Some Random Fixed-Point Theorems for Weakly Contractive Random Operators in a Separable Banach Space

Benkirane, Kenza; Adraoui, Abderrahim EL; Marhrani, El Miloudi

doi:https://doi.org/10.1155/2021/9484771

International Journal of Mathematics and Mathematical Sciences

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Research Article | Open Access

Volume 2021 | Article ID 9484771 | https://doi.org/10.1155/2021/9484771

Some Random Fixed-Point Theorems for Weakly Contractive Random Operators in a Separable Banach Space

Kenza Benkirane,¹Abderrahim EL Adraoui,¹and El Miloudi Marhrani¹

Academic Editor: Nawab Hussain

Received15 May 2021

Accepted19 Nov 2021

Published07 Dec 2021

Abstract

The aim of this paper is to prove a common random fixed-point and some random fixed-point theorems for random weakly contractive operators in separable Banach spaces. A random Mann iterative process is introduced to approximate the fixed point. Finally, the main result is supported by an example and used to prove the existence and the uniqueness of a solution of a nonlinear stochastic integral equation system.

1. Introduction

The fixed-point theory has been revealed as a very powerful and important tool in the study of different mathematical models expressed in the forms of differential equations [1], integral equations [2, 3], fractional differential equations [4, 5], matrix equations [6], etc. Also, its applications are very useful and interesting in economics, in game theory, in computer science, and in other domains.

Probabilistic functional analysis is one of the essential mathematical disciplines that are applied to solving problems, characterized with uncertainties, known as probabilistic models. The random fixed-point theorems are stochastic generalizations of classical fixed-point theorems which are known as deterministic results and are required for the theory of random equations, random matrices, random partial differential equations, and various classes of random operators.

The theory of random fixed point was initiated by the Prague School of Probability in the 1950s. The random fixed-point theory finds its roots in the work of Špaček [7] and Hanš [8, 9]. They established a stochastic generalization of Banach contraction principle (BCP), and they applied their results to study the existence of a solution of random linear Fredholm integral equations. In 1976, Bharucha-Reid published his review article [10] which has attracted the attention of several researchers and which has led to the development of random fixed-point theory. In 1979, Itoh [11] extended Špaček’s and Hanš’s theorems to multivalued contraction random mappings. The result obtained by Itoh in [11] was applied to solve a random differential equation in Banach space. In the recent past, random differential equations and random integral equations have been solved by random fixed-point theorems (see, for example, [12–16]). For some important contributions in the random fixed-point theory, we invite the reader to consult [17–25] and the references therein.

It is necessary to mention that the BCP is the first fundamental deterministic fixed-point theorem in a metric space.

Theorem 1 (see [26]). If is a complete metric space and is a self-mapping such thatfor all and , then has a unique fixed point.

Among the generalizations of this principle, we find the following theorem established in 2001 by Rhoades.

Theorem 2 (see [27]). If is a complete metric space and is a self-mapping such thatfor all , and is a continuous and nondecreasing function such that if and only if , then has a unique fixed point.

Several interesting weak contractions were considered in various frameworks (see, for example, [28–31] and references therein). Among all these weak contractions, we are interested in the one studied by Eslamian and Abkar in [32]. We state the result in the following.

Theorem 3 (see [32]). If is a complete metric space and is a -weakly contractive self-mapping, i.e.,then has a unique fixed point.Here, the three functions , called control functions, satisfy the following conditions:(a) and are continuous(b) is lower semicontinuous(c) is increasing(d)For , if and only if (e)For all , if and only if (f)For all , In this study, we prove a common random fixed-point theorem and some random fixed-point theorems for random -weakly contractive operators in a separable Banach space, where the three control functions satisfy all the conditions except the condition (d) which is replaced by the weak condition:

As an application, we show the existence and the uniqueness of a random solution for a system of nonlinear integral equations. To prove our main results, we need to recall the following concepts and results. For more details, the reader may consult [33, 34].

2. Preliminaries

Let be a separable Banach space, be the -algebra of all Borel subsets of , and be a complete probability measure space with the measure and be the -algebra of subsets of . Let be a nonempty subset of .

Definition 1 (see [33, 35]). (i)A mapping is said to be a random variable with values in if the inverse image under the mapping of every Borel set of belongs to , that is, for all (ii)A mapping is called a random variable with values in if for all

Definition 2 (see [33, 34]). (i)A mapping is said to be a -valued random variable if it is constant on each of a finite number of disjoint sets and equal to 0 on .(ii)A mapping is said to be a simple random variable if it is finitely valued.(iii)A mapping is said to be a strong (or Bochner) random variable if there exists a sequence of simple random variables which converges to almost surely, that is, there exists a set with such thatNext, we introduce the notion of a weak random variable.

Definition 3 (see [33]). A mapping is said to be a weak (or Pettis) random variable if the functions are real-valued random variables for each , where denotes the first normed dual space of .

Remark 1 (see [33]). In this work, we restrict our attention to the case where is a separable Banach space. In this setting, the concept of weak and strong random variables is equivalent.
The following definition of the mode of convergence for Banach space-valued random variables, which we use in the sequel, is borrowed from [33].

Definition 4. Let and be -valued random variables. The sequence converges to in strongly almost surely if there exists a set with such thatWe recall the following results from the study by Joshi and Bose ([34], Theorem 6.1.2).

Theorem 4. Let and be two strong random variables and and be two constants. Then, the following statements hold:(i) is a strong random variable(ii)If is a real-valued random variable and is a strong random variable, then is a strong random variable(iii)If is a sequence of strong random variables converging strongly to almost surely, then is a strong random variable

Definition 5 (see [35]). (i)A mapping is said to be random operator if for each , the mapping is measurable(ii)A random operator is continuous if the set of all for which is continuous has measure oneThroughout this paper, we denote as the set of all valued random variables and we adopt the following definition of the random fixed point given by Joshi and Bose in [34].

Definition 6. Let . is said to be a random fixed point of if

3. Main Results

In this section, we prove a common random fixed-point theorem and some random fixed-point theorems for weakly contractive mappings in a separable Banach space.

Definition 7. Let be a separable Banach space and be a complete probability measure space. The mapping is called a -weakly contractive random operator if satisfies the following inequality:almost surely, for all .

Theorem 5. Let be a separable Banach space, be a nonempty closed subset of , and be a complete probability measure space. Let be two continuous random operators satisfyingalmost surely, for all . Then, there exists a unique common random fixed point of and .

Proof. Let . Let and . Since and are two continuous random operators in a separable space, it follows that and are in . Now, consider the sequence defined, for each , by and .
By induction, is a sequence in . Consider the set such thatFor all , we denote by the set of elements such thatLet . As stated, and , for each . Then, . Let . For all , we haveThis implies thatIndeed, let us assume that there exists such thatSince is increasing, we getThen,which is a contradiction. Hence, for all , we haveIt follows that the sequence is decreasing and consequently there exists such thatSinceand by using the continuity of and and the lower semicontinuity of , we obtain that , which is a contradiction unless .
Hence,Now, fix in and let us prove that is a Cauchy sequence in . For this, it is sufficient to show that the subsequence is a Cauchy sequence. If we assume the contrary, thenFurthermore, corresponding to , we can choose in such a way that it is the smallest integer with satisfyingConsequently,By using the triangular inequality, we obtainHence,Also,Letting , we obtainBy the same way, we haveThen,We have, for all ,By passing to the upper limit, we get , which is a contradiction since . This shows that is a Cauchy sequence in , for each . Using (20), it is easy to check that is a Cauchy sequence in , for each .
Since is a closed subset of the Banach space , then is complete, which implies that, for all , the sequence converges by norm in . Let be the mapping such that , for each . Since the sequence converges strongly almost surely to , then, according to ([33], Theorem 1.6), is a -valued random variable.
Let . For all , we have and .
Since and by using the continuity of and , we get , for each .
Hence,It means that is a common random fixed point of and .
To prove the uniqueness of this common fixed point, let be another common random fixed point of and . Consider the two setsThen, . Let . For each , we haveThis implies that . Therefore, almost surely. This proves the uniqueness of the common random fixed point of and .
If in Theorem 5, we obtain the following random fixed-point theorem for -weakly contractive random mapping.

Corollary 1. Let be a separable Banach space, be a nonempty closed subset of , and be a complete probability measure space. Let be a continuous -weakly contractive random mapping. Then, there exists a unique random fixed point of .

Example 1. Let with the norm 1 defined, for all , as follows: and . Let be a -algebra of Lebesgue measurable subsets of . Consider the three functions defined for all as follows:Consider the random operator defined by , where and .
Let . We haveThen,Consequently, for all and for each ,All conditions of Corollary 1 are satisfied and has a random fixed point which isIn Corollary 1, if , for all , we obtain the following corollary which is an improvement of ([36], Theorem 5.2) in a separable Banach space.

Corollary 2. Let be a separable Banach space, be a nonempty closed subset of , and be a complete probability measure space. Let be a continuous random operator satisfyingalmost surely, for all . Then, there exists a unique random fixed point of .

In Corollary 1, if , we obtain the following random fixed-point theorem for -weakly contractive random mapping.

Corollary 3. Let be a separable Banach space, be a nonempty closed subset of , and be a complete probability measure space. Let be a continuous random operator satisfying the following condition:almost surely, for all . Then, there exists a unique random fixed point of .

In Corollary 1, if and , for all and for some , we obtain the following random fixed-point theorem for Banach’s contraction.

Corollary 4. Let be a separable Banach space, be a nonempty closed subset of , and be a complete probability measure space. Let be a continuous random operator satisfying the following condition:almost surely, for all and . Then, there exists a unique random fixed point of .

4. Random Mann Iteration Scheme

In the following, we investigate the convergence of random Mann iteration scheme applied to a -weakly contractive random operator.

Definition 8. (random Mann iteration scheme [35]). Let be a random operator, where is a nonempty convex subset of a separable Banach space . A random Mann iteration scheme is the sequence of -valued random variables defined, for all , bywhere and , for all .
In particular, if , for all , the sequence is said to be a random Picard iteration scheme.

Theorem 6. Let be a separable Banach space, be a nonempty closed convex subset of , and be a complete probability measure space.

Let be a continuous -weakly contractive random operator. Assume that is convex. Then, the following two statements hold:(i)There exists a unique random fixed point of (ii)The random Mann iteration scheme converges strongly almost surely to the unique random fixed point of

Proof. (i)From Corollary 1, has a unique random fixed point . Consider the set . Then, .(ii)Consider the set .Let be the set of elements such thatAs stated, , where .
Let . We claim that, for all ,Indeed, let us assume that there exists such thatSince is increasing, we getThis is a contradiction, since for each , .Hence, for all ,Then, for all ,It follows that the sequence is decreasing and consequently there exists such thatSince is a random Mann iteration scheme, we have, for all ,Then,Since is nondecreasing and convex,Then,
By passing to the upper limit, we obtainThen,This is a contradiction, since for each , . Then, .
Then, for all ,Consider the setSince and , then . This shows that the sequence of the -valued random variable converges strongly almost surely to the unique random fixed point .

5. Applications to Nonlinear Stochastic Integral Equations System

In this section, we give an application of Theorem 5 to show the existence and the uniqueness of a solution of a nonlinear stochastic integral equations system (NSIE) presented as follows:where we have the following:(a) is the locally compact real space with the usual norm of reals and is the Lebesgue measure on (b), where is the supporting set of the probability measure space (c)For all , and are two unknown elements in (d) is the stochastic free term defined for (e) is the stochastic kernel defined for and in (f) and are two real-valued functions

Remark 2 (see [37]). The topological space is the union of a countable family of compact subsets having the properties that and that for any other compact set in , there is a which contains it.
Let be the space of all continuous functions from into the space with the topology of uniform convergence on compact sets of . Note that is a locally convex space (see [38]) and so it can be endowed with a topology induced by a countable family of seminorms defined by , for each and .
Here,Note that furthermore, since is complete, is complete with respect to this topology.
We assume that, for each pair , and denote the norm in byAlso, we suppose the following:(i)For almost all , the function is continuous from into (ii)The function is -integrable, for each in and almost all (iii)There exists a real-valued function defined -a.e. on , such that is -integrable and for each pair ,Consider a random operator defined on such that, for all and ,However, for each , the function is -integrable; then, a.s. From (iii) and by using Lebesgue’s dominated convergence theorem, is a continuous in mean square, so a.s.
Let and be two Banach spaces. The pair is said to be admissible with respect to the linear operator if a.s.

Lemma 1 (see [39]). (1)The linear operator is continuous from into itself a.s.(2)If are two Banach spaces stronger than the space such that is admissible with respect to , then is continuous from to a.s.

Definition 9. By a random solution of NSIE, we will mean a pair of functions in which satisfies the two equations of NSIE -a.e.
Let . We consider the two functions defined, for each and , by and
Let be a positive real number. Consider the ball is centered on and the neutral element in is

Theorem 7. The following conditions hold:(1) and are Banach spaces stronger than so that is admissible by respect to the integral operator imposed by (66).(2)The functions and maps from into and there exists three control functions , , and such that for any ,(3).(4) a.s. andwhere the norm of is denoted by .

Then, if the above conditions hold, system NSIE admits a unique solution in .

Proof. Consider the mappings such that for all ,Let . Sincethen a.s. By the same argument, we prove that
Let . We haveThen,which shows that and are -weakly contractive mappings almost surely on . Then, the two operators are -weakly contractive random operators. Therefore, by Theorem 5, there is a unique common random fixed point of random operators and , which is the unique stochastic solution of NSIE. This completes the proof.

6. Conclusion

In this paper, we have the following: Theorem 5 is a random generalization of the main result of [32] in a separable Banach space Corollary 1 generalizes and improves ([36], Theorem 2.5) in the setting of Banach spaces Corollary 4 is a random version of Banach contraction principle in a separable Banach space Theorem 6 extends ([36], Theorem 5.3) to -weakly contractive random operators Theorem 7 shows the utility of our main result in solving a system of nonlinear stochastic integral equations

This work will open the door for other deterministic results that can be randomized, for example, [29].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

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Copyright © 2021 Kenza Benkirane 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.

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