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Journal of Function Spaces
Volume 2016, Article ID 5161682, 18 pages
http://dx.doi.org/10.1155/2016/5161682
Research Article

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

1Department of Mathematical Sciences, Bayero University, Kano, Nigeria
2Department of Science and Technology Education, Bayero University, Kano, Nigeria

Received 19 May 2016; Accepted 25 July 2016

Academic Editor: Adrian Petrusel

Copyright © 2016 Bashir Ali and M. H. Harbau. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Bregman -mapping for a finite family of Bregman quasiasymptotically nonexpansive mappings and show the fixed point set of the Bregman -mapping is the set of common fixed points of . Using the Bregman -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.