Global optimization of cyclic Kannan nonexpansive mappings in nonreflexive Banach spaces

Abstract

Consider a self-mapping T defined on a union of two subsets A and B of a Banach space such that T(A) ⊆ B and T(B) ⊆ A. In this work we survey the existence of an optimal approximate solution, known as a best proximity point for a class of cyclic mappings, called cyclic Kannan nonexpansive mappings, in Banach spaces under appropriate sufficient conditions. In this order, the notion of T-uniformly semi-normal structure is introduced and used to investigate the existence of best proximity points. As an application of the existence theorem, we conclude an old fixed point problem in Banach spaces which are not reflexive necessarily. Examples are given to support the usability of our main conclusions.

Keywords: Best proximity point, fixed point, cyclic Kannan nonexpansive mapping, T-uniformly semi-normal structure, uniformly convex Banach space