New Relation-Theoretic Fixed Point Theorems in Fuzzy Metric Spaces with an Application to Fractional Differential Equations

Samera M Saleh,Salvatore Sessa,Waleed M Alfaqih,Ferdinando Di Martino

doi:10.3390/axioms11030117

Samera M Saleh, Salvatore Sessa + Show 2 more

Open Access

https://doi.org/10.3390/axioms11030117

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Abstract

In this paper, we introduce the notion of fuzzy R−ψ−contractive mappings and prove some relevant results on the existence and uniqueness of fixed points for this type of mappings in the setting of non-Archimedean fuzzy metric spaces. Several illustrative examples are also given to support our newly proven results. Furthermore, we apply our main results to prove the existence and uniqueness of a solution for Caputo fractional differential equations.

Highlights

Introduction and PreliminariesThe concept of fuzzy sets was initially presented by Zadeh [1] in 1965, wherein he defined a fuzzy set as: a fuzzy set M on a non-empty set X is a function from X to [0, 1]
The fuzzy fixed point theory was started by Grabiec [4] in 1988, wherein he presented the concepts of G-Cauchy sequences and G-complete fuzzy metric spaces and provided a fuzzy metric version of Banach’s contraction principle
The present paper aims to introduce the concept of fuzzy R − ψ−contractive mappings and prove some relevant results on the existence and uniqueness of fixed points for such mappings in the setting of non-Archimedean fuzzy metric spaces which extended and generalized the results in [6,19]

Summary

Introduction and Preliminaries

The concept of fuzzy sets was initially presented by Zadeh [1] in 1965, wherein he defined a fuzzy set as: a fuzzy set M on a non-empty set X is a function from X to [0, 1]. A fuzzy metric space (X , M, ∗) which is endowed with a binary relation R is said to be R-complete if every R-Cauchy sequence is convergent in X. The present paper aims to introduce the concept of fuzzy R − ψ−contractive mappings and prove some relevant results on the existence and uniqueness of fixed points for such mappings in the setting of non-Archimedean fuzzy metric spaces (in Kramosil and Michalek’s sense as well as George and Veeramani’s sense) which extended and generalized the results in [6,19]. We apply our newly fixed point results to prove the existence and uniqueness of solutions for Caputo fractional differential equations

Main Results

Application to Nonlinear Fractional Differential Equations

Conclusions