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Fixed points of some new contractions on intuitionistic fuzzy metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 168 (2013)
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.
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 2.1 Let be a complete triangular intuitionistic fuzzy metric space, and let be a continuous mapping satisfying the contractive condition
for all . Then T has 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 2.2 Let be a complete triangular intuitionistic fuzzy metric space and let be a selfmap which satisfies the contractive condition
for all . Then T has 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 2.3 Let be a complete triangular intuitionistic fuzzy metric space, with and let be a continuous mapping which satisfies the contractive condition
for all . Then T has 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 2.1 Provided that the conditions of Theorem 2.3 are fulfilled, suppose that p is the unique fixed point of T. Then the Picard iteration is T-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 2.1 Let us consider to be a real number and to be a sequence of positive numbers such that . If is 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.
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Ionescu, C., Rezapour, S. & Samei, M.E. Fixed points of some new contractions on intuitionistic fuzzy metric spaces. Fixed Point Theory Appl 2013, 168 (2013). https://doi.org/10.1186/1687-1812-2013-168
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DOI: https://doi.org/10.1186/1687-1812-2013-168