Common Fixed-Point Theorems in G-complete Fuzzy Metric Spaces
Naser Abbasi^{*}, Mahmood Shakori, Hamid Mottaghi Golshan
Department of Mathematics, Lorestan University, Khoramabad, Iran
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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 ﬁxed 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