An Approach of Integral Equations in Complex-Valued  b -Metric Space Using Commuting Self-Maps

Rashad A R Bantan,Shahid Mehmood,Waleed Almutiry,Saif Ur Rehman,Mohammed Elgarhy,Amani Abdullah Alahmadi,Santosh Kumar

doi:10.1155/2022/5862251

Abstract

This paper is aimed at establishing some unique common fixed point theorems in complex-valued b -metric space under the rational type contraction conditions for three self-mappings in which the one self-map is continuous. A continuous self-map is commutable with the other two self-mappings. Our results are verified by some suitable examples. Ultimately, our results have been utilized to prove the existing solution to the two Urysohn integral type equations. This application illustrates how complex-valued b -metric space can be used in other types of integral operators.

Highlights

In 1922, Banach [1] proved a fixed point theorem (FPtheorem), which is stated as the following: “a single-valued contractive type mapping on a complete metric space has a unique fixed point.” After the publication of the Banach FPtheorem, many researchers have contributed their ideas to the theory of FP
Bakhtin [7] introduced the idea of b-metric space, while Czerwik [8] proved some fixed point results for nonlinear set-valued contractive type mappings in b-metric spaces
In 2011, the concept of complex-valued metric space was given by Azam et al [21], and they proved some common FP (CFP)-theorems for self-mappings

Summary

Introduction

In 1922, Banach [1] proved a fixed point theorem (FPtheorem), which is stated as the following: “a single-valued contractive type mapping on a complete metric space has a unique fixed point.” After the publication of the Banach FPtheorem, many researchers have contributed their ideas to the theory of FP. They extended some results in the framework of b-metric-like spaces They presented examples and established the application of their main results. In 2011, the concept of complex-valued metric space was given by Azam et al [21], and they proved some CFP-theorems for self-mappings. Abbas et al [23] presented some generalized CFP-results by using cocyclic mappings in complexvalued metric space They provided examples to indicate the authenticity of his expressions. Our results have been utilized to prove the existing solution to the two Urysohn integral type equations This application is illustrative of how complex-valued b -metric space can be used in other integral type operators.

Preliminaries

Main Result

Applications

Conclusions