Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings

Abstract

Let A and B be two nonempty subsets of a Banach space X. A mapping T : $A\cup B\to A\cup B$ is said to be cyclic relatively nonexpansive if T(A) $\subseteq B$ and T(B) $\subseteq A$ and $∥Tx-Ty∥\le ∥x-y∥$ for all ($x\text{,}y$) $\in A\times B$. 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 : $A\cup B\to A\cup B$ 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.