Some Compatible and Weakly-Compatible Four Self-Mapping Results Approach to Nonlinear Integral Equations in Fuzzy Cone Metric Spaces

Saif Ur Rehman,Hawraa Akram Yazbek,Mohammed Elgarhy,Rashad A R Bantan,John R Akeroyd

doi:10.1155/2021/7641303

Abstract

This paper is aimed at proving some unique common fixed point theorems by using the compatible and weakly-compatible four self-mappings in fuzzy cone metric (FCM) space. We prove the results under the generalized rational contraction conditions in FCM spaces with the help of one self-map are continuous. Furthermore, we prove some rational contraction results with the weaker condition of the self-mapping continuity. Ultimately, our theoretical work has been utilized to prove the existence solution of the two nonlinear integral equations. This is an illustrative application of how FCM spaces can be used in other integral type operators.

Highlights

The theory of fixed-point theory was introduced by Banach [1]
Many researchers have been generalized this principle in many directions and proved different contractive type FP and common fixed point (CFP) for single-valued and multivalued mappings in the context of metric spaces
We proved some unique CFP theorems by using the compatible and weakly-compatible four selfmappings in fuzzy cone metric (FCM) space

Summary

Introduction

The theory of fixed-point theory was introduced by Banach [1]. He proved a “Banach contraction principle,” which is stated as follows: “A self-mapping on a complete metric space verifying the contraction condition has a unique fixed point (FP).” Later on, many researchers have been generalized this principle in many directions and proved different contractive type FP and common fixed point (CFP) for single-valued and multivalued mappings in the context of metric spaces. Many researchers have been generalized this principle in many directions and proved different contractive type FP and common fixed point (CFP) for single-valued and multivalued mappings in the context of metric spaces. Journal of Function Spaces covered by Huang and Zhang [22] They proved the convergence properties and FP theorems for nonlinear contractive type mappings. In 2015, the notion of fuzzy cone metric space (FCM space) was introduced by Oner et al [29] They proved the key attributes of FCM space and a “fuzzy cone Banach contraction theorem for FP” in FCM space. We prove our main results for unique CFP-theorems under the generalized rational contraction conditions in FCM spaces by using compatibility and weakly-compatibility of four selfmappings.

Preliminaries

Main Results

Supportive Application

Conclusion