Abstract and Applied Analysis

Volume 2014 (2014), Article ID 484857, 9 pages

http://dx.doi.org/10.1155/2014/484857

Research Article

Coupled Coincidence Points for Mixed Monotone Random Operators in Partially Ordered Metric Spaces

Binghua Jiang,¹ Shaoyuan Xu,² and Lu Shi¹

¹School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China

²Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521041, China

Received 17 December 2013; Accepted 8 April 2014; Published 29 April 2014

Academic Editor: Sehie Park

Copyright © 2014 Binghua Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The aim of this work is to prove some coupled random coincidence theorems for a pair of compatible mixed monotone random operators satisfying weak contractive conditions. These results are some random versions and extensions of results of Karapınar et al. (2012). Our results generalize the results of Shatanawi and Mustafa (2012).

1. Introduction

Random coincidence point theorems are stochastic generalizations of classical coincidence point theorems and play an important role in the theory of random differential and integral equations. Random fixed point theorems for contractive mapping on complete separable metric space have been proved by several authors (see [1–8]). Fixed point theorems for monotone operators in ordered Banach spaces have been investigated and found various applications. Since then, fixed point theorems for mixed monotone mappings in partially ordered metric spaces are of great importance and have been utilized for matrix equations, ordinary differential equations, and the existence and uniqueness of solutions for some boundary value problems (see [9–17]).

Ćirić and Lakshmikantham [18] and Zhu and Xiao [19] proved some coupled random fixed point and coupled random coincidence results in partially ordered complete metric spaces. Moreover coupled random coincidence results in partially ordered complete metric spaces were considered in [20–22]. Following Karapınar et al. [17] and Shatanawi and Mustafa [21], we improve these results for a pair of compatible mixed monotone random mappings and , where and satisfy some weak contractive conditions. Presented results are also referred to the extensions and improve the corresponding results in [19, 21] and many other authors’ work.

2. Preliminaries

Let be a partially ordered set. The concept of a mixed monotone property of the mappings and has been introduced by Lakshmikantham and Ćirić in [16].

Definition 1 (see [16]). Let be a partially ordered set and a mapping. Then the map is said to have mixed -monotone property if is monotone -nondecreasing in and is monotone -nonincreasing in ; that is, for any ,

Definition 2 (see [16]). An element is called a coupled coincidence point of the mapping and if and .

Definition 3 (see [22]). The mappings and are said to be compatible if

where and are sequences in such that and for all being satisfied.

Theorem 4 (see [17]). Let be a partially ordered set and suppose that there exists a metric on such that is a complete metric space. Let and be two mappings such that has the mixed -monotone property and satisfies

for all with and , where and . Let and , , be continuous and let and be compatible mappings. If there exist such that and , then and have a coupled coincidence point in .

Denote as the set of functions satisfying the following:

(i)is continuous,(ii) for all and if and only if .

Let be a measurable space with sigma algebra of subsets of and let be a metric space. A mapping is called -measurable if, for any open subset of , . In what follows, when we speak of measurability, we will mean -measurability. A mapping is called a random operator if, for any is measurable. A measurable mapping is called a random fixed point of a random function , if , for every . A measurable mapping is called a random coincidence of and if for each .

Definition 5 (see [22]). Let be a separable metric space and a measurable space. Then and are said to be compatible random operators if

where and are sequences in such that and for all and for all being satisfied.

Theorem 6 (see [21]). Let be a partially ordered set, a complete separable metric space, and a measurable space. Let and be mappings such that there are two nonnegative real numbers and with such that

for all with and for all . Assume that and satisfy the following conditions:

are continuous, for all , are measurable, for all and , respectively,, for each , is continuous and commutes with and also suppose that either

is continuous or has the following properties:

if a nondecreasing sequence , then , for all ,if a nonincreasing sequence , then , for all .

If there exist measurable mappings such that and , then there are measurable mappings such that and for all ; that is, and have a coupled random coincidence.

Now, we state our main results as follows.

3. Main Results

In this section, we study coupled random coincidence and coupled random fixed point theorems for a pair of random mappings and . Then we will prove some results for random mixed monotone mappings, which are the extensions of corresponding results for deterministic mixed monotone mappings of Karapınar et al. [17].

Theorem 7. Let be a partially ordered set, a complete separable metric space, a measurable space, and and mappings such that

for all with and for all , where and . Assume that and satisfy the following conditions:

are continuous, for all ,, are measurable, for all and , respectively,, for each , is continuous and commutes with and also suppose that either

is continuous or has the following properties:

if a nondecreasing sequence , then , for all ,if a nonincreasing sequence , then , for all .

If there exist measurable mappings such that and , then there are measurable mappings such that and , for all ; that is, and have a coupled random coincidence.

Proof. Let be a family of measurable mappings. Define a function as . Since is continuous, for all , we conclude that is continuous, for all . Also, since is measurable, for all , we conclude that is measurable, for all (see [23, page 868]). Thus, is the Caratheodory function. Thus, if is measurable mapping, then is also measurable (see [24]). Also, for each , the function defined by is measurable; that is, .

Now we are going to construct two sequences of measurable mappings and in and two sequences and in as follows. Let be such that and , for all . Since , by a sort of Filippov measurable implicit function theorem (see [25, 26]), there is such that . Similarly, as , there is such that . Thus and are well defined now. Again, since

there are such that

Continuing this process we can construct sequences and in such that

for all . Now, we use mathematical induction to prove that

for all . Let , and by assumption we have

Since

we have

Therefore, (10) holds for . Suppose (10) holds for some fixed number . Then, since

and is monotone -nondecreasing in its first argument, we have

Also, since and . and is monotone -nonincreasing in its second argument, we have

Thus, from (9), we get

Thus, by mathematical induction, we conclude that (10) holds for all . Now, we prove that and are Cauchy sequences. Let , and, by (6)–(10), we have

which implies that

Similarly, we have

which implies that

From (19) and (21), we get that

Since , for all , by (22), we have

Set , then is a nonincreasing sequence of positive real numbers. Thus, there is such that

Suppose that ; letting in two sides of (22) and using the properties of , we have

which is a contradiction. Hence ; that is,

We will show that and are Cauchy sequences. Suppose, to the contrary, that at least one of or is not a Cauchy sequence. This means that there exists an for which we can find subsequences of and of with such that

Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfies (27). Then,

Using the triangle inequality and (28), we have

By (27) and (29), we obtain

Letting , in the inequalities above, we get

By the triangle inequalities, we have

By the above inequalities and (27), we have

Again, by the triangle inequality, we obtain

Therefore,

Taking in (33) and (35), we have

Since , and . Then, from (6)–(10), we get

Similarly,

From (37) and (38), we arrive at

Letting in the above inequality and using (26), (27), and the properties of , we have

which is a contradiction. This means that and are Cauchy sequences.

Since is complete, for all , there exist the functions and such that

Thus,

Since and are compatible mappings, we have

Suppose at first that assumption (a) holds. Taking the limit as in the following inequalities

and using (9) and the continuity of , , we get

This implies . Similarly, we can show that for each . The proof is complete.

Suppose now that (b) holds. From (9), we have

Since and , we have

Taking in the above inequality and using (46) and the properties of , we have

Hence .

Similarly, one can show that .

The proof is complete.

Remark 8. Taking , for all , , and , we have

where . Obviously, . Moreover, the conditions that

if a nondecreasing sequence , then , for all , if a nonincreasing sequence , then , for all , are weaker than the conditions that is monotone mapping and if a nondecreasing sequence , then , for all , if a nonincreasing sequence , then for all . Therefore, Theorem 7 generalizes Theorem 6 and [18, Theorem 2.2] and the following corollary is obtained.

Corollary 9. Let be a partially ordered set, a complete separable metric space, a measurable space, and and mappings such that

(i) is continuous, for all ,(ii) are measurable for all and , respectively,(iii) has the mixed -monotone property for each and

for all with and for all , where and . Suppose that for each , is monotone, and and are compatible random operators. Also suppose that has the following property:

(a)if a nondecreasing sequence , then , for all ,(b)if a nonincreasing sequence , then , for all .

If there exist measurable mappings such that

then there are measurable mappings such that

for all ; that is, and have a coupled random coincidence.

Remark 10. Comparing with [21, Theorem 2.6], we find that the monotone of is essential. Also the condition that is unnecessary and the proof of case (2) in [21, Theorem 2.6] was irrational. So our Corollary 9 generalizes and improves [21, Theorem 2.6].

Theorem 11. Let be a partially ordered set, a separable metric space, a measurable space, and and mappings such that

(i) are measurable, for all and , respectively;(ii) has the mixed -monotone property for each and

for all with and for all , where and . Suppose that and is complete subspace of for each . Also suppose that has the following property:

(a)if a nondecreasing sequence , then , for all ,(b)if a nonincreasing sequence , then , for all .

If there exist measurable mappings such that

then there are measurable mappings such that

for all ; that is, and have a coupled random coincidence.

Proof. Construct two sequences and as in Theorem 7. According to the proof of Theorem 7, and are Cauchy sequences. Since is complete, there exist , such that

Since is nondecreasing sequence and and is nonincreasing sequence and , by the assumption, we have and such that

On taking in the above inequality and using (57), we obtain

This means that . Similarly, it can be shown that . Thus, and have a coupled coincidence point in .

The proof is complete.

Remark 12. Following Theorem 7 and Corollary 9, we replace the continuity and monotone of , the compatibility of and , and the completeness of by assuming that is a complete subspace of . Moreover, by the measurable space, our random fixed point theorems generalize the main results in [17].

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgment

The research is partially supported by Doctoral Initial Foundation of Hanshan Normal University, China (no. QD20110920).

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