Fixed points and approximate solutions for nonlinear operator equations
type: Item
instance of: scholarly article
title: Fixed points and approximate solutions for nonlinear operator equations
zbMATH Open document ID: 0954.47038
DOI: 10.1016/S0377-0427(99)00240-X
author: Alberto Cabada
author: Juan J. Nieto
published in: Journal of Computational and Applied Mathematics
publication date: 2000-02-03T00:00:00Z
review text: The equation \[ Lu= Nu,\tag{1} \] where \(L\) and \(N\) are linear and nonlinear operators defined on a Hilbert space, respectively, is considered. If \(L\) is invertible, then the equation (1) is equivalent to \[ u= L^{-1}(Nu).\tag{2} \] Thus the problem of existence of solutions of (1) leads to the problem of finding fixed points for the operator \(L^{-1}\circ N\). It may happen that \(L\) is not invertible but for some \(s\in R\), there exists \((L+ sI)^{-1}\). Then the equation (1) is equivalent to \[ u= (L+ sI)^{-1} (Nu+ su).\tag{3} \] To solve such equation one may use the monotone iterative technique for increasing operators on the right-hand side of (3) to find fixed points. In this paper the authors present some new technique for operators which can be written as difference of monotone operators. They present the method of finding lower and upper solutions of (1) as well as the criteria for the existence of the unique solution of the equation (1). The last part of the paper contains applications of the concept of lower and upper solutions to the second-order boundary value problems and to the \(n\)th-order periodic bundary value problems.
reviewed by: Stefan Czerwik
Mathematics Subject Classification ID: 47H07
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47J25
zbMATH DE Number: 1398392
zbMATH Keywords: monotone iterative technique
zbMATH Keywords: increasing operators
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: second-order boundary value problems
zbMATH Keywords: \(n\)th-order periodic bundary value problems
Wikidata QID: Q60199470
MaRDI profile type: Publication
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full work available at URL: s0377-0427(99)00240-x
OpenAlex ID: W2017122043
description: scientific article; zbMATH DE number 1398392