New Fuzzy Fixed Point Results in Generalized Fuzzy Metric Spaces With Application to Integral Equations

Muhammad Sohail Ashraf,Rashid Ali,Nawab Hussain

doi:10.1109/access.2020.2994130

Abstract

In this paper, we introduce the notion of generalized fuzzy metric space. Many topological spaces like fuzzy metric spaces, fuzzy $b$ -metric spaces and dislocated fuzzy metric spaces have been generalized by this new generalized fuzzy metric space. We prove the Banach contraction principle and Ciric’s quasi-contraction theorem in the setting of generalized fuzzy metric space and furnish an example to illustrate our results. As consequences of our results we obtain Jleli and Samet and many other authors recent results as corollaries. An application related to our main result for nonlinear integral equation is also presented.

Highlights

The concept of a metric space was given by Fréshet in 1906
In this article motivated by Jleli and Samet [8], we introduce the concept of generalized fuzzy metric space that generalizes the notions of fuzzy metric space, fuzzy b-metric space and dislocated fuzzy metric space
MAIN RESULTS Motivated by Jleli and Samet [8] and following George and Veeramani [11], we introduce the notion of a generalized fuzzy metric spaces as follows: Definition 7: Consider a nonempty set X and a mapping G : X × X × (0, ∞) → [0, 1]

Summary

INTRODUCTION

After that many researchers generalized this concept and proved some fixed point results. Many fixed point theorems have been proved by many researchers in the field of b-metric spaces and its various extensions. In 2015, Jleli and Samet [8] introduced the notion of a generalized metric space. Mehmood et al [14] introduced the concept of extended fuzzy b-metric space and generalized the Banach contraction principle in this space. In this article motivated by Jleli and Samet [8], we introduce the concept of generalized fuzzy metric space that generalizes the notions of fuzzy metric space, fuzzy b-metric space and dislocated fuzzy metric space. We have proved some fixed point results in the setting of generalized fuzzy metric space.

PRELIMINARIES

APPLICATION

CONCLUSION