Generalized variational-like inequalities with pseudomonotone set-valued mappings

Abstract

In this paper, authors introduce the concepts of hY h-pseudo-monotone and hY h-demi-monotone mappings and prove some existence results of solutions for a new class of generalized variational-like inequalities with pseudo-monotone set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces. 1. Introduction and preliminaries. Since the appearance of the papers of Minty [20] and [21], Hartman and Stampacchia [14] and Browder [3], the theory of monotone (nonlinear) operators in general and the variational inequality in particular have generated a tremendous interest amongst mathematicians. This is because of the wide applicability of the variational inequality in nonlinear elliptic boundary value problems, obstacle problems, complementarity problems, mathematical programming, mathematical economics and in many other areas. Papers which concern differential equations and variational inequality are too many to cite. We will only cite [3], [14], [17] and [26]. Readers will find many further references cited in these. In [16] Kamaradian showed that the problem of complementarity can be reduced to the variational inequality, while the relationship between mathematical programming and the variational inequality was shown by Mancino and Stampacchia [19], between the variational inequality and convex functions by Rockafeller [28] and Moreau [22], and between the variational inequality and the equilibrium point of Walrasian economy in Riesz spaces in [1]. In [32] we have given a proof of the theorem on the existence of a solution of the variational inequality in topological vector space by applying a fixed point theorem of ours and recently in [33] we have generalized this fixed point theorem and shown that this generalized fixed point theorem is equivalent to Fan-Knaster-KuratowskiMazurkiewicz theorem [10] and we have applied this fixed point theorem to obtain variational inequalities under more general conditions [34]. In [8] and [9] we have dealt with variational inequalities of set-valued mappings. The present paper can be viewed as a further contribution in this direction.Let E and F be two vector spaces over a scalar field F (either the real field or the complex field). We shall denote by 2 F the family of all subsets of F and by fE the family of all

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