Weak contractive inequalities and compatible mixed monotone random operators in ordered metric spaces

The aim of this work is to define the notion of compatible random operators in a partially ordered metric space and prove some coupled random coincidence theorems for a pair of compatible mixed monotone random operators satisfying -weak contractive conditions. These results present random versions and extensions of recent results of Ćirić and Lakshmikantham (Stoch. Anal. Appl. 27:1246-1259, 2009), Choudhury and Kundu (Nonlinear Anal. 73:2524-2531, 2010), Alotaibi and Alsulami (Fixed Point Theory Appl. 2011:44, 2011) and many others.

Random coincidence point theorems are stochastic generalizations of classical coincidence point theorems. Some random fixed point theorems play an important role in the theory of random differential and random integral equations (see [1,2]). Random fixed point theorems for contractive mappings on separable complete metric spaces have been proved by several authors [3-8]. Sehgal and Singh [9] have proved different stochastic versions of the well-known Schauder fixed point theorem. Fixed point theorems for monotone operators in ordered Banach spaces have been investigated and have found various applications in differential and integral equations (see [10-12] and references therein). 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 for the existence and uniqueness of solutions for some boundary value problems (see [13-19]).

Recently Ćirić and Lakshmikantham [20] and Zhu and Xiao [21] proved some coupled random fixed point and coupled random coincidence results in partially ordered complete metric spaces. The purpose of this article is to improve these results for a pair of compatible mixed monotone random mappings and , where F and g satisfy the -weak contractive conditions. Presented results are also the extensions and improvements of the corresponding results in [22-24] and many others.

Recall that if is a partially ordered set and is such that for , implies , then a mapping F is said to be non-decreasing. Similarly, a non-increasing map may be defined. Bhaskar and Lakshmikantham [25] introduced the following notions of a mixed monotone mapping and a coupled fixed point.

Definition 2.1 ([25])

Let be a partially ordered set and . The mapping F is said to have the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any ,

and

Definition 2.2 ([25])

An element is called a coupled fixed point of the mapping if

The concept of the mixed monotone property is generalized in [24].

Definition 2.3 ([24])

Let be a partially ordered set and and . The mapping F is said to have the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, for any ,

and

Definition 2.4 An element is called a coupled fixed point of the mapping and if

Definition 2.5 The mappings F and g, where and , are said to be compatible if

and

whenever , are sequences in X such that and for all are satisfied.

Using the concept of compatible maps and the mixed g-monotone property, Choudhury and Kundu [23] proved the following theorem.

Theorem 2.6Letbe a partially ordered set, and let there be a metricdonXsuch thatis a complete metric space. Letbe such thatandfor all. Letandbe two mappings such thatFhas the mixedg-monotone property and satisfy

for all, for whichand. Let, gbe continuous and monotone increasing andFandgbe compatible mappings. Also, suppose either

(a) Fis continuous or

(b) Xhas the following properties:

(i) if a non-decreasing sequence, thenfor all,

(ii) if a non-increasing sequence, thenfor all.

If there exist, such thatand, then there existsuch thatand, that is, Fandghave a coupled coincidence point inX.

As in [17], let Φ denote all functions which satisfy

1. ϕ is continuous and non-decreasing,

2. if and only if ,

3. , ,

and let Ψ denote all the functions which satisfy for all and .

Alotaibi and Alsulami in [22] proved the following coupled coincidence result for monotone operators in partially ordered metric spaces.

Theorem 2.7Letbe a partially ordered set and suppose there is a metricdonXsuch thatis a complete metric space. Letbe a mapping having the mixedg-monotone property onXsuch that there exist two elementswith

Suppose there existandsuch that

for allwithand. Suppose, gis continuous and compatible withFand also suppose either

(a) Fis continuous or

(b) Xhas the following property:

(i) if a non-decreasing sequence, thenfor alln,

(ii) if a non-increasing sequence, thenfor alln.

Then there existssuch that

that is, Fandghave a coupled coincidence point inX.

Let be a measurable space with Σ being a sigma algebra of subsets of Ω, and let be a metric space. A mapping is called Σ-measurable if for any open subset U of X, . In what follows, when we speak of measurability, we will mean Σ-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 every .

Definition 3.1 Let be a separable metric space and be a measurable space. Then and are said to be compatible random operators if

and

whenever , are sequences in X, such that and for all and are satisfied.

As in [23], let be such that and for all .

Now, we state our main result.

Theorem 3.2Letbe a complete separable partially ordered metric space, be a measurable space, andandbe mappings such that

(i) is continuous for all;

(ii) , are measurable for allandrespectively;

(iii) has the mixed-monotone property for eachand

for all, for whichandfor all.

Supposefor each, gis monotone increasing, andFandgare compatible random operators. Also suppose either

(a) is continuous for allor

(b) Xhas the following property:

If there exist measurable mappingssuch thatandfor all, then there are measurable mappingssuch thatandfor all, that is, Fandghave a coupled random coincidence point.

Proof Let be a family of measurable mappings. Define a function as follows:

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 [26], p.868). Thus, is the Caratheodory function. Therefore, if is a measurable mapping, then is also measurable (see [27]). Also, for each , the function defined by is measurable, that is, .

Now, we shall construct two sequences of measurable mappings and in Θ, and two sequences and in X as follows. Let be such that and for all . Since by an appropriate Filippov measurable implicit function theorem [1,20,28,29], there is such that . Similarly, as , there is such that . Now and are well defined. Again from , there are such that and . Continuing this process, we can construct sequences and in X such that

for all .

We shall prove that

and

The proof will be given by mathematical induction. Let . By assumption we have and . Since and , we have

Therefore, (5) and (6) hold for .

Suppose now that (5) and (6) hold for some fixed . Then, since and and as F is monotone g-non-decreasing in its first argument, from (2) and (4), we have

Similarly, from (3) and (4), as and ,

Now from (7), (8), and (4), we get

and

Thus, by mathematical induction we conclude that (5) and (6) hold for all .

Denote for each

We show that

Since from (5) and (6) we have and , therefore from (4) and (1), we get

Similarly, from (4) and (1), as and ,

By adding (12) and (13), and dividing by 2, we obtain (11).

From (11), since for , it follows that is the monotone decreasing sequence of positive reals. Therefore, there is some such that

We show that . Suppose, to the contrary, that . Taking the limit in (11) when and having in mind that we assume that for all , we have

a contradiction. Thus, .

Now we prove that for each , and are Cauchy sequences. Suppose, to the contrary, that at least one, or , is not a Cauchy sequence. Then there exist an and two subsequences of positive integers , , with

for .

We may also assume

By choosing to be the smallest number exceeding for which (14) holds, such for which (15) holds exists, because . From (14), (15) and by the triangle inequality, we have

Taking the limit as , we get

Inequality (14) and the triangle inequality imply now

Hence,

From (5) and (6), we conclude that and .

Now (1) and (4) imply that

Also, from (1) and (4), as and ,

Inserting (18) and (19) in (17), we obtain

Letting , we get by (16)

a contradiction. Therefore, our supposition (14) was wrong. Thus, we proved that and are Cauchy sequences in X.

Since X is complete and , there exist such that and . Since and are measurable, therefore the functions and , defined by and are measurable. Thus,

Since F and g are compatible mappings, we have by (21)

Next, we prove that

and

Let (a) hold. We have

Taking the limit as , using (4), (21), and (22) and the fact that F and g are continuous, we have

Similarly, from (4), (21), and (23) and the continuity of F and g, we have

Combining the above two results, we obtain

and

for each .

Next, suppose that (b) holds. From (5), (6), and (21), we have is non-decreasing and is non-increasing sequence and

So, from (2) and (3), we have for all

Since F and g are compatible mappings and g is continuous, by (22) and (23) we have

and

Now, we have

Taking the limit as in the above inequality, using (4) and (25), we have

Since the mapping g is monotone increasing, by (1), (24), and the above inequality, we have for all

Using (21) and the property of a φ-function, we obtain

That is,

And similarly, by the virtue of (4), (21), and (26), we obtain

This proves that F and g have a coupled random coincidence point. □

Corollary 3.3Letbe a complete separable partially ordered metric space, be a measurable space, andandbe mappings such that

(i) is continuous for all;

(ii) , are measurable for allandrespectively;

(iii) has the mixed-monotone property for eachand for some

for all, for whichandfor all.

Supposefor each, gis monotone increasing, andFandgare compatible random operators. Also suppose either

(a) is continuous for allor

(b) Xhas the following property:

(i) if a non-decreasing sequence, thenfor alln,

(ii) if a non-increasing sequence, thenfor alln.

If there exist measurable mappingssuch thatandfor all, then there are measurable mappingssuch thatandfor all, that is, Fandghave a coupled random coincidence point.

Proof Taking with in Theorem 3.2, we obtain the result. □

The following theorem presents the stochastic version of Theorem 2.7 and generalizes the recent results in [20].

Theorem 3.4Letbe a complete separable partially ordered metric space, be a measurable space, andandbe mappings such that

(i) is continuous for all;

(ii) , are measurable for alland, respectively;

(iii) andare such thathas the mixed-monotone property for each; and suppose there existand, satisfying conditions of Theorem 2.7, such that

for all, for whichandfor all.

Supposefor each, gis monotone increasing, andFandgare compatible random operators. Also suppose either

(a) is continuous for allor

(b) Xhas the following property:

If there exist measurable mappingssuch thatandfor all, then there are measurable mappingssuch thatandfor all, that is, Fandghave a coupled random coincidence.

Proof Let be a family of measurable mappings. Define a function as follows:

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 [26], p.868). Thus, is the Caratheodory function. Therefore, if is a measurable mapping, then is also measurable (see [27]). Also, for each , the function defined by is measurable, that is, . Now, we shall construct two sequences of measurable mappings and in ⊖, and two sequences and in X as follows. Let such that and for all . Since , by an appropriate Filippov measurable implicit function theorem [1,20,28,29], there is such that . Similarly, as , there is such that . Now, and are well defined. Again, from , there are such that and . Continuing this process, we can construct sequences and in X such that

for all .

We shall prove that

and

The proof will be given by mathematical induction. Let . By assumption, we have and . Since and , we have and . Therefore, (31) and (32) hold for . Suppose now that (31) and (32) hold for some fixed . Then and as F is monotone g-non-decreasing in its first argument, from (28) and (30),

Similarly, from (29) and (30), as and , we have

Now, from (30), (33), and (34), we get

and

Thus, by mathematical induction we conclude that (31) and (32) hold for all .

Therefore,

and

Since and , using (27) and (30), we have

Similarly, since and , using (27) and (30), we also have

Using (39) and (40), we have

From the property (iii) of ϕ, we have

Using (41) and (42), we have

Since ψ is a non-negative function, therefore we have

Using the fact that ϕ is non-decreasing, we get

Let

Now, we show that as . It is clear that the sequence is decreasing; therefore, there is some such that

We shall show that . Suppose, to the contrary, that . Then taking the limit as on both sides of (43) and as for all and ϕ is continuous, we have

a contradiction. Thus, , that is

Now, we will prove that , are Cauchy sequences. Suppose, to the contrary, that at least one of or is not a Cauchy sequence. Then there exists an for which we can find subsequences of positive integers , with such that

for . We may also assume

by choosing in such a way that it is the smallest integer with and satisfying (46). Using (46), (47), and the triangle inequality, we have

Letting and using (45), we get

By the triangle inequality,

Using the property of ϕ, we have

Since , hence and . Using (27) and (30), we get

By the same way, we also have

Putting (50) and (51) in (49), we have

Taking and using (45) and (48), we get

a contradiction. This shows that and are Cauchy sequences.

Since X is complete and , there exist such that and . Since and are measurable, then the functions and , defined by and , are measurable. Thus,

Using the compatibility of F and g and the technique of the proof of Theorem 3.2, we obtain the required conclusion. □

Corollary 3.5Letbe a complete separable partially ordered metric space, be a measurable space, andandbe mappings such that

(i) is continuous for all;

(ii) , are measurable for allandrespectively;

(iii) has the mixed-monotone property for each; and suppose there existandsuch that

for all, for whichandfor all.

Supposefor each, gis continuous and monotone increasing, andFandgare compatible mappings. Also suppose either

(a) is continuous for allor

(b) Xhas the following property:

(i) if a non-decreasing sequence, thenfor alln,

(ii) if a non-increasing sequence, thenfor alln.

If there exist measurable mappingssuch thatandfor all, then there are measurable mappingssuch thatandfor all, that is, Fandghave a coupled random coincidence.

Proof Take in Theorem 3.4. □

Corollary 3.6Letbe a complete separable partially ordered metric space, be a measurable space, andandbe mappings such that

(i) is continuous for all;

(ii) , are measurable for allandrespectively;

(iii) has the mixed-monotone property for each; and suppose there existssuch that

for all, for whichandfor all.

Supposefor each, gis monotone increasing, andFandgare compatible random operators. Also suppose either

(a) is continuous for allor

(b) Xhas the following property:

(i) if a non-decreasing sequence, thenfor alln,

(ii) if a non-increasing sequence, thenfor alln.

If there exist measurable mappingssuch thatandfor all, then there are measurable mappingssuch thatandfor all, that is, Fandghave a coupled random coincidence point.

Proof Take in Corollary 3.5. □

Remark 3.7 By defining as for all in Theorem 3.2-Corollary 3.6, we obtain corresponding coupled random fixed point results.

The authors declare that they have no competing interests.

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

The first and second author gratefully acknowledge the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.

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