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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.

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Let $(pi,,H)$ be a unitary representation of $G$. Our interest to us here is the Banach space $WAP(pi)$; in particular,
we state some relations between $pi$ and any subrepresentation of $pi$ in this regard.