Some Convergence Results for a Class of Generalized Nonexpansive Mappings in Banach Spaces

Thabet Abdeljawad,Muhammad Naveed Khan,Kifayat Ullah,Manuel De La Sen,Junaid Ahmad,Remi Léandre

doi:10.1155/2021/8837317

Abstract

This paper investigates fixed points of Reich-Suzuki-type nonexpansive mappings in the context of uniformly convex Banach spaces through an M ∗ iterative method. Under some appropriate situations, some strong and weak convergence theorems are established. To support our results, a new example of Reich-Suzuki-type nonexpansive mappings is presented which exceeds the class of Suzuki-type nonexpansive mappings. The presented results extend some recently announced results of current literature.

Highlights

Fixed point theory and applications played an important role in many areas of applied sciences and solved many problems rising in engineering, mathematical economics and optimization
When the problem cannot be solved by ordinary analytical methods, we transform it to the form of fixed point problems and apply an appropriate iterative method to obtain the required fixed point
In 1922, Banach [1] proved that if T is contraction mapping on a closed subset K of a Banach space, that is, kTs − Ts′k ≤ αks − s′k for a fixed α ∈ 1⁄20, 1Þ and s, s′ ∈ K, T possesses a unique fixed point, which can be obtained by using the Picard [2] iteration process

Summary

Introduction

Fixed point theory and applications played an important role in many areas of applied sciences and solved many problems rising in engineering, mathematical economics and optimization. The first basic result concerning the existence of a fixed point for the class of nonexpansive mappings was independently proved by Browder [4] and Göhde [5] They proved that any nonexpansive mapping on a closed bounded convex subset of a uniformly convex Banach space (in short UCBS) always possesses a fixed point. Ullah and Arshad [19] introduced a new iterative process called M∗ iterative process, as follows: s1 ∈ K, zn = ð1 − ρnÞsn + ρnTsn, They proved that M∗ iterative process can be used for finding fixed points of Suzuki-type nonexpansive mappings. We extend their results to the general setting of Reich-Suzuki-type nonexpansive mappings

Preliminaries

Convergence Theorems in Uniformly Convex Banach Spaces

Numerical Example