Compositions and convex combinations of Bregman weakly relatively nonexpansive operators in reflexive Banach spaces

Abstract

In this paper, we investigate the right and left Bregman weakly relatively nonexpansive operators in reflexive Banach spaces. We first introduce the notions of convex hull demi-closedness principle, strictly strongly convex hull demi-closedness principle, strongly demi-closedness principle, and composition demi-closedness principle of a family of nonlinear mappings using another new notions of convex asymptotic fixed points, and composition asymptotic fixed points for such mappings in a Banach space E. We analyze, in particular, compositions and convex combinations of Bregman weakly relatively nonexpansive operators, and prove the convergence of the Mann iterative method for operators of these types. Finally, we use our results to approximate common zeros of maximal monotone mappings and solutions to convex feasibility problems. To support our results, we include nontrivial examples in the paper. Therefore, our results improve and generalize many known results in the current literature.