Abstract and Applied Analysis

Volume 2012 (2012), Article ID 832807, 7 pages

http://dx.doi.org/10.1155/2012/832807

## Set-Valued Fixed-Point Theorems for Generalized Contractive Mappings on Fuzzy Metric Spaces

^{1}Department of Mathematics and Statistics, Faculty of Science, Taif University, El-Haweiah, P.O. Box 888, Taif 21974, Saudi Arabia^{2}Department of Mathematics, Faculty of Science, Menofiya University, Shebin Elkom 32511, Egypt^{3}Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey^{4}Institute of Space Sciences, P.O. Box MG-23, 76900 Magurele-Bucharest, Romania

Received 6 February 2012; Accepted 17 March 2012

Academic Editor: Simeon Reich

Copyright © 2012 S. K. Elagan and Dumitru Baleanu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The purpose of this paper is to introduce new types of asymptotically ()-contractions which generalize the Binayak S. Choudhury type contraction on fuzzy metric spaces and prove some fixed-point theorems for single- and multivalued mappings on fuzzy metric spaces. Hence, our results can be viewed as a generalization and improvement of many recent results.

#### 1. Introduction and Preliminaries

The concept of fuzzy metric space was introduced in different ways by some authors (see, i.e., [1, 2]) and further,the fixed-point theory in this kind of spaces has been intensively studied (see [3–5]). Contraction mappings in probabilistic and fuzzy metric spaces have considered by many authors. Singh and Chauhan were the first to introduce contraction mapping principle in probabilistic metric space [6]. The result has been known as Seghal contraction. The structures of these spaces allow to extend the contraction mapping principle to these spaces in more than one inequivalent ways. One such concept is -contraction which was originally introduced by O. Hadžic in [7] and subsequently studied and generalized in several works like [2, 8, 9]. In [7] O. Hadžic introduced the notion of a -contraction in probabilistic metric space. In [10], Golet introduced the concept of -contraction where is a bijective function to generalize the Hicks-type contraction. Some other works may be noted in [4, 11–20].

In this paper we establish two coincidence point results for three mappings. For this purpose we consider fuzzy -contraction and fuzzy asymptotically -contraction, and prove several important fixed-point theorems for single- and multivalued mappings.

*Definition 1.1 (see [18]). *A binary operation is said to be a continuous -norm if it is satisfies the following conditions: (a) is associative and commutative;(b) is continuous;(c) for all ;(d) whenever and for each .

In [8], Kramosil and Michálek gave the following definition of fuzzy metric space.

*Definition 1.2. *The 3-tuple is said to be a fuzzy metric space if is an arbitrary set, is a continuous -norm, and is a fuzzy set on , satisfying the following conditions. For all and ,(M1);(M2) if and only if ;(M3);(M4);(M5) is left-continuous.

*Example 1.3 (see [21]). *Let be a metric space. Define (or ) and for all and ,
Then is a fuzzy metric space. We call this fuzzy metric induced by the metric the standard fuzzy metric. On the other hand, note that there exists no metric on satisfying (1.1).

Lemma 1.4. *Let be a continuous -norm according to Definition 1.1. Then the condition
**
holds if and only if
*

*Proof. *Suppose (1.2) holds. Let such that . Then
Therefore, for all , we have . Suppose (1.3) holds. Then clearly
and so (1.2) holds.

*Definition 1.5 (see [8]). *Let be a fuzzy metric space: a sequence in is said to be convergent to a point (denoted ) if

*Definition 1.6 (see [21]). * Let be a fuzzy metric space: a sequence in is called a Cauchy sequence if and only if for any , there exists such that
for all .

*Definition 1.7 (see [8]). *Let be a fuzzy metric space: a sequence in is called a Cauchy sequence if

*Definition 1.8. *Let and be two fuzzy metric spaces. A mapping is said to be (uniformly) continuous if for each , , there exist , such that
for each .

*Definition 1.9. *Let be a fuzzy metric space and let be a self-mapping on . The mapping is called asymptotically regular at if

#### 2. Main Results

Let be the class of all mappings with the following properties:(i) is strictly increasing;(ii) is right-continuous;(iii) for all .

Lemma 2.1. *For all , where is the -iteration of .*

* Proof. *Suppose that for some . By the monotonicity and right continuity of , we have
which is a contradiction.

*Definition 2.2. *Let be a fuzzy metric space and . We say that the mapping is a fuzzy -contraction if there exists a bijective function such that for every , the following implication holds:

Note that, if for , then (2.2) is a -contraction in the sense of Golet [10]. On the other hand, if is an identity function, then (2.2) is called --contraction due to Mihet [9]. Thus, our definition -contraction is a generalization of the Golet and Mihet's type contraction principle in the fuzzy settings.

*Definition 2.3. *Let be a fuzzy metric space and let . Let , and are three mappings defined on with values into itself and let one take as asymptotically regular at . Then is called fuzzy asymptotically -contraction with respect to if
Note that, if for , then (2.3) is asymptotically -contraction with respect to in the sense of Binayak S. Choudhury.

Lemma 2.4. *Let satisfy the condition given in Definition 2.2. Then is a continuous mapping on with values into itself.*

*Proof. *Let be a sequence in such that in under the fuzzy metric ; this implies that as for all . By definition, it follows that
as for all , which implies that
as for all , which implies that,
as for all . This shows that is continuous mapping on with values into itself.

Theorem 2.5. *Let be three mappings defined on complete fuzzy metric space with values into itself where is bijective, is asymptotically regular at , and is fuzzy asymptotically -contraction with respect to with at , and is continuous in . Then .*

*Proof. *Since is fuzzy asymptotically -contraction with respect to , we get for ,
where , this implies that
which implies that
where ; this implies that
which implies that
which implies that
which implies that
Continuing this process, we get
For every and there exists such that whenever . Thus, we have
which implies that . Let be the sequence defined as . Now, taking and , then we get from the above inequality
for every . This implies that is a fuzzy Cauchy sequence in . Since is complete fuzzy metric space, then also is complete fuzzy metric space, there exists such that () under . Again, is continuous on , it follows that , which implies that .

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