Krasnoselskii's fixed point theorem for weakly continuous maps.
type: Item
instance of: scholarly article
title: Krasnoselskii's fixed point theorem for weakly continuous maps.
zbMATH Open document ID: 1042.47035
DOI: 10.1016/S0362-546X(03)00208-6
author: Cleon S. Barroso
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-11-16T00:00:00Z
review text: The author proves the following Krasnosel'skiĭ-type theorem: Let \(X\) be a Banach space, let \(M\subset X\) be weakly compact and convex and \(A:M\to X\) weakly continuous. Assume that there is a continuous linear operator \(B:X\to X\) such that \(\| B^p\| \leq1\) for some \(p\geq1\) and such that \(x=\lambda Bx+Ay\) with \(y\in M\) and \(0<\lambda<1\) implies \(x\in M\). The conclusion is that the fixed point sets of \(A\) and of \(A+B\) are nonempty subsets of \(M\). There is an application to a Dirichlet problem in \(\mathbb{R}^n\).
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 2003967
zbMATH Keywords: fixed point
zbMATH Keywords: weakly continuous map
zbMATH Keywords: Dirichlet problem
reviewed by: Christian Fenske
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2003967