Abstract
In this paper, we introduce a new concept of probabilistic metric space, which is a generalization of the Menger probabilistic metric space, and we investigate some topological properties of this space and related examples. Also, we prove some fixed point theorems, which are the probabilistic versions of Banach’s contraction principle. Finally, we present an example to illustrate the main theorems.
MSC:54E70, 47H25.
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1 Introduction and preliminaries
Let ℝ be the set of all real numbers, be the set of all nonnegative real numbers, Δ denote the set of all probability distribution functions, i.e., Δ = { is left continuous and nondecreasing on ℝ, and }.
Definition 1.1 ([1])
A mapping is called a continuous t-norm if T satisfies the following conditions:
-
(1)
T is commutative and associative, i.e., and , for all ;
-
(2)
T is continuous;
-
(3)
for all ;
-
(4)
whenever and for all .
From the definition of T it follows that for all .
Two simple examples of continuous t-norm are and for all .
In 1942, Menger [2] developed the theory of metric spaces and proposed a generalization of metric spaces called Menger probabilistic metric spaces (briefly, Menger PM-space).
Definition 1.2 A Menger PM-space is a triple , where X is a nonempty set, T is a continuous t-norm and F is a mapping from ( denotes the value of F at the pair ) satisfying the following conditions:
(PM-1) for all and if and only ;
(PM-2) for all and ;
(PM-3) for all and .
The idea of Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. Since Menger, many authors have considered fixed point theory in PM-spaces and its applications as a part of probabilistic analysis (see [1, 3–14]).
In 1963, Gähler [15] investigated the concept of 2-metric spaces and he claimed that a 2-metric is a natural generalization of an ordinary metric space (for more detailed results, see the books [16, 17]). But some authors pointed out that there are no relations between 2-metric spaces and ordinary metric spaces [18]. Later, Dhage [19] introduced a new class of generalized metrics called D-metric spaces. However, as pointed out in [20], the D-metric is also not satisfactory.
Recently, Mustafa and Sims [21] introduced a new class of metric spaces called generalized metric spaces or G-metric spaces as follows.
Definition 1.3 ([21])
Let X be a nonempty set and be a function satisfying the following conditions:
(G1) if for all ;
(G2) for all with ;
(G3) for all with ;
(G4) for all ;
(G5) for all .
Then G is called a generalized metric or a G-metric on X and the pair is a G-metric space.
It was proved that the G-metric is a generalization of ordinary metric (see [21]). Recently, some authors studied G-metric spaces and obtained fixed point theorems on G-metric spaces [22–24]. Similar work can be found in [25–29].
It is well known that the notion of a PM-space corresponds to the situation that we may know probabilities of possible values of the distance although we do not know exactly the distance between two points. This idea leads us to seek a probabilistic version of G-metric spaces defined by Mustafa and Sims [21].
Definition 1.4 A Menger probabilistic G-metric space (shortly, PGM-space) is a triple , where X is a nonempty set, T is a continuous t-norm and is a mapping from into ( denotes the value of at the point ) satisfying the following conditions:
(PGM-1) for all and if and only if ;
(PGM-2) for all with and ;
(PGM-3) (symmetry in all three variables);
(PGM-4) for all and .
Remark 1.5 Golet introduced a concept of probabilistic 2-metric (or 2-Menger space) [30] based on 2-metric [15] defined by Gähler. In the concept of probabilistic 2-metric, a 2-t-norm is used. Our definition of a Menger probabilistic G-metric space is different from the one of Golet. The metric of Golet is not continuous in two arguments although it is continuous in any one of its three arguments. But is continuous in any two arguments as shown in Theorem 2.5.
Example 1.6 Let H denote the specific distribution function defined by
and D be a distribution function defined by
For any , define a function by
where G is a G-metric as in Definition 1.3. Set . Then is a probabilistic G-metric.
Proof It is easy to see that satisfies (PGM-1)-(PGM-3). Next we show for all and all . In fact, we only need show that D satisfies
Since , we have
Furthermore, we have
which, from (1.2) and (1.3), shows that
This implies (1.1) since D is nondecreasing. □
Example 1.7 Let be a PM-space. Define a function by
for all and . Then is a probabilistic G-metric.
Proof It is obvious that satisfies (PGM-1), (PGM-2), and (PGM-3). To prove that satisfies (PGM-4), we need to show that, for all and all ,
i.e.,
Now, from
and
we conclude that (1.5), i.e., (1.4) holds. Therefore, satisfies (PGM-4) and hence is a probabilistic G-metric. □
Example 1.8 Let be a PM-space. Define a function by
for all and . Then is a PGM-space.
Proof In fact, the proofs of (PGM-1)-(PGM-3) are immediate. Now, we show that satisfies (PGM-4). It follows that
Thus is a PGM-space. □
The following remark shows that the PGM-space is a generalization of the Menger PM-space.
Remark 1.9 For any function , the function defined by
is a probabilistic metric. It is easy to see that F satisfies the conditions (PM-1) and (PM-2).
Next, we show F satisfies (PM-3). Indeed, for any and , we have
It follows from (PGM-4) that
Since
and
it follows from (PGM-4) that
and
Therefore, we have
This shows that F satisfies (PM-3).
2 Topology, convergence, and completeness
In this section, we first introduce the concept of neighborhoods in the PGM-spaces. For the concept of neighborhoods in PM-spaces, we refer the readers to [1, 3].
Definition 2.1 Let be a PGM-space and be any point in X. For any and δ with , an -neighborhood of is the set of all points y in X for which and . We write
This means that is the set of all points y in X for which the probability of the distance from to y being less than ϵ is greater than .
Lemma 2.2 If and , then .
Proof Suppose that , so and . Since F is monotone, we have
and
Therefore, by the definition, . This completes the proof. □
Theorem 2.3 Let be a Menger PGM-space. Then is a Hausdorff space in the topology induced by the family of -neighborhoods.
Proof We show that the following four properties are satisfied:
-
(A)
For any , there exists at least one neighborhood, , of and every neighborhood of contains .
-
(B)
If and are neighborhoods of , then there exists a neighborhood of , , such that .
-
(C)
If is a neighborhood of and , then there exists a neighborhood of y, , such that .
-
(D)
If , then there exist disjoint neighborhoods, and , such that and .
Now, we prove that (A)-(D) hold.
-
(A)
For any and , since for any .
-
(B)
For any and , let
and
be the neighborhoods of . Consider
Clearly, and, since and , by Lemma 2.2, and , so
-
(C)
Let be the neighborhood of . Since ,
Now, is left-continuous at , so there exist and such that
Let , where and is chosen such that
Such a exists since T is continuous, for all and .
Now, suppose that , so that
Then, since is monotone, it follows from (PGM-4) that
Similarly, we also have . This shows and hence .
-
(D)
Let . Then there exist and , with such that and . Let
and
where and are chosen such that , , where . Such and exist since T is continuous, monotone, and .
Now, suppose that there exists a point such that
Then, by (PGM-4), we have
and
which are contradictions. Therefore, and are disjoint. This completes the proof. □
Next, we give the definition of convergence of sequences in PGM-spaces.
Definition 2.4
-
(1)
A sequence in a PGM-space is said to be convergent to a point (write ) if, for any and , there exists a positive integer such that whenever .
-
(2)
A sequence in a PGM-space is called a Cauchy sequence if, for any and , there exists a positive integer such that whenever .
-
(3)
A PGM-space is said to be complete if every Cauchy sequence in X converges to a point in X.
Theorem 2.5 Let be a PGM-space. Let , and be sequences in X and . If , and as , then, for any , as .
Proof For any , there exists such that . Then, by (PGM-4), we have
and
Letting in the above two inequalities and noting that T is continuous, we have
and
Letting in above two inequalities, since is left-continuous, we conclude that
for any . This completes the proof. □
3 Fixed point theorems
In [31], Sehgal extended the notion of a Banach contraction mapping to the setting of Menger PM-spaces. Later on, Sehgal and Bharucha-Raid [32] proved a fixed point theorem for a mapping under the contractive condition in a complete Menger PM-space. Before proving our fixed point theorems, we first introduce a new concept of contraction in PGM-spaces, which is a corresponding version of Sehgal’s contraction in PM-spaces.
Definition 3.1 Let be a PGM-space. A mapping is said to be contractive if there exists a constant such that
for all and .
The mapping f satisfying the condition (3.1) is called a λ-contraction.
Let T be a given t-norm. Then (by associativity) a family of mappings for each is defined as follows:
for any .
Definition 3.2 ([33])
A t-norm T is said to be of Hadzić-type if the family of functions is equicontinuous at , that is, for any , there exists such that
for each .
The t-norm is a trivial example of t-norm of Hadzić-type.
Lemma 3.3 Let be a Menger PGM-space with T of Hadžić-type and be a sequence in X. Suppose that there exists satisfying
for any and . Then is a Cauchy sequence in X.
Proof Since , by induction, we have
Since X is a Menger PGM-space, we have as , so
for any .
Now, let and . We show, by induction, that, for any ,
For , since is a real number, for all . Hence, , which implies that (3.3) holds for . Assume that (3.3) holds for some . Then, since T is monotone, it follows from (PGM-4) that
so we have the conclusion.
Now, we show that is a Cauchy sequence in X, i.e., for any . To this end, we first prove that for any . Let and be given. By hypothesis, is equicontinuous at 1 and , so there exists such that, for any ,
for all . From (3.2), it follows that . Hence there exists such that for any . Hence, by (3.3) and (3.4), we conclude that for any . This shows for any . By (GPM-4), we have
Therefore, by the continuity of T, we conclude that
for any . This shows that the sequence is a Cauchy sequence in X. This completes the proof. □
From Example 1.6 and Lemma 3.3 we get the following corollary.
Corollary 3.4 ([33])
Let be a PM-space with T of Hadžić-type and be a sequence. If there exists a constant such that
then is a Cauchy sequence.
Proof Define for all and all . Example 1.6 shows that is a PGM-space. Since and , implies for all and . By Lemma 3.3 we conclude that is a Cauchy sequence in the sense of PGM-space . That is, for every and , there exists a positive integer such that for all . By the definition of , we have
This shows that is a Cauchy sequence in the sense of PM-space . □
Theorem 3.5 Let be a complete Menger PGM-space with T of Hadžić-type. Let and be a λ-contraction. Then, for any , the sequence converges to a unique fixed point of T.
Proof Take an arbitrary point in X. Construct a sequence by for all . By (3.1), for any , we have
Lemma 3.3 shows that is a Cauchy sequence in X. Since X is complete, there exists a point such that as . By (3.1), it follows that
Letting , since and as , we have
for any . Hence .
Next, suppose that y is another fixed point of f. Then, by (3.1), we have
Letting , since X is a Menger PGM-space, as , so
for any , which implies that . Therefore, f has a unique fixed point in X. This completes the proof. □
Theorem 3.6 Let be a complete Menger PGM-space with T of Hadžić-type. Let be a mapping satisfying
for all , where . Then, for any , the sequence converges to a unique fixed point of f.
Proof Take an arbitrary point in X. Construct a sequence by for all . By (3.5), for any , we have
This shows that
Lemma 3.3 shows that is a Cauchy sequence in X. Since X is complete, there exists a point such that as . By (3.5), it follows that
Letting , since and as , we have, for any ,
i.e.,
Hence .
Next, suppose that y is another fixed point of f. Then, by (3.5), we have, for any ,
This shows that . Therefore, f has a unique fixed point in X. This completes the proof. □
Finally, we give the following example to illustrate Theorem 3.5 and Theorem 3.6.
Example 3.7 Set and for all . Define a function by
for all , where . Then G is a G-metric (see [24]). It is easy to check that satisfies (PGM-1)-(PGM-3). Since for all , we have
This shows that satisfies (PGM-4). Hence is a PGM-space.
(1) Let . Define a mapping by for all . For any , we have
and
Therefore, we conclude that f is a λ-contraction and f has a fixed point in X by Theorem 3.5. In fact, the fixed point is .
(2) Let . Define a mapping by for all . For any and all , since
and
we conclude that
and hence f has a fixed point in X by Theorem 3.6. In fact, the fixed point is .
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Acknowledgements
The first author is supported by the National Natural Science Foundation of China (Grant Number: 11371002,41201327) and Specialized Research Fund for the Doctoral Program of Higher Education (Grant Number:20131101110048) and Natural Science Foundation of Hebei Province (Grant Number: A2013201119). The second author is supported by the Fundamental Research Funds for the Central Universities (Grant Number: 13MS109). The authors S. Alsulami and L. Ciric thanks the Deanship of Scientific Research (DSR), King Abdulaziz University for financial support ( Grant Number: 130-037-D1433).
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Zhou, C., Wang, S., Ćirić, L. et al. Generalized probabilistic metric spaces and fixed point theorems. Fixed Point Theory Appl 2014, 91 (2014). https://doi.org/10.1186/1687-1812-2014-91
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DOI: https://doi.org/10.1186/1687-1812-2014-91