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Tripled common fixed point theorems under probabilistic φ-contractive conditions in generalized Menger probabilistic metric spaces
Fixed Point Theory and Applications volume 2014, Article number: 158 (2014)
Abstract
In this paper, the new concepts of generalized Menger probabilistic metric spaces and tripled common fixed point for a pair of mappings and are introduced. Utilizing the properties of the pseudo-metric and the triangular norm, some tripled common fixed point problems of hybrid probabilistic contractions with a gauge function φ are studied. The obtained results generalize some coupled common fixed point theorems in the corresponding literature. Finally, an example is given to illustrate our main results.
MSC:47H10, 46S50.
1 Introduction
Coupled fixed points were considered by Bhaskar and Lakshmikantham [1]. Recently, some new results for the existence and uniqueness of coupled fixed points were presented for the cases of partially ordered metric spaces, cone metric spaces and fuzzy metric spaces (see [2–12]). The concept of probabilistic metric space was initiated and studied by Menger which is a generalization of the metric space notion [13]. Many results on the existence of fixed points or solutions of nonlinear equations under various types of conditions in Menger spaces have been extensively studied by many scholars (see [14–18]). In 2010, Jachymski established a fixed point theorem for probabilistic φ-contractions and give a characterization of function φ having the property that there exists a probabilistic φ-contraction, which is not a probabilistic k-contraction () [19]. In 2011, Xiao et al. obtained some common coupled fixed point results for hybrid probabilistic contractions with a gauge function φ in Menger probabilistic metric spaces and in non-Archimedean Menger probabilistic metric spaces without assuming any continuity or monotonicity conditions for φ [20].
The purpose of this paper is to introduce the concept of generalized Menger probabilistic metric spaces and tripled common fixed point for a pair of mappings and . Utilizing the properties of the pseudo-metric and the triangular norm, some tripled common fixed points problems for pairs of commutative mappings under hybrid probabilistic contractions with a gauge function φ are studied in generalized Menger PM-spaces and in generalized non-Archimedean Menger PM-space, respectively. The obtained results generalize some coupled common fixed point theorems in corresponding literatures. Finally, an example is given to illustrate our main results.
2 Preliminaries
Consistent with Menger [13] and Zhang [14], the following results will be needed in the sequel.
Denote by R the set of real numbers, the nonnegative real numbers, and the set of all positive integers.
If is a function such that , then φ is called a gauge function. If , then denotes the n th iteration of and .
A mapping is called a distribution function if it is nondecreasing and left-continuous with , .
We shall denote by the set of all distribution functions while H will always denote the specific distribution function defined by
Definition 2.1 ([14])
A function is called a triangular norm (for short, a t-norm) if the following conditions are satisfied for any :
(Δ-1) , ;
(Δ-2) ;
(Δ-3) ;
(Δ-4) .
Two typical examples of t-norms are and for all .
We now introduce the definition of generalized Menger probabilistic metric space.
Definition 2.2 A triplet is called a generalized Menger probabilistic metric space (for short, a generalized Menger PM-space) if X is a non-empty set, Δ is a t-norm and ℱ is a mapping from into (we shall denote the distribution function by and will represent the value of at ) satisfying the following conditions:
(GPM-1) ;
(GPM-2) for all if and only if ;
(GPM-3) for all and ;
(GPM-4) for all and .
is called a generalized non-Archimedean Menger PM-space if it is a generalized Menger PM-space satisfying the following condition:
(GPM-5) for all and .
Remark 2.1 In 1942, Menger [13] proposed a generalization of a metric space called a Menger probabilistic metric space (briefly a Menger PM-space). Our definition of a generalized Menger PM-space is different from the one of Menger, since the t-norm we used here is an associative function of three variables rather than a function of two variables. Note that Definition 2.1 is first used by Chang to define a probabilistic 2-metric space. Our definition is also different from the one of Chang since the distribution function of the latter is from to .
Example 2.1 Suppose that . Define by
for . It is easy to verify that is a generalized Menger PM-space. Now, assume that and . Then we have
Hence is a generalized non-Archimedean Menger PM-space.
Proposition 2.1 Let be a generalized Menger PM-space and Δ be a continuous t-norm. Then is a Hausdorff topological space in the -topology , i.e., the family of sets
is a base of neighborhoods of a point x for , where
Proof It suffices to prove that:
-
(i)
For any , there exists an such that .
-
(ii)
For any given and , there exist and , such that .
-
(iii)
For any , there exist and , such that .
-
(iv)
For any , , there exist and , such that .
It is easy to check that (i)-(iii) are true. Now we prove that (iv) is also true. In fact, suppose that and . Then there exist and , such that . Let
where and (since Δ is continuous and , such b exists). Now suppose that there exists a point , which implies that and . Take . Then we have
which is a contradiction. Thus the conclusion (iv) is proved. This completes the proof. □
Definition 2.3 Let be a generalized Menger PM-space, Δ be a continuous t-norm.
-
(i)
A sequence in X is said to be -convergent to if for all .
-
(ii)
A sequence in X is said to be a -Cauchy sequence, if for any given and , there exists a positive integer , such that whenever .
-
(iii)
is said to be -complete, if each -Cauchy sequence in X is -convergent to some point in X.
Definition 2.4 A t-norm Δ is said to be of H-type if the family of functions is equicontinuous at , where
Definition 2.5 Let X be a non-empty set, and be two mappings. A is said to be commutative with T, if for all . A point is called a tripled common fixed point of T and A, if .
Imitating the proof in [9], we can easily obtain the following lemma.
Lemma 2.1 Let be a generalized Menger PM-space. For each , define a function by
Then the following statements hold:
-
(1)
if and only if ;
-
(2)
for all and ;
-
(3)
if and only if ;
-
(4)
for all and .
The following lemmas play an important role in proving our main results in Sections 3 and 4.
Lemma 2.2 ([17])
Suppose that . For any , let be nondecreasing, and satisfy for all . If for all , then for all .
Lemma 2.3 Let X be a nonempty set, and be two mappings. If , then there exist three sequences , , and in X, such that , , and .
Proof Let , , be any given points in X. Since , we can choose , such that , , and . Continuing this process, we can construct three sequences , , and in X, such that , , and . □
3 Tripled common fixed point results in generalized PM-spaces
Lemma 3.1 Let be a generalized Menger PM-space, be a family of pseudo-metrics on X defined by (2.1). If Δ is a t-norm of H-type, then for each , there exists such that for all and ,
Proof Since Δ is a t-norm of H-type, is equicontinuous at , and so for each , there exists , such that
For any given and , we write (). For any , it is evident that . By Lemma 2.1, we have
It follows from (3.1)-(3.2), and (GPM-4) that
Using Lemma 2.1 again, we have . By the arbitrariness of ε, we have
This completes the proof. □
Theorem 3.1 Let be a complete generalized Menger PM-space with Δ a t-norm of H-type, be a gauge function such that , , and for any . Let and be two mappings satisfying
for all and , where , A is continuous and commutative with T. Then T and A have a unique tripled common fixed point in X.
Proof By Lemma 2.3, we can construct three sequences , , and in X, such that , , and .
From (3.3), for all we have
and
Denote . From (3.4)-(3.6), we have
which implies that
and
Since and for each , using Lemma 2.2, we have
Thus
We claim that for any and ,
and
In fact, by (3.7)-(3.9), it is easy to see that (3.12)-(3.14) hold for . Assume that (3.12)-(3.14) hold for some k. Since , by (3.4) we have . By (3.3) and (3.12)-(3.14), we have
Hence, by the monotonicity of Δ, we have
Similarly, we have and . Therefore, by induction, (3.12)-(3.14) hold for all and .
Suppose that is given. Since Δ is a t-norm of H-type, there exists such that
By (3.11), there exists , such that for all . Hence, from (3.12)-(3.15), we get , , for all , . Therefore , , and are Cauchy sequences.
Since is complete, there exist , such that , and . By the continuity of A, we have
The commutativity of A with T implies that . From (3.3) and , we obtain
Letting in (3.16), we have . Hence, . Similarly, we can show that and .
Next we show that , , and . In fact, from (3.3), for all we have
and
Denote . By (3.17)-(3.19), we have , and hence for all
Thus, for all we have
Since and for all , by Lemma 2.2, we conclude that
This shows that , , and . Hence, , , and . Finally, we prove that . By (3.3), for all we have
which implies that . Using Lemma 2.2, we have for all , i.e., . Similarly, we can show that . Hence, there exists , such that .
Finally, we show the uniqueness of the tripled common fixed point of T and A. Suppose that is another tripled common fixed point of T and A, i.e., . By (3.3), for all we have
which implies that for all . Using Lemma 2.2, we have for all , i.e., . This completes the proof. □
Corollary 3.1 Let be a complete generalized Menger PM-space with Δ a t-norm of H-type and , be a gauge function such that and , for each . Let and be two mappings satisfying
for all and , where , A is continuous and commutative with T. Then T and A have a unique tripled common fixed point in X.
Letting (I is the identity mapping) in Corollary 3.1, we can obtain the following corollary.
Corollary 3.2 Let be a complete generalized Menger PM-space with Δ a t-norm of H-type and , be a gauge function such that and , for any . Let be a mapping satisfying
for all and . Then T has a unique fixed point in X.
Letting () in Corollary 3.1, we can obtain the following corollary.
Corollary 3.3 Let be a complete generalized Menger PM-space with Δ a t-norm of H-type and , and be two mappings satisfying
for all and , where , , A is continuous and commutative with T. Then T and A have a unique tripled common fixed point in X.
Letting (I is the identity mapping) in Theorem 3.1, we can obtain the following corollary.
Corollary 3.4 Let be a complete generalized Menger PM-space with Δ a t-norm of H-type, be a gauge function such that , , and for any . Let be a mapping satisfying
for all and . Then T has a unique fixed point in X.
Letting () in Theorem 3.1, we can obtain the following corollary.
Corollary 3.5 Let be a complete generalized Menger PM-space with Δ a t-norm of H-type, and be two mappings satisfying
for all and , where , , A is continuous and commutative with T. Then T and A have a unique tripled common fixed point in X.
From the proof of Theorem 3.1, we can similarly prove the following result.
Theorem 3.2 Let be a complete generalized Menger PM-space with Δ a t-norm of H-type, be a gauge function such that , , and for any . Let and be two mappings satisfying
for all and , where , A is continuous and commutative with T. Then T and A have a unique tripled common fixed point in X.
Letting (I is the identity mapping) in Theorem 3.2, we can obtain the following corollary.
Corollary 3.6 Let be a complete generalized Menger PM-space with Δ a t-norm of H-type, be a gauge function such that , , and for any . Let be a mapping satisfying
for all and . Then T has a unique fixed point in X.
4 Tripled common fixed point results in generalized non-Archimedean PM-spaces
In this section, we will use the results in Section 3 to get some corresponding results in generalized non-Archimedean Menger spaces.
Lemma 4.1 Let be a complete generalized non-Archimedean Menger PM-space, be a family of pseudo-metrics on X defined by (2.1). If Δ is a t-norm of H-type, then for each , there exists , such that for all and ,
Proof Since Δ is a t-norm of H-type, is equicontinuous at , and so for each , there exists such that
For any given , and , write (). For any , we have . It follows from (4.1) and (GPM-5) that
By Lemma 2.1, we have . By the arbitrariness of ε, we have
This completes the proof. □
Theorem 4.1 Let be a complete generalized non-Archimedean Menger PM-space such that and , be a gauge function such that and for any . Let and be two mappings satisfying
for all and , where , A is continuous and commutative with T. Suppose that there exist , such that for any ,
Then T and A have a unique tripled common fixed point in X.
Proof Take , , and . By Lemma 2.3, we can construct three sequences , , and in X, such that , , and .
From (4.2), for all , we have
and
Denote . From (4.4)-(4.6), and , we obtain
which implies that
Thus, by (4.4)-(4.7), we have
Suppose that and . By (4.3), there exists , such that
and
for all and .
Hence, it follows from (4.8) and (GPM-4) that
This shows that is a Cauchy sequence. Similarly, we can show that and are Cauchy sequences.
Since is complete, there exist , such that , , and . By the continuity of A, we have
From (4.2) and the commutativity of A with T, we have
Letting in (4.11), we have . Hence, . Similarly, we have and .
Next we claim that , , and . In fact, by (4.2), we have
and
Denote . It follows from (4.12)-(4.14) that
and thus
Since , we have
as . From (4.15), we have
Hence, , , and , i.e., , , and . Now we prove that . In fact, by (4.2), we have
which implies that . Letting , we have for all , i.e., . Similarly, we can show that . Hence, there exists , such that .
Finally, we show the uniqueness of the tripled common fixed point of T and A. Suppose that is another tripled common fixed point of T and A, i.e., . By (4.2), for all , we have
which implies that for all . Letting , we have for all , i.e., . This completes the proof. □
Letting (I is the identity mapping) in Theorem 4.1, we can obtain the following corollary.
Corollary 4.1 Let be a complete generalized non-Archimedean Menger PM-space such that and , be a gauge function such that and for any . Let be a mapping satisfying
for all and . Suppose that there exist , such that for any ,
Then T has a unique fixed point in X.
In a similar way, we can obtain the following result.
Theorem 4.2 Let be a complete generalized non-Archimedean Menger PM-space such that Δ is a t-norm of H-type, be a gauge function such that and for any . Let and be two mappings satisfying
for all and , where , A is continuous and commutative with T. Then T and A have a unique tripled common fixed point in X.
Letting (I is the identity mapping) in Theorem 4.2, we can obtain the following corollary.
Corollary 4.2 Let be a complete generalized non-Archimedean Menger PM-space such that Δ is a t-norm of H-type, be a gauge function such that and for any . Let be a mapping satisfying
for all and . Then T has a unique fixed point in X.
Letting () in Theorem 4.2, we can obtain the following corollary.
Corollary 4.3 Let be a complete generalized non-Archimedean Menger PM-space such that Δ is a t-norm of H-type. Let and be two mappings satisfying
for all and , where , , A is continuous and commutative with T. Then T and A have a unique tripled common fixed point in X.
Remark 4.1 If is a generalized non-Archimedean Menger PM-space, then the hypotheses concerning gauge functions can be weakened. Let us note that in Theorem 4.2 the gauge function only satisfies for all , and it does not necessarily satisfy for all .
5 An application
In this section, we shall provide an example to show the validity of the main results of this paper.
Example 5.1 Suppose that , . Then is a t-norm of H-type and . Define by
We claim that is a generalized Menger PM-space. In fact, it is easy to verify (GPM-1), (GPM-2), and (GPM-3). Assume that for any and ,
Then we have , , and so . It follows that
Hence (GPM-4) holds. It is obvious that is complete. Suppose that . For , define as follows:
Then for each and , we have
and so
Thus, all the conditions of Corollary 3.2 are satisfied. Therefore, T has a unique fixed point in X.
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Acknowledgements
The authors would like to thank the editor and the referees for their constructive comments and suggestions. The research was supported by the Natural Science Foundation of China (11361042, 11071108, 11326099), the Natural Science Foundation of Jiangxi Province of China (20132BAB201001, 20142BAB211016, 2010GZS0147), and the Youth Foundation of the Education Department of Jiangxi (GJJ13012).
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Luo, T., Zhu, C. & Wu, Z. Tripled common fixed point theorems under probabilistic φ-contractive conditions in generalized Menger probabilistic metric spaces. Fixed Point Theory Appl 2014, 158 (2014). https://doi.org/10.1186/1687-1812-2014-158
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DOI: https://doi.org/10.1186/1687-1812-2014-158