Abstract
The purpose of this paper is to study the solvability for a class of generalized strong vector variational-like inequalities in reflexive Banach spaces. Firstly, utilizing Brouwer's fixed point theorem, we prove the solvability for this class of generalized strong vector variational-like inequalities without monotonicity assumption under some quite mild conditions. Secondly, we introduce the new concept of pseudomonotonicity for vector multifunctions, and prove the solvability for this class of generalized strong vector variational-like inequalities for pseudomonotone vector multifunctions by using Fan's lemma and Nadler's theorem. Our results give an affirmative answer to an open problem proposed by Chen and Hou in 2000, and also extend and improve the corresponding results of Fang and Huang (2006).
Highlights
Introduction and preliminariesIn 1980, Giannessi [3] initially introduced and considered a vector variational inequality in a finite-dimensional Euclidean space, which is the vector-valued version of the variational inequality of Hartman and Stampacchia
The purpose of this paper is to study the solvability for a class of generalized strong vector variational-like inequalities in reflexive Banach spaces
We introduce the new concept of pseudomonotonicity for vector multifunctions, and prove the solvability for this class of generalized strong vector variational-like inequalities for pseudomonotone vector multifunctions by using Fan’s lemma and Nadler’s theorem
Summary
In 1980, Giannessi [3] initially introduced and considered a vector variational inequality in a finite-dimensional Euclidean space, which is the vector-valued version of the variational inequality of Hartman and Stampacchia. Vector variational inequalities have been extensively studied and generalized in infinite-dimensional spaces since they have played very important roles in many fields, such as mechanics, physics, optimization, control, nonlinear programming, economics and transportation equilibrium, engineering sciences, and so forth. On account of their very valuable applicability, the vector variational inequality theory has been widely developed throughout over last 20 years; see [1, 2, 4, 5, 7,8,9,10,11,12,13,14] and the references therein
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