Pure and Applied Mathematics Journal
Volume 4, Issue 6, December 2015, Pages: 255-258

Common Fixed-Point Theorems in G-complete Fuzzy Metric Spaces

Naser Abbasi*, Mahmood Shakori, Hamid Mottaghi Golshan

Department of Mathematics, Lorestan University, Khoramabad, Iran

Email address:

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

To cite this article:

Naser Abbasi, Mahmood Shakori, Hamid Mottaghi Golshan. Common Fixed-Point Theorems in G-complete Fuzzy Metric Spaces.Pure and Applied Mathematics Journal.Vol.4, No. 6, 2015, pp. 255-258. doi: 10.11648/j.pamj.20150406.15


Abstract: Following the approach of Gregori and Sapena, in this paper we introduced a new class of contractions and we establish some common fixed point theorems in G-complete fuzzy metric. Also a theorem on the equivalency related to completeness is given. The results are a genuine generalization of the corresponding results of Gregori and Sapena.

Keywords: G-complete, Fuzzy Metric Spaces, Common Fixed Point


1. Introduction

Several authors [1,4,5,10] have proved fixed point theorems for contractions in fuzzy metric spaces, using one of the two di erent types of completeness: in the sense of Grabiec [4], or in the sense of Schweizer and Sklar [3,9]. Gregori and Sapena [5] introduced a new class of fuzzy contraction mappings and proved several fixed point theorems in fuzzy metric spaces. Gregori and Sapena's results extend classical Banach fixed point theorem and can be considered as a fuzzy version of Banach contraction theorem. In this paper, following the results of [5] we give a new common fixed point theorem in the two different types of completeness and by using the recent definition of contractive mapping of Gregori and Sapena [5] in fuzzy metric spaces.

Recall [9] that a continuous t-norm is a binary operation such that  is an ordered Abelian topological monoid with unit 1. The two important t-norms, the minimum and the usual product, will be denoted by min and, respectively.

Definition 1.1 ([3]). 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.

Remark 1.2 ([3]). In fuzzy metric space, is non-decreasing for all.

Definition 1.3 ([4]). A sequence inis said to be convergent to a point x in (denoted by), if ,for all .

Definition 1.4 ([3,5]). Let be a fuzzy metric space.

a)  A sequence  is called G-Cauchy if for each and,. The fuzzy metric space  is called G-complete if every G-Cauchy sequence is convergent.

b)  A sequence  is called Cauchy sequence if for each  and each  there exists such that or all. The fuzzy metric space  is called complete if every Cauchy sequence is convergent.

Remark 1.5 ([7]). Let  be a fuzzy metric space then is a continuous function on.

2. Main Results

In this section, we extend common fixed point theorem of generalized contraction mapping in fuzzy metric spaces, our work is closely related to [1,2,5]. Gregori and Sepena introduced the notions of fuzzy contraction mapping and fuzzy contraction sequence as follows:

Definition 2.1 ([5]). Let be a fuzzy metric space.

a)  We call the mapping  is fuzzy contractive mapping, if there exists  such that

for each and .

b)  Let  be a fuzzy metric space. A sequence is called fuzzy contractive if there exists  such that

for every .

Theorem 2.2. Let be a G-complete fuzzy metric space endowed with minimum t-norm and  be a family of self-mappings of. If there exists a fixed such that for each

(2.1)

for some  and for each. Then all  have a unique common fixed point and at this point each  is continuous.

Proof. Let, and be arbitrary. Consider a sequence, defined inductivelyfor all. From (2.1) we get

(2.2)

Since

(2.3)

we have

hence, as  we get

Similarly, we get that

So is fuzzy contractive, thus, by proposition [5, Proposition 2.4] is G-Cauchy. Since X is G-complete,  converges to .. for some. From (2.1) we have

Taking the limit as infinity we obtain

Thus, hence, .

Now we show that u is a xed point of all . Let α J. From (2.1) we have

Hence, since  is arbitrary all have a common point.

Suppose that is also a fixed point of. Similarly, as above, is a common fixed point of all. Form (2.1) we get

Thus  is a unique common fixed point of all. It remains to show each  is continuous at. Let  be a sequence in X such that as. From (2.1) we have

and by

we deduce

So, as, for all . Thus  is continuous at a fixed point.

Theorem 2.3. Let  be a G-complete fuzzy metric space endowed with minimum t-norm. The following property is equivalent to completeness of:

If is any non-empty closed subset of and is any generalized contraction mapping then  has a fixed point in .

Proof. The sufficient condition follows from Theorem 2.2. Suppose now that the property holds, but  is not complete. Then there exists a Chuchy sequence in which does not converge. We may assume that  for all and for some. For any define

Clearly for all we have , as  has not a convergent subsequence. Let . We choose a subsequence  of  as follows. We define inductively a subsequence of positive integer greater than and such that

for all. This can done, as  is a Chuchy sequence.

Now define for all . Then for any we have

Thus is a general contraction mapping on . Clearly,  is closed and  has not a fixed point in . Thus we get the contraction.

Theorem 2.4. Let be a complete non-Archimedean fuzzy metric space endowed with minimum t-norm and be a family of self-mappings of. If there exists a fixed such that for each

for some  and for each. Then all  have a unique common fixed point and at this point each  is continuous.

Proof. The proof is very similar to the Theorem 2.2. Instead of the equation (2.3) we have

Proceed as the proof of the Theorem 2.2 then we conclude sequence is fuzzy contractive, thus by [5, Proposition 2.4] and [6, Lemma 2.5], converges to u for some. Proceed as the proof of the Theorem 2.2.

3. Conclusion

Motivated by a celebrated result of Gregori and Sapena we introduced a new class of contractions having common fixed points on every G-complete fuzzy metric space and then we prove a new result for completeness.


References

  1. N. Abbasi, H. Mottaghi Golshan, M. Shakori, Fixed-point theorems in G-complete fuzzy metric spaces. Pure and applied mathematics journal, 4 (4) (2015) 159-163.
  2. L.J. Ciric, On a family of contractive maps and fixed points. Publ. Inst. Math. (Beograd) (N.S.) 17(31) (1974), 45-51.
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