On generalized vector variational-like inequalities

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Abstract

In this paper, we consider a vector version of Minty's lemma and obtain existence theorems for two kinds of variational-like inequality. A fixed point theorem is also discussed.

Introduction

Giannessi [5] was the first to introduce the vector variational inequality in a finite-dimensional Euclidean space, which is the vector-valued version of the variational inequality of Hartman and Stampacchia [8]. Later on, many authors see [2], [3], [6], [7], [10], [12], [13], [19] have investigated vector variational inequalities in abstract spaces.

On the other hand, Minty's lemma [1], [9] has been shown to be an important tool in the variational field including variational inequality problems, obstacle problems, confined plasmas, free boundary problems, and stochastic optimal control problems when the operator is monotone and the domain is convex.

In 1999, Lee et al. [15] obtained a vector version of Minty's lemma using Nadler's result [16], and with their result they considered two kinds of vector variational-like inequalities for multifunctions under certain new pseudomonotonicity condition and certain new hemicontinuity condition, respectively, different from the conditions that in [11], [14], [17].

In this paper, a vector version of Minty's lemma is obtained and two kinds of vector variational-like inequalities for multifunctions are considered. We show the existence of solutions to a kind of vector variational like inequality for a multifunction under certain pseudomonotonic condition different from that in [11], [14], [17].

By using the vector version of Minty's lemma and the vector variational-like inequality, we prove the existence of solutions for another type of vector variational-like inequality for a compact-valued multifunction under certain hemicontinuity condition different from that in [11], [14], [17]. Further, in last section, an appropriate fixed point theorem is used to obtain an existence result of solution for vector variational-like inequality problems.

Section snippets

Preliminaries

In this section, we give some definitions and results, which are essential for our main results.

Definition 2.1

Yuan [20]

Let D be a subset of a topological vector space X. Then a multifunction F: D 2X is called KKM if for each nonempty finite subset N of D, CoNF(N), where Co denotes the convex hull and F(N)={F(u):uN}.

Definition 2.2

A multivalued mapping T:D2X is called convex if for any u,vD, 0<α<1, T(αu+(1-α)v)αT(u)+(1-α)T(v).

Lemma 2.1

Fan [4]

Let D be an arbitrary nonempty subset of a Hausdorff topological vector space. Let the multivalued

Main results

In this section, we state and prove the following vector version Minty's lemma under the conditions different from that in [2], [3], [10], [19].

Theorem 3.1

Let X and Y be real Banach space, D be a nonempty closed convex subset of X, and {C(u);uD} a family of closed convex solid cones of Y. Let T:D2L(X,Y) be a nonempty compact-valued multifunction such that for any u,vD. H(T(u+λ(v-u)),T(u))0as0+,where H is Hausdorff metric defined on L(X,Y) and η:D×DX an operator, suppose that the following conditions

Fixed point theorem

In this section, we used the vector version Minty lemma and an appropriate fixed point theorem to obtain an existence results of solution for a vector variational-like inequality.

Theorem 4.1

Let D be a nonempty closed and convex subset of real Banach space X. Let T:D2L(X,Y) be a nonempty compact-valued multifunction such that for any u,vD, H(T(u+λ(v-u)),T(u))0asλ0+,where H is a Hausdorff metric defined on L(X,Y) an η:D×DX an operator, suppose that the following hold:

  • (i)

    A:L(X,Y)L(X,Y) is a continuous

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