Existence and convergence theorems for Bregman best proximity points in reflexive Banach spaces

Abstract

In recent years, there has been considerable interest in the study of best proximity points. In this paper, using Bregman functions and Bregman distances, we first prove the existence of Bregman best proximity points in a reflexive Banach space. We then prove convergence results of Bregman best proximity points for Bregman cyclic contraction mappings in the setting of Banach spaces. It is well known that the Bregman distance does not satisfy either the symmetry property or the triangle inequality which are required for standard distances. So, Bregman distances enable us to provide affirmative answers to two problems raised by Eldred and Veeramani (J Math Anal Appl 323:1001–1006, 2006) and Al-Thagafi and Shahzad (Nonlinear Anal 70:3665–3671, 2009) concerning the existence of best proximity points for a cyclic contraction map in a reflexive Banach space. This can be done in the absence of either symmetry property or the triangle inequality which are required for standard distances. Our results improve and generalize many known results in the current literature.