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)

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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.

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