Fixed points of discontinuous multivalued increasing operators in Banach spaces with applications.
type: Item
instance of: scholarly article
title: Fixed points of discontinuous multivalued increasing operators in Banach spaces with applications.
zbMATH Open document ID: 1051.47042
DOI: 10.1016/S0022-247X(03)00111-2
author: Shihuang Hong
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: Let \(E\) be a Banach space with a partial ordering \(\leq\) induced by a cone \(P\) of \(E\). Fix \(u_0\in E\) and put \(K=\{x\in E:x\geq u_0\}\). The author establishes two fixed point theorems for multivalued increasing operators with nonempty weakly closure values \(A:E\rightarrow E\) satisfying the following conditions: (i) \(Ax\) is a totally ordered subset of \(E\) for any \(x\in K\); (ii) \(Au_0\subset K\); (iii) if \(C=\{x_n\}\subset K\) is countable, totally ordered and \(C\) is contained either in the weak closure or in the strong closure of the set \(\{x_1\}\cup A(C)\), then \(C\) is weakly relatively compact. As application of one these fixed point theorems, the author obtains an existence theorem for the solutions of Hammerstein type integral inclusions of the form \(u(t)\in\int_0^T k(t,s) g(s,u(s))\,ds\) almost everywhere on \([0,T]\).
reviewed by: Mircea Balaj
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H04
zbMATH DE Number: 1956908
zbMATH Keywords: relatively weakly compact
zbMATH Keywords: increasing operator
zbMATH Keywords: fixed point
zbMATH Keywords: cone
zbMATH Keywords: partial ordering
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1956908