Existence results for operator equations in partially ordered sets and applications
type: Item
instance of: scholarly article
title: Existence results for operator equations in partially ordered sets and applications
zbMATH Open document ID: 1078.47028
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-02-23T00:00:00Z
review text: In this paper, the author improves a previous result from [\textit{S. Heikkilä}, J. Math. Anal. Appl. 292, 262-273 (2004; Zbl 1072.47057)] regarding the solution existence of abstract equations of the form \(Lu=Nu\), where \(L,N\) are operators between partially ordered sets satisfying some conditions. In the author's previous work [loc. cit.] \(L\) is a bijection with \(L^{-1}\) increasing, while here \(L\) is not necessarily injective. The author proves the main result (Theorem 2.1) by reducing it to the case from [op. cit.]. As the author explains in the next sections, the result in abstract form covers a large class of practical problems, for example, operator equations in ordered Banach spaces, discontinuous functional parabolic problems, or elliptic boundary value problems. Moreover, these nice examples show that the result is applicable in some cases when standard theories such as the theory of monotone operators, or the method of upper and lower solutions cannot be used under the hypotheses given in Theorem 2.1.
reviewed by: Cristinel Mortici
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 47H07
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 2137245
zbMATH Keywords: operator equation
zbMATH Keywords: ordered space
zbMATH Keywords: parabolic equations
zbMATH Keywords: elliptic equations
author: S. Heikkilä
MaRDI profile type: Publication
full work available at URL: j.na.2004.07.048
OpenAlex ID: W2077036856
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DOI: 10.1016/J.NA.2004.07.048
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description: scientific article; zbMATH DE number 2137245