Coupled coincidence point results in partially ordered generalized fuzzy metric spaces with applications to integral equations

Binayak S Choudhury,Pradyut Das

doi:10.1007/s40096-015-0173-6

Binayak S Choudhury, Pradyut Das

Open Access

https://doi.org/10.1007/s40096-015-0173-6

Copy DOI

Abstract

Here we prove a coupled coincidence point theorem in G-fuzzy metric spaces for compatible mappings using $$Had\check{z}i\acute{c}$$ type t-norms which is characterized by the equi-continuity of its iterates. We apply our result toward obtaining a result in G-metric spaces. Two supporting examples are also given. Some existing results are extended by our theorem. We also apply our result to a problem of an integral equation. We further assume the G-metric space to be equipped with a partial ordering.

Highlights

The program of this work is to establish coupled coincidence point results in generalized fuzzy metric spaces which are fuzzy extensions of generalized metric spaces
Here we prove a coupled coincidence point theorem in G-fuzzy metric spaces for compatible mappings using Hadzic type t-norms which is characterized by the equi-continuity of its iterates
Several fuzzified versions of the exiting mathematical structures were introduced in the literatures, the fuzzification of metric space followed through adoption of different approaches

Summary

Introduction

The program of this work is to establish coupled coincidence point results in generalized fuzzy metric spaces which are fuzzy extensions of generalized metric spaces (abbreviated as G-metric spaces in the literatures). This paper aimed at establishing a new coupled fixed point theorem in G-fuzzy metric spaces with a partial order. For this purpose we prove a lemma which establishes a Cauchy criterion for two sequences simultaneously. Definition 2.2 [22] The 3-tuple ðA; G; ÃÞ called G-fuzzy metric space if A is any non-empty set, Ã is a t-norm which is continuous and G is a fuzzy membership function on A3 Â ð0; 1Þ which satisfies z1; z2; z3; z4 2 A and t1; t2 [ 0: 1. Lemma 2.16 Let ðA; G; ÃÞ be a G-fuzzy metric space having a t-norm of Hadzic type for which Gðx; x; y; uÞ ! Note 2.17 Equi-continuous of the iterates is essential in the proof of the above lemma

Main results

Application to integral equations

Zt a