Let A, B be nonempty subsets of a Banach space X and \(\mathscr {F}\) be a commuting family of relatively nonexpansive mappings on \(A\cup B\) . A point \(x_0\in A\) is a common best proximity point of \(\mathscr {F}\) if \(\Vert x_0-Tx_0\Vert =\inf \{\Vert a-b\Vert :a\in A,b\in B\}\) , for all \(T\in \mathscr {F}\) . The known common best proximity point theorem for \(\mathscr {F}\) states that if A and B are nonempty compact convex subsets of a strictly convex Banach space, then \(\mathscr {F}\) has common best proximity point. In this manuscript, we provide sufficient conditions for the existence of a best proximity point of \(\mathscr {F}\) , where X need not be strictly convex and A, B are not necessarily compact.
