Let \(X\) and \(Y\) be any two topological spaces. A multifunction \(T:X\to 2^Y\) is said to be (i) upper semi-continuous if \(T^{-1}(B)= \{x\in X:(Tx)\cap B\neq\emptyset\}\) is closed in \(X\) whenever \(B\) is a closed subset of \(Y\); (ii) Kakutani multifunction if (a) \(T\) is upper semi-continuous, (b) either \(Tx\) is a singleton for each \(x\in X\) or \(Tx\) is a non-empty compact convex subset of \(Y\), assuming \(Y\) to be a non-empty convex set in a Hausdorff topological vector space; (iii) Kakutani factorizable if \(T\) can be expressed as a composition of finitely many Kakutani multifunctions. Let \(E\) be a Hausdorff locally convex topological vector space with a continuous seminorm \(p\). A non-empty subset \(A\) of \(E\) is said to be approximately \(p\)-compact if for each \(y\in E\) and each net \(\{x_\alpha\}\) in \(A\) satisfying \(d_p (x_\alpha,y)\to d_p(y,A)\equiv\inf\{p(y-a):a\in A\}\), there is a subset of \(\{ x_\alpha\}\) converging to an element of \(A\). In the present paper, the authors prove best proximity pair theorems which furnish sufficient conditions ensuring the existence of an element \(x_0\in A\) such that \[ d_p(g x_0,Tx_0)= d_p(A,B)\equiv \inf\{p(a-b): a\in A,b\in B\}, \] when \(A\) is a non-empty approximately \(p\)-compact convex subset, \(B\) a non-empty closed convex subset of \(E\), \(T:A\to 2^B\) is a Kakutani factorizable multifunction and \(g:A\to A\) is a single-valued function.