A New Approach to Fuzzy Differential Equations Using Weakly-Compatible Self-Mappings in Fuzzy Metric Spaces

Iqra Shamas,Abdu Gumaei,Saif Ur Rehman,Mabrook Al-Rakhami,Naeem Jan,Douadi Drihem

doi:10.1155/2021/6123154

Iqra Shamas, Abdu Gumaei + Show 4 more

Open Access

https://doi.org/10.1155/2021/6123154

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Abstract

The key objective of this research article includes the study of some rational type coincidence point and deriving common fixed point (CFP) results for rational type weakly-compatible three self-mappings in fuzzy metric (FM) space. The “triangular property of FM” is used as a fundamental tool. Moreover, some unique coincidence points and CFP theorems were presented for three self-mappings in an FM space under the conditions of rational type weakly-compatible fuzzy-contraction. In addition, some suitable examples are also given. Furthermore, an application of fuzzy differential equations is provided in the aid of the proposed work. Hence, the innovative direction of rational type weakly-compatible fuzzy-contraction with the application of fuzzy differential equations in FM space will certainly play a vital role in the related fields. It has the potential to be extended in any direction with different types of weakly-compatible fuzzy-contraction conditions for self-mappings with different types of differential equations.

Highlights

In 1922, Banach [1] proved a “Banach contraction principle for fixed point (FP)” which is stated as “A self-mapping in a complete metric space satisfy the contraction condition has a unique FP.”
By using the concept of [27], Rehman and Aydi [29] proved some rational type common fixed point (CFP) theorems in fuzzy cone metric (FCM) spaces with an application to Fredholm integral equations
We used the “triangular property of fuzzy metric” as an elementary tool and proved some unique coincidence points and CFP theorems under the rational type weakly compatible fuzzy-contraction conditions for three self-mappings in FM spaces with some suitable examples

Summary

Introduction

In 1922, Banach [1] proved a “Banach contraction principle for fixed point (FP)” which is stated as “A self-mapping in a complete metric space satisfy the contraction condition has a unique FP.” After the publication of the “Banach contraction principle,” many researchers contributed in flourishing the FP theory. By using the concept of [27], Rehman and Aydi [29] proved some rational type CFP theorems in FCM spaces with an application to Fredholm integral equations. We used the “triangular property of fuzzy metric” as an elementary tool and proved some unique coincidence points and CFP theorems under the rational type weakly compatible fuzzy-contraction conditions for three self-mappings in FM spaces with some suitable examples. The current novel direction of rational type weakly-compatible fuzzycontraction with the application of fuzzy differential equations in FM space will play a vital role in the related fields.

Preliminaries

Main Result

Application to the Fuzzy Differential Equations

Conclusion