On generalized vector quasivariational-like inequality problems
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We introduce a class of generalized vector quasivariational-like inequality problems in Banach spaces. We derive some new existence results by using KKM-Fan theorem and an equivalent fixed point theorem. As an application of our results, we have obtained as special cases the existence results for vector quasi-equilibrium problems, generalized vector quasivariational inequality and vector quasi-optimization problems. The results of this paper generalize and unify the corresponding results of several authors and can be considered as a significant extension of the previously known results. 1. Introduction Let K be a nonempty subset of a space X and f : K × K → R be a bifunction. The equilibrium problem introduced and studied by Blum and Oettli [4] in 1994 is defined to be the problem of finding a point x ∈ K such that f (x, y) ≥ 0 for each y ∈ K. If we take f (x, y) = T(x), y − x, where T : K → X ∗ (dual of X) and ·, · is the pairing between X and X ∗ then the equilibrium problem reduces to standard variational inequality, introduced and studied by Stampacchia [20] in 1964. In recent years this theory has become very powerful and effective tool for studying a wide class of linear and nonlinear problems arising in mathematical programming, optimization theory, elasticity theory, game theory, economics, mechanics, and engineering sciences. This field is dynamic and has emerged as an interesting and fascinating branch of applicable mathematics with wide range of applications in industry, physical, regional, social, pure, and applied sciences. The papers by Harker and Pang [9] and M. A. Noor, K. I. Noor, and T. M. Rassias [18, 19] provide some excellent survey on the developments and applications of variational inequalities whereas for comprehensive bibliography for equilibrium problems we refer to Giannessi [8], Daniele, Giannessi, and Maugeri [5], Ansari and Yao [3] and references therein. In the present paper, we consider a general type of variational inequality problem which contains equilibrium problems as a special case. So it is interesting to compare these two ways of the problem setting. We establish some existence results for solution to this type of variational inequality problem by using KKM-Fan theorem and an equivalent Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 243–255 DOI: 10.1155/FPTA.2005.243
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On generalized VQVLIP
fixed point theorem. From special cases, we obtain various known and new results for solving various classes of equilibrium problems, variational inequalities and related problems. Our results generalizes and improves the corresponding results in the literature. 2. Preliminaries Let X and Y be real Banach Spaces. A nonempty subset P of X is called convex cone if λP⊆ P for all λ ≥ 0 and P + P = P. A cone P is called pointed cone if P is a cone and P (−P) = {0}, where 0 denotes the zero vector. Also, a cone P is called proper if it is properly contained in X. Let K be a non-empty subset of X. We will denote by 2K the set of all non
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