New generalized fuzzy metrics and fixed point theorem in fuzzy metric space
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RESEARCH
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New generalized fuzzy metrics and fixed point theorem in fuzzy metric space Robert Plebaniak* * Correspondence: [email protected] Department of Nonlinear Analysis, Faculty of Mathematics and Computer Science, University of Łód´z, Banacha 22, Łód´z, 90-238, Poland
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 NJ on X. The paper includes also the comparison of our results with those existing in the literature. Keywords: fuzzy sets; fuzzy metric space; contraction of Banach type; fixed point; generalized fuzzy metrics; fuzzy metrics
1 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’ski˘ı 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. It is worth noticing that there exist at least five different concepts of a fuzzy metric space (see Artico and Moresco [], Deng [], George and Veeramani [], Erceg [], Kaleva and Seikkala [], Kramosil and Michalek []). In , Grabiec [] proved an analog of the Banach contraction theorem in fuzzy metric spaces (in the sense of Kramosil and Michalek []). In his proof, he used a fuzzy version of Cauchy sequence. It is worth noticing that in the literature in order to prove fixed point theorems in fuzzy metric space, authors used two different types of Cauchy sequences. For ©2014 Plebaniak; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and re
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