The existence theorems of solutions for generalized vector-valued variational-like inequalities
type: Item
instance of: scholarly article
title: The existence theorems of solutions for generalized vector-valued variational-like inequalities
zbMATH Open document ID: 0949.47052
DOI: 10.1016/S0898-1221(99)00082-6
author: Bevan Thompson
author: Shihsen Chang
author: George Xian-Zhi Yuan
publication date: 2000-01-20T00:00:00Z
review text: The generalized vector-valued variational-like inequalities include as special cases several known variational inequalities. Let \(Y\) be an ordered Banach space with respect to a closed pointed convex cone with nonvoid interior; \(z_{0}\in A\) is called a vector maximal point of \(A\) if \(\{ z\in A:z_{0}\leq z\), \(z\neq z_{0}\} =\emptyset\), and the set of all vector maximal points of \(A\) is denoted by \(\max A\). Let \(X\) be a Banach space, \(K\subset X\) a closed convex nonvoid set, \( C:K\rightarrow 2^{Y}\) so that \(C(x)\) is a closed pointed convex cone with \(\text{int }C(x)\neq \emptyset .\) For \(\eta :K\times K\rightarrow K\) continuous and \(T:X\rightarrow 2^{L(X,Y)}\) nonvoid compact valued mapping, the problem of the existence of \(x_{\ast }\in K\) such that \[ \max \left\langle T(x_{\ast }),\eta (y,x_{\ast })\right\rangle \subset Y\backslash (-\text{int }C(x_{\ast })),\quad \text{for all }y\in K \] is considered. Using KKM theory, the authors prove the existence of solutions of the above generalized vector-valued variational-like inequality for \(T\) \(\eta\)-pseudomonotone and hemicontinuous.
reviewed by: Mira-Cristiana Anisiu
Mathematics Subject Classification ID: 47J20
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 1391543
zbMATH Keywords: generalized vector-valued variational-like inequality
zbMATH Keywords: KKM mapping
zbMATH Keywords: vector maximal point
zbMATH Keywords: pseudomonotone
zbMATH Keywords: hemicontinuous
zbMATH Keywords: semicontinuous
MaRDI profile type: Publication
cites work: A generalization of Tychonoff's fixed point theorem
cites work: Q3944358
cites work: Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications
cites work: On some non-linear elliptic differential functional equations
cites work: Q3480789
cites work: Generalized vector variational inequality and fuzzy extension
cites work: General variational inequalities
cites work: Extension of the Generalized Complementarity Problem
cites work: On vector variational inequalities
cites work: Generalized convex functions and vector variational inequalities
cites work: Existence of solutions for a vector variational inequality: An extension of the Hartmann-Stampacchia theorem
cites work: A vector variational inequality and optimization over an efficient set
cites work: The vector complementary problem and its equivalences with the weak minimal element in ordered spaces
cites work: Vector variational inequality and its duality
cites work: Pseudo-monotone complementarity problems in Hilbert space
cites work: Generalized variational inequalities
cites work: Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives
cites work: Q4833571
cites work: On minimax inequalities on spaces having certain contractible subsets
cites work: A saddlepoint theorem for set-valued maps
cites work: Existence theorems for cone saddle points of vector-valued functions in infinite-dimensional spaces
cites work: Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1391543