The split common null point problem for Bregman generalized resolvents in two Banach spaces

• Published in: Optimization Vol. 70; no. 8; pp. 1725 - 1758
• Main Authors: Gazmeh, Hamid, Naraghirad, Eskandar
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 03.08.2021; Taylor & Francis LLC
• Subjects: Banach spaces; Bregman function and Bregman distance; Convergence; fixed point; Halpern iteration; maximal monotone operator; split common null point problem; uniformly convex function; uniformly smooth function
• ISSN: 0233-1934; 1029-4945
• DOI: 10.1080/02331934.2020.1751157

In this paper, we first consider the split common null point problem in two Banach spaces. Then, using the Bregman generalized resolvents of maximal monotone operators, we prove strong convergence theorems of Halpern type iteration for finding a solution of the split common null point problem in two Banach spaces. As an application of our result, we study the split equilibrium problem in general Banach spaces and approximate a solution of the problem for the first time. Our new technique is based on basic properties of a Bregman distance induced by a Bregman function without using Bregman projection or the requirement of Mosco convergence of the sequences produced by the method. 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, the results of the paper improve and extend many recent results in the literature.