Approximation of Fixed Points for Enriched Suzuki Nonexpansive Operators with an Application in Hilbert Spaces

Kifayat Ullah,Zhenhua Ma,Junaid Ahmad,Muhammad Arshad

doi:10.3390/axioms11010014

Abstract

In this article, we introduce the class of enriched Suzuki nonexpansive (ESN) mappings. We show that this new class of mappings properly contains the class of Suzuki nonexpansive as well as the class of enriched nonexpansive mappings. We establish existence of fixed point and convergence of fixed point in a Hilbert space setting under the Krasnoselskii iteration process. One of the our main results is applied to solve a split feasibility problem (SFP) in this new setting of mappings. Our main results are a significant improvement of the corresponding results of the literature.

Highlights

For many different types of problems in applied physics and mathematical engineering, one faces too many difficulties to guarantee the ability to find solutions using the already known analytical methods
To answer the Problem 1 in the affirmative, we introduce the concept of enriched Suzuki nonexpansive (ESN) mappings and show that these mappings are essentially more general than the concept of Suzuki nonexpansive and enriched nonexpansive mappings
Examples we show by examples that the class of ESN mappings properly includes the class of Suzuki nonexpansive and the class of enriched nonexpansive maps

Summary

Introduction

For many different types of problems in applied physics and mathematical engineering, one faces too many difficulties to guarantee the ability to find solutions using the already known analytical methods (see, e.g., [1,2,3,4,5] and others). Berinde [22] used the Krasnoselskii iteration process [7] for establishing convergence (weak and strong) and existence of fixed point for these mappings in a Hilbert space setting. He noted that enriched nonexpansive mappings are essentially continuous. The main result of the Berinde [22] is stated as follows This theorem extended Theorem 2 from nonexpansive mappings to the enriched nonexpansive mappings. In this paper, we shall use Krasnoselskii iteration [7] instead of Picard iteration [6], to study the existence of fixed point, fixed point set, weak and strong convergence theorems in a Hilbert space setting

Preliminaries

Enriched Suzuki Nonexpansive Mappings

Conclusions and Future Plan