Random Fixed Point Theorems of Random Comparable Operators and an Application

Abstract

We introduce the new concept of random comparable operators as a generalization of random monotone operators and prove several random fixed point theorems for such a class of operators in partially ordered Banach spaces. Part of the presented results generalize and extend some known results of random monotone operators. Finally, as an application, we consider the existence of the solution of a random Hammerstein integral equation.

1. Introduction and Preliminaries

In 1950s, Špaček [1] and Hanš [2] initiated the study of random fixed point theories. From then on, to study random fixed point theories had been a central topic of random theories. Moreover, the random theories played a main role in the developing theories of random differential equations and random integral equations and attracted much attention. For example, Sehgal and Waters [3] proved the random Rothe fixed point theorem in 1984 and Mukherjea [4] proved the random Schauder fixed point theorem in 1996. In recent years, random fixed point theories and their applications developed very rapidly (see Lin [5]; Xu [6]; Li and Debnath [7]; Shahzad [8]; Li and Duan [9]; Zhu and Yin [10]; and Kumam [11–26]). In particular in 2005, Li and Duan [9] proved the existence of fixed points for random monotone operators.

In this work, as a generalization of the concept of random monotone operators given by Li and Duan [9], we introduce the concept of random comparable operators and under different contractive conditions, we prove several random fixed point theorems for such operators in partially ordered Banach spaces. Some of our results generalize and extend the main results of Li and Duan [9].

Let be a separable real Banach space, a complete measure space, and a measurable space, where denotes the -algebra of all Borel subsets generated by all open subsets of . Suppose that is a nonempty subset of , and is a cone in . Cone defines a partial order as follows: for , . is called normal if there exists a constant such that implies . Let ; write if and . If , we call the set an order interval in .

is called measurable if for each Borel subset of .

is said to be a random operator if for each fixed , is measurable.

A random operator is said to be continuous if for any , is continuous.

A measurable map is called a random fixed point of a random operator if for almost every .

Definition 1. Suppose that are measurable. and are said to be random comparable if for any , or holds.

Assume that are random comparable. If for any , , then we write ; if , then we write .

Definition 2 (see [3]). A mapping is said to be a random endomorphism of if is an -valued random variable, where denotes the linear bounded operator space of .

Definition 3. A random operator is said to be random comparable if and are random comparable for any random comparable pair , .

Remark 4. The concept of random comparable operators generalizes the concept of random increasing (decreasing) operators given by Li and Duan [9].

Definition 5. A random comparable operator is said to be random -ordered contractive if there exists a random endomorphism such that for each and any measurable mappings , if and are random comparable, then

By the definition of random comparable operators, the following lemmas are easy, so we omit their proofs (wherein, we assume that , , , , are measurable, ).

Lemma 6. If for each , , are random comparable, then and are random comparable and

Lemma 7. If for each , and , and , and and are random comparable, then

Lemma 8. If for each and any positive integer , and are random comparable and , then and are random comparable.

Lemma 9. If for each and any positive integer , and are random comparable, and , , then and are random comparable.

2. Main Results

Theorem 10. Let be a real Banach space and a normal cone in with the normal constant . Let be a continuous random operator satisfying the following:(i)is a random -ordered contractive operator and , ;(ii)there exists such that for any , and are random comparable. Then has a random fixed point . Furthermore, the iterative sequence converges to and .

Proof. For any fixed , set Since and are random comparable, by the given condition (i), for any , and are random comparable and From the normality of , we have . As for each , , then is a Cauchy sequence in . Hence there exists such that . Since is continuous,
Now, we prove that is measurable. Since is measurable, that is, is measurable, from the measurable theorem of complex operators, it is easy to prove that is measurable for all . Hence , being the limit of a sequence of measurable mappings, is also measurable. So is a random fixed point of . Furthermore,

Theorem 11. Let be a real Banach space and a normal cone in with the normal constant , with and an order interval in . Suppose that is a continuous random -ordered contractive operator, where . Then has a unique random fixed point .

Proof. Define iterative sequences as follows: then , . Since and is a continuous -random ordered contractive operator, , are random comparable for each and From the normality of , we have Since , and are random comparable for any and By the normality of again, we get As , it is seen easily that is a Cauchy sequence in . Hence there exists such that . Similarly, we can prove that is also a Cauchy sequence in and there exists such that . It follows from (10) that So . Since is continuous, we have Let . Equation (14) together with (15) implies that In addition, by a proof similar to that of Theorem 10, we get that is a random fixed point of .
Next we prove that is the unique random fixed point of . Suppose that is another random fixed point of . By induction, one can prove that, for any , and are random comparable. Since , by Lemma 8, and are also random comparable. Because is continuous and -random ordered contractive, By the normality of , we have which implies that as . That is .

Remark 12. In the work of Li and Duan [9], the random operator in Theorems needs to be random increasing and random decreasing, respectively. Hence, Theorems 10-11 in this work generalize and extend the results of Theorems in [9], respectively.

Theorem 13. Let be a real Banach space and a normal cone in with the normal constant . Let be a continuous random comparable operator and satisfy the following:(i)there exists such that if and , and , and and are random comparable, then (ii)there exists such that and are random comparable.Then has a random fixed point . Furthermore, the iterative sequence converges to and .

Proof. For any fixed , set
By a similar approach as in the proof of Theorem 10, we obtain that and are random comparable and So From the normality of , we get . Since , is a Cauchy sequence in . Hence there exists such that . The continuity of implies that
By a proof similar to that of Theorem 10, we can easily prove that is measurable, so is a random fixed point . Furthermore,

Theorem 14. Let be a real Banach space and a normal cone in with the normal constant . Suppose that is a continuous random comparable operator satisfying the following:(i)if , are random comparable, then and , and are random comparable and there exists such that (ii)there exists such that for any , , are random comparable and , are random comparable.Then has a random fixed point . Furthermore, the iterative sequence converges to , where and

Proof. For any fixed , put Since and , and are random comparable, then according to (i), for any , and , and are random comparable; that is, for any , and , and are random comparable, and So The normality of implies that Since , is a Cauchy sequence in . Hence there exists such that . By the continuity of , it is easy to see that
Using a proof similar to that of Theorem 10, it is not difficult to prove that is a random fixed point of and

3. Applications

We consider the following random Hammerstein integral equation:

Suppose that(i)the kernel is nonnegative random continuous on satisfying (ii)for any bounded continuous functions , satisfying the following condition, there exists such that for any , Then (33) has a unique random solution .

Proof. Since the kernel is nonnegative random continuous on , is a random operator. Set and ; from (34), we get that . For any , put ; from (36), we obtain that is a random comparable operator. Thus we prove that (33) has a unique random solution by Theorem 11.

Remark 15. The operator defined by (33) is a random increasing operator (of course, it is random comparable), but just from Theorem 2.3 of Li and Duan [9], we cannot get the conclusion because does not satisfy the condition (ii) of Theorem 2.3 of Li and Duan [9]. However, by Theorem 11 of this work, we can easily get the conclusion. Thus, from this application, it is shown that some of the results in this work generalize and extend the corresponding results in [9] again.

Conflict of Interests

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

Acknowledgments

This work was supported by the Natural Science Foundation of Jiangxi Province of China (no. 20132BAB201009) and the National Natural Science Foundation of China (no. 11261039).