Best proximity pair theorems

Abstract

Let A be a non-empty approximately p-compact convex subset and B be a non-empty closed convex subset of a Hausdorff locally convex topological vector space E will a continuous semi-norm p. Given a Kakutani factorizable multifunction T: A --> 2(B) and a single valued function g: A --> A, best proximity pair theorems furnishing the sufficient conditions for the existence of an element x(0) is an element of A such that d(p) (gx(0), Tx(0)) = d(p) (A, B), are proved. Indeed, a generalization of Ky Fan's fixed point theorem for multifunctions is a consequence of a best proximity pair theorem. Also, a stochastic analogue of a best proximity pair theorem is proved.