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# Fixed points of some new contractions on intuitionistic fuzzy metric spaces

Cristiana Ionescu1*, Shahram Rezapour2 and Mohamad Esmaeil Samei2

Author Affiliations

1 Department of Mathematics, Azarbaijan University of Shahid Madani, Azarshahr, Tabriz, Iran

2 Faculty of Applied Sciences, University Politehnica of Bucharest, 313 Splaiul Independenţei, Bucharest, 060042, Romania

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Fixed Point Theory and Applications 2013, 2013:168  doi:10.1186/1687-1812-2013-168

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/168

 Received: 21 March 2013 Accepted: 3 June 2013 Published: 26 June 2013

© 2013 Ionescu et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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 space

### 1 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 quasi-metric spaces [1,2], quasi-partial metric spaces [3], convex metric spaces [4], cone metric spaces [5-7], partially ordered metric spaces [8-17], partial metric spaces [18], Menger spaces [19], G-metric spaces [20,21], and fuzzy metric spaces [22-25].

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, [29-31]. 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 [32-35].

For necessary notions to our results, such as continuous t-norm, 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

for all 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.

Proof Let . Put and for all .

If for some n, then we have nothing to prove.

Assume that for all n. Then

for all n.

Now, for each n, put .

If , then

which is a contradiction. Thus, for all n, and so

But

and for all n.

Thus,

Hence, for each , we obtain

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 non-increasing sequence and so it is convergent to some .

If , then by putting

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

for all n, we get and so . □

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.

In fact, for and , we have

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.

Proof Let . Put and for all .

If for some n, then we have nothing to prove.

Assume that for all n. Then

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 T-stable.

Corollary 2.1Provided that the conditions of Theorem 2.3 are fulfilled, suppose thatpis the unique fixed point ofT. Then the Picard iteration isT-stable.

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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