Alternative principles and their applications

In this paper we establish two alternative principles of the following type: If X and Y are convex subsets of two locally convex Hausdorff topological vector spaces and F, S : X Y are two set-valued mappings satisfying certain conditions, then either there exists x0 ∈ X such that F(x0) = ∅ or⋂x∈X S(x) = ∅. As first applications of the alternative principles we obtain two matching theorems of Ky Fan type. Next, are given several analytic alternatives and minimax inequalities. Finally we establish two very general alternative theorems concerning existence of solutions of a vector equilibrium problem.