Abstract
We introduce some new contractions on intuitionistic fuzzy metric spaces, and give fixed point results for these classes of contractions. A stability result is established.
Keywords:
contractive mapping; fixed point; intuitionistic metric space1 Introduction and preliminaries
The great interest in the study of various fixed point theories for different classes of contractions on some specific spaces is known. We underline studies on quasimetric spaces [1,2], quasipartial metric spaces [3], convex metric spaces [4], cone metric spaces [57], partially ordered metric spaces [817], partial metric spaces [18], Menger spaces [19], Gmetric spaces [20,21], and fuzzy metric spaces [2225].
The concept of fuzzy set was introduced by Zadeh in 1965 [26]. Ten years later, Kramosil and Michalek introduced the notion of fuzzy metric spaces [24] and George and Veeramani modified the concept in 1994 [27]. Also, they defined the notion of Hausdorff topology in fuzzy metric spaces [27].
In 2004, Park introduced the notion of intuitionistic fuzzy metric space. In his elegant article [28], he showed that for each intuitionistic fuzzy metric space , the topology generated by the intuitionistic fuzzy metric coincides with the topology generated by the fuzzy metric M.
Actually, Park’s notion is useful in modeling some phenomena where it is necessary to study the relationship between two probability functions. Some authors have introduced and discussed several notions of intuitionistic fuzzy metric spaces in different ways (see, for example, [2931]. Grabiec obtained a fuzzy version of the Banach contraction principle in fuzzy metric spaces in Kramosil and Michalek’s sense [22], and since then many authors have proved fixed point theorems in fuzzy metric spaces [3235].
For necessary notions to our results, such as continuous tnorm, intuitionistic fuzzy metric space and the induced topology, which is denoted by , we refer the reader to [28] and [36].
A sequence in an intuitionistic fuzzy metric space is said to be Cauchy sequence whenever, for each and , there exists a natural number such that and for all .
The space is called complete whenever every Cauchy sequence is convergent with respect to the topology .
Let be an intuitionistic fuzzy metric space. According to [32], the fuzzy metric is called triangular whenever
and
We shall use the above background to develop our new results in this article. Our results are stated on complete triangular intuitionistic fuzzy metric spaces. In this framework, we introduce some new classes of contractive conditions and give fixed point results for them.
2 Main results
Now, we are ready to state and prove our main results.
Theorem 2.1Letbe a complete triangular intuitionistic fuzzy metric space, and letbe a continuous mapping satisfying the contractive condition
for all. ThenThas a fixed point.
If for some n, then we have nothing to prove.
for all n.
which is a contradiction. Thus, for all n, and so
But
Thus,
Therefore, is a Cauchy sequence and so there exists such that . Since T is continuous, and so . □
Theorem 2.2Letbe a complete triangular intuitionistic fuzzy metric space and letbe a selfmap which satisfies the contractive condition
for all. ThenThas a fixed point.
Proof Let . Define the sequence by for all n. Then
for all n and . Therefore, is a nonincreasing sequence and so it is convergent to some .
we obtain and so , which is a contradiction. Thus, .
Note that
for all n. Thus, for each , we get
Now, we consider . Since , it follows that . Hence, is a Cauchy sequence and so it converges to some .
We claim that is a fixed point of T.
Since
The following example shows that there are discontinuous mappings which satisfy the conditions of Theorem 2.2.
Example 2.1 Let endowed with the usual distance . Consider and for all and . Define the selfmap T on X by
It is easy to check that T satisfies the conditions of Theorem 2.2.
and so
Theorem 2.3Letbe a complete triangular intuitionistic fuzzy metric space, withand letbe a continuous mapping which satisfies the contractive condition
for all. ThenThas a unique fixed point in X.
If for some n, then we have nothing to prove.
and so
for all n.
By using the triangular inequality, for each , we obtain
where . Thus, is a Cauchy sequence, therefore it converges to some . Since t is continuous, it follows , hence is a fixed point of T.
Now, suppose that T has another fixed point . Then we have
which is a contradiction. Hence, T has a unique fixed point. □
We would like to prove that the iterative process utilized above is stable [4,37]. More accurately, we need this definition.
Definition 2.1 On an intuitionistic fuzzy metric space , consider T a selfmap on X, with a fixed point p. For , consider the Picard iteration, , which converges to p. Let be an arbitrary sequence in X. If
we say that the Picard iteration is Tstable.
Corollary 2.1Provided that the conditions of Theorem 2.3 are fulfilled, suppose thatpis the unique fixed point ofT. Then the Picard iteration isTstable.
Proof Indeed, using the triangular condition, we get
and so
Now, we have to interpret this relation in terms of real sequences. For this purpose, we need the following result, [38].
Lemma 2.1Let us considerto be a real number andto be a sequence of positive numbers such that. Ifis a sequence of positive real numbers such that, then.
Using Lemma 2.1 it follows that , and the corollary is proved. □
3 Conclusion
In this work, we introduced some classes of contractive conditions on intuitionistic fuzzy metric spaces endowed with triangular metric. With additional condition of completeness, we introduced new fixed point results for these classes of mappings. A stability result is established.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors completed the paper together. All authors read and approved the final manuscript.
Acknowledgements
The authors thank the referees for their remarks on the first version of our article.
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