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Fixed point theorems for complex non-self mappings satisfying an implicit relation in Kaleva-Seikkala’s type fuzzy metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 254 (2013)
Abstract
In this paper, by using -type real functions, some fixed point theorems for complex non-self mappings satisfying an implicit relation in two fuzzy metric spaces in the sense of Kaleva and Seikkala are established. As applications, we obtain the corresponding fixed point theorems for complex non-self mappings satisfying an implicit relation in two Menger probabilistic metric spaces. Also, an example, which shows the validity of the hypotheses of our main results, is given.
MSC:47H10, 46S40, 54E70.
1 Introduction
In 1984, Kaleva and Seikkala [1] introduced the concept of a fuzzy metric space by setting the distance between two points to be a nonnegative fuzzy real number. From then on, some important results for mappings in fuzzy metric spaces, such as variational principles, coincidence theorems and various fixed point theorems, etc., were stated in subsequent work (see [2–7], etc.). It is well known that the Kaleva-Seikkala’s type fuzzy metric space is an important generalization of a Menger probabilistic metric space and a metric space as well (see [1, 4, 8]) and possesses rich structure with suitable choices of binary operations. Recently, Huang and Wu [9] investigated the completion of the Kaleva-Seikkala’ type fuzzy metric space. Previous work on this question is based on the t-norm min and the t-conorm max. Their meaningful work does away with this restriction. Shortly afterwards, Xiao et al. [10] proved some fixed point theorems of self-mapping for nonlinear contraction in a complete fuzzy metric space. Their main results do away with the restriction of the t-conorm max and are based on a generic class of binary operations.
On the other hand, fixed point theorems and their applications for complex non-self mappings in two metric spaces were considered by Fisher [11] in 1981. Telci [12] generalized the fixed point theorems by introducing the notion of the real functions ℱ. Recently, Aliouche and Fisher [13] proved two new fixed point theorems for complex non-self mappings by introducing the notion of the real functions ℱ satisfying an implicit relation in two metric spaces, which is a generalization of the Telci fixed point theorem in [12].
Inspired by the work of [9, 10, 13], in this paper we discuss the unique existence of fixed points for complex non-self mappings in two fuzzy metric spaces in the sense of Kaleva and Seikkala. In Sections 2-3, we first introduce the new real function class satisfying an implicit relation. Then, by using -type real functions, some fixed point theorems for complex non-self mappings satisfying an implicit relation in fuzzy metric spaces are established. Our main results do away with the restriction of the t-conorm max and are based on a generic class of binary operations. As some immediate consequences of this theorems, we obtain Theorem 3.4 and Theorem 3.5. It is a generalization of the main results in [13]. In Section 4, as applications, we obtain the corresponding fixed point theorems in Menger probabilistic metric spaces. Also, an example, which shows the validity of the hypotheses of our main results, is given.
2 Basic concepts and lemmas
Throughout this paper, let ℕ be the set of all positive integers, and . For the details of a fuzzy real number, we refer the reader to Kaleva and Seikkala [1], Dubois and Prade [14], Bag and Samanta [15].
Definition 2.1 (Dubois and Prade [14])
A mapping is called a fuzzy real number or a fuzzy interval, whose α-level set is denoted by , if it satisfies two axioms:
-
(1)
There exists such that ;
-
(2)
is a closed interval of ℝ for each , where .
Let us denote the set of all such fuzzy real numbers by G. If and whenever , then u is called a nonnegative fuzzy real number, and by we mean the set of all nonnegative fuzzy real numbers. If and are admissible, then, for the sake of clarity, u is called a generalized fuzzy real number. The sets of all generalized fuzzy real numbers or all generalized nonnegative fuzzy real numbers are denoted by and , respectively. In that case, if , for instance, then means the interval . Since each can be considered as a fuzzy real number defined by
ℝ can be embedded in G, and .
Lemma 2.1 (Xiao and Zhu [16])
Let , , and . Then
-
(1)
;
-
(2)
is a left-continuous and non-increasing function for ;
-
(3)
is a left-continuous and non-increasing function for .
Definition 2.2 (Kaleva and Seikkala [1])
Let X be a non-empty set, d be a mapping from into , and let the mappings be symmetric, nondecreasing in both arguments and satisfy and . For and , define the mapping
The quadruple is called a fuzzy metric space (briefly, FMS), and d is called a fuzzy metric on X if
-
(FM-1)
if and only if ;
-
(FM-2)
for all ;
-
(FM-3)
for all :
-
(FM-3L)
, whenever , and ;
-
(FM-3R)
, whenever , and .
-
(FM-3L)
If d is a mapping from into and satisfies (FM-1)-(FM-3), then is called a generalized fuzzy metric space (briefly, GFMS).
From Lemma 2.1 and Definition 2.2, we obtain the following consequences.
Lemma 2.2 Let be a FMS, for , where are any two fixed elements. Then
-
(1)
;
-
(2)
is a left-continuous and non-increasing function for ;
-
(3)
is a left-continuous and non-increasing function for .
Lemma 2.3 (Xiao and Zhu [17])
Let be a FMS, and suppose that
(R-1) ;
(R-2) for each , there exists such that for all ;
(R-3) .
Then (R-1) ⇒ (R-2) ⇒ (R-3).
Remark 2.1 (Xiao et al. [10, 17])
Since satisfies (R-2) and does not satisfy (R-1), (R-2) does not imply (R-1). Since satisfies (R-3) and does not satisfy (R-2), (R-3) does not imply (R-2).
Lemma 2.4 Let be a FMS. Then
-
(1)
(R-1) ⇒ for each , for all (cf. [9]);
-
(2)
(R-2) ⇒ for each , there exists such that for all (cf. [10]);
-
(3)
(R-3) ⇒ for each , there exists such that for all (cf. [16]).
Lemma 2.5 (Kaleva and Seikkala [1])
Let be a FMS with (R-3). Then the family of sets forms a basis for a Hausdorff uniformity on . Moreover, the sets
form a basis for a Hausdorff topology on X and this topology is metrizable.
According to Lemma 2.5, convergence in a FMS can be defined by sequences.
Definition 2.3 (Kaleva and Seikkala [1])
Let be a FMS.
-
(1)
A sequence in X is said to be convergent to x (we write or ) if , i.e., for all ;
-
(2)
A sequence in X is said to be a Cauchy sequence in X if , equivalently, for any given and , there exists such that , whenever ;
-
(3)
is said to be complete if each Cauchy sequence in X is convergent to some point in X.
Lemma 2.6 (Xiao et al. [10])
Let be a FMS with (R-2). Then, for each , is continuous at .
Definition 2.4 Let be a function and let denote the n th iteration of .
-
(1)
φ is said to satisfy condition if it is nondecreasing, right-continuous and for all .
-
(2)
φ is said to satisfy condition if it is nondecreasing, right-continuous and for all .
Remark 2.2 Obviously, if satisfies condition , then for all , i.e., . Since satisfies condition and does not satisfy condition , does not imply , i.e., . Also, if satisfies condition , then .
Definition 2.5 A function is called a real function satisfying an implicit relation if the following conditions are satisfied:
(A-1) F is right-lower semi-continuous;
(A-2) There exists or such that or for implies .
We denote by the collection of all real functions satisfying an implicit relation.
The following examples show that the is a largish class of real functions.
Example 1 Let or . We define the functions as follows:
respectively. Then .
In fact, since or , by the right-continuity of φ, we know that satisfies condition (A-1). If or , then we have for . This implies , and so . In addition, if , then . This shows that satisfies condition (A-2). Hence .
Similarly, we can prove .
Example 2 Let or . We define the functions , respectively, as follows:
where and . Then .
Obviously, satisfies condition (A-1).
Now suppose that or . If , then holds evidently. If , then we have or , which implies that or , and so . Hence, we have or , i.e., (A-2) holds. Therefore .
Similarly, it is easy to prove that .
Example 3 Let or . The function is defined by
then .
In fact, it is easy to see that satisfies condition (A-1). Now suppose that or . If , then holds evidently. If , then we have , which implies that , i.e., (A-2) holds. Therefore .
Example 4 Let or . The function is defined by
then .
In fact, it is easy to see that satisfies condition (A-1). Now suppose that or . If , then holds evidently. If , then we have , which implies that , and so . Hence, we have , i.e., (A-2) holds. Therefore .
Definition 2.6 (Aliouche and Fisher [13])
Let be a function satisfying the following conditions:
-
(F-1)
f is lower semi-continuous;
-
(F-2)
There exists such that or for implies .
We denote by ℱ the collection of all real functions satisfying the conditions of Definition 2.6.
Remark 2.3 Taking , , , we have . Then it is easy to see that , which implies that the implicit relation of Definition 2.5 is a generalization of Aliouche and Fisher [[13], implicit relation].
3 Main results
Theorem 3.1 Let and be two complete FMS s with R and satisfying (R-2). Let and be two non-self mappings, and . If there exist and such that
for all , and , then ST has a unique fixed point , and TS has a unique fixed point with , .
Proof For any given , we construct a sequence in X and a sequence in Y, respectively, as follows:
Obviously, we have , , for all .
For , , applying (3.1), we obtain for each
Note that , it is not difficult to see that
Again, for , , applying (3.2), we obtain for each
Note that , we have
Since , by Remark 2.2, (3.3) and (3.4), we can obtain
and
Using the inductive method, for , we have
and
In the next step, we show that is a Cauchy sequence in X. Since is with (R-2), it follows from Lemma 2.4(2) that for each , there exists such that
For and , by (3.7), (3.5) and Lemma 2.2(3), we have
Since , i.e., for all , it follows from (3.8) that is a Cauchy sequence in X. Hence, by the completeness of , there exists such that . By the similar reasoning process, from (3.6), (3.7), and Lemma 2.2(3), we can prove that is a Cauchy sequence in Y. Hence, by the completeness of , there exists such that .
Now we prove that is a fixed point of ST and is a fixed point of TS. For , , applying (3.1), we have for each
Let , by the lower semi-continuity of F and Lemma 2.6, we have for each
Note that and (A-2) of Definition 2.5, we can obtain for each , i.e., . Similarly, we can prove that . Hence, and , which imply that is a fixed point of ST and is a fixed point of TS.
Finally, we show the uniqueness of a fixed point. If is another fixed point of TS, then by (3.1) we have for each
Note that , it is not difficult to obtain that for each . We claim that . In fact, if , there exists such that . From (3.2), it follows that
Note that , we have . Since , by Remark 2.2, it is not difficult to obtain that
which is a contradiction. Hence, , i.e., for each . This shows that , i.e., the uniqueness of a fixed point for TS is true. Similarly, we can prove the uniqueness of a fixed point for ST. So, the proof of Theorem 3.1 is finished. □
Theorem 3.2 Let and be two complete FMS s with R and satisfying (R-2). Let and be two non-self mappings, and . If there exist and such that , and
for all , and , then ST has a unique fixed point , and TS has a unique fixed point with , .
Proof Now, we use inequality (3.9) to prove that inequality (3.1) holds. In fact, for each , and , if we set , , , , then for any , it is obvious that , , , . By (3.9), we have , which implies that
i.e.,
Then, by the arbitrariness of ε and the right-lower semi-continuity of F, we have
for each , and , i.e., inequality (3.1) holds for all and .
Similarly, by inequality (3.10), we can prove that inequality (3.2) also holds. Moreover, the other conditions in Theorem 3.1 are satisfied, thus by Theorem 3.1, the theorem is proved. □
In Theorem 3.2, taking , , we obtain the following corollary.
Corollary 3.1 Let be a complete FMS with R satisfying (R-2). Let be a self-mapping, and . If there exists such that and
for all and , then T has a unique fixed point in X.
Corollary 3.2 Let be a complete FMS with R satisfying (R-2), and let be a self-mapping. If there exists such that and
for all , and , then T has a unique fixed point in X.
Proof Taking , , , from Example 2, we obtain . Furthermore, for all and , by and (3.12), we have
which implies that (3.11) holds. Therefore, the conclusion follows from Corollary 3.1 immediately. □
Remark 3.1 Corollary 3.2 is a fuzzy version of the Boyd-Wong-type nonlinear contraction theorem (see [18]).
Theorem 3.3 Let and be two complete FMS s with R and satisfying (R-2). Let . Suppose that and are two continuous non-self mappings satisfying the following conditions:
-
(1)
There exists such that
(3.13)for all , and with ;
-
(2)
There exists such that
(3.14)for all , and with ;
-
(3)
There exists such that has an accumulation point in X.
Then ST has a unique fixed point , and TS has a unique fixed point with , .
Proof From condition (3), we can construct a sequence in X and a sequence in Y, respectively, as follows:
Obviously, we have , , for all . If for some , then is a fixed point of ST. So, we can assume that for all . Then, for all , it is obvious that .
Again by condition (3), we can assume that is an accumulation point of . Then there exists a subsequence of such that . Let . Next we show that is a fixed point of TS.
If , then by , we have , which implies that , i.e., . It follows from (3.13) that for all ,
Note that , it is not difficult to see that for all ,
Similarly, by and (3.14), we obtain for all
Note that , we have for all
Since , we know that there exists such that . From , (3.15) and (3.16), we can obtain
On the other hand, by (3.13) and (3.14), we can prove that and are non-increasing. In fact, applying (3.14), we obtain
Note that , we have
Similarly, by (3.13) and , we can obtain that
Then from (3.18) and (3.19), we have and , i.e., and are two non-increasing sequences, and so there exist such that and . Since T and S are continuous, by Lemma 2.6, we can obtain that
and
which imply that . This is a contradiction with (3.17). Hence, is a fixed point of TS.
Now we set , then , i.e., is a fixed point of ST.
Finally, we show the uniqueness of a fixed point. Assume that is another fixed point of TS with , then . By (3.13), we have for each
Note that , it is not difficult to obtain that for each . Since , there exists such that . From , it follows that
On the other hand, by , from (3.14) and , it is easy to obtain that . This is a contradiction with (3.20). Hence, the uniqueness of a fixed point for TS is true.
Assume that is also a fixed point of ST. Let , then . By the uniqueness of a fixed point for TS, we know . This shows that , i.e., the uniqueness of a fixed point for ST holds. This completes the proof. □
Let be a metric space and
Then is a FMS (cf. [1, 4]). It is easy to see that and are homeomorphic and for all .
Theorem 3.4 Let and be two complete metric spaces. Let and be two non-self mappings, and . If there exist and such that
for all , , then ST has a unique fixed point , and TS has a unique fixed point with , .
Proof Note that the topology and completeness of and , and the induced FMS and FMS are coincident respectively, as well as for all and , and for all and . Then it is not difficult to see that inequality (3.1) holds as a result of (3.21), and inequality (3.2) holds as a result of (3.22). Moreover, the other conditions of Theorem 3.1 are satisfied, thus by Theorem 3.1, the theorem is proved. □
Applying the same method, we can obtain the following theorem by virtue of Theorem 3.3.
Theorem 3.5 Let and be two complete metric spaces. Let . Suppose that and are two continuous non-self mappings satisfying the following conditions:
-
(1)
There exists such that
(3.23)for all , with ;
-
(2)
There exists such that
(3.24)for all , with ;
-
(3)
There exists such that has an accumulation point in X.
Then ST has a unique fixed point in X, and TS has a unique fixed point in Y with , .
Remark 3.2 Taking , , and in Theorems 3.4 and 3.5, we can obtain Theorems 3 and 4 in [13], respectively. This shows that our results improve and generalize Theorems 3-4 in [13] , and so the main results in [11, 12].
4 Applications to Menger probabilistic metric spaces and example
In this section, we first point out that our fixed point results for fuzzy metric spaces contain some corresponding results for Menger probabilistic metric spaces. After that, we give an example to discuss the validity of the hypotheses of Theorem 3.1.
Definition 4.1 A function is called a distribution function if it is nondecreasing and left-continuous with and .
If F is a distribution function which satisfies , then F is called a nonnegative distribution function. Let be the set of all nonnegative distribution functions. A special element of is the Heaviside function H defined by
Definition 4.2 (Hadz̆ić and Pap [19])
A function is called a triangular norm (for short, a t-norm) if the following conditions are satisfied for any :
-
(△-1) ;
-
(△-2) ;
-
(△-3) , for ;
-
(△-4) .
For each , the sequence is defined by and . A t-norm △ is said to be of H-type if the sequence of functions is equicontinuous at .
Lemma 4.1 (Xiao et al. [10])
Let △ be a t-norm for each , R be defined by , then
-
(1)
R is a symmetric and nondecreasing function such that ;
-
(2)
If △ is of H-type, then R satisfies (R-2).
Remark 4.1 (Xiao et al. [10])
Let be a symmetric and nondecreasing function such that and for all . Then satisfies (△-2) and (△-3). But does not necessarily satisfy (△-1) and (△-4). Hence is not necessarily a t-norm. From Remark 2.1 we see that satisfies (R-2); but is not a t-norm of H-type.
Definition 4.3 (Hadžić and Pap [19])
A triplet is called a Menger probabilistic metric space if X is a non-empty set, △ is a t-norm and F is a mapping from into satisfying the following conditions ( for is denoted by ):
-
(M-1)
for all if and only if ;
-
(M-2)
for all and ;
-
(M-3)
for all and .
Lemma 4.2 (Kaleva and Seikkala [1])
Let be a Menger probabilistic metric space, and . Let be a mapping defined by
Then and . Let be defined by and . Then is a FMS.
From Lemma 4.2 we see that each Menger probabilistic metric space can be considered as a special Kaleva-Seikkala’s type fuzzy metric space. But Remark 4.1 shows that in general a Kaleva-Seikkala’s type fuzzy metric space cannot be considered as a Menger probabilistic metric space. Hence, as direct consequences of our results, we can obtain the corresponding fixed point theorems in Menger probabilistic metric spaces. For example, from Theorem 3.1 and Lemmas 4.1-4.2 we can obtain the following consequence.
Theorem 4.1 Let and be two complete Menger probabilistic metric spaces such that △ and are two t-norms of H-type. Let and be two non-self mappings, and . If there exist and such that
for all , and , then ST has a unique fixed point , and TS has a unique fixed point with , .
Proof Let and be defined as in Lemma 4.2, respectively. Then and are two FMSs. Since △ and are of H-type, by Lemma 4.1, R and satisfy (R-2). Now we check that (3.1) holds.
From (4.1) we see that
For each , and , if we set , , , , then for any , it is obvious that , , , . By (4.4), we have , , , . Note (4.2), we can obtain that
which implies that
i.e.,
Then by the arbitrariness of ε and the right-lower semi-continuity of f, we have
for each , and , i.e., (3.1) holds for all and .
Similarly, by (4.3), we can prove that (3.2) also holds. Moreover, the other conditions in Theorem 3.1 are satisfied, thus by Theorem 3.1, the theorem is proved. □
In Theorem 4.1, taking , , we obtain the following corollary.
Corollary 4.1 Let be a complete Menger probabilistic metric space such that △ is a t-norm of H-type. Let be a self-mapping, and . If there exists such that
for all and , then T has a unique fixed point in X.
Corollary 4.2 Let be a complete Menger probabilistic metric space such that △ is a t-norm of H-type. Let be a self-mapping. If there exists such that
for all , and , then T has a unique fixed point in X.
Proof Taking , , , from Example 2, we obtain . Furthermore, for all and , by and (4.6), we have
which implies that (4.5) holds. Therefore, the conclusion follows from Corollary 4.1 immediately. □
Remark 4.2 Recently, in [20] Jachymski obtained the following result.
Theorem J Let be a complete Menger probabilistic metric space such that △ is a continuous t-norm of H-type. Let a function be such that, for any ,
Let be a mapping such that
Then there exists a unique such that .
Remark 4.3 Comparing Corollary 4.2 with Theorem J, it is not difficult to see the differences between them. Although Corollary 4.2 demands the function to imply that (4.7) holds by Definition 2.4(2), it does not require the t-norm of H-type to be continuous for all such that (4.8) holds.
Finally, we give an example to support the main results presented herein.
Example 5 Suppose that . Define by
Let be defined by and . Then is a complete FMS.
In fact, (FM-1), (FM-2) and (FM-3L) are easy to check. We only see (FM-3R). Since for all , and so for all . To prove (FM-3R), we assume that , and
Then we have , and so . It follows that
Hence (FM-3R) holds. It is clear that is complete.
In the same manner, if we take and given by (4.9), then is a complete FMS.
Define and by
Obviously, for each , for each , and , , , .
Next we check that T and S satisfy the conditions in Theorem 3.1. Obviously, we have
and
We set , , where , (). It is easily seen that , and . If , then (3.1) and (3.2) are easy to check. We only see . In the next step, we consider the following four cases.
Case 1. If and , then for each we have
Since , i.e., for all and , it follows that for each
which implies that (3.1) holds. Similarly, we have
Since , i.e., for all and , it follows that for each
which implies that (3.2) also holds.
Case 2. If and , then for each we have
Since , i.e., for all and , it follows that for each
which implies that (3.1) holds. Similarly, we have , which shows that (3.2) also holds.
Case 3. If and , then for each we have , which shows that (3.1) holds. In addition, for each we have
Since , i.e., for all and , then we can obtain that for each
which implies that (3.2) also holds.
Case 4. If and , then for each we have , , which imply that (3.1) and (3.2) hold.
Thus, all the conditions of Theorem 3.1 are satisfied. This shows the validity of the hypotheses of our main results.
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Acknowledgements
This work was supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant no. 13KJB110004) and Qing Lan Project of Jiangsu Province of China.
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Song, ML., Wang, Zq. Fixed point theorems for complex non-self mappings satisfying an implicit relation in Kaleva-Seikkala’s type fuzzy metric spaces. Fixed Point Theory Appl 2013, 254 (2013). https://doi.org/10.1186/1687-1812-2013-254
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DOI: https://doi.org/10.1186/1687-1812-2013-254