Several Fixed Point Theorems in Convex b-Metric Spaces and Applications

Lili Chen,Chaobo Li,Yanfeng Zhao,Radoslaw Kaczmarek

doi:10.3390/math8020242

Abstract

Our paper is devoted to indicating a way of generalizing Mann’s iteration algorithm and a series of fixed point results in the framework of b-metric spaces. First, the concept of a convex b-metric space by means of a convex structure is introduced and Mann’s iteration algorithm is extended to this space. Next, by the help of Mann’s iteration scheme, strong convergence theorems for two types of contraction mappings in convex b-metric spaces are obtained. Some examples supporting our main results are also presented. Moreover, the problem of the T-stability of Mann’s iteration procedure for the above mappings in complete convex b-metric spaces is considered. As an application, we apply our main result to approximating the solution of the Fredholm linear integral equation.

Highlights

In the last few decades one could observe a huge amount of interest for the development of the fixed point theory because of plenty of applications, especially in metric spaces [1,2]
In 1970, Takahashi [29] introduced the concepts of a convex structure and a convex metric space, and formulated some first fixed point theorems for nonexpansive mappings in the convex metric space
We firstly introduce the concept of the convex b-metric space by means of the convex structure

Summary

Introduction

In the last few decades one could observe a huge amount of interest for the development of the fixed point theory because of plenty of applications, especially in metric spaces [1,2]. In 1970, Takahashi [29] introduced the concepts of a convex structure and a convex metric space, and formulated some first fixed point theorems for nonexpansive mappings in the convex metric space. Goebel and Kirk [30] studied some iterative processes for nonexpansive mappings in the hyperbolic metric space, and in 1988, Xie [31] found fixed points of quasi-contraction mappings in convex metric spaces by Ishikawa’s iteration scheme. The Picard iteration algorithm is widely used in studying the fixed point problems for many kinds of contraction mappings and quasi-contraction mappings in b-metric spaces. We introduce the concept of weak T-stability of the iteration for mappings in complete metric spaces and discuss the problem of weak T-stability of Mann’s iteration procedure for above two kinds of mappings in complete convex b-metric spaces. We apply our main result to approximating the solution of the Fredholm linear integral equation

Preliminaries

Main Results

Applications