Existence of solutions for nonlinear inequalities in \(G\)-convex spaces
type: Item
instance of: scholarly article
title: Existence of solutions for nonlinear inequalities in \(G\)-convex spaces
zbMATH Open document ID: 0995.54045
DOI: 10.1016/S0893-9659(02)00035-6
author: Xieping Ding
author: Il Hyo Jung
author: Jong Yeoul Park
published in: Applied Mathematics Letters
publication date: 2002-09-03T00:00:00Z
review text: For a nonempty set \(Y\), \(2^Y\) and \(\left\langle Y\right\rangle\) denote the class of all subsets of \(Y\) and the class of all nonempty finite subsets of \(Y\), respectively. A generalized convex space (briefly, a \(G\)-convex X space) (\(X, \Gamma\)) consists of a topological space \(X\) and a map \(\Gamma : \left\langle X\right\rangle\rightarrow 2^X\) satisfying the following condition: for each \(A\in\left\langle X\right\rangle\) with \(|A|=n+1\), there exists a continuous mapping \(\Phi_A:\Delta_n\rightarrow\Gamma (A)\) such that for each \(J\in\left\langle A\right\rangle\), \(\Phi_A(\Delta_J)\subset\Gamma (J)\), where \(\Delta_n\) is the standard \(n\)-dimensional simplex and \(\Delta_J\) is the face of \(\Delta_n\) corresponding to \(J\). Given a \(G\)-convex space (\(X, \Gamma\)) and two bifunctions \(\Phi,\Psi:X\times X\rightarrow\mathbb R\), the authors of this paper study the following abstract nonlinear problem, of finding \(\overline{x}\in X\) such that \(\Phi (\overline{x},y)\geq\Psi (\overline{x},y)\) for all \(y\in Y\). Using a fixed-point theorem in \(G\)-convex spaces, the authors establish two existence theorems of solutions for the abstract nonlinear problem above. As consequence, an existence theorem for the perturbed saddle point problem in noncompact \(G\)-convex spaces is obtained.
reviewed by: Mircea Balaj
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47J20
zbMATH DE Number: 1794939
zbMATH Keywords: abstract nonlinear inequality
zbMATH Keywords: saddle point
MaRDI profile type: Publication
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full work available at URL: s0893-9659(02)00035-6
OpenAlex ID: W2045572774
description: scientific article; zbMATH DE number 1794939