Some Fixed Point Theorems in Generalized Probabilistic Metric Spaces

Abstract

We introduce the concepts of -contraction and probabilistic -contraction mappings in generalized probabilistic metric spaces and prove some fixed point theorems for such two types of mappings in generalized probabilistic metric spaces. Our results generalize and extend many comparable results in existing literature. Some examples are also given to support our results. Finally, an application to the existence of solutions for a class of integral equations is presented by utilizing one of our main results.

1. Introduction and Preliminaries

The notion of a probabilistic metric space was introduced and studied by Menger [1]. The idea of Menger was to use distribution functions instead of nonnegative real numbers to describe the distance between two points. It has become an active field since then and many fixed point results for mappings satisfying different contractive conditions have been studied [2–8].

On the other hand, Mustafa and Sims [9] defined the concept of a -metric space and presented the Banach fixed point theorem in the context of a complete -metric space. Following their results, some authors have obtained many fixed point theorems for contractive mappings in -metric spaces [9–16].

As a generalization, using PM-spaces and -metric spaces, Zhou et al. [17] defined the notion of generalized probabilistic metric spaces (or probabilistic -metric spaces). The purpose of this paper is to establish some fixed point theorems for two types of mappings satisfying the -contraction or probabilistic -contraction in generalized probabilistic metric spaces. As consequences, our results generalize and extend many comparable results (see, e.g., [6–8, 14, 16, 17]).

We introduce some useful concepts and lemmas for the development of our results.

Let denote the set of reals and the nonnegative reals. A mapping is called a distribution function if it is nondecreasing and left continuous with and . We will denote by the set of all distribution functions and let .

Let denote the specific distribution function defined by

Definition 1 (see [2]). The mapping is called a triangular norm (for short, a -norm) if the following conditions are satisfied: , for all ; ; , for ; .
Three sample examples of continuous -norms are , , and for all .

Definition 2 (see [17]). A Menger probabilistic -metric space (briefly, a Menger -space) is a triplet , where is a nonempty set, is a continuous -norm, and is a mapping from into ( denotes the value of at the point ) satisfying the following conditions:  for all and if and only if ;  for all with and ;  (symmetry in all three variables);  for all and .

Example 3. Let be an ordinary metric space. Define and , where . Then and are both Menger -spaces.

Example 4. Let be a distribution function, , and Then is a Menger -space.

Example 5. Let be an ordinary metric space, let be a distribution function different from , satisfying , and Then is a Menger -space.

Remark 6 (see [17]). Zhou et al. pointed out that if () is a Menger -space and is continuous, then () is a Hausdorff topological space in the -topology ; that is, the family of sets is a basis of neighborhoods of a point for , where .

Definition 7 (see [17]). Let be a -space.(1)A sequence in is said to be convergent to a point in (write ) if, for any and , there exists a positive integer such that , whenever .(2)A sequence in is called a Cauchy sequence if, for any and , there exists a positive integer , such that , whenever .(3)A -space is said to be complete if every Cauchy sequence in converges to a point in .(4)A mapping is said to be continuous at a point if is convergent to implying that is convergent to .

Definition 8 (see [3]). A -norm is said to be of -type if the family of functions is equicontinuous at , where and , .

Definition 9 (see [4]). A function is said to satisfy the condition if it is strictly increasing, right-continuous and , , as , where is the th iteration of .

Lemma 10 (see [6]). Suppose that . For every , let be nondecreasing and satisfy for any . If for any , then for any .

Lemma 11 (see [17]). Let be a Menger -space. Let , , and be sequences in and . If , , and as , then, for any , as .

2. Main Results

In this section, we will give some fixed point theorems for two types of mappings satisfying the -contraction or probabilistic -contraction in generalized probabilistic metric spaces. We first present some useful lemmas.

Lemma 12. Let be a Menger -space and be a continuous -norm. Then the following statements are equivalent:(i)the sequence is a Cauchy sequence;(ii)for any and , there exists such that , for all .

Proof. (i)(ii). This can be easily obtained from Definition 7 (2).
(ii)(i). Since is continuous, for every and , there exists , such that . Let . Then . Hence, from (ii), there exists , such that and , for all . Then we have . Thus, is a Cauchy sequence.

Lemma 13. Let be a Menger -space. For each , define a function by for . Then(1) if and only if ;(2) for all if and only if ;(3);(4)if , then, for every , .

Proof. It is not difficult to prove that , , and hold. Now, we prove that also holds. For any and , we have and . Hence, from and , we have which implies that . Letting , we have for all .

Lemma 14. Let be a Menger -space and let be a family of functions on defined by (4). If is a -norm of -type, then, for each , there exists , such that, for each , for all .

Proof. Since is a -norm of -type, there exists , such that, for all ,
For any given and , we put (). For every , it is evident that . By Lemma 13, for , we have
By (7) and (8), we have Again by Lemma 13, we get . Letting , for all , we have . Similarly, we have .

Inspired by the notion of -contraction mappings in [8], we introduce the notion of -contraction mappings in -spaces.

Definition 15. Let be a Menger -space, satisfy the condition and be a nondecreasing function, if and only if , . A mapping is said to be a -contraction mapping on , if, for any and , whenever . If for all , then is said to be a -contraction mapping.

Theorem 16. Let be a complete Menger -space and let be a continuous -norm. Let be a -contraction mapping satisfying (10). Then we have the following:(i) is continuous on ;(ii) has a unique fixed point in , and, for any given , the iterative sequence converges to this fixed point.

Proof. (i) Since is right continuous, strictly increasing and , we have and , for . Since , we get . Hence, for any given and , there exists , such that and .
Now, for each , suppose that is convergent to . Then, for , there exists , such that , whenever . Since is a -contraction mapping, we have , whenever . Thus, , which implies that is continuous at . By the arbitrariness of , we obtain that is continuous on .
(ii) For any given , define the sequence by . Let . Then , and it follows from (10) that for .
Since , we have Hence, for any and , there exists such that and , whenever . Hence, for , we have . Using Lemma 12, we know that is a Cauchy sequence. Since is complete, there exists such that . Since is continuous, we have .
Now, Suppose that is another fixed point of . Then, for any and , , which implies that . For any given and , there exists such that and , whenever . Hence, for , we have , which implies that . This completes the proof.

Now, we present an example to illustrate Theorem 16.

Example 17. Let . Define by for all and . It is easy to verify that is a complete Menger -space. Let for all , let   for all , and let   be defined by .
Now, if , then and for all .
If , then . For any , we have . Hence,
If and , then and Thus, all the conditions of Theorem 16 are satisfied. Hence, has a unique fixed point in . In fact, is the unique fixed point of .

Remark 18. Theorem 16 generalizes and extends fixed point theorems for -contraction mappings in [8] to the setting of generalized probabilistic metric spaces (take ).

Definition 19. Let be a given mapping and be a function. We say that is generalized -admissible; if , for all , , then for all .

Definition 20. Let be a -space and let be a strictly increasing function such that and for any . A mapping is said to be a probabilistic -contraction mapping on if there exists a function , for each , , where , .

Theorem 21. Let be a complete Menger -space such that is a -norm of -type and . Let be a probabilistic -contraction mapping satisfying (13). Suppose that the following hold:(i) is generalized -admissible;(ii)there exists such that and for all ;(iii)if is a sequence in such that , for all , , and , then and for all and .Then has a fixed point in . Moreover, let be the set of fixed points of ; if, for each , we have for all , then has a unique fixed point in .

Proof. Let and define the sequence by . Suppose that , for any (If not, there exists , such that and then the conclusion holds).
Since is -admissible, , and for all , we have and for all . By induction, we obtain that
For any , from (13) and (14), we have
Suppose that . Then, from the above inequalities, we get . If , then By Lemma 10, we have for any . Then, , which is in contradiction to .
Hence, . Then we have
Next, we show that is a Cauchy sequence. For every , suppose that . Then . From (17), for every , we get . By Lemma 13, for any , we have By Lemma 14, for every , there exists such that
Suppose that and are given. Since , there exists , such that for all . Thus, by (18) and (19), we have . Using Lemma 13, we have for all . From Lemma 12, we know that is a Cauchy sequence. Since is complete, there exists , such that .
Since , by (14) and (iii), we have and for all and . From (13), for any , we have Letting , since as , by Lemma 11, (20), we have Hence, By Lemma 10, we have and for all . Thus .
Moreover, let be the set of fixed points of ; if for each , we have for all . Now, suppose that is another fixed point of . Then for all . Hence, by (13), we have By Lemma 10, we get . This completes the proof.

Now, we present an example to illustrate Theorem 21.

Example 22. Let and . Then is a -norm of -type. Define by for all . We claim that is a Menger -space. In fact, - are easy to check. Next, we prove that holds.
Suppose that , , and . Then we have and so , which implies that .
Similarly, if , then we also have . Hence, holds. It is easy to prove that is complete. Let for , It is not difficult to prove that and satisfy (i), (ii), and (iii) of Theorem 21. Now, suppose that at least one of is in . Then and so inequality (13) holds. Hence, for , we have for all . Then, for , we have Thus, all the conditions of Theorem 21 are satisfied. Hence, has a unique fixed point in . In fact, is the unique fixed point of in .

Remark 23. Theorem 21 generalizes and extends Theorem  4.2 in [16] and Theorems 3.5 and   3.6 in [17] (take and ).

3. An Application

In this section, we will apply one of our main results to investigate the existence of solutions for a class of integral equations.

Consider the following class of integral equations: where , , and , , and are all continuous functions.

Let be the set of all real continuous functions defined on . We define by for all and . It is easy to verify that is a complete Menger -space.

Now, we define by and we write if and only if for all . Then, is a solution of (27) if and only if it is a fixed point of .

Theorem 24. Suppose that the following hypotheses hold:(i) is nondecreasing and is a strictly increasing function such that and for any ;(ii)one of the following conditions is satisfied:(a)for all , , and , we have (b)there exists a continuous function such that (iii)Then the integral equation (27) has a solution in .

Proof. Since , is a -norm of -type and . Let If , for all , , then . Since is nondecreasing, we have . Hence, . Thus, is generalized -admissible. Let . Then . Hence, and for all . Suppose that is a sequence in such that and for all , , and . Then . Thus, and for all and .
We now prove that , for . Suppose that or . Then . Hence, the above inequality holds.
If , then for all . Here we distinguish two cases.
Case (a). If , then .
If , then we have , , and . By (ii), we get , , and . Hence, .
Case (b). For any , , and , we have Hence, , for all , , and . Then, where . Thus, , for .
In all case, we obtain , for all , where is the same as the one in Definition 20. Thus, all the required conditions of Theorem 21 are satisfied. Therefore, has a fixed point ; that is, is a solution of the integral equation (27).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank the editor and the referees for their valuable comments and suggestions. The research was supported by the National Natural Science Foundation of China (11361042, 11326099, 11461045, and 11071108) and the Provincial Natural Science Foundation of Jiangxi, China (20132BAB201001, 20142BAB211016, and 2010GZS0147).