Minimum selections and fixed points of set-valued operators in Banach spaces with some uniform convexity
type: Item
instance of: scholarly article
title: Minimum selections and fixed points of set-valued operators in Banach spaces with some uniform convexity
zbMATH Open document ID: 1231.47052
author: Xing-Hua Zhu
author: Jian-Zhong Xiao
published in: Applied Mathematics and Computation
publication date: 2011-03-08T00:00:00Z
review text: Let \(X\) be a topological space and \(Y\) a real Banach space. For a set-valued operator \(F\), the single-valued operator \(f\) is called a selection of \(F\) if \(fx\in Fx\) for some \(x\in X\). A~single-valued operator \(m: X\to Y\) is called a minimum selection of \(F\) if it satisfies \(m(x)\in F(x)\), \(\| m(x)\|= m_F(x)\) for all \(x\in X\), where \(m_F(x)= \text{inf}\{\| y\|: y\in Fx\}\). In Part~2 of the paper under review, the following theorem of minimum continuous selection is proved. Theorem 2.4. Let \(X\) be a topological space and \(Y\) a Banach real space. Let \(F: X\to Y\) be a continuous set-valued operator. Suppose that one of the following conditions is satisfied: (a) \(Y\) is uniformly convex and \(F\) is with closed convex values, (b) \(Y\) is locally uniformly convex and \(F\) is with weakly compact values. Then the minimum selection \(m\) of \(F\) is continuous. In the last part of the paper, as an applicaton, some problem concerning the minimum right inverses for linear operators and minimum fixed points for condensing set-valued nonlinear operators are discussed. Also, the existence of minimum solutions for an integral inclusion is proved.
reviewed by: Valeriu Popa
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 47J22
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 5862543
zbMATH Keywords: set-valued operator
zbMATH Keywords: topological space
zbMATH Keywords: Banach space
zbMATH Keywords: minimum selection
zbMATH Keywords: weakly uniformly convex space
MaRDI profile type: Publication
full work available at URL: j.amc.2010.12.082
OpenAlex ID: W2000818106
cites work: Set-valued analysis
cites work: Q3782303
cites work: Q3324260
cites work: Continuous selections. I
cites work: Q4489863
cites work: Geometric properties of Banach spaces and nonlinear iterations
cites work: Fixed Point Theory for Lipschitzian-type Mappings with Applications
cites work: Q3998720
cites work: Continuous selections and reflexive Banach spaces
cites work: Condensing mappings and the Leray-Schauder boundary condition
cites work: Q5551837
cites work: Q3917065
cites work: Sadovskii's fixed point theorem without convexity
cites work: Q3706042
cites work: New fixed point theorems for 1-set-contractive operators in Banach spaces
DOI: 10.1016/J.AMC.2010.12.082
description: scientific article; zbMATH DE number 5862543