[EN] In this paper we survey some results on contractive sequences in fuzzy metric spaces in the sense of George and Veeramani.This research is supported by Ministry of Economy and Competitiveness of Spain under Grant MTM2015-64373-P.Gregori Gregori, V.; Sapena Piera, A. (2017). Fuzzy contractive sequences. En Proceedings of the Workshop on Applied Topological Structures. Editorial Universitat Politècnica de València. 77-84. http://hdl.handle.net/10251/128042OCS778

Gregori Gregori, Valentín

Sapena Piera, Almanzor

[EN] In this paper we survey some results on contractive sequences in fuzzy metric spaces in the sense of George and Veeramani.This research is supported by Ministry of Economy and Competitiveness of Spain under Grant MTM2015-64373-P.Gregori Gregori, V.; Sapena Piera, A. (2017). Fuzzy contractive sequences. En Proceedings of the Workshop on Applied Topological Structures. Editorial Universitat Politècnica de València. 77-84. http://hdl.handle.net/10251/128042778

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WATS'17Workshop on Applied Topological StructuresValencia, 11-12 July 2017pp. 77 – 84Fuzzy contractive sequencesValent´ın Gregori 1 and Almanzor SapenaInstituto Universitario de Matema´tica Pura y Aplicada, Universitat Polite`cnica de Vale`ncia, Spain(vgregori@mat.upv.es, alsapie@mat.upv.es)AbstractIn this paper we survey some results on contractive sequences in fuzzymetric spaces in the sense of George and Veeramani.Keywords: Contractive mapping; contractive sequence; fuzzy metricspaceMSC:54A40; 54D35; 54E501. Introduction and preliminariesGeorge and Veeramani [1] gave the following definition of fuzzy metric which is aslight modification of the one given by Kramosil and Michalek [11].Definition 1. A fuzzy metric space is an ordered triple (X,M, ∗) such that X isa (non-empty) set, ∗ is a continuous t-norm andM is a fuzzy set on X×X×]0,∞[satisfying the following conditions, for all x, y, z ∈ X , s, t > 0:1This research is supported by Ministry of Economy and Competitiveness of Spain underGrant MTM2015-64373-P.This work is licensed under a Creative Commons License CC BY-NC-ND 4.0Editorial Universitat Polite`cnica de Vale`ncia77V. Gregori and A. Sapena(GV1) M(x, y, t) > 0;(GV2) M(x, y, t) = 1 if and only if x = y;(GV3) M(x, y, t) =M(y, x, t);(GV4) M(x, y, t) ∗M(y, z, s) ≤M(x, z, t+ s);(GV5) M(x, y, ) :]0,∞[→]0, 1] is continuous.It is also said that M is a fuzzy metric on X .If we define M(x, y, 0) = 0 and (GV2) and (GV5) are replaced by(KM2) M(x, y, t) = 1 for all t > 0 if and only if x = y;(KM5) M(x, y, ) : [0,∞[→ [0, 1] is left continuous,then (X,M, ∗) is a KM -fuzzy metric space.From both fuzzy metric spaces on can deduce on X a topology τM which has asa base the family of open sets of the form B(x, ε, t) = {x ∈ X, 0 < ε < 1, t > 0}where B(x, ε, t) = {y ∈ X :M(x, y, t) > 1− ε} for all ε ∈]0, 1[ and t > 0.If (X, d) is a metric space then (X,Md, ∗) is a fuzzy metric space which is calledstandard fuzzy metric space deduced from (X, d) whereMd(x, y, t) =tt+ d(x, y)Definition 2. Let (X,M, ∗) be a fuzzy metric space. A sequence {xn} is calledCauchy if given ε ∈]0, 1[ and t > 0 there exists n0 ∈ N, which depends onε and t, such that M(xm, xn, t) > 1 − ε for all m,n ≥ n0 or, equivalently,limm,nM(xm, xn, t) = 1 for all t > 0.(X,M, ∗), or simply M , is called complete if every Cauchy sequence in X is con-vergent in (X, τM ).2. Contractive sequencesLet (X, d) be a metric space. A sequence {xn} in X is called contractive if thereexists k ∈]0, 1[ such that d(xn+2, xn+1) ≤ k · d(xn+1, xn), n ∈ N.78Fuzzy contractive sequencesIt is well known that every contractive sequence is Cauchy and hence in a completemetric space every contractive sequence is convergent. Cauchy sequences, and socontractive sequences, are interesting because one can assert their convergence(in complete metric spaces) ignoring the point of convergence. Notice that insome cases to verify the contractive condition on a sequence it can be easier thanCauchyness’s condition.The concept of contractive sequence is strongly related with the theory of fixedpoint theorems initiated by the Banach Contraction Principle. Indeed, supposethat f is a self-contractive mapping (a contraction) of a complete metric space(X, d), that is there exists k ∈]0, 1[ such that d(f(x), f(y)) ≤ k · d(x, y). Then, foreach x0 ∈ X , the sequence of iterates {xn}, where xn = f(xn−1), n = 1, 2, . . . iscontractive and then {xn} is convergent to a point y ∈ X , which is the fixed pointfor f .The aim of this paper is to revise the results obtained on contractive sequences inour fuzzy setting.3. On fuzzy contractive sequencesIn order to obtain a fixed point theorem in fuzzy setting the authors gave thefollowing definition.Definition 3 ([5]). Let (X,M, ∗) be a GV-fuzzy metric space. A mapping f : X →X is called fuzzy contractive if there exists k ∈]0, 1[ such that 1M(f(x),f(y),t) − 1 ≤k(1M(x,y,t) − 1)for all x, y ∈ X and t > 0.Accordingly to this definition the authors also gave the following concept.Definition 4. A sequence {xn} in X is called fuzzy contractive if there existsk ∈]0, 1[ such that.(1)1M(f(xn+2), f(nn+1), t)− 1 ≤ k(1M(xn+1, xn, t)− 1)for all n ∈ N and t > 0.79V. Gregori and A. SapenaRadu [15] rewrote the notion of fuzzy contractive mapping in the equivalent form(2) M(f(x), f(y), t) ≥M(x, y, t)M(x, y, t) + k(1−M(x, y, t))which is more convenient that (1) because it remains valid in the context of KM−fuzzy metric spaces in which M(x, y, t) can take the value 0.Accordingly, a sequence {xn} is fuzzy contractive in a fuzzy metric spaces (X,M, ∗)if(3) M(xn+2, xn+1, t) ≥M(xn+1, xn, t)M(xn+1, xn, t) + k(1−M(xn+1, xn, t))The given concept of fuzzy contractive sequence is appropriate in the sense thatif (X, d) is a metric space then {xn} is contractive in (X, d) if and only if {xn}is fuzzy contractive in (X,Md, ·). Consequently, in (X,Md, ∗) a fuzzy contractivesequence is Cauchy. In order to prove that this assertion is true for any fuzzymetric space, the authors posed in [5] the following question, which is the subjectof this paper, stated in GV -fuzzy metric spaces.Question 5. Is a fuzzy contractive sequence a Cauchy sequence (in George andVeeramani’s sense)?A negative response to this question, but in the context of KM -fuzzy metricspaces was given by D.Mihet, which we reproduce, in a slight different way, in thefollowing example.Example 6 ([12]). Let X = [0,+∞[ and d(x, y) = |x − y|. Then (X, d) is acomplete metric space. DefineM(x, y, t) ={0 if t ≤ d(x, y)1 if t > d(x, y)Then (X,M, ∗) is a KM -fuzzy metric space for the t-norm minimum and con-sequently for every continuous t-norm, but clearly M is not a GV -fuzzy metric.Further, τM agrees with the topology on X deduced from d, and (X,M, ∗) iscomplete.80Fuzzy contractive sequencesNow, in this case it is easy to verify that a sequence {xn} is fuzzy contractive if andonly if M(xn+2, xn+1, t) ≥ M(xn+1, xn, t) for all n ∈ N and t > 0, or equivalentlyif and only if d(xn+2, xn+1) ≤ d(xn+1, xn, ) for all n ∈ N.Consider the sequence {xn} where xn = n for all n ∈ N. Then {xn} is fuzzycontractive since it fulfills the last condition, and obviously {xn} is not Cauchysince limm,nM(xn, xm,12 ) = 0.4. Affirmative partial responses to Question 5In the following we will see two recent partial affirmative responses to Question 5.Definition 7 ([13]). Let Ψ be the class of all mappings ψ :]0, 1]→]0, 1] such thatψ is continuous, non-decreasing and ψ(s) > s for all s ∈]0, 1]. Let (X,M, ∗) be afuzzy metric space and ψ ∈ Ψ. A mapping f : X → X is called ψ-contractive ifM(f(x), f(y), t) ≥ ψ(M(x, y, t)) for all x, y ∈ X and t > 0.A sequence {xn} in X is called fuzzy ψ-contractive if M(xn+2, xn+1, t) ≥ψ(M(xn+1, xn, t)) for all x, y ∈ X and t > 0.In order to obtain fixed point theorems in GV -fuzzy metric spaces, in [7] theauthors gave the following results.Proposition 8 ([7], Corollary 3.8). Let (X,M, ∗) be a GV -fuzzy metric space suchthat∧t>0M(x, y, t) > 0 for each x, y ∈ X. Then every fuzzy ψ-contractive sequenceis Cauchy.Proposition 9 ([7], Lemma 3.12). Let (C,M, ∗) be a strong GV -fuzzy metricspace. Then, every fuzzy ψ-contractive sequence is Cauchy.It is easy to observe, after seeing equation (3), that every fuzzy contractive se-quence is fuzzy ψ-contractive for ψ(s) =ss+ k(1− s). Consequently, the last twopropositions are affirmative partial responses to Question 5.81

Fuzzy contractive sequences

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Fuzzy contractive sequences

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