Extended Mixed Vector Equilibrium Problems

Mijanur Rahaman,Rais Ahmad,Adem Kılıçman

doi:10.1155/2014/376759

Abstract

We study extended mixed vector equilibrium problems, namely, extended weak mixed vector equilibrium problem and extended strong mixed vector equilibrium problem in Hausdorff topological vector spaces. Using generalized KKM-Fan theorem (Ben-El-Mechaiekh et al.; 2005), some existence results for both problems are proved in noncompact domain.

Highlights

Giannessi [1] first introduced and studied vector variational inequality problem in a finite-dimensional vector space
A family {(Ci, Zi)}i∈I of pair of sets is said to be coercing for a mapping F : K → 2Y if and only if (i) for each i ∈ I, Ci is contained in a compact convex subset of K and Zi is a compact subset of Y; (ii) for each i, j ∈ I, there exists k ∈ I such that Ci ∪ Cj ⊆ Ck; (iii) for each i ∈ I, there exists k ∈ I with ⋂x∈Ck F(x) ⊆ Zi
Let X be a Hausdorff topological vector space, Y a convex subset of X, K a nonempty subset of Y, and F : K → 2Y a KKM mapping with compactly closed values in Y (i.e., for all x ∈ K, F(x) ∩ Z is closed for every compact set Z of Y)

Summary

Introduction

Giannessi [1] first introduced and studied vector variational inequality problem in a finite-dimensional vector space. A family {(Ci, Zi)}i∈I of pair of sets is said to be coercing for a mapping F : K → 2Y if and only if (i) for each i ∈ I, Ci is contained in a compact convex subset of K and Zi is a compact subset of Y; (ii) for each i, j ∈ I, there exists k ∈ I such that Ci ∪ Cj ⊆ Ck; (iii) for each i ∈ I, there exists k ∈ I with ⋂x∈Ck F(x) ⊆ Zi. Remark 5.

Objectives

Results

Conclusion