Chinese Journal of Mathematics

Volume 2014, Article ID 416794, 14 pages

http://dx.doi.org/10.1155/2014/416794

## On Quadruple Random Fixed Point Theorems in Partially Ordered Metric Spaces

Department of Mathematics, University of Assiut, P.O. Box 71516, Assiut, Egypt

Received 18 July 2013; Accepted 8 October 2013; Published 30 January 2014

Academic Editors: F. Bobillo, M. Coppens, Z. Gong, and S. Zhang

Copyright © 2014 R. A. Rashwan and D. M. Al-Baqeri. 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

We prove some quadruple random coincidence and quadruple random fixed point theorems under a set of conditions. We give examples to support our result. Our results are a generalization of the recent paper of Ćirić and Lakshmikantham (2009).

#### 1. Introduction

Random nonlinear analysis is an important mathematical discipline which is mainly concerned with the study of random nonlinear operators and their properties and is needed for the study of various classes of random equations. Random fixed point theorems are stochastic generalizations of classical fixed point theorems. Random fixed point theorems for contraction mappings on separable complete metric spaces have been proved by several authors [1–12].

Existence of fixed points in partially ordered metric spaces was first investigated by Turinici [13], where he extended the Banach contraction principle in partially ordered sets. In 2004, Ran and Reurings [14] presented some applications of Turinici’s theorem to matrix equations.

Bhaskar and Lakshmikantham [15] introduced the notion of a coupled fixed point and proved some coupled fixed point theorems for mappings satisfying a mixed monotone property. They discussed the problem of uniqueness of coupled fixed point and applied their theorems to problems of existence and uniqueness of solution for a periodic boundary value problem. Lakshmikantham and Ćirić [16] introduced the concept of mixed -monotone mapping and proved coupled coincidence and coupled common fixed point theorems for commuting mappings, extending the theorems due to Bhaskar and Lakshmikantham [15]. Recently, Ćirić and Lakshmikantham [17] studied coupled random coincidence and coupled random fixed point theorems for a pair of random mappings and , where is a complete separable metric space and is a measurable space, under some contractive conditions. Very recently, Berinde and Borcut [18] introduced the concept of tripled fixed point and proved some related theorems. In a natural fashion, Karapinar and others [19–23] used the concept of quadruple fixed point and proved some fixed point theorems on the topic.

Following the above studies, we establish the existence and uniqueness of quadruple random coincidence and quadruple random fixed point theorems for a pair of random mappings which extend Theorems 2.2 and 2.3 of Ćirić and Lakshmikantham results [17].

#### 2. Preliminaries

The concept of a mixed monotone property of the mapping has been introduced by Bhaskar and Lakshmikantham [15] by the following definitions.

*Definition 1 (see [15]). *Let be a partially ordered set and . The map has the mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in ; that is, for any ,

*Definition 2 (see [15]). *An element is called a coupled fixed point of a mapping if

The concept of mixed monotone property is generalized to the concept of a mixed -monotone property in [16].

*Definition 3 (see [16]). *Let be a partially ordered set and and . The mapping is said to have the mixed -monotone property if is monotone -nondecreasing in and is monotone -nonincreasing in ; that is, for any ,

It is clear that if is the identity mapping, then Definition 3 reduces to Definition 1.

*Definition 4 (see [16]). *An element is called a coupled coincidence point of a mapping and if

Berinde and Borcut [18] introduced the following partial order on the product space :
where . Regarding this partial order, we state the following definition.

*Definition 5 (see [18]). *Let be a partially ordered set and . The map has the mixed monotone property if is monotone nondecreasing in and and is monotone nonincreasing in ; that is, for any ,

*Definition 6 (see [18]). *An element is called a tripled fixed point of a mapping if

Karapinar [19] introduced the concept of quadruple fixed point. He introduced the following partial order on the product space :
where . Regarding this partial order, we state the following definition.

*Definition 7 (see [19]). *Let be a partially ordered set and . The map has the mixed monotone property if is monotone nondecreasing in and and is monotone nonincreasing in ; that is, for any ,

*Definition 8 (see [19]). *An element is called a quadruple fixed point of a mapping if

*Definition 9 (see [20]). *Let be a partially ordered set and and . The map has the mixed -monotone property if is monotone -nondecreasing in and and is monotone -nonincreasing in ; that is, for any ,

*Definition 10 (see [20]). *An element is called a quadruple coincidence point of a mapping and if

*Definition 11 (see [20]). *Let and be mappings. We say and are commutative if
for all .

Let denote the all functions which are continuous and satisfy that(i),
(ii) for each .Let be a measurable space with , a sigma algebra of subsets of , and let be a metric space. A mapping is called measurable if for any open subset of . A mapping is said to be random mapping if for each fixed , the mapping is measurable. A measurable mapping is called a random fixed point of the random mapping if for each . A measurable mapping is called a random coincidence of and if for each .

Ćirić and Lakshmikantham in [17] proved coupled random fixed point theorems for a pair of random mappings and . They proved new results for random mixed -monotone mappings, which extend the corresponding results for deterministic mixed monotone mappings of [16].

Theorem 12 (see [17]). *Let be a complete separable metric space, let be a measurable space, and let and be mappings such that *(1)*, are continuous for all ,*(2)*, are measurable for all and , respectively,*(3)* and are such that has the mixed -monotone property and
**for all for which and for all . Suppose for each and is continuous and commutes with and also suppose either*(a)* is continuous or*(b)* has the following property:(1) if a nondecreasing sequence then for all ,(2)if a nonincreasing sequence then for all .*

*If there exist measurable mappings such that*

*for all , then there are measurable mappings such that*

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

#### 3. Main Results

The following theorem is our main result.

Theorem 13. *Let be a complete separable metric space, and let be a measurable space and . Let and be mappings such that *(1)*, are continuous for all ,*(2)*, are measurable for all and , respectively,*(3)* and are such that has the mixed -monotone property and
**for all for which , , , and for all . Suppose for each and is continuous and commutes with and also suppose either*(a)* is continuous or*(b)* has the following property:(1) if a nondecreasing sequence then for all ,(2)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 quadruple 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 Wagner [11], page 868). Thus, is the Caratheodory function. Therefore, if is a measurable mapping, then is also measurable (see [9]). Also, for each , the function defined by is measurable; that is, .

Now, we will construct four sequences of measurable mappings , , , and in and four sequences , , , and in as follows: by assumption (18), let such that
for all . Since , then by a sort of Filippov measurable implicit function theorem [1, 5, 6, 24], we can choose such that
Again taking into account that and continuing this process, we can construct sequences , , , and in such that
We claim for all that
Indeed, we will use mathematical induction. By using (18) and (21) together, we obtain
for all . Therefore (23) holds for .

Suppose that (23) holds for some . Then, since has the mixed -monotone property and by (22) we have
Thus, (23) holds for all .

Denote
We show that
Due to (17), (22), and (23)
Similarly,
Summing up (28)–(31) and dividing by 4, we get (27). Since for all , then for all ; that is, is the monotone decreasing sequence of positive reals and so there is some such that
We will prove that . Suppose, to the contrary, that . Taking the limit in (27) as and using the assumption that for each , we obtain
which is a contradiction; thus .

Now, we will prove that , , and are Cauchy sequences. Suppose, to the contrary, that at least one of , , , or is not Cauchy sequence. Then, there exists an for which we can find subsequences of positive integers , with such that
Further, corresponding to , we can choose such that it is the smallest integer with and satisfying (34). Then
By triangle inequality with (34) and (35), we get
Letting in (36), we get
Again using triangle inequality,
Since , then from (23), we have
Hence from (17) and (22), we obtain
By the same way, we have
Inserting (40)–(43) in (38), we get
Letting , we get
a contradiction. This shows that , , , and are Cauchy sequences. Since is complete and , then there exist such that
Since , , , and are measurable, then the functions , , , and , defined by
are measurable too. Thus
Since is continuous, (48) implies that
From (22) and commutativity of and , we have
Now, we will show that if the assumption (a) or (b) holds, then
for all . Suppose (a) holds; then from (48), (49), (50), and the continuity of , we obtain
and similarly
Thus, we proved that is a quadruple random coincidence of and .

Suppose, now, the assumption (b) holds. Since and are nondecreasing and , and also and are nonincreasing and , , then by the assumption (b), we have