Approximating Fixed Points of Bregman Generalized α-Nonexpansive Mappings

Kanikar Muangchoo,Sakulbuth Ekvittayaniphon,Poom Kumam,Sompomg Dhompongsa,Yeol Je Cho

doi:10.3390/math7080709

Abstract

In this paper, we introduce a new class of Bregman generalized α -nonexpansive mappings in terms of the Bregman distance. We establish several weak and strong convergence theorems of the Ishikawa and Noor iterative schemes for Bregman generalized α -nonexpansive mappings in Banach spaces. A numerical example is given to illustrate the main results of fixed point approximation using Halpern’s algorithm.

Highlights

In 1967, Bregman [1] discovered an effective technique using the so-called Bregman distance function D f in the process of designing and analyzing feasibility and optimization algorithms.This opened a growing area of research in which Bregman’s technique was applied in various ways in order to design and analyze some algorithms for solving feasibility and optimization problems, and algorithms for solving variational inequality problems, equilibrium problems, and fixed point problems for nonlinear mappings.In recent years, several authors have been constructing algorithms for finding fixed points of nonlinear mappings by using the Bregman distance and the Bregman projection
Motivated by the aforementioned results, we investigate the new class of Bregman generalized α-nonexpansive mappings
We prove the existence of fixed points for such mappings under some conditions, and establish weak and strong convergence theorems regarding those fixed points

Summary

Introduction

In 1967, Bregman [1] discovered an effective technique using the so-called Bregman distance function D f in the process of designing and analyzing feasibility and optimization algorithms. A Banach space E is said to satisfy Opial’s property if, for any sequence { xn }n∈N in E that converges weakly to x ∈ E, we have lim sup k xn − x k < lim sup k xn − yk, ∀y ∈ E \ { x }. Working with a Bregman distance D f with respect to f , the following Opial-like inequality holds [16]: for any Banach space E and sequence { xn }n∈N in E, we have lim sup D f ( xn , x ) < lim sup D f ( xn , y), n→∞.

A Banach space E is said to be strictly convex if

Approximating Fixed Points

Bregman Noor’s Type Iteration for Bregman Generalized α-Nonexpansive Mappings

Numerical Examples

Conclusions