Equilibria and fixed points of set-valued maps with nonconvex and noncompact domains and ranges

Let CC be a nonempty subset of a topological vector space EE. We state and prove new various fixed point theorems of Fan–Browder type for set-valued maps F:C→2EF:C→2E such that C⊂F(C)C⊂F(C) (called expansive), without assuming that the sets CC and F(C)F(C) are convex or compact or equal, and EE is Hausdorff. Let KK be a convex subset of EE and let CC be a nonempty subset of KK. Our proofs use a technique based on the investigations of the images of maps and restated for maps f:C×K→R∪{−∞,+∞}f:C×K→R∪{−∞,+∞} of G.X.-Z. Yuan’s results concerning the existence of equilibrium points and minimax inequalities for maps f:K×K→R∪{−∞,+∞}f:K×K→R∪{−∞,+∞}. Examples are provided.