*3.1. 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 {(*xn*, *yn*)} ∈ *Q* be such that

$$(\mathfrak{x}\_{\mathsf{n}}, y\_{\mathsf{n}}) \to (\mathfrak{x}, y) \qquad w.p.1.1$$

Using Lemmas 1–3, then applying stochastic Lebesgue dominated convergence Theorem [9], we can obtain

*L*.*i*.*mn*→∞*L*(*xn*, *yn*)=(*L*.*i*.*mn*→∞*L*1*xn*, *L*.*i*.*mn*→∞*L*2*yn*) = (*L*.*i*.*mn*→∞{*x*<sup>0</sup> − *τ* 0 *<sup>h</sup>*1(*s*, *xn*(*s*))*dW*(*s*) + *<sup>t</sup>* 0 *f*1(*s*, *yn*(*φ*1(*s*)))*ds*}, *L*.*i*.*mn*→∞{*y*<sup>0</sup> − *η* 0 *h*2(*s*, *yn*(*s*))*ds* + *t* 0 *f*2(*s*, *xn*(*φ*2(*s*)))*dW*(*s*)}) = (*x*<sup>0</sup> − *τ* 0 *<sup>h</sup>*1(*s*, *<sup>L</sup>*.*i*.*mn*→∞*xn*(*s*))*dW*(*s*) + *<sup>t</sup>* 0 *f*1(*s*, *L*.*i*.*mn*→∞*yn*(*φ*1(*s*)))*ds*, *y*<sup>0</sup> − *η* 0 *h*2(*s*, *L*.*i*.*mn*→∞*yn*(*s*))*ds* + *t* 0 *f*2(*s*, *L*.*i*.*mn*→∞*xn*(*φ*2(*s*)))*dW*(*s*)) = (*x*<sup>0</sup> − *τ* 0 *<sup>h</sup>*1(*s*, *<sup>x</sup>*(*s*))*dW*(*s*) + *<sup>t</sup>* 0 *f*1(*s*, *y*(*φ*1(*s*)))*ds*, *y*<sup>0</sup> − *η* 0 *h*2(*s*, *y*(*s*))*ds* + *t* 0 *f*2(*s*, *x*(*φ*2(*s*)))*dW*(*s*)) = (*L*1*x*, *L*2*y*) = *L*(*x*, *y*).

This proves that the operator *L* : *Q* → *Q* is continuous.

Then by the Arzela–Ascoli Theorem [9], the closure of *LQ* is a compact subset of *X*, then applying Schauder Fixed Point Theorem [9], 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)).
