Random Coupled Hilfer and Hadamard Fractional Differential Systems in Generalized Banach Spaces

This article deals with some existence and uniqueness result of random solutions for some coupled systems of Hilfer and Hilfer–Hadamard fractional differential equations with random effects. Some applications are made of generalizations of classical random fixed point theorems on generalized Banach spaces.


Introduction
Fractional calculus is an extension of the ordinary differentiation and integration to arbitrary non-integer order.In recent years, this theory has become an important object of investigations due to its demonstrated applications in different areas of physics and engineering (see, for example, [1,2] and the references therein).In particular, time fractional differential equations are used when attempting to describe transport processes with long memory.Recently, the study of time fractional ordinary and partial differential equations has received great attention from many researchers, both in theory and in applications; we refer the reader to the monographs of Abbas et al. [3][4][5], Samko et al. [6], and Kilbas et al. [7], and the papers [8][9][10][11][12][13][14] and the references therein.On the other hand, the existence of solutions of initial and boundary value problems for fractional differential equations with the Hilfer fractional derivative have started to draw attention.For the related works, see for example [1,[15][16][17][18][19][20] and the references therein.
Functional differential equations with random effects are differential equations with a stochastic process in their vector field [21][22][23][24][25].They play a fundamental role in the theory of random dynamical systems.

Preliminaries
We denote by C; the Banach space of all continuous functions from I into R m with the supremum (uniform) norm • ∞ .As usual, AC(I) denotes the space of absolutely continuous functions from I into R m .By L 1 (I), we denote the space of Lebesgue-integrable functions v : I → R m with the norm: By C γ (I) and C 1 γ (I), we denote the weighted spaces of continuous functions defined by: with the norm: and: with the norm: we denote the product weighted space with the norm: Now, we give some definitions and properties of fractional calculus.Definition 1. [4,6,7] The left-sided mixed Riemann-Liouville integral of order r > 0 of a function w ∈ L 1 (I) is defined by: Notice that for all r, r 1 , r 2 > 0 and each w ∈ C, we have I r 0 w ∈ C, and: (I r 1 0 I r 2 0 w)(t) = (I r 1 +r 2 0 w)(t); f or a.e.t ∈ I.
w exists and is in L 1 (I), then: 4. If D γ 0 w exists and is in L 1 (I), then: Then, the Cauchy problem: has the following unique solution: Definition 5. Let A × β R m be the direct product of the σ-algebras A and β R m those defined in Ω and R m , respectively.A mapping T : A random operator is a mapping T : Ω × R m → R m such that T(w, u) is measurable in w for all u ∈ R m , and it expressed as T(w)u = T(w, u); we also say that T(w) is a random operator on R m .The random operator T(w) on E is called continuous (resp.compact, totally bounded, and completely continuous) if T(w, u) is continuous (resp.compact, totally bounded, and completely continuous) in u for all w ∈ Ω.The details of completely continuous random operators in Banach spaces and their properties appear in Itoh [26].Definition 7. [27] Let P (Y) be the family of all nonempty subsets of Y and C be a mapping from Ω into P (Y).A mapping T : {(w, y) : w ∈ Ω, y ∈ C(w)} → Y is called a random operator with stochastic domain C if C is measurable (i.e., for all closed A ⊂ Y, {w ∈ Ω, C(w) ∩ A = ∅} is measurable), and for all open D ⊂ Y and all y ∈ Y, {w ∈ Ω : y ∈ C(w), T(w, y) ∈ D} is measurable.T will be called continuous if every T(w) is continuous.For a random operator T, a mapping y : Ω → Y is called a random (stochastic) fixed point of T if for P-almost all w ∈ Ω, y(w) ∈ C(w) and T(w)y(w) = y(w), and for all open D ⊂ Y, {w ∈ Ω : y(w) ∈ D} is measurable.
called random Carathéodory if the following conditions are satisfied: (i) The map (t, w) → f (x, y, u, w) is jointly measurable for all u ∈ R m and (ii) The map u → f (t, u, w) is continuous for all t ∈ I and w ∈ Ω.
Definition 9. Let X be a nonempty set.By a vector-valued metric on X, we mean a map d : X × X → R m with the following properties: We call the pair (X, d) a generalized metric space with d(x, y) Notice that d is a generalized metric space on X if and only if d i ; i = 1, . . ., m are metrics on X.For r = (r 1 , . . ., r m ) ∈ R m and x 0 ∈ X, we will denote by: the open ball centered in x 0 with radius r and: the closed ball centered in x 0 with radius r.We mention that for generalized metric spaces, the notations of open, closed, compact, convex sets, convergence, and Cauchy sequence are similar to those in usual metric spaces.Definition 10. [28,29] A square matrix of real numbers is said to be convergent to zero if and only if its spectral radius ρ(M) is strictly less than one.In other words, this means that all the eigenvalues of M are in the open unit disc, i.e., |λ| < 1; for every λ ∈ C with det(M − λI) = 0; where I denotes the unit matrix of M m×m (R).(1) b = c = 0, a, d > 0, and max{a, d} < 1.
In the sequel, we will make use of the following random fixed point theorems: Theorem 1. [23-25] Let (Ω, F ) be a measurable space, X a real separable generalized Banach space, and F : Ω × X → X a continuous random operator, and let M(w) ∈ M n×n (R + ) be a random variable matrix such that for every w ∈ Ω, the matrix M(w) converges to zero and: then there exists a random variable x : Ω → X that is the unique random fixed point of F.
Theorem 2. [23-25] Let (Ω, F ) be a measurable space, X be a real separable generalized Banach space, and F : Ω × X → X be a completely continuous random operator.Then, either: (i) the random equation F(w, x) = x has a random solution, i.e., there is a measurable function x : Ω → X such that F(w, x(w)) = x(w) for all w ∈ Ω or (ii) the set M = {x : Ω → X is measurable : λ(w)F(w, x) = x} is unbounded for some measurable function λ : Ω → X with 0 < λ(w) < 1 on Ω.
Furthermore, we will use the following Gronwall lemma: Lemma 1. [23] Let u : I → [0, ∞) be a real function and u(•) a nonnegative, locally-integrable function on I.
Assume that there exist constants c > 0 and r < 1 such that: then, there exists a constant K := K(r) such that: for every t ∈ I.

Coupled Hilfer Fractional Differential Systems
In this section, we are concerned with the existence and uniqueness results of the system (1) and (2).Definition 11.By a solution of the problem (1) and (2), we mean coupled measurable functions (u, v) ∈ C γ 1 × C γ 2 , which satisfy the Equation (1) on I, and the conditions ( The following hypotheses will be used in the sequel. (H 1 ) The functions f i ; i = 1, 2 are Carathéodory.(H 2 ) There exist measurable functions p i , q i : Ω → (0, ∞); i = 1, 2 such that: f or a.e.t ∈ I, and each u i , v i ∈ R m , i = 1, 2. (H 3 ) There exist measurable functions a i , b i : Ω → (0, ∞); i = 1, 2 such that: First, we prove an existence and uniqueness result for the coupled system (1)-( 2) by using Banach's random fixed point theorem in generalized Banach spaces.Theorem 3. Assume that the hypotheses (H 1 ) and (H 2 ) hold.If for every w ∈ Ω, the matrix: converges to zero, then the coupled system (1) and ( 2) has a unique random solution.

Proof. Define the operators N
and: Consider the operator N : C × Ω → C defined by: Clearly, the fixed points of the operator N are random solutions of the system (1) and ( 2).Let us show that N is a random operator on C. Since f i ; i = 1, 2 are Carathéodory functions, then w → f i (t, u, v, w) are measurable maps.We concluded that the maps: w → (N 1 (u, v))(t, w) and w → (N 2 (u, v))(t, w), are measurable.As a result, N is a random operator on C × Ω into C.We show that N satisfies all conditions of Theorem 1.
For any w ∈ Ω and each (u 1 , v 1 ), (u 2 , v 2 ) ∈ C, and t ∈ I, we have: Then, Furthermore, for any w ∈ Ω and each (u 1 , v 1 ), (u 2 , v 2 ) ∈ C, and t ∈ I, we get: Thus, where: Since for every w ∈ Ω, the matrix M(w) converges to zero, then Theorem 1 implies that the operator N has a unique fixed point, which is a random solution of system (1) and ( 2).Now, we prove an existence result for the coupled system (1) and ( 2) by using the random nonlinear alternative of the Leray-Schauder type in generalized Banach space.Theorem 4. Assume that the hypotheses (H 1 ) and (H 3 ) hold.Then, the coupled system (1) and ( 2) has at least one random solution.
Proof.We show that the operator N : C × Ω → C defined in ( 8) satisfies all conditions of Theorem 2. The proof will be given in four steps.
For any w ∈ Ω and each t ∈ I, we have: Since f 1 is Carathéodory, we have: On the other hand, for any w ∈ Ω and each t ∈ I, we obtain: Furthermore, from the fact that f 2 is Carathéodory, we get: Hence, N(•, •, w) is continuous.
Let B R be the ball defined in Step 2. For each t 1 , t 2 ∈ I with t 1 ≤ t 2 and any (u, v) ∈ B R and w ∈ Ω, we have: Furthermore, we get: As a consequence of Steps 1-3, with the Arzela-Ascoli theorem, we conclude that N(•, •, w) maps B R into a precompact set in C.
Theorem 6. Assume that the hypotheses (H 1 ) and (H 3 ) hold.Then, the coupled system (3) and ( 4) has at at least a random solution.