Convergence Theorems for BregmanK-Mappings and Mixed Equilibrium Problems in Reflexive Banach Spaces

Abstract

We introduce a new mixed equilibrium problem with a relaxed monotone mapping in a reflexive Banach space and prove the existence of solution of the equilibrium problem. Using Bregman distance, we introduce the concept of BregmanK-mapping for a finite family of Bregman quasiasymptotically nonexpansive mappings and show the fixed point set of the BregmanK-mapping is the set of common fixed points of{Ti}i=1N. Using the BregmanK-mapping, we introduce an iterative sequence for finding a common point in the set of a common fixed points of the finite family of Bregman quasiasymptotically nonexpansive mappings and the set of solutions of some mixed equilibrium problems. Strong convergence of the iterative sequence is proved. Our results generalise and improve many recent results in the literature.

Inertial hybrid self-adaptive subgradient extragradient method for fixed point of quasi$-\phi-$nonexpansive multivalued mappings and equilibrium problem

“…Due to their importance, various methods have been imployed to approximate solutions of equilibrium and xed point problems (see, for example, [3,18,19,35,36] and the references contained therein). One of the common methods use is the proximal point method in which the convergence analysis has been considered when the bifunction g is monotone see [26].…”