The aim of this paper is to extend the results of Harjani and Sadarangani and some other authors and to prove a new fixed point theorem of a contraction mapping in a complete metric space endowed with a partial order by using altering distance functions. Our theorem can be used to investigate a large class of nonlinear problems.As an application, we discuss the existence of a solution for a periodic boundary value problem.

A new hybrid Bregman projection method is considered for finding common solutions of the set of common fixed points of an infinite family of closed, uniformly asymptotic regular and uniformly Bregman totally quasi-D-asymptotically nonexpansive mappings, the set of solutions to a variational inequality problem and the set of common solutions to a system of generalized mixed equilibrium problems, strong convergence theorems of common elements are proved by using new analysis techniques and Bregman mappings in the setting of uniformly smooth and 2-uniformly convex real Banach spaces. Our results improve and generalize many important known recent results in the current literature, because Bregman projection mapping generalizes the generalized projection mapping and the metric projection mapping. c 2016 All rights reserved. Keywords: Hybrid Bregman projection method, Bregman totally quasi-D-asymptotically nonexpansive mapping, variational inequality problem, generalized mixed equilibrium problem, uniformly smooth Banach space, 2-uniformly convex Banach space. 2010 MSC: 47J20, 47H09, 47H10.

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