Coupled fixed points for multivalued mappings in fuzzy metric spaces
© Qiu and Hong; licensee Springer. 2013
Received: 11 January 2013
Accepted: 4 June 2013
Published: 25 June 2013
In this paper, we establish two coupled fixed point theorems for multivalued nonlinear contraction mappings in partially ordered fuzzy metric spaces. The theorems presented extend some corresponding results due to ordinary metric spaces. An example is given to illustrate the usability of our results.
In 1969, Nadler  extended the famous Banach contraction principle from single-valued mappings to multivalued mappings and proved the existence of fixed points for contractive multivalued mappings in complete metric spaces. Since then, the existence of fixed points for various multivalued contractive mappings has been studied by many authors under different conditions. For details, we refer to [2–10] and the references therein. For instance, in  Ćirić has proved a fixed point theorem for the single-valued mappings satisfying some contractive condition. Samet and Vetro  extended this result to multivalued mappings and proved the existence of a coupled fixed point theorem for the multivalued contraction.
One of the most important problems in fuzzy topology is to obtain an appropriate concept of fuzzy metric spaces. This problem has been investigated by many authors from different points of view. In particular, George and Veeramani [11, 12] introduced and studied the notion of fuzzy metric M on a set X with the help of continuous t-norms introduced in , and from now on, when we talk about fuzzy metrics, we refer to this type. Fuzzy metric spaces have many applications. In particular, on the fuzzy metric space, by using some topological properties induced by this kind of fuzzy metrics, there are several fixed point results established. Some instances of these works are in [14–23]. In fact, fuzzy fixed point results are more versatile than the regular metric fixed point results. This is due to the flexibility which the fuzzy concept inherently possesses. For example, the Banach contraction mapping principle has been extended in fuzzy metric spaces in two inequivalent ways in [17, 24]. Fuzzy fixed point theory has a developed literature and can be regarded as a subject in its own right (see ).
In recent times, the existence of common or coupled fixed points of a fuzzy version for multiple mappings has attracted much attention. We mention that the coupled fixed point results were proved by Sedghi et al. , which is a fuzzy version of the result of . Choudhury  further extended the result of  and provided the existence results of coupled coincidence points for compatible mappings in partially ordered fuzzy metric spaces. After that common coupled fixed point results in fuzzy metric spaces were established by Hu . However, to the best of our knowledge, few papers were devoted to fixed point problems of multivalued mappings in fuzzy metric spaces (see [26, 27]).
The aim of this paper is to present two new coupled fixed point theorems for two multivalued mappings in the fuzzy metric space. The idea of the present paper originates from the study of an analogous problem examined by Samet et al.  in regular partially ordered metric spaces due to Ćirić . Our results give a significant extension of some corresponding results. An example is also given to illustrate the suitability of our results.
To set up our main results in the sequel, we recall some necessary definitions and preliminary concepts in this section.
Definition 2.1 
for all ,
whenever and for each ,
∗ is continuous, associative and commutative.
In this sequel, we further assume that ∗ satisfies
(T4) for all .
For examples of t-norm satisfying the conditions (T1)-(T4), we enumerate , and for , respectively.
Definition 2.2 
for all if and only if ,
In this sense, is called a fuzzy metric on X.
Example 2.3 
Let X be the set of all real numbers and d be the Euclidean metric. Let for all . For each , , let . Then is a fuzzy metric space.
A subset is called open if for each , there exist and such that . Let denote the family of all open subsets of X. Then is a topology on X induced by the fuzzy metric . This topology is metrizable (see ). Therefore, a closed subset B of X is equivalent to if and only if there exists a sequence such that topologically converges to x. In fact, the topological convergence of sequences can be indicated by the fuzzy metric as follows.
Definition 2.4 
A sequence in X is said to be convergent to a point if for any .
A sequence in X is called a Cauchy sequence if for any and a positive integer p.
A fuzzy metric space , in which every Cauchy sequence is convergent, is said to be complete.
Lemma 2.5 If , if and only if for all .
Proof Since , there exists a sequence such that . Let , we have . From it follows that .
Conversely, if , we have for any . This implies that for all . □
At the end of this section, we introduce the following necessary notions.
The function is said to be uniformly upper semi-continuous with respect to on if implies that for and all .
3 Main results
In order to prove our main results, we need the following hypothesis.
with is uniformly upper semi-continuous with respect to on .
Theorem 3.1 Let be a complete fuzzy metric space with the partial order ⪯ and . Let be a Δ-symmetric mapping and satisfy that
the condition (H) holds and
- (ii)for any , if , then there exist and with(2)
where, , the function is nonincreasing, for and .
Then F admits a coupled fixed point on .
a contradiction. Thus for all .
Let , we have . By induction, we get that the first inequality in (11) is true. The proof of the second inequality in (11) is analogous.
From Lemma 2.5 it follows that and , that is, is a coupled fixed point of F. The proof is completed. □
Theorem 3.2 Let be a complete fuzzy metric space endowed with a partial order ⪯, , and let be a Δ-symmetric multivalued mapping satisfying that
the (H) holds and
- (ii)for any , there exist and with and(12)
where for , the function is nonincreasing and satisfies that for and for each .
Then F admits a coupled fixed point on .
for any and , which implies that . Moreover, one has for all .
we obtain , a contradiction. Hence, . If , repeating the above proceeding, we can obtain the contradiction of . Consequently, .
Finally, by , it is easy to see that the sequences and satisfy (10). Following the lines of the arguments of Theorem 3.1, we can obtain that and are Cauchy sequences. The rest of the proof is the same as that of Theorem 3.1. This completes our proof. □
4 An example
In this section, we conclude the paper with the following example.
Conclusion F admits a coupled fixed point in .
This yields that the inequality (3) is valid. (3) is obviously true if . Now Theorem 3.1 guarantees that F admits a coupled fixed point on . □
The authors wish to express their hearty thanks to the anonymous referees for their valuable suggestions and comments. Supported by Natural Science Foundation of Zhejiang Province (LY12A01002).
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