theorems follow characterizing a ribbon as a vortex nerve, and a vortex nerve with $k>1$ cycles contains $k-1$ ribbons, and a vortex nerve with $k>2$ cycles contains $k-2$ ribbon nerves.
Next the author introduces the approximate descriptive closeness of cell complexes in a CW space. Then, in this context, a collection $K$ of ribbon complexes can be equipped with an approximate descriptive near relation, so that the corresponding pair forms an approximate descriptive proximity space. As main results for ribbon complexes the author summarizes the following theorems: A ribbon in the interior of a finite, bounded region of the plane divides the region into three disjoint bounded regions. A planar ribbon divides the plane into three open sets and provides a boundary of each of the three planar regions. Every continuous map from $\mathbb{R}^n$ to itself has a fixed point. A map on a nonempty CW complex to itself has a fixed point (Brouwer fixed points on ribbons).
At the end of the paper the author notes that ``there are three basic types of Betti numbers that have intuitive meaning, namely $B_0$ (number of cells in a complex), $B_1$ (number of cycles in a complex), and $B_2$ (number of holes in a complex). In terms of ribbons and ribbon nerves in CW spaces, Betti numbers that enumerate fundamental shape structures are useful.'' Analogously he introduces ribbon Betti number, Vergili ribbon complex Betti number, ribbon nerve Betti number. At last the author presents three theorems for homotopy types of ribbon complexes and ribbon nerves based on the Edelsbrunner-Harer theorem for homotopy types, i.e.: Let $F$ be a finite collection of closed convex sets in Euclidean space. Then the nerve of $F$ and the union of the sets in \(F\) have the same homotopy type.
Reviewer: Dieter Leseberg (Berlin)Some Caristi-type fixed point theorems.https://zbmath.org/1460.540342021-06-15T18:09:00+00:00"Aslantas, Mustafa"https://zbmath.org/authors/?q=ai:aslantas.mustafa"Sahin, Hakan"https://zbmath.org/authors/?q=ai:sahin.hakan"Turkoglu, Duran"https://zbmath.org/authors/?q=ai:turkoglu.duranSummary: In [\textit{J. Caristi}, Trans. Am. Math. Soc. 215, 241--251 (1976; Zbl 0305.47029)], Caristi fixed point theorem was proved for ordinary metric spaces. After that, it was shown that this result is equivalent to Ekeland variational principle which has a great number of applications in many branches of mathematics. In this paper, we first introduce a new concept so called strong \(M_b\)-metric to remedy lackness of continuity of \(M_b\)-metric. Then, we investigate whether Caristi fixed point theorem can be extended to this space. Next, we obtain Caristi-type fixed point theorem and some generalizations on strong \(M_b\)-metric spaces. Also, we provide some illustrative and interesting examples showing that our theorems extend the results existing in the literature. Finally, we present some applications of our results to ordinary metric spaces.Answer to Kirk-Shahzad's question on Zhang-Jiang's fixed point theorem.https://zbmath.org/1460.540492021-06-15T18:09:00+00:00"Nguyen Van Dung"https://zbmath.org/authors/?q=ai:nguyen-van-dung.Summary: In this note we give a positive answer to \textit{W. Kirk} and \textit{N. Shahzad}'s question on Zhang-Jiang's fixed point theorem [Fixed point theory in distance spaces. Cham: Springer (2014; Zbl 1308.58001), question on page 18].Fixed point results for orbital contractions in complete gauge spaces with applications.https://zbmath.org/1460.540542021-06-15T18:09:00+00:00"Petruşel, Adrian"https://zbmath.org/authors/?q=ai:petrusel.adrian"Petruşel, Gabriela"https://zbmath.org/authors/?q=ai:petrusel.gabriela"Wong, Mu-Ming"https://zbmath.org/authors/?q=ai:wong.mu-mingSummary: In this paper, we will present fixed point results for orbital contractions in complete gauge spaces. We discuss existence and approximation of the fixed point (in the local and the global case), data dependence, well-posedness, Ostrowski stability property and Ulam-Hyers stability for the fixed point equation. An application to an initial value problem associated to a first order differential equation is given.Some fixed point theorems in $G_b$-cone metric space.https://zbmath.org/1460.540432021-06-15T18:09:00+00:00"Goyal, Komal"https://zbmath.org/authors/?q=ai:goyal.komal"Prasad, Bhagwati"https://zbmath.org/authors/?q=ai:prasad.bhagwatiSummary: The intent of the paper is to introduce a $G_b$-cone metric space and study its properties. Some fixed point theorems for the maps satisfying a general contractive condition are established in this setting. Some of the well-known existing results are obtained as special cases.
For the entire collection see [Zbl 1429.00019].Perov type theorems for orbital contractions.https://zbmath.org/1460.540552021-06-15T18:09:00+00:00"Petruşel, Adrian"https://zbmath.org/authors/?q=ai:petrusel.adrian"Petruşel, Gabriela"https://zbmath.org/authors/?q=ai:petrusel.gabriela"Yao, Jen-Chih"https://zbmath.org/authors/?q=ai:yao.jen-chihSummary: In this paper, we will present a study of the fixed point equation \(x=f(x)\), \(x\in X\) (where \((X,d)\) is a generalized metric space in the sense that \(d(x,y)\in\mathbb{R}_+^m\) and \(f:X\to X\) is an orbital contraction) by the following perspectives: existence, uniqueness, approximation, data dependence of the operator perturbation, well-posedness, and Ulam-Hyers stability. The non-self case is also discussed and some applications are given.Paratopological groups: local versus global.https://zbmath.org/1460.540252021-06-15T18:09:00+00:00"Li, Piyu"https://zbmath.org/authors/?q=ai:li.piyu"Mou, Lei"https://zbmath.org/authors/?q=ai:mou.lei"Xu, Yanqin"https://zbmath.org/authors/?q=ai:xu.yanqinAn example is given in order to show that locally metrizable paratopological groups might be neither paracompact nor normal. This provides negative answers to questions previously posed in the field by \textit{A. Arhangel'skii} and \textit{M. Tkachenko} [Topological groups and related structures. Hackensack, NJ: World Scientific; Paris: Atlantis Press (2008; Zbl 1323.22001)] for paratopological groups. The authors also consider the conditions under which 2-pseudocompactness is a three space property in paratopological groups.
Reviewer: Watchareepan Atiponrat (Chiang Mai)Fixed points of hybrid generalized weakly contractive mappings in metric spaces.https://zbmath.org/1460.540622021-06-15T18:09:00+00:00"Xue, Zhiqun"https://zbmath.org/authors/?q=ai:xue.zhiqunSummary: Let \((E,d)\) be a complete metric space and \(S,T:E\to E\) be two self-mappings such that
\[
\varphi (F(d(Sx,Ty)))\leq \psi (F(M(x,y))),
\]
for all \(x,y\in E\), where
\begin{itemize}
\item[(i)] \(F: [0,+\infty)\to [0,+\infty)\) is a continuous function with \(F(0) = 0\) and \(F(t)>0\) for all \(t >0\);
\item[(ii)] \(\psi,\varphi : [0,+\infty)\to [0,+\infty)\) are two functions with \(\psi (0) =\varphi (0) = 0\) and \(\varphi (t)> \psi (t)\) and \(\lim_{\tau\to t}\inf\varphi (\tau)>\lim_{\tau\to t}\sup\psi (\tau)\) for all \(t >0\).
\end{itemize}
Then \(S\) and \(T\) have a unique common fixed point.Some extension results on cone \(b\)-metric spaces over Banach algebras via \(\varphi\)-operator.https://zbmath.org/1460.540642021-06-15T18:09:00+00:00"Zhu, Xiaolin"https://zbmath.org/authors/?q=ai:zhu.xiaolin"Zhai, Chengbo"https://zbmath.org/authors/?q=ai:zhai.chengboSummary: We first state the concepts of cone \(b\)-norm and cone \(b\)-Banach space as generalizations of cone \(b\)-metric spaces, and then we give a definition of \(\varphi\)-operator and obtain some new fixed point theorems in cone \(b\)-Banach spaces over Banach algebras by using \(\varphi\)-operator.Remark on contraction principle in \(\mathrm{cone}_{tvs}\) \(b\)-metric spaces.https://zbmath.org/1460.540362021-06-15T18:09:00+00:00"Bajović, Dušan"https://zbmath.org/authors/?q=ai:bajovic.dusan"Mitrović, Zoran D."https://zbmath.org/authors/?q=ai:mitrovic.zoran-d"Saha, Mantu"https://zbmath.org/authors/?q=ai:saha.mantuSummary: In this paper we obtain the Banach contraction principle theorem in bipolar \(\mathrm{cone}_{tvs}\) \(b\)-metric space. Our results improve certain results that could be found in literatures. Some examples are given in support of our established results.Approximating fixed points of generalized \(\alpha\)-Reich-Suzuki nonexpansive mapping in CAT(0) space.https://zbmath.org/1460.650602021-06-15T18:09:00+00:00"Uddin, Izhar"https://zbmath.org/authors/?q=ai:uddin.izhar"Khatoon, Sabiya"https://zbmath.org/authors/?q=ai:khatoon.sabiya"Colao, Vittorio"https://zbmath.org/authors/?q=ai:colao.vittorioSummary: The aim of this paper is to study the convergence behaviour of \(K^\ast\) iteration process for generalized \(\alpha\)-Reich-Suzuki nonexpansive mapping in CAT\((0)\) space. Besides proving some convergence results, we provide a numerical example to illustrate our results. In this process, several existing results in the literature are generalized and improved, in particular, results of \textit{K. Ullah} and \textit{M. Arshad} [J. Linear Topol. Algebra 7, No. 2, 87--100 (2018; Zbl 1413.47146)] and \textit{R. Pandey} et al. [Result. Math. 74, No. 1, Paper No. 7, 24 p. (2019; Zbl 06996045)].On hybrid contractions via simulation function in the context of quasi-metric spaces.https://zbmath.org/1460.540452021-06-15T18:09:00+00:00"Karapinar, Erdal"https://zbmath.org/authors/?q=ai:karapinar.erdal"Fulga, Andreea"https://zbmath.org/authors/?q=ai:fulga.andreeaSummary: In this manuscript, we aim at investigating the existence of a fixed point theorem for the mappings that satisfy hybrid contraction in the setting of quasi-metric spaces. We provide examples to indicate the validity of the observed results.Coincidence points for mappings under generalized contraction.https://zbmath.org/1460.540412021-06-15T18:09:00+00:00"Gairola, Ajay"https://zbmath.org/authors/?q=ai:gairola.ajay"Joshi, Mahesh C."https://zbmath.org/authors/?q=ai:joshi.mahesh-chandraSummary: In this paper we establish some results on the existence of coincidence and fixed points for multi-valued and single-valued mappings extending the result of \textit{Y.-Q. Feng} and \textit{S.-Y. Liu} [J. Math. Anal. Appl. 317, No. 1, 103--112 (2006; Zbl 1094.47049)] and \textit{Z.-Q. Liu} et al. [Fixed Point Theory Appl. 2010, Article ID 870980, 18 p. (2010; Zbl 1207.54062)]. It is also proved by counterexample that our results generalize and extend some well-known results.Common fixed point theorem in metric spaces of Fisher and Sessa.https://zbmath.org/1460.540612021-06-15T18:09:00+00:00"Valipour Baboli, Alireza R."https://zbmath.org/authors/?q=ai:valipour-baboli.alireza-r"Ghaemi, Mohammad Bagher"https://zbmath.org/authors/?q=ai:ghaemi.mohammad-bagherSummary: In this paper, it is shown that \(T\) and \(I\) have a unique common fixed point on a compact subset \(C\) of a metric space \(X\), where \(T\) and \(I\) are two self maps on \(C\), \(I\) is non-expansive and the pair \((T,I)\) is weakly commuting. In [Int. J. Math. Math. Sci. 9, 23--28 (1986; Zbl 0597.47036)], \textit{B. Fisher} and \textit{S. Sessa} verified the same problem but with \(C\) closed. Further, we show this result by replacing compatibility with weakly commutativity pair \((T,I)\) and continuity with non-expansiveness of \(I\).Coincidence results for compositions of multivalued maps based on countable compactness principles.https://zbmath.org/1460.540512021-06-15T18:09:00+00:00"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: We present a general Mönch coincidence type result for set-valued maps on Hausdorff topological spaces.Fixed point results for locally contraction with applications to fractals.https://zbmath.org/1460.540532021-06-15T18:09:00+00:00"Petruşel, Adrian"https://zbmath.org/authors/?q=ai:petrusel.adrian"Petruşel, Gabriela"https://zbmath.org/authors/?q=ai:petrusel.gabriela"Wong, Mu-Ming"https://zbmath.org/authors/?q=ai:wong.mu-mingSummary: The purpose of this paper is to connect the fixed point results for uniformly locally contractions in complete metric and \(\varepsilon\)-chainable metric spaces to coupled fixed point problems and to fractal and coupled fractal theory.Fixed point theorems in quasi-metric spaces and the specialization partial order.https://zbmath.org/1460.540592021-06-15T18:09:00+00:00"Shahzad, Naseer"https://zbmath.org/authors/?q=ai:shahzad.naseer"Valero, Oscar"https://zbmath.org/authors/?q=ai:valero.oscarSummary: In this paper, we present a new fixed point theorem in quasi-metric spaces which captures the spirit of Kleene's fixed point theorem. To this end, we explore the fundamental assumptions in the aforesaid result when we consider quasi-metric spaces endowed with the specialization partial order. Thus, we introduce an appropriate notion of order-completeness and order-continuity that ensure the existence of fixed point with distinguished properties. Moreover, some fixed point theorems are derived as a particular case of our main result when the self-mappings under consideration satisfy, in addition, any type of Banach contractive condition under different quasi-metric notions of completeness.An extension of two fixed point theorems of Fisher to partial metric spaces.https://zbmath.org/1460.540522021-06-15T18:09:00+00:00"Pavlović, Vladimir"https://zbmath.org/authors/?q=ai:pavlovic.vladimir"Ilić, Dejan"https://zbmath.org/authors/?q=ai:ilic.dejan-b"Rakočević, Vladimir"https://zbmath.org/authors/?q=ai:rakocevic.vladimirSummary: In [in: Papers on general topology and applications. Papers from the 8th summer conference at Queens College, New York, NY, 1992. New York, NY: The New York Academy of Sciences. 183--197 (1994; Zbl 0911.54025)], \textit{S. G. Matthews} introduced and studied the concept of partial metric space and obtained a Banach type fixed point theorem on complete partial metric spaces. The present paper forms part of the study of possible extensions of metric fixed point results to the context of partial metric spaces, in particular, two theorems due to \textit{B. Fisher} [Proc. Am. Math. Soc. 75, 321--325 (1979; Zbl 0411.54049)]. The theory is illustrated by some examples.Generalized Meir-Keeler type contractions and discontinuity at fixed point.https://zbmath.org/1460.540382021-06-15T18:09:00+00:00"Bisht, Ravindra K."https://zbmath.org/authors/?q=ai:bisht.ravindra-kishor"Rakočević, Vladimir"https://zbmath.org/authors/?q=ai:rakocevic.vladimirSummary: In this paper, we show that generalized Meir-Keeler type contractive definitions are strong enough to generate a fixed point but do not force the mapping to be continuous at the fixed point. Thus we provide more answers to the open question posed by \textit{B. E. Rhoades} [Contemp. Math. 72, 233--245 (1988; Zbl 0649.54024)].Fixed point structures, invariant operators, invariant partitions, and applications to Carathéodory integral equations.https://zbmath.org/1460.540562021-06-15T18:09:00+00:00"Petruşel, A."https://zbmath.org/authors/?q=ai:petrusel.adrian"Rus, I. A."https://zbmath.org/authors/?q=ai:rus.ioan-a"Şerban, M.-A."https://zbmath.org/authors/?q=ai:serban.marcel-adrianThe aim of the paper is to present a theory of the fixed point partitions with respect to an operator and a fixed point structure. We will focus only on one result of this paper. We give it in the reviewer's formulation.
Theorem 6. Let \((X, d)\) be a complete metric space, \(S(X)\) be the family of all nonempty closed subsets of \(X\) and
\[
M(U) = \{f: U \to U \mid f \text{ is a contraction} \}
\]
for every nonempty \(U \subset X\). Suppose that \ \(Y \subset X\), \(A: Y \to Y\) is an operator and \(Y = \bigcup_{\lambda \in \Lambda} Y_\lambda\) is a partition of \(Y\) such that
\[
A(Y_{\lambda}) \subset Y_\lambda, \;Y_{\lambda} \in S(X) \text{ and } A|_{Y_\lambda} \in M(Y_\lambda)
\]
for \(\lambda \in \Lambda\) and let \(l_\lambda\) be the contraction constant for \(A|_{Y_\lambda}\). Then we have \((\alpha)\) \(\{x \in X\mid A(x) = x \}\cap Y_\lambda = \{x^*_\lambda\}\) for \(\lambda \in \Lambda\), \((\beta)\) \(A^n(x) \to x^*_\lambda\) as \(n \to \infty\) for \(x \in Y_\lambda\), \(\lambda \in \Lambda\), \((\gamma)\) \(d(x,x^*_\lambda) \leq \frac{1}{1-l_\lambda}d(x,A(x))\) for \(x \in Y_\lambda\), \(\lambda \in \Lambda\), \((\delta)\) \( \lambda \in \Lambda\), \(y_n \in Y_\lambda\), \(n \in \mathbb{N}\), \(d(y_{n+1}, A(y_n)) \to 0\) as \(n \to \infty\) imply that \( y_n \to x^*_\lambda\) as \(n \to \infty\). \((\epsilon)\) \( \lambda \in \Lambda\), \(y_n \in Y_\lambda\), \(n \in \mathbb{N}\), \(d(y_n, A(y_n)) \to 0\) as \(n \to \infty\) imply that \(y_n \to x^*_\lambda\) as \(n \to \infty\).
This theorem is applied to study a class of Carathéodory integral equations.
For the entire collection see [Zbl 1355.00026].
Reviewer: Andrzej Smajdor (Kraków)Coupled fixed points of monotone mappings in a metric space with a graph.https://zbmath.org/1460.540302021-06-15T18:09:00+00:00"Alfuraidan, Monther Rashed"https://zbmath.org/authors/?q=ai:alfuraidan.monther-rashed"Khamsi, Mohamed Amine"https://zbmath.org/authors/?q=ai:khamsi.mohamed-amineSummary: In this work, we define the concept of mixed $G$-monotone mappings defined on a metric space endowed with a graph. Then we obtain sufficient conditions for the existence of coupled fixed points for such mappings when a weak contractivity type condition is satisfied.