New fixed point theorems for 1-set-contractive operators in Banach spaces
type: Item
instance of: scholarly article
title: New fixed point theorems for 1-set-contractive operators in Banach spaces
zbMATH Open document ID: 1124.47038
author: Shaoyuan Xu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-05-30T00:00:00Z
review text: This article presents a list of sufficient conditions for the existence of fixed points of semi-closed \(1\)-set-contractive operators in real Banach spaces. All results presented here are evident corollaries of the following analogue for semi-closed \(1\)-set-contractive operators of the Leray-Schauder fixed point principle: {If \(A\) is a semi-closed \(1\)-set-contractive operator such that \[ Ax - x_0 \neq k(x - x_0) \;\text{for some} \;x_0 \in \Omega \;\text{and for all} \;x \in \partial\Omega, \;1 < k < \infty,\tag{*} \] then \(A\) has at least one fixed point in \(\overline{\Omega}\)}. In turn, this result is an evident corollary of the statement that \(\deg (I - A,\Omega) = 1\) provided that \(A\) is semi-closed \(1\)-set-contractive operator satisfying the condition \(Ax - x_0 \neq k(x - x_0)\) for some \(x_0 \in \Omega\) and for all \(x \in \partial\Omega\), \(1 \leq k < \infty\). The main part of the article is devoted to inequalities that imply (*); among others, the inequality \[ \| Ax - x\| ^\alpha \geq \| Ax\| ^{\alpha+\beta} - \| x\| ^\alpha, \quad x \in \partial\Omega, \] (\(\alpha > 1\), \(\beta \geq 0\)) is treated.
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 5159599
zbMATH Keywords: semi-closed \(1\)-set-contractive operators
zbMATH Keywords: real Banach spaces
zbMATH Keywords: topological degree
zbMATH Keywords: fixed point theorems
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: j.na.2006.06.051
OpenAlex ID: W2053484131
cites work: Q3994493
cites work: Q5185568
cites work: Q3706042
cites work: Q4421936
cites work: Remarks on condensing and \(k\)-set-contractive mappings
cites work: Degree theory for local condensing maps
cites work: Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces
cites work: The fixed point index and asymptotic fixed point theorems for $k$-set-contractions
cites work: The Fixed-Point Index and the Fixed-Point Theorems of 1-Set-Contraction Mappings
cites work: Approximation theorems and fixed point theorems for various classes of 1-set-contractive mappings in Banach spaces
cites work: Random fixed points for several classes of 1-ball-contractive and 1-set-contractive random maps
cites work: Random Approximations and Random Fixed Point Theorems for Continuous 1-Set- Contractive Random Maps
DOI: 10.1016/J.NA.2006.06.051
Recommended article: Fixed point theorems of 1-set-contractive operators in Banach spaces
Recommended article: The fixed point theorems of 1-set-contractive operators in Banach space
Recommended article: Q3452431
Recommended article: Fixed Point Theory for 1-Set Weakly Contractive Operators in Banach Spaces
Recommended article: Q4920979
Recommended article: Fixed point theorems of convex-power 1-set-contraction operators in Banach spaces
Recommended article: New fixed point results for 1-set weakly contractive operators on unbounded domains
Recommended article: Q5126499
Recommended article: New fixed point theorems for 1-set-contractive operators and variational iterative method
Recommended article: Approximation theorems and fixed point theorems for various classes of 1-set-contractive mappings in Banach spaces
description: scientific article; zbMATH DE number 5159599