A new approach to relatively nonexpansive mappings

In this paper we study the nonexpansivity of the so-called relatively nonexpansive mappings. A relatively nonexpansive mapping with respect to a pair of subsets $(A,B)$ of a Banach space $X$ is a mapping defined from $A\cup B$ into $X$ such that $\Vert Tx-Ty\Vert\le \Vert x-y\Vert$ for $x\in A$ and $y\in B$. These mappings were recently considered in a paper by Eldred et al. (Proximinal normal structure and relatively nonexpansive mappings, Studia Math. 171 (3) (2005), 283-293) to obtain a generalization of Kirk's Fixed Point Theorem. In this work we show that, for certain proximinal pairs $(A,B)$, there exists a natural semimetric for which any relatively nonexpansive mapping with respect to $(A,B)$ is nonexpansive. This fact will be used to improve one of the two main results from the aforementioned paper by Eldred et al. At that time we will also obtain several consequences regarding the strong continuity properties of relatively nonexpansive mappings and the relation between the two main results from the same work.