A new approach to relatively nonexpansive mappings
Rafa Espínola · Proceedings of the American Mathematical Society · 2008
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 ) (A,B) of a Banach space X X is a mapping defined from A ∪ B A\cup B into X X such that ‖ T x − T y ‖ ≤ ‖ x − y ‖ \|Tx-Ty\|\le \|x-y\| for x ∈ A x\in A and y ∈ B 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 ) (A,B) , there exists a natural semimetric for which any relatively nonexpansive mapping with respect to ( A , B ) (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.