Optimality Conditions for Solving Nonconvex Set-Valued Equilibrium Problems

In this paper, sufficient conditions ensuring the existence of solutions for set-valued equilibrium problems are obtained. The convexity assumption on the whole domain is not necessary and just the closure of a quasi-self-segment-dense subset of the domain is convex. Using a KKM theorem and a notion of Q-selected preserving $R_{-}^{*}$-intersection ($R_{-}^{*}$-inclusion) for set-valued mapping, the existence results are proved in real Hausdorff topological vector spaces.

Keywords:

Set-valued equilibrium problem , Quasi-self-segment-dense set , Quasiconvexity , Generalized semicontinuity , Topological vector space.