Leader type contractions, periodic and fixed points and new completivity in quasi-gauge spaces with generalized quasi-pseudodistances
type: Item
instance of: scholarly article
title: Leader type contractions, periodic and fixed points and new completivity in quasi-gauge spaces with generalized quasi-pseudodistances
zbMATH Open document ID: 1258.47075
author: Kazimierz Włodarczyk
author: Robert Plebaniak
published in: Topology and its Applications
publication date: 2012-10-11T00:00:00Z
review text: Definition 1. Let \(X\) be a nonempty set. (i) A quasi-pseudometric on \(X\) is a map \(p: X\times X\to [0,\infty)\) such that: \((P_1)\): \(\forall x\in X\{p(x,x)= 0\}\), and \((P_2)\): \(\forall x,y,z\in X\{p(x,z)\leq p(x,y)+ p(y,z)\}\). For given quasi-pseudometric \(p\) on \(X\), a pair \((X,p)\) is called a quasi-pseudometric space. A quasi-pseudometric space \((X,p)\) is called Hausdorff if \(\forall x,y\in X\{x\neq y\Rightarrow p(x,y)> 0 \vee p(y,x)> 0\}\). (ii) Each family \({\mathcal P}= \{p_\alpha: \alpha\in{\mathcal A}\}\) of quasi-pseudometrics \(p_\alpha: X\times X\to [0,\infty)\), \(\alpha\in{\mathcal A}\), is called a quasi-gauge space. (iii) Let the family \({\mathcal P}= \{p_\alpha: \alpha\in{\mathcal A}\}\) be a quasi-gauge on \(X\). The topology \(\tau({\mathcal P})\), having as a subbase the family \({\mathcal B}(D)= \{B(x,\varepsilon_\alpha) : x\in X\), \(\varepsilon_\alpha> 0\), \(\alpha\in{\mathcal A}\}\) of all balls \(B(x,\varepsilon_\alpha)= \{y\in X: p_\alpha(x,y)< \varepsilon_\alpha\), \(x\in X\), \(\alpha> 0\), \(\alpha\in{\mathcal A}\}\), is called the topology induced by \({\mathcal P}\) on \(X\). (iv) A topological space \((X,\tau)\) such that there is a quasi-gauge \({\mathcal P}\) on \(X\) with \(\tau= \tau({\mathcal P})\) is called a quasi-gauge space and is denoted by \((X,{\mathcal P})\). (v) A quasi-gauge space \((X,{\mathcal P})\) is called Hausdorff if the quasi-gauge \({\mathcal P}\) has the property: \(\forall x,y\in X\{x\neq y\Rightarrow\exists\alpha \in{\mathcal A}\{ p_\alpha(x,y)> 0 \vee\,p_\alpha(y,x)> 0\}\}\). Recently, the concept of generalized pseudodistance was introduced by the authors [Appl. Math. Lett. 24, No. 3, 325-328 (2011; Zbl 1206.54068); Fixed Point Theory Appl. 2011, Article ID 712706 (2011; Zbl 1213.81161)]. In the present paper, the authors introduce the notions of generalized quasipseudodistances in quasi-gauge spaces. Definition 2. Let \((X,{\mathcal P})\) be a quasi-gauge space. The family \({\mathca…
reviewed by: Valeriu Popa
Mathematics Subject Classification ID: 47H99
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 37C25
zbMATH DE Number: 6092995
zbMATH Keywords: periodic point
zbMATH Keywords: fixed point
zbMATH Keywords: contraction of leader type
zbMATH Keywords: quasi-gauge space
zbMATH Keywords: quasi-pseudometric space
zbMATH Keywords: quasi-metric space
zbMATH Keywords: generalized quasi-pseudodistance
MaRDI profile type: Publication
full work available at URL: j.topol.2012.08.013
OpenAlex ID: W1999806076
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DOI: 10.1016/J.TOPOL.2012.08.013
description: scientific article; zbMATH DE number 6092995