Abstract
Recently Huang et al. (Math. Comput. Model. 43:1267-1274, 2006)introduced a class of parametric implicit vector equilibrium problems (for shortPIVEP) and they presented some existence results for a solution of PIVEP. Also,they provided two theorems about upper and lower semi-continuity of the solutionset of PIVEP in a locally convex Hausdorff topological vector space. The paperextends the corresponding results obtained in the setting of topological vectorspaces with mild assumptions and removing the notion of locally non-positivenessat a point and lower semi-continuity of the parametric mapping.
Highlights
1 Introduction and preliminaries Equilibrium problems have been extensively studied in recent years, the origin of which can be traced back to Takahashi [, Lemma ], Blum and Oettli [ ], and Noor and Oettli [ ]
In, Huang et al [ ] considered the implicit vector equilibrium problem which consists of finding x ∈ E such that f g(x), y ∈/ – int C(x), ∀y ∈ E, where f : E × E → Y and g : E → E, are mappings, X and Y are two Hausdorff topological vector spaces, E is a nonempty closed convex subset of X and C : E → Y be a set-valued mapping such that for any x ∈ E, C(x) is a closed and convex cone with C(x) ∩ –C(x) = { }, that is pointed, with nonempty interior. They continued their research and introduced the parametric implicit vector equilibrium problem, which consists of finding x∗ ∈ K(λ), for each given (λ, ) ∈ × such that f, g x∗, y ∈/ – int C x∗, ∀y ∈ K(λ), where i (i =, ) are Hausdorff topological vector spaces, K :
This paper is motivated and inspired by the recent paper [ ] and its aim is to extend the results to the setting of Hausdorff topological vector spaces with mild assumptions and removing the condition of being locally non-positive at a point has been applied in Proposition . of [ ] and lower semi-continuity of the parametric mapping used in Theorem . of [ ]
Summary
Introduction and preliminariesEquilibrium problems have been extensively studied in recent years, the origin of which can be traced back to Takahashi [ , Lemma ], Blum and Oettli [ ], and Noor and Oettli [ ].
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