On a Coupled System of Stochastic It ˆ o -Differential and the Arbitrary (Fractional) Order Differential Equations with Nonlocal Random and Stochastic Integral Conditions

: The fractional stochastic differential equations had many applications in interpreting many events and phenomena of life, and the nonlocal conditions describe numerous problems in physics and ﬁnance. Here, we are concerned with the combination between the three senses of derivatives, the stochastic Itˆ o -differential and the fractional and integer orders derivative for the second order stochastic process in two nonlocal problems of a coupled system of two random and stochastic differential equations with two nonlocal stochastic and random integral conditions and a coupled system of two stochastic and random integral conditions. We study the existence of mean square continuous solutions of these two nonlocal problems by using the Schauder ﬁxed point theorem. We discuss the sufﬁcient conditions and the continuous dependence for the unique solution.


Introduction
The existence and uniqueness of solutions to stochastic differential equations driven by Brownian motion have been studied by many authors (see [1][2][3] ).
The results are important since they cover non-local generalizations of fractional stochastic differential equations (FSDE), more applications are arising in fields such as heat conduction, electromagnetic theory and dynamic system (see for example [9,10]).
Many authors have been interested to study the fractional stochastic differential equations see [11][12][13][14] and investigate their results all the time. Let (Ω, G, µ) be a probability space see [15]. The motive of this work is to generalize the scope results of A.M.A. El-Sayed [16,17] on the stochastic fractional operators and the solution of non-local coupled systems of stochastic differential equations see [7,16].
Also, we study the existence of solutions of a coupled system of Itô-differential equation and arbitrary (fractional)orders random differential equation subject to two coupled systems of non-local random and stochastic integral conditions. The effect of random functions and data which ensures the continuous dependence of the solution has been proved.
Let α, β ∈ (0, 1] and T ≥ 1. Here we prove the existence of solutions X, Y ∈ C([0, T], L 2 (Ω)) of the two nonlocal fractional coupled system of the two Itô-deferential and arbitrary orders, differential equations and with the stochastic and random integral conditions and the coupled system of the two stochastic and random non-local integral conditions where X o and Y o are two second order random variables. The existence of solutions X, Y ∈ C(I, L 2 (Ω)) of the problems (1)-(3) and (1)-(2) and (4) are proved. The continuous dependence of the unique solutions X, Y ∈ C(I, L 2 (Ω)) on X o and Y o , h 1 and h 2 and the solution Y ∈ C(I, L 2 (Ω)) on D α X(t) will be studied.
Proof. Consider the set Q 1 such that and define the mapping F 1 where Let Y ∈ Q 1 , then This proves the equicontinuity of the class This proves that the operator F 1 : Q 1 → Q 1 is continuous. Consequently, the closure of {F 1 Q 1 } is compact and (see [15]) and integral Equation (11) has a solution Y ∈ C([0, T], L 2 (Ω)).
Let the set Q 2 be such that and define the mapping F 2 X such that Let X ∈ Q 2 , then Using Theorem 1, we get Applying Schauder Fixed Point Theorem [15], (10) has a solution X ∈ C(I, L 2 (Ω)).
Proof. Let Y 1 and Y 2 be two solutions of (11), then then the solution of (11) is unique. Consequently, the solution of (10) is unique.

Theorem 5. The solution of (1)-(3) depends continuously on D α X(t) = u(t).
Proof. LetX,Ŷ be the solution of By the same way which is complete the proof.

Solutions of the Problem (1)-(2) and (4)
Let Λ = C(I, L 2 (Ω)) × C(I, L 2 (Ω)) be the set of ordered pairs (X, Y), X, Y ∈ C and Define the mapping P(X, Y) = (P 1 Y, P 2 X) where P 1 Y, P 2 X are given by Consider the set Q,

Uniqueness of the Solution
Similarly, we can obtain Hence from (28) and (29) This implies that

Conclusions
Here, we have combined between the three senses of derivatives, the stochastic Itodifferential and the fractional and integer orders derivatives for the second order stochastic process in two non-local problems of the coupled system of the two random and stochastic differential Equations (1) and (2) with the stochastic and random integral conditions (3) and the coupled system of the two stochastic and random nonlocal integral conditions (4) where X 0 and Y 0 are two second order random variables.
The existence of solutions X, Y ∈ C(I, L 2 (Ω)) of the problems (1)-(3) and (1)-(2) and (4) are proved. The unique solution and the sufficient conditions are discussed. The continuous dependence of the solution X, Y ∈ C(I, L 2 (Ω)) on the two random variables X 0 and Y 0 and on the two random functions h 1 and h 2 and the continuous dependence of the solution Y ∈ C(I, L 2 (Ω)) on the fractional order derivative D α X(t) are be studied. An example is given.