Abstract
In this paper, we introduce a notion of α-continuity of fuzzy mappings and some generalized contractive conditions for α-level sets. Then we prove some theorems on the existence of common α-fuzzy fixed points for a pair of fuzzy mappings. Consequently, we obtain some results on metric spaces endowed with binary relations, and graphs. Further, using α-fuzzy fixed point techniques we obtain common fixed point results for multi-valued mappings on metric spaces.
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1 Introduction
In 1965, Zadeh firstly introduced and studied the notion of fuzzy set in his seminal paper [1], which opened an avenue for further development of analysis in this field. Afterward, it was developed extensively by many researchers, which also included many interesting applications of this theory in different fields such as neural network theory, stability theory, mathematical programming, modeling theory, engineering sciences, medical sciences (medical genetics, nervous systems), image processing, control theory, communication, etc. In 1981, Heilpern [2] proved the fuzzy Banach contraction principle for fuzzy contractive mappings on a complete linear metric space provided with -metric for fuzzy sets. This result is an improvement and a generalization of the well known Nadler contraction principle [3]. Further, Frigon and O’Regan [4] extended Heilpern’s result under a contractive condition for 1-level sets of a fuzzy contraction on a complete metric space, where 1-level sets are not assumed to be convex and compact. In 2009, Azam and Beg [5] proved existence theorems of common fixed points for a pair of fuzzy mappings under Edelstein, Alber and Guerr-Delabriere’s type contractive conditions in a linear metric space. Later, Azam et al. [6] showed existence theorems of fixed points for fuzzy mappings satisfying Edelstein locally contractive conditions on a compact metric space provided with the -metric for fuzzy sets. Besides, there are many results about fixed points of fuzzy mappings with different contractive contractions.
In 2013, Azam and Beg [7] established a common α-fuzzy fixed point result for a pair of fuzzy mappings on a complete metric space under a generalized contractivity condition for α-level sets via Hausdorff metrics for fuzzy sets, which generalized the results proved by Azam and Arshad [8], Bose and Sahani [9] and Vijayaraju and Marudai [10], among others. Recently, Phiangsungnoen et al. [11] extended the Azam and Beg’s main results [7] by using the concept of -admissible pairs which is a generalization of the notion of β-admissible pairs for multi-valued mappings due to Mohammadi et al. [12]. For the supremum metric spaces and fixed points of fuzzy mappings, see also [13–16]. Other notions of fixed point in the fuzzy ambient can be found in [17, 18] (see also [1, 10, 15, 16, 19–31]).
In this work, we introduce the notion of α-continuity of fuzzy mappings and we present some generalized contractive conditions for α-level sets via Hausdorff metrics for fuzzy sets. We establish common α-fuzzy fixed point theorems under such conditions and we also show some consequences of our results on metric spaces endowed with an arbitrary binary relation and on metric spaces endowed with graphs. Finally, we use α-fuzzy fixed point techniques to deduce common fixed point results for multi-valued mappings. Our results improve, extend, and generalize many results existing in the literature.
2 Preliminaries
Throughout this paper, we denote by X a nonempty set and stands for the collection of all nonempty subsets of X. A fuzzy set on X is a function with domain X and values in . If is a fuzzy set and , then the function-value is called the grade of membership of x in . Given , the α-level set of is the set . If X is endowed with a topology, then the 0-level set of is , where denotes the closure of .
In the sequel, assume that is a metric space. For a point x in X and a nonempty subset , the distance from x to A is
It is clear that , and if, and only if, .
Let denote the collection of all fuzzy sets in X (provided with the metric topology) and let be the family of nonempty closed bounded subsets of . We denote by the fuzzy set , where is the characteristic function of the crisp set . Notice that there exists an injective mapping that associates to each , and that lets us see as a subset of .
For , we define the Hausdorff distance between A and B by
which is symmetric in A and B. It is well known that is a metric space.
Definition 1 Let X be a nonempty set and Y be a metric space. A mapping T is said to be a fuzzy mapping if T is a mapping from X into .
Remark 2 The function-value is the grade of membership of y in Tx.
Definition 3 Let be a metric space, let and let S and T be fuzzy mappings from X into . A point z in X is called an α-fuzzy fixed point of T if . The point z is called a common α-fuzzy fixed point of S and T if . When , it is called a common fixed point of S and T.
Remark 4 Notice that the notion of common α-fuzzy fixed point of S and T depends on the level sets and . In this sense, it would be more appropriate to call it a common α-level set-valued fuzzy fixed point of S and T. However, in order not to complicate the notation, we follow the original notation introduced in [11].
Lemma 5 Let be a metric space and , then for each and all
Proposition 6 If in a metric space and , then .
Proof It follows from the fact that, for all and , . Taking the infimum on , it follows that for all . Now, letting , we conclude that . □
Lemma 7 (Nadler [3])
Let be a metric space and , then for each , , there exists an element such that
Definition 8 ([19])
Let X be a nonempty set and let and be two mappings. We say that T is β-admissible if
Definition 9 ([32])
Let X be a nonempty set and let and be two mappings. We say that T is -admissible if
where
Definition 10 ([12])
Let X be a nonempty set and let and be two mappings. We say that T is β-admissible whenever for each and with , we have for all .
Definition 11 ([11])
Let be a metric space and let , and be four mappings. The ordered pair is said to be -admissible if it satisfies the following conditions:
-
(i)
for each and , with , we have for all ;
-
(ii)
for each and , with , we have for all .
If then T is called -admissible.
It is easy to see that if is -admissible, then is also -admissible.
Definition 12 ([11])
Let be a metric space, and . The ordered pair is said to be β-admissible if it satisfies the following conditions:
-
(i)
for each , with , we have for all ;
-
(ii)
for each , with , we have for all .
Remark 13 It is easy to prove that is β-admissible if, and only if, is β-admissible. If , then G is called β-admissible (this notion was introduced by Mohammadi et al. in [12]).
We will use the following property.
Lemma 14 Let be a sequence in a metric space and assume that there exists such that
Then is a Cauchy sequence.
Proof By induction, it follows from (2) that
Therefore, if verify , then
d’Alembert’s ratio test for series of real numbers guarantees that the series and are convergent. As a consequence, and is a Cauchy sequence. □
3 Common α-fuzzy fixed point theorems
In [11], Phiangsungnoen et al. proved the following theorem.
Theorem 15 ([11])
Let be a complete metric space and let , and be four mappings such that the following properties are fulfilled.
-
(a)
For all , we have .
-
(b)
There exist such that and either or , verifying
(3)
for all .
-
(c)
is a -admissible pair.
-
(d)
There exist and such that .
-
(e)
If and is a sequence in X such that and for all , then for all .
Then there exists such that , that is, there exists a point which is an -fuzzy fixed point of T and S.
In this section, we introduce the notion of α-continuous fuzzy mapping and some generalized contractive conditions for α-level sets via control functions and the Hausdorff metric for fuzzy sets. Also, we give existence theorems of common α-fuzzy fixed points for a pair of fuzzy mappings satisfying such conditions.
Definition 16 A mapping is an α-continuous fuzzy mapping if, for all sequences such that , we see that .
Let denote by Φ the family of all functions such that there exist verifying and
for all . Examples of functions in Φ are
Next we present the main result of this paper.
Theorem 17 Let be a complete metric space and let , and be four mappings such that the following properties are fulfilled.
-
(a)
For , we have .
-
(b)
There exists verifying, for ,
(5) -
(c)
is a -admissible pair.
-
(d)
There exist and such that .
-
(e)
At least one of the following properties holds.
-
(e.1)
T and S are α-continuous fuzzy mappings.
-
(e.2)
If and is a sequence in X such that and for all , then for all .
-
(e.1)
Then there exists such that , that is, there exists a point which is an -fuzzy fixed point of T and S.
Proof Since , there exist verifying and (4). Let us define
Condition implies that
If , then ϕ only takes the value zero. Therefore, as , condition (5) yields
This implies that and thus
Since and is -admissible, we get . From this inequality and (5), we get
This implies that
So and the proof is finished. Next, assume that , that is, . Let be the sequence of positive real numbers given by
Starting from the points and such that , and using repeatedly Lemma 7, we can determine successive points in X verifying the following properties:
By induction, we can construct a sequence in X verifying, for all ,
Since the pair is -admissible, then
By induction, it follows that
This implies that
In such a case, using (4), (5), (6), and (8), we have, for all ,
Therefore,
If the maximum in (10) is , then
and if the maximum in (10) is , then
In any case, as , we deduce, combining (11) and (12), that for all ,
Next, we analyze the other indices. Using a similar reasoning,
Therefore,
If the maximum in (14) is , then
and if the maximum in (14) is , then
In any case, as , we deduce, combining (15) and (16), that for all ,
And combining (13) and (17), we conclude that
By Lemma 14, is a Cauchy sequence in . As the metric space X is complete, there exists such that . We will show that z verifies distinguishing the cases of hypothesis (d).
Case (e.1). Assume that T and S are α-continuous fuzzy mappings. In such a case,
By (9) and Lemma 5,
Letting in the previous inequality, we deduce that
which is only possible when . Furthermore, as
we also deduce that , so .
Case (e.2). Assume that if and is a sequence in X such that and for all , then for all . In this case, we have for all , so
Let apply Lemma 5 and the contractive condition (5) to obtain
Letting , by Proposition 6 we have
Taking into account that , the previous inequality leads to , that is, . Similarly, using
it is possible to prove that , and the proof is finished. □
Example 18 Let be endowed with the Euclidean metric for all . Clearly, is a complete metric space. Let be three real numbers such that . Let consider the mappings , and be given by
The following facts are easy to check.
-
For all ,
As a result, and are nonempty, closed, bounded subsets of .
-
The pair is a -admissible pair. To prove it, let and be such that . Hence and . As a result, if , then . In the other hand, let and be such that . Therefore, and . If , then .
-
If we take and , then .
-
We claim that T and S are α-continuous fuzzy mappings. Indeed, let be a sequence such that . Hence
Similarly, as
for all , then
-
Let us show that T and S satisfy the contractivity condition (5) using the function given by
for all . Indeed, let be such that . It follows that . Then
As , then
-
The function β does not satisfy condition (e) in Theorem 15. Indeed, if for all , then and for all . However, for all .
As a consequence of the last bullet item, Theorem 15 is not applicable to T and S. However, Theorem 17 guarantees that there exists which is an -fuzzy fixed point of T and S (in this case, ).
Notice that, as the previous example illustrates, one of the main advantages of the contractivity condition (5) versus (3) is that we only have to prove it for pairs such that , but not necessarily for all .
In the following example, we use similar arguments but involving nonlinear mappings.
Example 19 Let X be the real interval endowed with the Euclidean metric for all . Given five real numbers such that and , let consider the mappings , , and given by
The following properties hold.
-
For all , and , which are nonempty, closed, bounded subsets of .
-
The pair is a -admissible pair. To prove it, let and be such that . Hence and . If , then because . In the other hand, let and be such that . Therefore, and . If , then .
-
If we take and , then .
-
We claim that T and S are α-continuous fuzzy mappings. Indeed, let be a sequence such that . Hence and for all . As a result,
-
Let us show that . Recall that
(19)
Hence, for all such that ,
As , then .
-
We claim that T and S satisfy the contractivity condition (5) using the function . Indeed, let be such that . It follows that . Notice that
(20)
and also
Then, by (19), (20), and (21),
-
The function β does not satisfy condition (e) in Theorem 15. Indeed, if for all , then and for all . However, for all .
As a consequence of the last bullet item, Theorem 15 is not applicable to T and S. However, Theorem 17 guarantees that there exists which is an -fuzzy fixed point of T and S (in this case, ).
The following consequence is another way to interpret the contractivity condition that can be useful.
Corollary 20 Let be a complete metric space and let , and be four mappings such that the following properties are fulfilled.
-
(a)
For , we have .
-
(b)
There exists verifying
(22)
for all .
-
(c)
is a -admissible pair.
-
(d)
There exist and such that .
-
(e)
At least one of the following properties holds.
-
(e.1)
T and S are α-continuous fuzzy mappings.
-
(e.2)
If and is a sequence in X such that and for all , then for all .
-
(e.1)
Then there exists such that , that is, there exists a point which is an -fuzzy fixed point of T and S.
Proof It is easy to see that condition (22) implies condition (5). Indeed, if are such that , then, by hypothesis (b), we obtain
Therefore, all hypotheses of Theorem 20 are satisfied, and the desired result follows immediately from this theorem. □
Corollary 21 If in assumption of Theorem 15, then Theorem 15 follows from Corollary 20.
Proof Letting and , we have
and, for all such that ,
Using for all , we conclude that condition (3) implies (5). □
Remark 22 If we have supposed , then Theorem 15 and Corollary 20 (case (e.2)) would have been equivalent because, in such a case, condition (5) also implies (3). Indeed, if we take and , then
Therefore, the best thing to do to take advantage of condition (5) is that we only suppose
In the following results, we present several contractivity conditions that can be reduced to (3).
Corollary 23 Let be a complete metric space and let , and be four mappings such that the following properties are fulfilled.
-
(a)
For , we have .
-
(b)
There exist and verifying
(23)
for all .
-
(c)
is a -admissible pair.
-
(d)
There exist and such that .
-
(e)
At least one of the following properties holds.
-
(e.1)
T and S are α-continuous fuzzy mappings.
-
(e.2)
If and is a sequence in X such that and for all , then for all .
-
(e.1)
Then there exists such that , that is, there exists a point which is an -fuzzy fixed point of T and S.
Proof We will show that condition (23) implies condition (5) in Theorem 17. Suppose that are such that
By using (23), we get
This implies that
Therefore, condition (5) in Theorem 17 holds. By Theorem 17, we get the desired result. □
Corollary 24 Let be a complete metric space and let , and be four mappings such that the following properties are fulfilled.
-
(a)
For , we have .
-
(b)
There exist and verifying
(24)
for all .
-
(c)
is a -admissible pair.
-
(d)
There exist and such that .
-
(e)
At least one of the following properties holds.
-
(e.1)
T and S are α-continuous fuzzy mappings.
-
(e.2)
If and is a sequence in X such that and for all , then for all .
-
(e.1)
Then there exists such that , that is, there exists a point which is an -fuzzy fixed point of T and S.
Proof It is easy to see that condition (24) implies condition (5) in Theorem 17. Indeed, if are such that
by using (24), we have
As the exponential function is strictly increasing when , we get
This shows that condition (5) in Theorem 17 holds. By Theorem 17, we get the desired result. □
4 Consequences
In this section, we present some consequences of our main results applied to very different contexts: metric spaces endowed with arbitrary binary relations, metric spaces endowed with graphs.
4.1 α-Fuzzy fixed point theorems on metric spaces endowed with arbitrary binary relations
In this section, we present α-fuzzy fixed point theorems on metric spaces endowed with arbitrary binary relations. The following notions and definitions are needed.
Let be a metric space and ℛ be a binary relation over X. Let denote
that is, is the symmetric relation on X such that, for all ,
Next we introduce the notion of -comparative pair of two fuzzy mappings.
Definition 25 Let ℛ be a binary relation over metric space and let and be three mappings. The ordered pair is said to be -comparative if satisfies the following conditions:
-
(i)
for each and such that , we have for all ;
-
(ii)
for each and such that , we have for all .
Here we show a α-fuzzy fixed point theorem for -comparative pair on metric spaces endowed with a binary relation.
Theorem 26 Let be a complete metric space, ℛ be a binary relation over X and let and be three mappings such that the following properties are fulfilled.
-
(A)
For , we have .
-
(B)
There exists verifying
(25)
for all for which .
-
(C)
is a -comparative pair.
-
(D)
There exist and such that .
-
(E)
At least one of the following properties holds.
(E.1) T and S are α-continuous fuzzy mappings.
(E.2) If and is a sequence in X such that and for all , then for all .
Then there exists such that , that is, there exists a point which is an -fuzzy fixed point of T and S.
Proof Consider the mapping defined by
By using (25), for all , we get
This implies that condition (5) in Theorem 17 holds using mapping β. Since is a -comparative pair, it is also a -admissible pair. From (D) and definition of β, we find that there exist and such that . Furthermore, it is easy to see that condition (E.2) implies condition (e.2). Therefore, all hypotheses of Theorem 17 are satisfied. As a consequence, we can find a point which is an -fuzzy fixed point of T and S. This completes the proof. □
4.2 α-Fuzzy fixed point theorems on metric spaces endowed with graph
In this section, we study existence of α-fuzzy fixed points on a metric space endowed with graph. To do that, let be a metric space. The subset is called the diagonal of the Cartesian product and it is denoted by Δ. Consider a graph G such that the set of its vertices coincides with X and the set of its edges contains all loops, i.e., . We assume G has no parallel edges, so we can identify G with the pair . Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices.
Next we introduce the notion of -edge pair of fuzzy mappings.
Definition 27 Let be a metric space endowed with a graph G and let and be three mappings. The ordered pair is said to be -edge if it satisfies the following conditions:
-
(i)
for each and , with , we have for all ;
-
(ii)
for each and , with , we have for all .
Now we state and prove an α-fuzzy fixed point theorem for -edge pairs on metric spaces endowed with graphs.
Theorem 28 Let be a complete metric space endowed with a graph G and let and be three mappings such that the following properties are fulfilled.
-
(A)
For , we have .
-
(B)
There exists verifying
(26)
for all for which .
-
(C)
is a -edge pair.
-
(D)
There exist and such that .
-
(E)
At least one of the following properties holds.
(E.1) T and S are α-continuous fuzzy mappings.
(E.2) If and is a sequence in X such that and for all , then for all .
Then there exists such that , that is, there exists a point which is an -fuzzy fixed point of T and S.
Proof This proof is similar to the proof of Theorem 26 by considering the mapping defined by
□
5 Application to a common fixed point for multi-valued mappings
In this section, we study some relationships between multi-valued mappings and fuzzy mappings. Here, using the concept of β-admissible pair of multi-valued mappings due to Phiangsungnoen et al. [11] (recall Definition 12), we indicate that Theorem 17 can also be employed to derive some common fixed point results for multi-valued mapping.
Theorem 29 Let be a complete metric space and let and be three mappings such that the following properties are fulfilled.
(⋆1) There exists verifying for ,
(⋆2) is a β-admissible pair.
(⋆3) There exist and such that .
(⋆4) At least one of the following properties holds.
(⋆4.1) F and G are continuous mappings.
(⋆4.2) If and is a sequence in X such that and for all , then for all .
Then there exists such that , that is, F and G have a common fixed point.
Proof Let be an arbitrary mapping. Consider two fuzzy mappings defined by
and
By definition of S and T, we get
and
Hence condition (27) turns into condition (5) in Theorem 17. Also, we find that the other conditions in Theorem 17 hold. Therefore, Theorem 17 can be applied to obtain such that
that is, u is a common fixed point of F and G. This completes the proof. □
Remark 30 Theorem 29 improves Theorem 4.3 of Phiangsungnoen et al. [11]. Also, Theorem 29 extends and generalizes Corollary 7 of Azam and Beg in [7]. Moreover, Theorem 29 is a complementary result of the famous Nadler contraction mapping principle [3].
By using Corollaries 20, 23, and 24, we get the following results.
Corollary 31 Let be a complete metric space and let and be three mappings such that the following properties are fulfilled.
(⋆1) There exists verifying
for all .
(⋆2) is a β-admissible pair.
(⋆3) There exist and such that .
(⋆4) At least one of the following properties holds.
(⋆4.1) F and G are continuous mappings.
(⋆4.2) If and is a sequence in X such that and for all , then for all .
Then there exists such that , that is, F and G have a common fixed point.
Corollary 32 Let be a complete metric space and let and be three mappings such that the following properties are fulfilled.
(⋆1) There exist and verifying
for all .
(⋆2) is a β-admissible pair.
(⋆3) There exist and such that .
(⋆4) At least one of the following properties holds.
(⋆4.1) F and G are continuous mappings.
(⋆4.2) If and is a sequence in X such that and for all , then for all .
Then there exists such that , that is, F and G have a common fixed point.
Corollary 33 Let be a complete metric space and let and be three mappings such that the following properties are fulfilled.
(⋆1) There exist and verifying
for all .
(⋆2) is a β-admissible pair.
(⋆3) There exist and such that .
(⋆4) At least one of the following properties holds.
(⋆4.1) F and G are continuous mappings.
(⋆4.2) If and is a sequence in X such that and for all , then for all .
Then there exists such that , that is, F and G have a common fixed point.
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Acknowledgements
The authors thank the referees and the editor for their insightful comments that helped to improve this manuscript. The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.
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Latif, A., Roldán-López-de-Hierro, AF. & Sintunaravat, W. On common α-fuzzy fixed points with applications. Fixed Point Theory Appl 2014, 234 (2014). https://doi.org/10.1186/1687-1812-2014-234
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DOI: https://doi.org/10.1186/1687-1812-2014-234