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Coupled fixed points for multivalued mappings in fuzzy metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 162 (2013)
Abstract
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.
1 Introduction
In 1969, Nadler [1] 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 [3] Ćirić has proved a fixed point theorem for the single-valued mappings satisfying some contractive condition. Samet and Vetro [4] 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 [13], 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 [16]).
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. [15], which is a fuzzy version of the result of [25]. Choudhury [16] further extended the result of [25] 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 [19]. 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. [4] in regular partially ordered metric spaces due to Ćirić [3]. Our results give a significant extension of some corresponding results. An example is also given to illustrate the suitability of our results.
2 Preliminaries
To set up our main results in the sequel, we recall some necessary definitions and preliminary concepts in this section.
Definition 2.1 [13]
The binary operation is called a continuous t-norm if the following properties are satisfied:
-
(T1)
for all ,
-
(T2)
whenever and for each ,
-
(T3)
∗ 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 [11]
The triplet is said to be a fuzzy metric space if X is an arbitrary nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on satisfying the following conditions for all , :
-
(F1)
,
-
(F2)
for all if and only if ,
-
(F3)
,
-
(F4)
and
-
(F5)
is continuous.
In this sense, is called a fuzzy metric on X.
Example 2.3 [11]
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.
Let be a fuzzy metric space. For and , the open ball with center is defined by
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 [12]). 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 [11]
Let be a fuzzy metric space.
-
(i)
A sequence in X is said to be convergent to a point if for any .
-
(ii)
A sequence in X is called a Cauchy sequence if for any and a positive integer p.
-
(iii)
A fuzzy metric space , in which every Cauchy sequence is convergent, is said to be complete.
By we denote the collection consisting of all nonempty closed subsets of X. For and , we define
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 . □
Definition 2.6 An element is a coupled fixed point of if
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 .
Let X be a nonempty set endowed with a partial order ⪯ and let be a given mapping. We recall the set
and the binary relation ℛ on defined by
Definition 2.7 Let be a given mapping. We say that F is a Δ-symmetric mapping if and only if
3 Main results
In order to prove our main results, we need the following hypothesis.
(H) The function defined by
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
-
(i)
the condition (H) holds and
-
(ii)
for any , if , then there exist and with
(2)
such that
where, , the function is nonincreasing, for and .
Then F admits a coupled fixed point on .
Proof If , then it is easy to see that is a coupled fixed point of F. Otherwise, and the definition of φ guarantees that for each . By virtue of the definition of , there exist and for any given such that
This implies that, for each , there exist and such that (2) holds. Thus, for arbitrarily given , either , then we get our desired result, or . In this case, by the condition (ii), we can choose and such that
From (4) and (5) it yields that
Note that F is a Δ-symmetric mapping. By Definition 2.7 we have , which further implies . If , then is a coupled fixed point of F and our desired result comes out. Otherwise, , which implies that . Again, the condition (ii) guarantees that there exist and such that
Hence, we obtain and
Continuing this process, we can choose the sequences such that either , which implies that is a coupled fixed point of F, or , which implies . In this case, we have
for and
It follows that the sequence is increasing from (8) and upper bounded from the definition of f. Therefore, this deduces is convergent, say,
We assert for . Suppose that this is not true, then there exists such that . Note that , we have for each since φ is nonincreasing. By means of this, taking the limit on both sides of (8) with and having in mind the assumptions of φ, we have
a contradiction. Thus for all .
We are in a position to verify that both and are Cauchy sequences in . Let
Note that is nondecreasing, we obtain that is nonincreasing. Hence, β exists. Since , the properties of φ guarantee that . Let the real number p with and be any fixed. For any with , there exists such that
On the other hand, we easily see that there exists such that
Substituting this inequality in (8), we have
For any , we inductively obtain
Let . By means of this, together with (7) and (9) with , one has
By virtue of , we have when is large enough. In addition, the hypothesis of ε implies that for . Consequently, we can choose that is large enough such that
which implies that, by the arbitrariness of ε,
In order to prove that and are Cauchy sequences, for any given , it is sufficient to check
for any and . Virtually, for any , if , then (11) is valid by (10) with . Inductively suppose that (11) is valid for , each and . Then, in view of the hypotheses (F4) and (T4), we have
Now, the inductive assumption shows that and the first inequality in (10) guarantees . Hence we have
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.
Finally, by means of the completeness of , there exists such that
In what follows, we prove that z is a coupled fixed point of F. Since f is upper semi-continuous, from we get
This implies ; moreover, we have
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
-
(i)
the (H) holds and
-
(ii)
for any , there exist and with and
(12)
such that
where for , the function is nonincreasing and satisfies that for and for each .
Then F admits a coupled fixed point on .
Proof If , then it is easy to see that and . In this case, is a coupled fixed point of F if and . Otherwise, and the definition of φ guarantees that for each . As an analogy of the argument of Theorem 3.1, we can construct the sequences such that either , which implies that is a coupled fixed point of F, or , which implies . In this case, we have
for and
Again, (16) guarantees that the sequence is increasing and upper bounded. Therefore, there exists such that
On the other hand, note that the sequence with is bounded, we have
for some . By virtue of (14), combining (1), we have
for any and , which implies that . Moreover, one has for all .
In what follows, we prove that . To this end, we first prove for all . Suppose, to the contrary, that there exists such that . The monotonicity of the sequence and yields that and for all . In addition, there exists the subsequence of , , such that . In view of the hypothesis of φ, we have . By means of this and the property of φ, taking the limit on both sides of the following inequality:
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.
Example 4.1 Let , for all and
Then is a fuzzy metric space (see [28]). Let X be endowed with the usual order ≤. Define the function by for any , where c is a positive constant. Then . Define the multivalued function by
Conclusion F admits a coupled fixed point in .
Proof It is not hard to see that X is complete and F is a Δ-symmetric mapping. Moreover,
Adding the above two formulae together, we obtain
f is obviously uniformly upper semi-continuous with respect to , i.e., (H) is satisfied. Let the function be defined by
For any , we take , , then , and
Therefore, u, v satisfy the inequality (2) and
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 . □
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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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Qiu, Z., Hong, S. Coupled fixed points for multivalued mappings in fuzzy metric spaces. Fixed Point Theory Appl 2013, 162 (2013). https://doi.org/10.1186/1687-1812-2013-162
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DOI: https://doi.org/10.1186/1687-1812-2013-162