A strong convergence theorem for Bregman asymptotically quasi-nonexpansive mappings in the intermediate sense

Yukino Tomizawa

doi:10.1186/1687-1812-2014-154

Abstract

The purpose of this paper is to introduce a new class of Bregman asymptotically quasi-nonexpansive mappings in the intermediate sense. A strong convergence theorem of the shrinking projection method with the modified Mann iteration is established to find fixed points of the mappings in reflexive Banach spaces. This theorem generalizes some known results in the current literature.

Highlights

Fixed point theory is an important branch of nonlinear analysis and has been applied in numerous studies of nonlinear phenomena
Many authors have considered the problem of iterative algorithms for mappings of nonexpansive type which converge to some fixed points
The purpose of this paper is to prove strong convergence theorems for asymptotically quasi-nonexpansive mappings with respect to Bregman distances in the intermediate sense by using the shrinking projection method

Summary

Introduction

Fixed point theory is an important branch of nonlinear analysis and has been applied in numerous studies of nonlinear phenomena. Motivated by the results above, we design a new hybrid iterative scheme for finding fixed points of the mapping in reflexive Banach spaces This iterative method is expected to be applicable to many other problems in nonlinear functional analysis relating to Bregman distances. Motivated by [ , ], we design a new hybrid iterative scheme for finding a fixed point of mappings in the new class by using the shrinking projection method with respect to Bregman distances in reflexive Banach spaces. We introduce a new class of mappings; the mapping T is said to be Bregman asymptotically quasi-nonexpansive in the intermediate sense if F(T) = ∅ and lim sup sup Df p, Tnx – Df (p, x) ≤.

This implies

We have