On a coupled system of random and stochastic diﬀerential equations with nonlocal stochastic integral conditions

Here we are concerning with two problems of a coupled system of random and stochastic nonlinear diﬀerential equations with two coupled systems of nonlinear nonlocal random and stochastic integral conditions. The existence of solutions will be studied. The suﬃcient condition for the uniqueness of the solution will be given. The continuous dependence of the unique solution on the nonlocal conditions will be proved.

Let T ≥ 1. In this paper we study the existence of solutions (x, y) ∈ C([0, T ], L 2 (Ω)) of the problem of the coupled system of random and stochastic differential equations subject to each one of the two nonlinear nonlocal stochastic integral conditions for all x ∈ L 2 (Ω) and continuous in x ∈ L 2 (Ω) for all t ∈ [0, T ]. There exist two bounded measurable functions k i : [0, T ] → R and two positive constants c i such that Now, integrating the two random and stochastic differential equations (1)-(2) (see [2], [3], [7]- [11]) and using the nonlocal conditions (3)and (4) the following Lemma can be proved.  (3) and (1) -(2) with conditions (4) are given by and respectively.

Existence Theorem
Now, we have the following existence theorem Theorem 1. Let the assumptions (A1) -(A5) be satisfied, then there exists at least one solution (x, y) ∈ X of the problem (1)-(3).
Proof. Let (x n , y n ) ∈ Q be such that (x n , y n ) → (x, y) w.p.1. Using lemmas 1-3, then applying stochastic Lebesgue dominated convergence Theorem [1], we can obtain This proves that the operator F : Q → Q is continuous. Then by Arzela-Ascoli Theorem [1], the closure of F Q is a compact subset of X, then applying Schauder Fixed Point Theorem [1], there exists at least one solution (x, y) ∈ X of the problem (1)- (3) such that x, y ∈ C([0, T ], L 2 (Ω)).

Uniqueness Theorem
Replace the assumptions (A1) and (A2) by (A * 1) and (A * 2) respectively such that for all x ∈ L 2 (Ω) and satisfy Lipschitz condition with respect to the second argument T ] for all x ∈ L 2 (Ω) and satisfy Lipschitz condition with respect to the second argument Remark. Let the assumptions (A * 1) and (A * 2) be satisfied, then we can get where This implies that then (x 1 , y 1 ) = (x 2 , y 2 ) and the solution of the problem (1)-(3) is unique.

Continuous Dependence
Theorem 3. Let the assumptions of Theorem 2 be satisfied. Then the solution (15) of the problem (1) -(3) depends continuously on the two random data (x 0 , y 0 ). Proof. Let (x,ŷ) be the solution of the coupled system such that (x 0 , y 0 ) − (x 0 ,ŷ 0 ) X < δ 1 , then This implies that which completes the proof. Proof. Let (x,ŷ) be the solutions of the coupled system Similarly we can get This implies that which completes the proof.

Existence Theorem
Now, we have the following existence theorem Theorem 5. Let the assumptions (A1) -(A5) be satisfied, then there exists at least one solution (x, y) ∈ X of the problem (1)- (2) and (4).
Proof. Let {(x n , y n )} ∈ Q be such that (x n , y n ) → (x, y) w.p.1. This proves that the operator L : Q → Q is continuous. Then by the Arzela-Ascoli Theorem [1], the closure of LQ is a compact subset of X, then applying Schauder Fixed Point Theorem [1], there exists at least one solution (x, y) ∈ X of the problem (1)- (2) and (4) such that x, y ∈ C([0, T ], L 2 (Ω)).

Conclusions
Here, we proved the existence of solutions of a coupled system of random and stochastic nonlinear differential equations with coupled nonlocal random and stochastic nonlinear integral conditions. The sufficient condition for the uniqueness of the solution have been given. The continuous dependence of the unique solution have been studied.