Abstract
Let A and B be two nonempty subsets of a Banach space X. A mapping T :T : AUB -> AUB is said to be cyclic relatively nonexpansive if T(A) subset of B and T(B) subset of A and parallel to x - y parallel to for all (x, y) is an element of A x B. In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach space X. It is shown that if (A, B) is a nonempty, weakly compact, and convex pair and (A, B) has seminormal structure, then a cyclic relatively nonexpansive mapping T : AUB -> AUB has a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings.