Chebyshev center, best proximity point theorems and fixed point theorems

Brodskii and Milman proved that there exists a point in C(A), the set of all Chebyshev centers of A, which is fixed by every surjective isometry from A into A whenever A is a nonempty weakly compact convex set having normal structure in a Banach space. Motivated by this result, Lim et al. proved that every isometry from A into A has a fixed point in C(A) whenever A is a nonempty weakly compact convex set having normal structure in a Banach space. In this paper, we prove that every relatively isometry map T : A ∪ B → A[B, satisfying T(A) ⊆ B and T(B) ⊆ A, has a best proximity point in CA(B), the set of all Chebyshev centers of B relative to A, whenever the nonempty weakly compact convex proximal pair (A,B) has proximal normal structure and rectangle property. Also, we prove that, under suitable assumptions, an analogous result of Brodskii and Milman for relatively isometry mappings holds. In case of A = B, we obtain the results of Brodskii and Milman, and Lim et al. as a particular case of our results.