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 with a continuous semi-norm $p$. Given a multifunction $T\colon A\to2^B$ and a single valued function $g\colon A\to A$, a best proximity pair theorem which provides sufficient conditions ascertaining the existence of an element $x_0\in A$ such that $d_p(Tx_0,gx_0)=d_p(A,B)$ is established. In the setting of normed linear spaces, this theorem reduces to a fixed point theorem for multifunctions if $T$ is a self-multifunction on $A$ and $g$ is the identity function on $A$. Further, a best approximation theorem is proved for continuous Kakutani factorizable multifunctions which are not necessarily convex valued.

AMS Subject Classification (1991): 47H10, 54H25

Keyword(s): Kakutani factorizable multifunction, Best proximity pair, Best approximation, Pseudo affinity, Quasi affinity

Received November 23, 1996. (Registered under 6116/2009.)