Abstract

In this paper we give sufficient conditions for the existence of solutions of Problem (P1) (resp. Problem (P2)) of finding a point (z0, x0) ∈ B(z0, x0) × A(x0) such that F(z0, x0, x) ⊂ C(z0, x0, x0) (resp. F(z0, x0, x0) ⊂ C(z0, x0, x)) for all x ∈ A(x0), where A, B, C, F are set-valued maps between locally convex Hausdorff spaces. Some known existence theorems are included as special cases of the main results of the paper.