Some theorems of random operator equations
type: Item
instance of: scholarly article
title: Some theorems of random operator equations
zbMATH Open document ID: 1007.60069
DOI: 10.1155/S0161171202111252
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2002-08-12T00:00:00Z
full work available at URL: https://eudml.org/doc/49843
full work available at URL: S0161171202111252.html
review text: This note is devoted to the existence of random solutions of the equation \[ A(\omega ,x)=\mu x,\tag{1} \] where \(A:\Omega\times\overline{D}\to X\) is a 1-set-contractive random operator, \(X\) is a closed convex cone in a Banach space, \(D\subset X\) is open in \(X\), bounded and such that \(0\in D\), and \(\mu\geq 1\). The authors assume that for every \(\omega\in\Omega\) the operator \(A(\omega ,\cdot)\) satisfies some inequalities on the boundary of \(D\). The proofs are based on a result of the first author [Adv. Math. Beijing 27, No. 5, 464-468 (1998; Zbl 1054.47520)], and on some elementary inequalities. Reviewer's remark: The beginning of the proof of Theorem 3 is not clear: If the equation (1) has no random solution on \(\partial D\), then we cannot deduce that \(A(\omega ,x)\neq\mu x\) a.s. for every \(x\in\partial D\).
reviewed by: Andrzej Nowak
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H40
zbMATH DE Number: 1779133
zbMATH Keywords: random operator equation
zbMATH Keywords: 1-set-contractive operator
zbMATH Keywords: semiclosed operator
author: Chuanxi Zhu
author: Zongben Xu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1779133