On Generalized Strong Vector Variational-Like Inequalities in Banach Spaces
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Research Article On Generalized Strong Vector Variational-Like Inequalities in Banach Spaces Lu-Chuan Ceng, Yen-Cherng Lin, and Jen-Chih Yao Received 8 December 2006; Accepted 9 February 2007 Recommended by Yeol Je Cho
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). Copyright © 2007 Lu-Chuan Ceng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and preliminaries 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. Ever since then, 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, 7–14] and the references therein.
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Journal of Inequalities and Applications
Let X and Y be two real Banach spaces, let K ⊆ X be a nonempty, closed, and convex set, and let C ⊆ Y be a closed, convex, and pointed cone with apex at the origin. Recall that C is said to be a closed, convex, and pointed cone with apex at the origin if and only if C is closed and the following conditions hold: (i) λC ⊆ C, for all λ > 0; (ii) C + C ⊆ C; (iii) C ∩ (−C) = {0}. Given a closed, convex, and pointed cone C with apex at the origin in Y , we can define relations “≤C ” and “≤C ” as follows: x ≤C y ⇐⇒ y − x ∈ C,
x ≤C y ⇐⇒ y − x ∈ C.
(1.1)
Clearly “≤C ” is a partial order. In this case, (Y , ≤C ) is called an ordered Banach space ordered by C. Let L(X,Y ) denote the space of all continuous linear maps from X into Y . Let T : K → 2L(X,Y ) be a multifunction, where 2L(X,Y ) denotes the collection of all nonempty subsets of L(X,Y ). Let A : L(X,Y ) → L(X,Y ), h : K → Y , and η : K × K → X be three mappings. Definition 1.1
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