Let and be any two topological spaces. A multifunction is said to be
(i) upper semi-continuous if is closed in whenever is a closed subset of ;
(ii) Kakutani multifunction if (a) is upper semi-continuous, (b) either is a singleton for each or is a non-empty compact convex subset of , assuming to be a non-empty convex set in a Hausdorff topological vector space;
(iii) Kakutani factorizable if can be expressed as a composition of finitely many Kakutani multifunctions.
Let be a Hausdorff locally convex topological vector space with a continuous seminorm . A non-empty subset of is said to be approximately -compact if for each and each net in satisfying , there is a subset of converging to an element of .
In the present paper, the authors prove best proximity pair theorems which furnish sufficient conditions ensuring the existence of an element such that
when is a non-empty approximately -compact convex subset, a non-empty closed convex subset of , is a Kakutani factorizable multifunction and is a single-valued function.