Pure and Applied Mathematics Journal
Volume 4, Issue 4, August 2015, Pages: 159-163

Fixed-point Theorems in G-complete Fuzzy Metric Spaces

Naser Abbasi, Hamid Mottaghi Golshan, Mahmood Shakori

Department of Mathematics, Lorestan University, Khoramabad, Iran

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(N. Abbasi)
(H. M. Golshan)
(M. Shakori)

To cite this article:

Naser Abbasi,Hamid Mottaghi Golshan, Mahmood Shakori. Fixed-point Theorems in G-complete Fuzzy Metric Spaces. Pure and Applied Mathematics Journal. Vol. 4, No. 4, 2015, pp. 159-163. doi: 10.11648/j.pamj.20150404.14


Abstract: In the present paper we introduce generalized contraction mapping in fuzzy metric space and some fixed-point theorems for G-complete fuzzy metric space are proved. Our results generalize and extend many known results in metric spaces to a (non-Archimedean) fuzzy metric space in the in the sense of George and Veeramani [George A, Veeramani P, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 1994;64:395-9].

Keywords: Fuzzy Metric Spaces, Generalized Contraction Mapping, G-Complete


1. Introduction

Kramosil and Michalek [7] generalized the concept of probabilistic metric space and studied an interesting notion of fuzzy metric space with the help of continuous t-norm. Later on, in order to construct a Hausdorff topology on the fuzzy metric space, George and Veeramani [4] modied the concept of fuzzy metric space introduced by Grabiec [5] and Kramosil and Michalek [7]. Fixed point theory has important applications in diverse disciplines of mathematics, statistics, engineering and economics in dealing with problems arising in: approximation theory, potential theory, game theory, mathematical economics, etc. Fixed-point theory for contraction type mappings in fuzzy metric space is closely related to the fixed-point theory for the same type of mappings in probabilistic metric space of Menger type (see [9,13]). Many authors [3,5,6,8,11] have proved some fixed point theorems for various generalizations of contraction mappings in probabilistic and fuzzy metric spaces. In this paper we discuss generalized contraction mapping in fuzzy metric space in the sense George and Veeramani [4]. To de ne the fuzzy metric space we have to state several concepts as follows:

Definition 1.1 ([12]). A binary operation  is continuous t-norm if satisfies the following conditions:

(a)        is commutative and associative;

(b)        is continuous;

(c)        for all ;

(d)        whenever and and .

The t-norm minimum will be denoted by min, the usual product by. These two t-norms are the most commonly used in fuzzy logic.

Definition 1.2 ([4]). A fuzzy metric space is an ordered triple such that is a nonempty set,  is a continuous t-norm and is a fuzzy set of satisfying the following conditions, for all:

(FM1) ;

(FM2) if and only if;

(FM3) ;

(FM4) ;

(FM5) is continuous.

If, in the above definition, the triangular inequality (FM4) is replaced by (NAF)

,

then the triple  is called a non-Archimedean fuzzy metric space.

George and Veeramani [4] proved that every fuzzy metric space  on  generates a Hausdorff first countable topology on which has as a base the family of sets of the form, where for all.

Remark 1.3 ([4]). In fuzzy metric space, is non decreasing for all .

Definition 1.4 ([5]). A sequence inis said to be convergent to a point x in (denoted by), if ,for all .

Definition 1.5 ([4]). (a) A sequence  in a fuzzy metric space  is a Cauchy sequence if for each  and each  there exists such that for all.

(b) We say fuzzy metric space  is complete if every Cauchy sequence is convergent with respect to .

Remark 1.6 ([10]). Let  be a fuzzy metric space then is a continuous function on.

Example 1.7. Let be an ordinary metric space and  be a t-norm.

(1) (See [4]) Let be a fuzzy set on  define as follow:

        

for all for all. Then  is a fuzzy metric space and called induced fuzzy metric space. If in equation (1.1) we take then we have

.

This fuzzy metric space  is called standard fuzzy metric space.

(2) (See [11]) It is immediate to show that  is a non-Archimedean metric space if and only if  is a non-Archimedean fuzzy metric space.

Further examples and results for fuzzy metric spaces may be found in [4, 5, 7, 10, 12].

2. Main Results

Gregori and Sepena introduced the notions of fuzzy contraction mapping and fuzzy contractive sequence as follows:

Definition 2.1 ([6]). Let  be a fuzzy metric space. We call the the mapping  is fuzzy contractive mapping, if there exists  such that

for each and , ( is called the contractive constant of ).

Recall that a sequence  in a metric space  is said to be contractive if there exists such that, for all .

Definition 2.2 ([6]). Let  be a fuzzy metric space. A sequence is called fuzzy contractive if there exists  such that

for every .

Definition 2.3 ([6]). Let be a fuzzy metric space. A sequence is called  G-Cauchy if for each and, .

A fuzzy metric space in which every G-Cauchy sequence is convergent is called G-complete.

The following proposition is justified the above definitions.

Proposition 2.4 ([6]). (a) The sequence  in the metric space  is contractive iff is fuzzy contractive in the induced fuzzy metric space.

(b) The standard fuzzy metric space  is complete iff the metric space is complete.

(c) If sequence  is fuzzy contractive in  then it is G-Cauchy.

A continuous t-norm  is of Hadžić-type if there exists a strictly increasing sequence such that for all , is an example of such t-norm.

Lemma 2.5 ([8]). Each complete non-Archimedean fuzzy metric space with of Hadžić-type is G-complete.

Let we recall the generalized contractions mapping in metric spaces due to Ćirić [1].

Theorem 2.6. Let  be a complete metric space and be a self-mapping on such that for each Equation Chapter (Next) Section 2.

   

where  are functions from  into [0, 1) such that

Then  has a unique fixed point.

Mapping which satisfies called generalized contractions. As observed in [1], a self-mapping on a metric space is a generalized contraction if and only if satisfies the following condition:

      

where. Later on Ćirić take the term  instead of  and by a new method of proof deduce theorem 2.6 for following which he called it quasicontraction mapping

    

The following is quasi-contraction theorem for non-Archimedean fuzzy metric spaces:

Theorem 2.7. Let be a G-complete fuzzy metric space where the continuous t-norm  is defined as min and be self-mapping on such that for each

       

Proof. Let and  be arbitrary and consider a sequence of Pickard iterations, defined inductively by for each, we will show that is fuzzy contractive. From (2.4) by replacing we get

    

By our choice of t-norm  and triangular inequality for the last parenthesis in (2.5) we have

      

from (2.5) we get

hence,

where , this implies

    

so sequence is fuzzy contractive sequence. Since is a complete fuzzy metric space so by proposition 2.4 and lemma 2.5, sequence converges to  for some. Now we show is fixed point of, from (2.4) we have

   

taking the limit as we obtain

Since, we have, thus,, by (2.9) we find fixed point is unique.

Theorem 2.8. Let be a complete non-Archimedean fuzzy metric space where the continuous t-norm  is defined as min and  be a self-mapping on such that for each.

   

where and, Thenhas a unique fixed point.

Proof. The proof is very similar to the theorem 2.8. In stead of the equation (2.5) and (2.6) we find

and

respectively. Proceed as the proof of the Theorem 2.8 then we find sequence  is fuzzy contractive, Since  is a complete fuzzy metric space so by proposition 2.4 and lemma 2.5 there exists  such that . Instead of (2.8) we find

taking the limit as we obtain

Since, we have, thus,, by (2.9) we find fixed point is unique.

Remark 2.9. As observed in [2], by a similar proof we find the generalized contraction condition (2.4) and (2.9) are equivalent to following:

and

respectively, where.

Remark 2.10. For complete metric space (X, d), consider standard fuzzy metric space  in example 1.7 then by the following equality

        

and above remark we find contractions (2.2) and (2.4) are equivalent. Also by proposition 2.4-(a) and (b) and following the same line as Theorem 2.7, we automatically deduce fuzzy contractive sequence (2.7) is convergent, so theorem 2.6 is special case of Theorem 2.7. In next theorem the assumption * = min in theorem 2.7 is not needed.

Theorem 2.11. Let be a G-complete fuzzy metric space and be a self mapping on such that for each

  

where and . Then has a unique fixed point.

Proof. The assumption t-norm in theorem 2.7 is applied only in equation (2.6), so the proof is very similar to theorem 2.7 and it is omitted.

Corollary 2.12. Let be a complete standard fuzzy metric space and be a self-mapping on such that inequality (2.11) holds. Then has a unique fixed point.

Acknowledgements

This research was partially supported by Lorestan University, Khoramabad.


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