Inequalities and solution of an operator equation
type: Item
instance of: scholarly article
title: Inequalities and solution of an operator equation
zbMATH Open document ID: 1153.47045
published in: Applied Mathematics Letters
publication date: 2008-05-21T00:00:00Z
review text: The article deals with fixed points of an operator \(A:\overline{D} \to E\), where \(D\) is a bounded open convex set in a real Banach space \(E\), \(0 \in D\). It is assumed that \(I - A\) is a closed operator and \(\alpha(AD) < \alpha(D)\), where \(\alpha\) denotes the noncompact measure (such operators are called by the authors \textit{semi-closed \(1\)-set-contractive}). It is proved that \(A\) has a fixed point in \(D\) if one of the following conditions holds: \[ \| Ax + x\| ^{n+1} + \| Ax - x\| ^{n+1} + \| x\| \leq 2[\| Ax\| ^{n+1} + \| x\| ^{n+1}] + e \ln \| x\| , \quad x \in \partial D, \] with \(n \in {\mathbb N}\), \(n > 1\); \[ 2\| Ax - x\| ^p - \| Ax + x\| ^p \geq \| Ax\| ^p - \| x\| ^p, \quad x \in \partial D, \] with \(p > 1\); \[ \| Ax - x_0\| \leq \| x - x_0\| , \quad x \in \partial D, \] with \(x_0 \in D\). All these statements are direct corollaries of the analogue of Schauder's fixed point principle for semi-closed \(1\)-set-contractive operators. Moreover, the definition of these operators (cited above) is incorrect.
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 5277570
zbMATH Keywords: real Banach space
zbMATH Keywords: semi-closed 1-set-contractive operator
zbMATH Keywords: operator equation
zbMATH Keywords: homotopy invariance property
zbMATH Keywords: open convex subset
author: Chuanxi Zhu
author: Zongben Xu
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: j.aml.2007.07.013
OpenAlex ID: W2046763483
cites work: The Fixed-Point Index and the Fixed-Point Theorems of 1-Set-Contraction Mappings
cites work: On the number of solutions of nonlinear equations in ordered Banach spaces
cites work: Q3776015
cites work: Q3685668
cites work: Positive solutions for nonlinear eigenvalue problems
cites work: Q4384592
cites work: Q3154175
cites work: Some new fixed point theorems in probabilistic metric spaces
cites work: Some theorems in the X-M-PN space
cites work: Some theorems on random operator equations of 1-set-contractive type.
cites work: Some theorems of random operator equations
DOI: 10.1016/J.AML.2007.07.013
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description: scientific article; zbMATH DE number 5277570