Random Fixed Point Theorems and Applications to Random First-Order Vector-Valued Differential Equations

El-Ghabi, Adil; Khchine, Abdelmjid; Taoudi, Mohamed Aziz

doi:https://doi.org/10.1155/2021/6648938

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Fixed Point Theory and Applications for Function Spaces

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Volume 2021 | Article ID 6648938 | https://doi.org/10.1155/2021/6648938

Random Fixed Point Theorems and Applications to Random First-Order Vector-Valued Differential Equations

Adil El-Ghabi,¹Abdelmjid Khchine,¹and Mohamed Aziz Taoudi¹

Academic Editor: Nawab Hussain

Received30 Oct 2020

Revised18 Dec 2020

Accepted30 Dec 2020

Published05 Mar 2021

Abstract

In this paper, we establish several random fixed point theorems for random operators satisfying some iterative condition w.r.t. a measure of noncompactness. We also discuss the case of monotone random operators in ordered Banach spaces. Our results extend several earlier works, including Itoh’s random fixed point theorem. As an application, we discuss the existence of random solutions to a class of random first-order vector-valued ordinary differential equations with lack of compactness.

1. Introduction

Random fixed point theorems are stochastic generalizations of deterministic fixed point theorems. In recent years, random fixed point theorems have assumed importance due to their applications to random differential and integral equations (see for instance [1–9] and the references therein). Since Bharucha-Reid published his survey paper [10], numerous works of great importance have appeared and contributed strongly to the enrichment of the random fixed point theory. We quote, for instance, the contributions by Itoh [11], Reich [12], Rybinski [13], Sehgal and Singh [14], Sehgal and Waters [15], Papageorgiou [16], Tan and Yuan [17], Hussain et al. [18], and many others.

In 1979, Itoh [11] proved the random version of Sadovskii’s well-known fixed point theorem [19]. Later on, Tan and Yuan [17] established an interesting result which serves as a bridge that links the random fixed point theory with the deterministic fixed point theory. Specifically, their result ensures the existence of a random fixed point for a continuous random operator under some compactness conditions provided that the corresponding deterministic fixed point problem is solvable. Recently, El-Ghabi and Taoudi [20] developed a new fixed point approach that combines the advantages of the strong topology with the advantages of the weak topology and which enables them to draw new and meaningful conclusions about random fixed points for a given random operator and to handle nonlinear random problems with lack of compactness.

In the present paper, we prove some new random fixed point theorems for (countably) convex-power condensing random operators in Banach spaces. Our results extend, generalize, and unify several known random fixed point results including the famous Itoh random fixed point theorem [11], Theorem 2.1. We should point out here that the concept of convex-power condensing operator was introduced in [21] as a generalization of the concept of condensing operator. We also prove some new random fixed point results for monotone (countably) convex-power condensing random operators in ordered Banach spaces. To illustrate our theoretical results, we investigate the solvability of a broad class of random first-order vector-valued ordinary differential equations. At this point, it is of significance to note that this class of random differential equations was already examined in [11] under some restriction on the constants used in the condition of measure of noncompactness and we have been successful in using our random fixed point results to remove this restriction (see Remark 34).

The paper is arranged as follows. In Section 2, we fix the notation and present some key tools that will be used to prove our main results. Section 3 is devoted to random fixed point theorems for (countably) convex-power condensing random operators and their corollaries. In Section 4, we prove some new random fixed point theorems for a class of monotone random operators in an ordered Banach space. Finally, in Section 5, we use some material from the previous sections to solve the random differential equation where belongs to a bounded and closed interval in the real line , belongs to a set endowed with a -algebra, and the functions have values in a Banach space .

2. Preliminaries

Throughout this paper, denotes a measurable space, where is a nonempty set and is a -algebra of subsets of . Let () be a Banach space, be a nonempty subset of E and be the closed unit ball of . Let , , and denote, respectively, the collections of all nonempty subsets, bounded subsets, and closed subsets of . In addition, let and denote, respectively, the classes of compact and weakly compact subsets of . We say that a mapping is hemicompact if each sequence in has a convergent subsequence whenever as . A mapping is said to be measurable if for every open subset of , . It is important to stress here that, in the case where for each , is measurable if and only if for each (cf. [22], Theorem 3]). A mapping is called a random operator if is measurable for each . A mapping is said to be: (i)a deterministic fixed point of a random operator if for every ,(ii)a random fixed point of a random operator if it is measurable and for every ω ∈ Ω,

Now, we present some basic facts regarding measures of (weak) noncompactness in Banach spaces, which we will be needed in the sequel. It is noteworthy that measures of noncompactness have played a substantial role in nonlinear functional analysis. They are very often used in the theory of functional equations, including ordinary differential equations, equations with partial derivatives, integral and integrodifferential equations, and optimal control theory. We also highlight that the interplay between fixed point theory and measures of noncompactness is very powerful and fruitful (see for instance [19, 23–29] and the references therein).

Definition 1 (see [30–32]). Let be a Banach space. A function is said to be a measure of (weak) noncompactness if it satisfies the following conditions: (1)The family is nonempty and is contained in the set of relatively (weakly) compact sets of E.(2) for all in .(3) for each , where denotes the closed convex hull of (4) for all and .(5)If is a sequence of nonempty (weakly) closed subsets of with bounded and such that , then is nonempty.The family is said to be the kernel of the measure of (weak) noncompactness . Note that the intersection set belongs to (see [33]). A measure of (weak) noncompactness is said to be (6)Full if if and only if is a relatively (weakly) compact set.(7)Nonsingular if for all and (8)Set additive or has the maximum property if (9)Homogeneous if (10)Subadditive if (11)Regular if is full, homogeneous, nonsingular, subadditive, and set additive.

The first important measure of weak noncompactness was defined by De Blasi [34] as follows: for each in . Notice that is regular (see [34]).

Also, the most important regular measures of noncompactness include the Kuratowski measure of noncompactness defined by here . For more details, we refer the reader to the interesting book by Banaś and Goebel [30].

Now, we give a short note about (countably) convex-power condensing mappings with respect to a measure of (weak) noncompactness. Let be a nonempty closed convex subset of and . Let be a bounded mapping (i.e., takes bounded sets into bounded ones). Following [33, 35], for any , we set

Definition 2. Let be a Banach space and be a measure of (weak) noncompactness on . Let be a nonempty closed convex subset of and be a bounded mapping. We say that is (countably) -convex-power condensing about and if for any (countable) bounded subset of with .

It is customary to say “(countably) -condensing” in place of “-convex-power condensing about and 1.” We also say that is (countably) -convex-power condensing if there exist and such that is (countably) -convex-power condensing about and .

3. Random Fixed Point Theorems for (Countably) Convex-Power Condensing Random Operators in Banach Spaces

Throughout this section, is a measurable space, is a Banach space and is a nonempty closed convex subset of . Let be a random operator. We say that is continuous (resp., uniformly continuous, hemicompact) if for each , is continuous (resp., uniformly continuous, hemicompact). If has values in , we say that is (countably) -convex-power condensing if is (countably) -convex-power condensing for every . We say also that is -lipschitzian (resp. a -contraction) if there exists a mapping (resp.) satisfying for all and all .

The following result guarantees the existence of a random fixed point for a continuous random operator provided that the corresponding deterministic fixed point problem is solvable.

Theorem 3 (see [17], Theorem 2.3). Let be a measurable space and be a nonempty separable complete subset of a Banach space . Suppose that is a continuous hemicompact random operator. Then, has a deterministic fixed point if and only if has a random fixed point.

Our main purpose in the immediate sequel is to prove some new random fixed point theorems for countably convex-power condensing mappings. Before proceeding further with the first main theorem, we obtain some auxiliary results.

Lemma 4. Let be a Banach space, be a closed convex subset of and be a regular measure of noncompactness on . Let be a uniformly continuous countably -convex-power condensing operator. If is bounded, then is hemicompact.

Proof. Let be a sequence of elements of such that

We show that for each , where is the ^th iterate of . To see this, notice first that if is uniformly continuous, then so is for each integer . Let be fixed and take any integer n. Then, there exists such that for all with , we have . By virtue of (10), there exists an integer such that for each , we have , so that, whenever . With this in mind, we can easily see that where is the closed unit ball of . From the properties of the measure of noncompactness, it follows that which amounts to

The arbitrariness of yields . Thus, is an increasing sequence and therefore for any integer . Let and such that is countably -convex-power condensing about and . By mathematical induction, we can see that for any integer . Going back to (15), we get

From our hypotheses, we know that and so is relatively compact. Thus, the sequence has a convergent subsequence and consequently is hemicompact.

Now, we are ready to state and prove the following deterministic fixed point theorem.

Theorem 5. Let be a Banach space and be a closed convex subset of . Let be a continuous mapping such that is bounded. Assume further that there are an integer and a vector such that for any countable subset , we have Then, has a fixed point.

Proof. Consider the iterative sequence of sets given by

By mathematical induction, it is easy to show that and is convex compact for each . Let . Plainly, is convex. Furthermore, observe that for each , is separable (since it is compact), and hence, there exists a countable set such that . Let us consider the countable subset of . It is readily apparent that , and so .

We claim that for each ,

We proceed by induction. For the base case , we can easily verify that

Let be fixed and suppose that

Then, so that

This proves our claim. From our hypotheses, we deduce that (and so ) is compact. Furthermore, it is not difficult to see that

An appeal to Schauder’s fixed point theorem yields a fixed point for .

Remark 6. It is of significance to note that condition (18) is fulfilled whenever is (countably) -convex-power condensing about and .

After these preparations, we are now ready to state the main result of this section. This result ensures the existence of a random fixed point for a uniformly continuous random operator under general compactness conditions.

Theorem 7. Let be a Banach space and be a regular measure of noncompactness on . Let be a separable closed convex subset of and let be a uniformly continuous and countably -convex-power condensing random operator. Assume that is bounded for each , then, has a random fixed point.

Proof. Invoking Theorem 5, we infer that the random operator has a deterministic fixed point. In addition, since for each , is a uniformly continuous countably -convex-power condensing mapping and is bounded, then by Lemma 4, is hemicompact. The desired result follows from Theorem 3.

Next, we turn our attention to the case when the boundary condition is of Rothe type. Before making a formal statement of our next random fixed point result, we need some auxiliary results. We first recall the following lemma.

Lemma 8 (see [15], Lemma 2). Let and be measurable mappings. Then, for any , the mapping defined by is measurable.

Recall that, if is a convex neighborhood of the origin in , then the Minkowski functional of is defined on by

It is well known [36], Lemma 5.12.1 that is continuous, subadditive, positively homogeneous, and satisfies

Note also that if has a nonempty interior, then is uniformly continuous.

Now, we state the following Browder-Fan type result [37], which is crucial for our purposes.

Theorem 9. Let be a separable closed convex subset of a Banach space with and let be a regular measure of noncompactness on . Let , be a mapping and be a uniformly continuous random operator. Assume that is bounded and is (countably) -convex-power condensing about and for each . Then, there exists a measurable mapping satisfying for each , where is the Minkowski functional of . Further, if for some , then .

Proof. Let be the Minkowski functional of and define the mapping by for each . Consider the mapping defined by

The reasoning in [15], Theorem 1 yields that is a continuous random operator such that for each , is bounded. We will show that satisfies all conditions of Theorem 7,. To this end, let be fixed. First notice that for any , so that, . In addition, it is easy to check that for all . Since and are uniformly continuous, so is . Furthermore, for all , we obtain

It follows that is uniformly continuous. Next, we will illustrate that is countably -convex-power condensing about and . To see this, take any bounded countable subset of with . We claim that for each . Indeed, for , by (30), we have , so that

Let be fixed and assume that .

Then,

This proves our claim. Thus,

Consequently, is countably -convex-power condensing about and . Now, by applying Theorem 7, we infer that there is a measurable mapping such that for each , we have , that is,

Now, for any , either (a) or (b) . In case of (a), it follows by (29) that and hence by (37), . If (b) holds, then by (29), we have and since for every , it follows by (37) that for every , we have

Then,

Since , the last inequality implies that and

Thus, (a) and (b) provide the desired conclusion.

With these preliminaries, we can proceed to the following interesting random fixed point theorem under Rothe-type boundary conditions.

Theorem 5. Let be a separable closed convex subset of a Banach space with and let be a regular measure of noncompactness on . Let be a mapping and be a uniformly continuous random operator. Let such that for each is bounded and is countably -convex-power condensing about and . In addition, assume that (Rothe boundary condition). Then, has a random fixed point.

Proof. Let be the Minkowski functional of . By Theorem 9, there is a measurable mapping satisfying (28). To prove that is a random fixed point of , it suffices to show that for each . Suppose that for some , we have , then . This implies that . Consequently, there is such that . Hence, by (28), we have

Thus, . This obviously yields that which contradicts the assumption.

Our next concern will be the existence of random fixed points for the sum of two random operators. We should mention here that the need for such random fixed point theorems arose out of the study of random differential equations. Specifically, the inversion of a random differential operator may yield the sum of a contraction and an operator satisfying some compactness conditions. Before, to state our next random fixed point result, we need to recall some basic facts.

Lemma 3 (see [20], Lemma 2.11). Let be a separable Banach space and be a -contraction random operator. Then, (i)for each there is a unique measurable function such that for each (ii)the mapping defined by is a Lipschitzian random operator

Lemma 4 (see [20], Lemma 2.12). Let be a closed and convex subset of a separable Banach space , be a continuous random operator, and be measurable. Then, the mapping is measurable.

Now, we are in a position to state the following random fixed point theorem for the sum of two random operators. For convenient purposes, we list some necessary definitions below for completeness. Let be a nonempty bounded closed convex subset of a Banach space and and be two random operators. Following [24, 32, 38], we set for any and for any subset of :

and

In the case when , we have and

Theorem 6. Let be a nonempty bounded closed convex subset of a separable Banach space and be a regular measure of noncompactness on . Let and be random operators satisfying the following conditions: (i) is uniformly continuous(ii)there are mappings and such that for every and for any countable subset of with ,(iii) is a −contraction(iv)for each , and Then, has a random fixed point.

Proof. Consider the mapping defined by where is as described in Lemma 3. Referring to Lemma 4, we see that every is measurable. Thus, is a random operator. In addition, the fact that is uniformly continuous and is Lipschitzian imply that is uniformly continuous. Now, let be fixed. We claim that . Indeed, by virtue of Lemma 3 (i), we infer that for any we have so that,

Hence, by (iv), we have . This proves our claim.

Furthermore, for any subset of , by (52), we obtain

Let , suppose that

We have

Hence, for each , we have

Thus, by (ii), we have for any countable subset of with . As a result, is countably -convex-power condensing about and . Consequently, is countably -convex-power condensing. Invoking Theorem 7, we see that has a random fixed point which is in turn a random fixed point of .

Corollary 10. Let be a nonempty bounded closed convex subset of a separable Banach space and be a measure of noncompactness on . Let and be random operators satisfying the following conditions: (i) is uniformly continuous(ii)there are mappings , and such that for every and for any countable subset of (iii) is a −contraction(iv)for each , and . Then, has a random fixed point

4. Random Fixed Point Theorems for Monotone Random Operators

In this section, we prove some random fixed point theorems for monotone random operators in ordered real Banach spaces. We combine the advantages of the strong topology (continuity of random operators with respect to the strong topology) with the advantages of the weak topology (the random operators will satisfy some compactness conditions relative to the weak topology) to draw new conclusions about random fixed points for a given monotone random operator. Our results are random versions of the results in [39].

We start this section by recalling some definitions and auxiliary results which will be used further on. Throughout this section, is a measurable space, is a real Banach space, and is a nonempty subset of .

Definition 11. A subset of is called an order cone if it satisfies the following conditions: (i) is closed, nonempty and ,(ii), (iii).An order cone permits to define a partial order in by

Conversely, let be a real Banach space with a partial order compatible with the algebraic operations in , that is,

The positive cone of is defined by

Let with , the order interval is defined by

Definition 12. (i)A subset is said order bounded if there exist such that (ii)The order cone is called normal if and only if there is a number such that

The least positive number (if it exists) satisfying (63) is called a normal constant.

Remarks 13. (1)If is normal, then every order interval is norm bounded.(2)Let be a Hausdorff space and be an ordered Banach space with normal cone . We denote by the Banach space of all continuous E-valued functions on equipped with the usual maximum norm. Plainly, is an ordered Banach space whose positive cone is given by and it is normal

Definition 14. Let be a subset of and be an operator. (i)The operator is said to be increasing if (ii)The operator is said to be decreasing if

Definition 15. Let be an ordered real Banach space with a normal order cone . A sequence is said to be totally ordered if for all .

The following lemmas are quite useful below.

Lemma 16 (see [27], Lemma 2.1). Let be an ordered real Banach space with a normal order cone . Suppose that is a monotone sequence which has a subsequence converging weakly to . Then, converges strongly to . Moreover, if is an increasing sequence then for each ; if is a decreasing sequence then for each .

Lemma 17 (see [39], Lemma 1.8). Let be an ordered real Banach space with a normal order cone . Suppose that is a totally ordered sequence which is contained in a relatively weakly compact set, then it converges strongly in .

Let be two measurable mappings. By (resp., ) on , we mean (resp., ) for every . If , the sector defined by is called a random interval in .

Let be a random operator. For all , , and , we denote by the value at of the iterate of the mapping . We say that a random operator satisfies the condition on if there is a mapping such that for each ,

We say that a random operator satisfies the condition on if satisfies with for every .

Definition 18. Let be a random interval in with . A random operator is called increasing (resp., decreasing) on if for each , is increasing (resp., decreasing) on .

Theorem 19. Let be an ordered separable real Banach space with a normal cone and be a continuous random operator. Let be a random interval in such that is increasing on and satisfies the condition . Assume that

Then, has a random fixed point in which can be obtained by monotone iterative procedure starting from or from .

Proof. Consider the iterate sequence of mappings defined by , for all and all . Let be fixed. We can show by induction that

Let us set . Then, with is a finite set of cardinal . Therefore, it follows from our hypotheses that is relatively weakly compact. Referring to Lemma 17, we see that the sequence converges strongly to some in . The continuity of yields that . Now, by using Lemma 4, we can show (via induction) that is measurable for every . The use of [8], Theorem 1.6 yields that is measurable; and therefore, is a random fixed point of . Similarly, we can show that has random fixed point obtained by monotone iterative procedure starting from .

As a convenient specialization of Theorem 19, we state the following result.

Corollary 20. Let be an ordered separable real Banach space with a normal cone and be a continuous random operator. Let be a random interval in such that is increasing on and satisfies the condition . Assume that

Then, has a random fixed point in which can be obtained by monotone iterative procedure starting from or from .

Proof. Apply Theorem 19 with for every .

Another consequence of Theorem 19 is the following.

Corollary 21. Let be an ordered separable real Banach space with a normal cone and be a nonsingular measure of weak noncompactness on . Let be a continuous random operator and let be a random interval in such that is increasing on and In addition, if there is a mapping , such that for each and any monotone sequence in with , we have Then, has a random fixed point in which can be obtained by monotone iterative procedure starting from or from .

Proof. Let be fixed and let be a monotone sequence of such that with is a finite set of cardinal . By (74), is relatively weakly compact. Thus, satisfies the condition () on . The arbitrariness of yields that satisfies the condition on the random interval . The desired result follows from Theorem 19.

Corollary 22. Let be an ordered separable real Banach space with a normal cone and be a measure of weak noncompactness on .
Let be a continuous random operator and let be a random interval in such that is increasing on and

In addition, assume that there is a mapping such that for each , is relatively weakly compact. Then, has a random fixed point in which can be obtained by monotone iterative procedure starting from or from .

5. Applications to Random Differential Equations

Throughout this section, denotes a measurable space and is a separable real Banach space. Let and . Denote by the Banach space of all continuous mappings equipped with the supremum norm and by the Banach space of all continuously differentiable mappings from to . Let (resp., ) be the Kuratowski measure of noncompactness on .

Definition 23. A mapping is said to satisfy the following conditions: (1) if for each , is continuous and for each is measurable(2) if for every and is measurable for every

If satisfies condition , then is considered a mapping of into . The following result, due to Itoh, discusses the measurability of .

Proposition 24 (see [11], Proposition 4.2). satisfies condition if and only if is measurable as a mapping of into .
Now, we consider the initial random differential equation where and the function is measurable. Our main purpose in the immediate sequel is to show the existence of a random solution to Eq. (77). Before doing so, it is appropriate to clarify the definition of solution we will consider.

Definition 25. A mapping is said to be a random solution of (77) if satisfies conditions and (77).
For convenience of later reference, we list some necessary results below for completeness.

Lemma 26 (see [30]). Let be a Banach space and be the space of continuous functions defined on with values in . If is bounded, then for any , where . Furthermore, if is equicontinuous, then is continuous on , for all t ∈ [0,T], where .

Lemma 27 (see [21]). Let be a Banach space. If is equicontinuous and , then is also equicontinuous in .

Lemma 28 (see [40]). Let be a bounded subset of . Then, for each , there exists a sequence such that

Lemma 29 (see [41], Lemma 2.7). For each , , and , we put

Then, .

Before we proceed further, we present the following useful lemma.

Lemma 30. Let be a Banach space and be a subset of . Let be a mapping such that there exists a nonnegative constant satisfying for any countable subset of . Then, for any subset of , we have

Proof. Let be a subset of and be fixed. In view of Lemma 28, there is a sequence of such that

Further, for each , there is such that and so

Linking (83) and (84), we obtain

The arbitrariness of yields that

If we replace the Kuratowski measure of noncompactness by the Hausdorff measure of noncompactness in Lemma 30, we obtain the following result.

Lemma 31. Let be a Banach space and be subset of . Let be a mapping such that there exists a nonnegative constant satisfying for any countable subset of . Then, for any subset of , we have

Proof. By virtue of [42], Lemma 2.9, the reasoning in the proof of Lemma 30 yields the desired result.

Now, we are in a position to state the following existence result.

Theorem 32. Let and be mappings satisfying the following assumptions: (i)For each is uniformly continuous on (ii)For each , is measurable(iii)There exists a function such thatfor all , , and for any countable subset of , where each (iv)(v) is measurable and

Then, there exists a random solution of (77) on , with .

Proof. Let where . Clearly, is a nonempty bounded closed convex equicontinuous subset of and for each .

Define the mapping by for all Ω, , and . We show that is a random operator. To see this, let be fixed. Since the mapping is measurable for each and the mapping is continuous for each , then the mapping , satisfies the condition . Thus, by Proposition 24, the mapping defined by for every is measurable. Further, observe that mapping for all and is linear and hence, by [1], Theorem 2.14, is measurable. Therefore, is measurable. The arbitrariness of yields that is a random operator. Now, we will prove that fulfills all conditions of Theorem 7. To achieve this, let be fixed. First, by (i) for , there exists such that for all satisfying and . Let such that

We obtain for each . Hence,

Thus, is uniformly continuous and, therefore, by arbitrariness of , is uniformly continuous. Next, we claim that is countably -convex-power condensing. Let be a countable subset of and let . We see that for there exists such that for all satisfying . Then, as independently to .

Hence, is equicontinuous. By using Lemma 26 and assumption (iii), we obtain for each

Let , we know that there is a continuous function : such that

So, for each t ∈ I₁, we have

Thus, where is the mapping defined by for every .

Now, since for , then is equicontinious. Thus, in view of (101), assumption (iii) and Lemmas 26–30, we obtain

Hence, by mathematical induction, for all and , we obtain where .

Using the equicontinuity of the on , we get

Since and , Lemma 29 yields that there exists a positive integer such that . Consequently, for any countable subset of with . Thus, is countably -convex-power condensing about and . Therefore, is countably -convex-power condensing. Now, Theorem 7 guarantees that has a random fixed point . By Proposition 24, the mapping satisfies condition as a mapping of into . Further, by (92), satisfies condition and is a random solution of problem (77).

As a convenient specialization of Theorem 32, we consider the particular case when and . We, therefore, obtain the following result.

Corollary 33. Let be a mapping satisfying the following assumptions: (i)For each is uniformly continuous on (ii)For each , is measurable(iii)There exists a function such thatfor all , , and for any countable subset of , where each (vi)Then, the random differential equation has a random solution on , with .

Remark 34. (1)Theorem 32 improves [11], Theorem 4.3. In our considerations, mapping is not subject to any restriction(2)Theorem 32 remains true if we replace the Kuratowski measure of noncompactness by the Hausdorff measure of noncompactness

In the following, we present a new approach to solve (77) based on Theorem 19. To achieve this, let be an ordered separable real Banach space with a normal cone and be the De Blasi measure of weak noncompactness on and on .

Theorem 35. Let be a mapping satisfying the following assumptions: (i)For each , is continuous on and maps bounded sets into bounded ones(ii)For all and , is measurable(iii)There exist two mappings satisfying condition and the following conditions hold(iv)There is a measurable mapping satisfyingwhenever , (v)there is a mapping such that for each , for any monotone sequence of and for all with we havewhere

Then, the random differential equation (77) has a random solution on which can obtained by monotone iterative procedure starting from or from .

Remark 36. Set , for . Then,
(G₁) For each , (G₂) For each , for any monotone sequence in and for all with , we have where

Proof. The random problem (77) is equivalent to the random problem which is equivalent to the random problem

Let us write (115) as a random integral equation

Define the mapping by for each and any

We will show that is a random operator. To this end, let be fixed. Notice first that for each , the mapping , is measurable, since , is continuous and is measurable. Let and . Using Lemma 8 together with assumptions (ii) and (iv) we infer that and are measurable. Furthermore, since is continuous for each , then the mapping , satisfies the condition . Thus, by Proposition 24, the mapping defined by for every , is measurable. Next, observe that the mapping defined by for all and is linear. Hence, by [1], Theorem 2.14, is measurable. By virtue of Proposition 24, we see that the mapping satisfies condition .

Accordingly, for each , the mapping is measurable. In particular, the mapping is measurable.

Therefore, is measurable. This is true for each . By a simple verification using assumption (i), we can show that is continuous for each . Then, the mapping satisfies condition . Finally by invoking Proposition 24, we conclude that is measurable. The arbitrariness of yields that is a random operator.

Now, we will prove that fulfills all conditions of Theorem 19. This will be achieved in three steps:

Step 1. We show that is continuous. To this end, and . Let be a sequence in which converges to . Then, there is an integer such that

Hence,

Now, we put and

Note that, for all and we have . Furthermore, the continuity of on implies that

The dominated convergence theorem yields

On the other hand, for each ,

Hence,

Therefore, and consequently is continuous at . The arbitrariness of implies that is continuous on .

Step 2. We illuminate that is increasing on the random interval

To do this, let be fixed and with . Using (G₁) we obtain

Now, (117) becomes

Thus,

Therefore, is increasing on .

Step 3. We show that the following conditions hold: To this end, let . We begin by setting Clearly, , and for each , we have Keeping in mind the fact that and , we deduce that for each have: Accordingly, .

Step 4. The reasoning in [39], Theorem 3.2 yields that each satisfies the condition . Therefore, the random operator satisfies the condition .

Now, by applying Theorem 19, we infer that has a random fixed point ξ which can obtained by monotone iterative procedure starting from (or ): . The mapping defined by is measurable as mapping from into . Hence, by Proposition 24, it satisfies the condition . Further, satisfies the random integral equation (116). Then, for every , we have and so satisfies condition . is a random solution of problem (77).

Data Availability

The authors received no data support for this research.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Adil El-Ghabi 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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