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Common fixed point theorem for a hybrid pair of mappings in Hausdorff fuzzy metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 225 (2012)
Abstract
In this paper, we prove a coupled fixed point theorem for a multivalued fuzzy contraction mapping in complete Hausdorff fuzzy metric spaces. As an application of the first theorem, a coupled coincidence and coupled common fixed point theorem has been proved for a hybrid pair of multivalued and single-valued mappings. It is worth mentioning that to find coupled coincidence points, we do not employ the condition of continuity of any mapping involved therein. Also, coupled coincidence points are obtained without exploiting any type of commutativity condition. Our results extend, improve, and unify some well-known results in the literature.
MSC:47H10, 47H04, 47H07.
1 Introduction and preliminaries
Bhaskar and Lakshmikantham [1] introduced the concept of a coupled fixed point of a mapping F from to X and established some coupled fixed point theorems in partially ordered sets. Later on some authors gave improved and generalized results in this context. For details, we refer to [2, 3].
The concept of fuzzy sets was initiated by Zadeh [4] in 1965. Fuzzy metric spaces were introduced by Kramosil and Michalek [5]. George and Veeramani [6, 7] modified the notion of fuzzy metric spaces by using continuous t-norm and generalized the concept of a probabilistic metric space to a fuzzy situation. Then a number of authors started the study of fixed point theory in fuzzy metric spaces; for a detailed survey, we refer to [8–18] and the references therein. Recently López and Romaguera [19] introduced a Hausdorff fuzzy metric on a set of nonempty compact subsets of a given fuzzy metric space. In 2011, Kiany et al. [20] proved fixed point and endpoint theorems for set-valued fuzzy contraction maps in fuzzy metric spaces.
Recently Abbas [21] introduced the concept of coupled fixed points of a mapping (a collection of all nonempty subsets of X) and coupled coincidence points of a hybrid pair F and . The aim of this paper is to obtain a coupled fixed point theorem for F and a coupled coincidence and coupled common fixed point theorem for a hybrid pair which satisfies a contractive condition in complete Hausdorff fuzzy metric spaces. It is to be noted that to find coupled coincidence points, we do not employ the condition of commutativity and continuity of any mapping involved therein. Our results unify, extend, and generalize various known comparable results given in existing literature (see, for example, [20] and some references therein).
Definition 1 [22]
A binary operation is called a continuous t-norm if
-
(1)
∗ is associative and commutative;
-
(2)
∗ is continuous;
-
(3)
for all ;
-
(4)
whenever and .
Definition 2 [6]
Let X be a nonempty set and ∗ be a continuous t-norm. If a mapping satisfies the following conditions:
(F1) ;
(F2) if and only if ;
(F3) ;
(F4) ;
(F5) is continuous;
for each and , then 3-tuple is called a fuzzy metric space.
Example 3 [6]
Let be a metric space. Define and
for all and . Then is a fuzzy metric space. We call this a fuzzy metric M, the standard fuzzy metric induced by d.
Definition 4 [6]
Let be a fuzzy metric space.
-
(i)
A sequence is said to be convergent to a point if for all .
-
(ii)
A sequence is said to be a Cauchy sequence if for all .
-
(iii)
A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.
-
(iv)
A subset is said to be closed if for each convergent sequence with and , we have .
-
(v)
A subset is said to be compact if each sequence in A has a convergent subsequence. The set of all compact subsets of X will be denoted by .
Lemma 5 [10]
For all , is nondecreasing.
Definition 6 Let be a fuzzy metric space, M is said to be continuous on if
whenever is a sequence in which converges to a point ; that is,
Lemma 7 [10]
M is a continuous function on .
Kiany et al. [20] introduced the following lemma in fuzzy metric spaces.
Lemma 8 [20]
Let be a fuzzy metric space satisfying
for every , , and . Suppose is a sequence in X satisfying
for all and . Then is a Cauchy sequence.
Lemma 9 [19]
Let be a fuzzy metric space. Then, for each , , and , there is a such that , where
Definition 10 [19]
Let be a fuzzy metric space. For each and , set
The 3-tuple is called a Hausdorff fuzzy metric space.
Lemma 11 [23]
Let X be a nonempty set and be a mapping. Then there exists a subset such that and is one-to-one.
Theorem 12 [20]
Let be a complete fuzzy metric. Suppose is a multivalued mapping such that
for each and , where satisfying , for all , and . Furthermore, assume that satisfies (1) for some and . Then F has a fixed point.
We also need the following definitions given in [21].
Definition 13 [21]
Let X be a nonempty set, (a collection of all nonempty subsets of X) and . An element is called
(C1) a coupled fixed point of F if and ;
(C2) a coupled coincidence point of a hybrid pair if and ;
(C3) a coupled common fixed point of a hybrid pair if and .
We denote the set of coupled coincidence points of mappings F and g by . Note that if , then is also in .
Definition 14 [21]
Let be a multivalued mapping and g be a self-map on X. The hybrid pair is called w-compatible if whenever .
Definition 15 [21]
Let be a multivalued mapping and g be a self-mapping on X. The mapping g is called F-weakly commuting at some point if and .
2 Coupled fixed and coincidence point theorems
In the following theorem, we obtain a coupled fixed point for a multivalued mapping satisfying a contractive condition.
Theorem 16 Let be a complete fuzzy metric space and let be a set-valued mapping satisfying
for each , . Suppose that and is a mapping satisfying
for all . Furthermore, assume that satisfies (1) for some and . Then F has a coupled fixed point.
Proof Let be arbitrary. Choose and . Since F is compact valued, then by Lemma 9 there exists such that
Since F is compact valued, there exists such that
Continuing this process, we obtain a sequence and in X such that and satisfying
That is, we obtain
Similarly,
Inequalities (3) and (4) show that the sequences and are nondecreasing. Thus, and are nonnegative nonincreasing and so they are convergent, say, to and . Since, by the given assumption,
then there exist , , and such that
Since is nondecreasing, then (3), (4), and (6) yield
Then we obtain
Hence, by Lemma 8, and are Cauchy sequences. Since is a complete fuzzy metric space, then there exist x and y in X such that and , we have
Thus
Since
then there exist , such that , , and such that
Now, we show that and . Consider
On taking limit as , we obtain
So that
Similarly, we can obtain
Since and , from (7) and (8) we obtain
There exist sequences and such that
for each . Now, for each , we have
On taking limit as , we get
Similarly, for each , we have
On taking limit as , we get
-
(9)
and (10) imply
Since and are compact, we get and . □
Corollary 17 Let be a complete fuzzy metric space and let be a mapping satisfying
for each , , and . Suppose that satisfies (1) for some and . Then F has a coupled fixed point.
Corollary 18 Let be a complete fuzzy metric space and let be a set-valued mapping satisfying
for each , . Suppose that and is a mapping satisfying
for all . Furthermore, assume that satisfies (1) for some and . Then F has a coupled fixed point.
Corollary 19 Let be a complete fuzzy metric space and let be a mapping satisfying
for each , , and . Suppose that satisfies (1) for some and . Then F has a coupled fixed point.
Example 20 Let and be defined as
Hence, is a metric space. Consider
for and satisfies (1). Define as follows:
For and and for and , we have
for any . Hence, (11) is satisfied. For the rest of cases,
and
Hence, for all , , and , (11) holds. All the conditions of Corollary 17 and Theorem 16 with are satisfied. Moreover, and are coupled fixed points of F.
Example 21 Let be endowed with the usual metric and for each . Let and let for each . Then we have
for each and . Then by Corollary 18, F has a coupled fixed point ( is a coupled fixed point of F).
Example 22 Let be endowed with the usual metric and let for each . Let and let for each . Then we have
for each and . Then by Corollary 19, f has a coupled fixed point ( is a coupled fixed point of f).
Now, as an application of the above theorem, we obtain a coupled coincidence and common fixed point theorem for a hybrid pair of multivalued and single-valued mappings.
Theorem 23 Let be a complete fuzzy metric space and let and be mappings satisfying
for each , . Suppose that and is a mapping satisfying
for all . Furthermore, assume that and satisfies (1) for some and . Then F and g have a coupled coincidence point. Moreover, F and g have a coupled common fixed point if one of the following conditions holds:
-
(a)
F and g are w-compatible, and for some , , and g is continuous at u and v.
-
(b)
g is F-weakly commuting for some , and gx and gy are fixed points of g, that is, and .
-
(c)
g is continuous at x, y for some and for some , and .
Proof By Lemma 11 there exists such that is one-to-one and . Now, define a mapping by
and since g is one-one on E, so is well defined. Further,
Hence, satisfies (2) and all the conditions of Theorem 16. By using Theorem 16 with a mapping , it follows that has a coupled fixed point . Finally, it is left to prove that F and g have a coupled coincidence point. Since has a coupled fixed point , we get
Since , so there exist such that and . Thus, it follows from (15),
This implies that is a coupled coincidence point of F and g. Hence, is nonempty. Suppose now that (a) holds. Then, for some ,
where . Since g is continuous at u and v, we have that u and v are fixed points of g. As F and g are w-compatible, so
That is, for all ,
Using (12), we obtain
On taking limit as , we get
This implies . Similarly, . Consequently, and . Hence, is a coupled common fixed point of F and g. Suppose now that (b) holds. If for some , g is F-commuting and and , then
Hence, is a coupled common fixed point of F and g. Suppose now that (c) holds and assume that for some and for some , and . By the continuity of g at x and y, we get
Hence, is a coupled common fixed point of F and g. □
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Acknowledgements
The authors would like to thank the editor and anonymous reviewers for their helpful comments that helped to improve the presentation of this paper. The third author was partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran and by a grant from IPM (No. 91470412).
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Abbas, M., Ali, B. & Amini-Harandi, A. Common fixed point theorem for a hybrid pair of mappings in Hausdorff fuzzy metric spaces. Fixed Point Theory Appl 2012, 225 (2012). https://doi.org/10.1186/1687-1812-2012-225
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DOI: https://doi.org/10.1186/1687-1812-2012-225