New generalized fuzzy metrics and fixed point theorem in fuzzy metric space

Robert Plebaniak

doi:10.1186/1687-1812-2014-241

Abstract

In this paper, in fuzzy metric spaces (in the sense of Kramosil and Michalek (Kibernetika 11:336-344, 1957)) we introduce the concept of a generalized fuzzy metric which is the extension of a fuzzy metric. First, inspired by the ideas of Grabiec (Fuzzy Sets Syst. 125:385-389, 1989), we define a new G-contraction of Banach type with respect to this generalized fuzzy metric, which is a generalization of the contraction of Banach type (introduced by M Grabiec). Next, inspired by the ideas of Gregori and Sapena (Fuzzy Sets Syst. 125:245-252, 2002), we define a new GV-contraction of Banach type with respect to this generalized fuzzy metric, which is a generalization of the contraction of Banach type (introduced by V Gregori and A Sapena). Moreover, we provide the condition guaranteeing the existence of a fixed point for these single-valued contractions. Next, we show that the generalized pseudodistance J:X×X→[0,∞) (introduced by Włodarczyk and Plebaniak (Appl. Math. Lett. 24:325-328, 2011)) may generate some generalized fuzzy metric N J on X. The paper includes also the comparison of our results with those existing in the literature.

Highlights

A number of authors generalize Banach’s [ ] and Caccioppoli’s [ ] result and introduce the new concepts of contractions of Banach and study the problem concerning the existence of fixed points for such a type of contractions; see e.g. Burton [ ], Rakotch [ ], Geraghty [, ], Matkowski [ – ], Walter [ ], Dugundji [ ], Tasković [ ], Dugundji and Granas [ ], Browder [ ], Krasnosel’skiı et al [ ], Boyd and Wong [ ], Mukherjea [ ], Meir and Keeler [ ], Leader [ ], Jachymski [, ], Jachymski and Jóźwik [ ], and many others not mentioned in this paper.In, Kramosil and Michalek [ ] introduced the concept of fuzzy metric spaces
Fixed point theory for contractive mappings in fuzzy metric spaces is closely related to the fixed point theory for the same type of mappings in probabilistic metric spaces of Menger type; see Hadžić [ ], Sehgal and Bharucha-Reid [ ], Schweizer et al [ ], Tardiff [ ], Schweizer and Sklar [ ], Qiu and Hong [ ], Hong and Peng [ ], Mohiuddine and Alotaibi [ ], Wang et al [ ], Hong [ ], Saadati et al [ ], and many others not mentioned in this paper
In fuzzy metric spaces, we introduce the concept of a generalized fuzzy metric which is the extension of a fuzzy metric

Summary

Introduction

A number of authors generalize Banach’s [ ] and Caccioppoli’s [ ] result and introduce the new concepts of contractions of Banach and study the problem concerning the existence of fixed points for such a type of contractions; see e.g. Burton [ ], Rakotch [ ], Geraghty [ , ], Matkowski [ – ], Walter [ ], Dugundji [ ], Tasković [ ], Dugundji and Granas [ ], Browder [ ], Krasnosel’skiı et al [ ], Boyd and Wong [ ], Mukherjea [ ], Meir and Keeler [ ], Leader [ ], Jachymski [ , ], Jachymski and Jóźwik [ ], and many others not mentioned in this paper. (III) [ ] A fuzzy metric space in which every GV-Cauchy sequence is convergent is called complete in George and Veeramani’s sense (GV-complete for short). (III) A fuzzy metric space is called N -G-complete if each N -G-Cauchy sequence (xm : m ∈ N) in X is N -convergent to some x ∈ X and (NC) ∀t> {limm→∞ N(xm, x, t) = limm→∞ N(x, xm, t) = }. (II) A fuzzy metric space is called N -GV-complete, if each N -GV-Cauchy sequence (xm : m ∈ N) in X is N -convergent to some x ∈ X and ∀t> {limm→∞ N(xm, x, t) = limm→∞ N(x, xm, t) = }. Using similar arguments to the corresponding ones appearing in Section and in the paper of Gregori and Sapena [ ] we may conclude the following fixed point theorem in George and Veeramani’s fuzzy metric space.

We obtain

Let NJ be defined by t

Findings

By we know that