On the fixed points of correspondences in modular spaces
type: Item
instance of: scholarly article
title: On the fixed points of correspondences in modular spaces
zbMATH Open document ID: 1229.47080
DOI: 10.5402/2011/530254
author: Fatemeh Lael
author: Kourosh Nourouzi
published in: ISRN Geometry
publication date: 2011-07-22T00:00:00Z
review text: Let \(\rho\) be a modular on a real linear space \(X\), and let \(f\) be a correspondence from \(X\) to \(Y\). \(f: C\to C\) is called a \(k\)-contraction \((0\leq k< 1)\) if for all \(x,y\in C\) and \(p\in f(x)\), \(q\in f(y)\), there holds \(\rho(p-q)\leq k\rho(x-y)\). The growth function, defined for \(t\in [0,\infty)\), is defined as \[ w_\rho(t)= \text{inf}\{w: \rho(tx)\leq w\rho(x),\, x\in X,\,0<\rho(x)\}, \] and is applied to obtain the following fixed point theorems in modular spaces \((X,\rho)\). 1. Every \(k\)-contraction \(f: B\to B\) with \(w_\rho(2)^2 k<{1\over 2}\) has a fixed point. 2. Let \(X_\rho\) be a complete modular space, \(B\) a compact and starshaped subset of \(X_\rho\). If \(w_\rho(\beta) w^2_\rho(2)<{1\over 2}\) for every \(\beta\in (0,1)\) and \(f\) is nonexpansive, then \(f\) has a fixed point.
reviewed by: Julian Musielak
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 46A80
zbMATH DE Number: 5931060
zbMATH Keywords: modular space
zbMATH Keywords: contractive correspondence
zbMATH Keywords: fixed point theorem
Wikidata QID: Q58689645
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.5402/2011/530254
OpenAlex ID: W1993176230
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description: scientific article; zbMATH DE number 5931060