Convergence theorems for Bregman \(K\)-mappings and mixed equilibrium problems in reflexive Banach spaces (Q324643)

Summary: We introduce a new mixed equilibrium problem with a relaxed monotone mapping in a reflexive Banach space and prove the existence of a solution of the equilibrium problem. Using Bregman distance, we introduce the concept of Bregman \(K\)-mapping for a finite family of Bregman quasiasymptotically nonexpansive mappings and show the fixed point set of the Bregman \(K\)-mapping is the set of common fixed points of \(\{T_i \}_{i = 1}^N\). Using the Bregman \(K\)-mapping, we introduce an iterative sequence for finding a common point in the set of 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.