This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

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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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M1">View MathML</a>, the topology generated by the intuitionistic fuzzy metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M2">View MathML</a> 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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M3">View MathML</a>, we refer the reader to [28] and [36].

A sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M4">View MathML</a> in an intuitionistic fuzzy metric space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M5">View MathML</a> is said to be Cauchy sequence whenever, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M6">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M7">View MathML</a>, there exists a natural number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M8">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M9">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M10">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M11">View MathML</a>.

The space <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M12">View MathML</a> is called complete whenever every Cauchy sequence is convergent with respect to the topology <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M3">View MathML</a>.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M12">View MathML</a> be an intuitionistic fuzzy metric space. According to [32], the fuzzy metric <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M2">View MathML</a> is called triangular whenever

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M16">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M17">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M18">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M7">View MathML</a>.

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 2.1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M12">View MathML</a>be a complete triangular intuitionistic fuzzy metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M21">View MathML</a>and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M22">View MathML</a>be a continuous mapping satisfying the contractive condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M23">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M24">View MathML</a>. ThenThas a fixed point.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M25">View MathML</a>. Put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M26">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M27">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M28">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M29">View MathML</a> for some n, then we have nothing to prove.

Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M30">View MathML</a> for all n. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M31">View MathML</a>

for all n.

Now, for each n, put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M32">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M33">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M34">View MathML</a>

which is a contradiction. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M35">View MathML</a> for all n, and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M36">View MathML</a>

But

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M37">View MathML</a>

and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M38">View MathML</a> for all n.

Thus,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M39">View MathML</a>

Hence, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M40">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M41">View MathML</a>

Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M42">View MathML</a> is a Cauchy sequence and so there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M43">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M44">View MathML</a>. Since T is continuous, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M45">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M46">View MathML</a>. □

Theorem 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M12">View MathML</a>be a complete triangular intuitionistic fuzzy metric space and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M48">View MathML</a>be a selfmap which satisfies the contractive condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M49">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M24">View MathML</a>. ThenThas a fixed point.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M25">View MathML</a>. Define the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M42">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M53">View MathML</a> for all n. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M54">View MathML</a>

for all n and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M7">View MathML</a>. Therefore, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M56">View MathML</a> is a non-increasing sequence and so it is convergent to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M57">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M58">View MathML</a>, then by putting

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M59">View MathML</a>

we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M60">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M61">View MathML</a>, which is a contradiction. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M62">View MathML</a>.

Note that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M63">View MathML</a>

for all n. Thus, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M64">View MathML</a>, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M65">View MathML</a>

Now, we consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M66">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M67">View MathML</a>, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M68">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M69">View MathML</a> is a Cauchy sequence and so it converges to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M43">View MathML</a>.

We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M71">View MathML</a> is a fixed point of T.

Since

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M72">View MathML</a>

for all n, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M73">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M74">View MathML</a>. □

The following example shows that there are discontinuous mappings which satisfy the conditions of Theorem 2.2.

Example 2.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M75">View MathML</a> endowed with the usual distance <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M76">View MathML</a>. Consider <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M77">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M78">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M79">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M80">View MathML</a>. Define the selfmap T on X by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M81">View MathML</a>

It is easy to check that T satisfies the conditions of Theorem 2.2.

In fact, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M82">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M83">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M84">View MathML</a>

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M85">View MathML</a>

Theorem 2.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M12">View MathML</a>be a complete triangular intuitionistic fuzzy metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M87">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M88">View MathML</a>and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M48">View MathML</a>be a continuous mapping which satisfies the contractive condition

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M90">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M24">View MathML</a>. ThenThas a unique fixed point in X.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M25">View MathML</a>. Put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M26">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M27">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M28">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M29">View MathML</a> for some n, then we have nothing to prove.

Assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M30">View MathML</a> for all n. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M98">View MathML</a>

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M99">View MathML</a>

for all n.

By using the triangular inequality, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M100">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M101">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M102">View MathML</a>. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M69">View MathML</a> is a Cauchy sequence, therefore it converges to some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M43">View MathML</a>. Since t is continuous, it follows <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M74">View MathML</a>, hence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M71">View MathML</a> is a fixed point of T.

Now, suppose that T has another fixed point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M107">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M108">View MathML</a>

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 <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M109">View MathML</a>, consider T a selfmap on X, with a fixed point p. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M110">View MathML</a>, consider the Picard iteration, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M111">View MathML</a>, which converges to p. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M112">View MathML</a> be an arbitrary sequence in X. If

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M113">View MathML</a>

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

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M114">View MathML</a>

and so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M115">View MathML</a>

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 consider<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M116">View MathML</a>to be a real number and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M117">View MathML</a>to be a sequence of positive numbers such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M118">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M119">View MathML</a>is a sequence of positive real numbers such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M120">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M121">View MathML</a>.

Using Lemma 2.1 it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M122">View MathML</a>, 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.

References

  1. Caristi, J: Fixed point theorems for mapping satisfying inwardness conditions. Trans. Am. Math. Soc.. 215, 241–251 (1976)

  2. Hicks, TL: Fixed point theorems for quasi-metric spaces. Math. Jpn.. 33, 231–236 (1988)

  3. Shatanawi, W, Pitea, A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl.. 2013, Article ID 153 (2013)

  4. Olatinwo, MO, Postolache, M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput.. 218(12), 6727–6732 (2012). Publisher Full Text OpenURL

  5. Altun, I, Durmaz, G: Some fixed point results in cone metric spaces. Rend. Circ. Mat. Palermo. 58, 319–325 (2009). Publisher Full Text OpenURL

  6. Shatanawi, W: Some coincidence point results in cone metric spaces. Math. Comput. Model.. 55, 2023–2028 (2012). Publisher Full Text OpenURL

  7. Shatanawi, W: On w-compatible mappings and common coincidence point in cone metric spaces. Appl. Math. Lett.. 25, 925–931 (2012). Publisher Full Text OpenURL

  8. Agarwal, RP, El-Gebeily, MA, O’Regan, D: Generalized contractions in partially ordered metric spaces. Appl. Anal.. 87, 109–116 (2008). Publisher Full Text OpenURL

  9. Altun, I: Some fixed point theorems for single and multivalued mappings on ordered non-Archimedean fuzzy metric spaces. Iranian J. Fuzzy Syst.. 7(1), 91–96 (2010)

  10. Aydi, H, Shatanawi, W, Postolache, M, Mustafa, Z, Tahat, N: Theorems for Boyd-Wong type contractions in ordered metric spaces. Abstr. Appl. Anal.. 2012, Article ID 359054 (2012)

  11. Aydi, H, Karapınar, E, Postolache, M: Tripled coincidence point theorems for weak φ-contractions in partially ordered metric spaces. Fixed Point Theory Appl.. 2012, Article ID 44 (2012)

  12. Berinde, V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal.. 74, 7347–7355 (2011). Publisher Full Text OpenURL

  13. Chandok, S, Postolache, M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl.. 2013, Article ID 28 (2013)

  14. Rezapour, Sh, Amiri, P: Some fixed point results for multivalued operators in generalized metric spaces. Comput. Math. Appl.. 61, 2661–2666 (2011). Publisher Full Text OpenURL

  15. Shatanawi, W, Postolache, M: Common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces. Fixed Point Theory Appl.. 2013, Article ID 60 (2013)

  16. Shatanawi, W, Samet, B: On <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M123">View MathML</a>-weakly contractive condition in partially ordered metric spaces. Comput. Math. Appl.. 62(8), 3204–3214 (2011). Publisher Full Text OpenURL

  17. Zhilong, L: Fixed point theorems in partially ordered complete metric spaces. Math. Comput. Model.. 54, 69–72 (2011). Publisher Full Text OpenURL

  18. Shatanawi, W, Postolache, M: Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces. Fixed Point Theory Appl.. 2013, Article ID 54 (2013)

  19. Menger, K: Statistical metrics. Proc. Natl. Acad. Sci. USA. 28, 535–537 (1942). PubMed Abstract | Publisher Full Text | PubMed Central Full Text OpenURL

  20. Aydi, H, Postolache, M, Shatanawi, W: Coupled fixed point results for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/168/mathml/M125">View MathML</a>-weakly contractive mappings in ordered G-metric spaces. Comput. Math. Appl.. 63(1), 298–309 (2012). Publisher Full Text OpenURL

  21. Shatanawi, W, Postolache, M: Some fixed point results for a G-weak contraction in G-metric spaces. Abstr. Appl. Anal.. 2012, Article ID 815870 (2012)

  22. Grabiec, M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst.. 27, 385–389 (1988). Publisher Full Text OpenURL

  23. Gregori, V, Sapena, A: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst.. 125, 245–252 (2002). Publisher Full Text OpenURL

  24. Kramosil, O, Michalek, J: Fuzzy metric and statistical metric spaces. Kybernetica. 11, 326–334 (1975)

  25. Rafi, M, Noorani, MSM: Fixed point theorem on intuitionistic fuzzy metric spaces. Iranian J. Fuzzy Syst.. 3(1), 23–29 (2006)

  26. Zadeh, LA: Fuzzy sets. Inf. Control. 8, 338–353 (1965). Publisher Full Text OpenURL

  27. George, A, Veeramani, P: On some results in fuzzy metric spaces. Fuzzy Sets Syst.. 64, 395–399 (1994). Publisher Full Text OpenURL

  28. Park, JH: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals. 22, 1039–1046 (2004). Publisher Full Text OpenURL

  29. Alaca, C, Turkoghlu, D, Yildiz, C: Fixed points in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals. 29, 1073–1078 (2006). Publisher Full Text OpenURL

  30. Atanassov, K: Intuitionistic fuzzy sets. Fuzzy Sets Syst.. 20, 87–96 (1986). Publisher Full Text OpenURL

  31. Coker, D: An introduction to intuitionistic fuzzy metric spaces. Fuzzy Sets Syst.. 88, 81–89 (1997). Publisher Full Text OpenURL

  32. Di Bari, C, Vetro, C: A fixed point theorem for a family of mappings in a fuzzy metric space. Rend. Circ. Mat. Palermo. 52, 315–321 (2003). Publisher Full Text OpenURL

  33. Karayilan, H, Telci, M: Common fixed point theorem for contractive type mappings in fuzzy metric spaces. Rend. Circ. Mat. Palermo. 60, 145–152 (2011). Publisher Full Text OpenURL

  34. Miheţ, D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst.. 144, 431–439 (2004). Publisher Full Text OpenURL

  35. Park, JS, Kwun, YC, Park, JH: A fixed point theorem in the intuitionistic fuzzy metric spaces. Far East J. Math. Sci.. 16, 137–149 (2005)

  36. Schweizer, B, Sklar, A: Statistical metric spaces. Pac. J. Math.. 10, 314–334 (1960)

  37. Haghi, RH, Postolache, M, Rezapour, S: On T-stability of the Picard iteration for generalized φ-contraction mappings. Abstr. Appl. Anal.. 2012, Article ID 658971 (2012)

  38. Berinde, V: On stability of some fixed point procedures. Bul. Stiint. Univ. Baia Mare Ser. B Fasc. Mat.-Inform.. 18, 7–14 (2002)