A Sequential Random Airy Type Problem of Fractional Order: Existence, Uniqueness and β−Di erential Dependance

In this work, a new class of sequential random di erential equations of Airy type is introduced. The existence and uniqueness criteria for stochastic process solutions for the introduced class are discussed. Some notions on β−di erential dependance are also introduced. Then, new results on the β−dependance are discussed. At the end, some illustrative examples are discussed.


Introduction
The theory of fractional calculus has been distinguished in dierent elds of applied mathematics and many investigations have been stated as modeling, existence of solutions and various methods for solving fractional dierential problems [2,4,5,8,9,10,15,18,20,21,22,24,27,29,33,34]. Recently, the concept of fractional calculus and random dierential equations have appeared as important and interesting subjects; this new random fractional theory has become very interesting to many researchers. To cite some papers related to this subject, we cite [1,36,19,38,39]. To investigate the theory of fractional dierential equations with randomness, we have to use the mean square calculus because of its importance for stochastic processes, see [16,17,31]. Many of random problems have been formulated by the Airy equation (and its solutions called Airy functions) which is given by Z − tZ = 0, t ∈ R.
In [11], the authors have been concerned with the initial value problems for space-time-fractional Airy problem given by: with u(x 1 , 0) = 1 6 x β 1 . In [30], M.D. Ovidio and E. Orsingher have expressed the law of the stable process H v (t), t > 0 in terms of Airy functions. In [28], the authors have been concerned with the M-Wright function in time-fractional diusion process and they have shown that the auxiliary functions can be expressed in terms of the Airy functions. An example from quantum mechanics is given in the paper [26] where the exact solution of Schrodinger equation, for the motion of a particle in a homogeneous external eld, can be expressed in terms of the Airy functions. Solutions of the Schrodinger equation involving the Airy functions are also given in [35]. For some other applications of Airy equations (and Airy functions) in elasticity theory, uid mechanics and quantum physics, the reader is invited to see the research works [3,6,23,25]. We cite also the paper [7], where the authors have studied the following random fractional initial value problem of Airy type: Motivated by the above works, in the present paper we shall study a very important class of random fractional problem that generalizes the classical Airy-type dierential equations both in the random and in the fractional senses. Specically, we will deal with the following sequential random fractional generalized Airy-type problem: where: D (·) represents the mean square derivative in the sense of Caputo, Y (·) is a second random function, f i : J × L 2 (Ω) → L 2 (Ω), i = 1, 2, 3, X i are second random variables i = 0, . . . , n − 1, and A 1 , A 2 , A 3 are also second random variables. a 1 , a 2 , a k are real positive numbers. It is to note that if n = 2, α 1 = α 2 = 1, a 2 = a 3 = 0, then we obtain the standard Airy equation. Under some other considerations on the input data of (2), we can obtain the generalized Airy problem of [7]. These are two reasons that have motivated the study of the above problem. And, to the best of our knowledge, there is no paper dealing with such random Airy type problem. This paper is organized as follows: In section 2, we recall some denitions and lemmas that we need in the rest of the paper. In section 3, we present our main results for problem (2). Section 4 is devoted to introduce new concepts on random data dependence. In section 5, we provide some examples of applications to illustrate our theoretical results. At the end, a conclusion follows.
In view of this lemma, we can easily conrm that

Existence and Uniqueness Criteria
We begin our main result by proving the following random integral lemma.
Lemma 3.1. The random dierential problem (2) has the following random integral representation Proof. To prove the result, we begin by considering the following homogenous linear dierential problem: where, Applying the mean square Riemann-Liouville integral of order α 1 , to (4), we can write Again, thanks to the square Riemann-Liouville integral of order α 2 , we can state that Consequently, Using the same arguments as before, we get the following formula where, γ i ∈ R, i = 1 . . . , n.
Let also introduce the random integral operator H : F → F : To facilitate the fastidious calculation, we consider the following notations and assumptions: (H1) : There are three real positive numbers K 1 , K 2 , K 3 > 0, such that for all Y 1 , Y 2 ∈ L 2 (Ω), t ∈ J, the following inequalities are valid: (H2) : There exist three positive real numbers 0 ≤ r 1 , r 2 , r 3 , such that Now, we prove the existence of a unique stochastic process solution for our above Airy type problem. Proof. To prove this theorem, we shall consider an arbitrary real positive number r, such that We begin rst by showing that HB r ⊂ B r , where So, let t ∈ J, Y ∈ B r . It is clear that by denition, we have Using both (H.1) and (H.2), we can state that With the same arguments, we get Therefore, it yields that So, we obtain On the other side, we can write Thanks to (10) and (11), we can deduce that We have thus proved that HB r ∈ B r . Now, we prove that H is contractive.
which leads to Thanks to (H.1), we have the following estimate Some easy calculation will allow us to state that The inequalities (12) and (13) allow us to say that At the end of this proof, we can conclude that problem (2) has a unique stochastic process solution on J.

Random Data and β−Dependance
Using the introduced norm of the above Banach space, we shall be concerned with introducing some random dependance denitions for the above fractional Airy type problem. Then, we prove some random variables data dependance results for the same problem. To do this, we shall rst consider the following auxiliary problem: We introduce the following rst denition. Denition 4.1. The solution Y of (2) is continuously and β-dierentially dependent on the random data X i , i = 0, . . . , n − 1, n ∈ N * , if At this moment, we are able to present to the reader the following main result. Proof. Let Y andỸ be the unique random solution of (2) and (14), where: We have So, we get Consequently, we obtain With the same arguments as before, we have By the inequalities (22) and (23), we get This leads to where R = max(φ 1 , σ 1 ). The proof is thus complete.

Applications
This section deals with two examples to review the main results by a numerical point of view.
Thanks to Theorem 3.2, the problem (22) has a unique stochastic process solution on J = [0, 7].