Chinese Journal of MathematicsVolume 2014 (2014), Article ID 416794, 14 pageshttp://dx.doi.org/10.1155/2014/416794

Research Article

On Quadruple Random Fixed Point Theorems in Partially Ordered Metric Spaces

R. A. Rashwan and D. M. Al-Baqeri

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 Therefore, by the triangle inequality, (50), and (17), we have and since , we have Letting and by (49), we get . Hence Similarly, we can show that for all .Thus we showed that is a quadruple random coincidence of and .

As a consequence of Theorem 13, we can obtain the following theorem.

Theorem 14. Let be a complete separable metric space, and let be a measurable space, and has the mixed monotone property such that (1) are continuous for all ,(2) are measurable for all ,(3)there exists a such that satisfies the following: for all for which , , , and . 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, has a quadruple random fixed point.

Proof. Taking with and defining by for all in Theorem 13, then we obtain Theorem 14.

In the following we prove the uniqueness of quadruple random fixed point.

We say that is equal to if and only if , , , and .

Theorem 15. In addition to the hypothesis of Theorem 13, assume that for all measurable mappings and , there exist measurable mappings such  that is comparable to Then and have a unique quadruple common random fixed point.

Proof. By Theorem 13, the set of quadruple common random fixed points of and is not empty. Suppose that are quadruple common random fixed points of and ; that is, We prove that and are equal. By assumption, there exists such that is comparable to Define sequences , , and such that , , , and for all , and Since (69) is comparable with (68), we can assume that ; then, we get Using (72) and (17) Set From (73) which implies Since and then for all . Therefore (69) implies that Similarly, we can show that Combining (77) and (78) and using triangle inequality, Therefore By commutativity of and and using (67), we get Set Then, (81) becomes Thus, is a quadruple common random coincidence point of and . Putting in (80), we have From (82), (84), and (83), we have Thus, is a quadruple common random fixed point of and . Due to (80), it is unique.

4. Examples

In this section we give some examples to show that our results are effective.

Example 16. Let with the usual ordering and usual metric. Let and let   be the sigma algebra of Lebesgue’s measurable subset of .Define and as follows., , , and for all . We will check that the contraction (17) is satisfied for all satisfying , , , and for all .The left-hand side (L.H.S.) of (17) is The right-hand side (R.H.S.) of (17) is Then .; that is, contraction (17) is satisfied. It is obvious that the other hypotheses of Theorem 13 are satisfied. We deduce that (0, 0, 0, 0) is the unique quadruple common random fixed point of and .

Example 17. Let with the usual ordering and usual metric. Let and let   be the sigma algebra of Lebesgue’s measurable subset of .Define , and as follows: L.H.S R.H.S. Thus it is verified that the functions , , and satisfy all the conditions of Theorem 13, and (0, 0, 0, 0) is the quadruple random coincidence of and in.

Conflict of Interests

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

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