**2. Solutions of the Problem (1)–(3)**

Define the mapping (*F*(*x*, *y*))(*t*)=(*F*1*y*, *F*2*x*)(*t*), *t* ∈ [0, *T*] where (*F*1*y*)(*t*), (*F*2*x*)(*t*) are given by the following stochastic integral equations

$$(F\_1 y)(t) = -\chi\_0 - \int\_0^\tau h\_1(s, y(s)) dW(s) + \int\_0^t f\_1(s, y(\phi\_1(s))) ds,\tag{12}$$

$$(\mathsf{F}\_2 \mathsf{x})(t) \quad = \quad y\_0 - \int\_0^\eta h\_2(s, \mathsf{x}(s)) ds + \int\_0^t f\_2(s, \mathsf{x}(\phi\_2(s))) dW(s). \tag{13}$$

Consider the set *Q* such that

$$Q = \{ \mathbf{x}, \mathbf{y} \in \, \mathrm{C}([0, T], L\_2(\Omega)), (\mathbf{x}, \mathbf{y}) \in X : ||(\mathbf{x}, \mathbf{y})||\_X = \max \{ ||\mathbf{x}(t)||\_{2^\star} ||\mathbf{y}(t)||\_2 \} \le r \}$$

Now, we have the following two lemmas

**Lemma 2.** *F* : *Q* → *Q*.

**Proof.** Let *y* ∈ *Q*, *y*(*t*)<sup>2</sup> ≤ *r*1, then

$$\begin{split} \|(\mathbb{F}y)(t)\|\_{2} &\leq \|\mathbb{x}\_{0}\|\_{2} + \|\int\_{0}^{\tau}h\_{1}(s,y(s))dW(s)\|\_{2} + \|\int\_{0}^{t}f\_{1}(s,y(\rho\_{1}(s)))ds\|\_{2} \\ &\leq \|\mathbb{x}\_{0}\|\_{2} + \sqrt{\int\_{0}^{\tau}\|h\_{1}(s,y(s))\|\_{2}^{2}ds} + \int\_{0}^{t}\|f\_{1}(s,y(\rho\_{1}(s)))\|\_{2}ds \\ &\leq \|\mathbb{x}\_{0}\|\_{2} + \sqrt{\int\_{0}^{\tau}(|h\_{1}(s)| + c\_{1}\|y(s)\|\_{2})^{2}ds} + \int\_{0}^{t}(|m\_{1}(S)| + b\_{1}\|y(s)\|\_{2})ds \\ &\leq \|\mathbb{x}\_{0}\|\_{2} + (K + cr\_{1})\sqrt{T} + (M + br\_{1})T < \|\mathbb{x}\_{0}\|\_{2} + (K + cr\_{1})T + (M + br\_{1})T = r\_{1} \\ &\text{where} \end{split}$$

$$r\_1 = \frac{||\varkappa\_0||\_2 + KT + MT}{1 - (b + c)T} > 0.5$$

$$\text{Let } \mathfrak{x} \in \mathbb{Q}\_{\prime} \quad ||\mathfrak{x}(t)||\_{2} \le r\_{2\prime} \text{ then}$$

$$\begin{split} \|(\mathsf{f}\_{2}\mathsf{x})(t)\|\_{2} &\leq \quad \|y\_{0}\|\_{2} + \|\int\_{0}^{\eta}h\_{2}(\mathsf{x},\mathsf{x}(s))ds\|\_{2} + \|\int\_{0}^{t}f\_{2}(\mathsf{x},\mathsf{x}(\phi\_{2}(s)))dW(s)\|\_{2} \\ &\leq \quad \|y\_{0}\|\_{2} + \int\_{0}^{\eta} \|h\_{2}(\mathsf{x},\mathsf{x}(s))\|\_{2}ds + \sqrt{\int\_{0}^{t} \|f\_{2}(\mathsf{x},\mathsf{x}(\phi\_{2}(s)))\|\_{2}^{2}ds} \\ &\leq \quad \|y\_{0}\|\_{2} + \int\_{0}^{\eta} (|k\_{2}(s)| + c\_{2} \|\mathsf{x}(s)\|\_{2})ds + \sqrt{\int\_{0}^{t} (|m\_{2}(t)| + b\_{2} \|\mathsf{x}\|\_{2})^{2}ds} \\ &\leq \quad \|y\_{0}\|\_{2} + (K + cr\_{2})T + (M + br\_{2})T < \|y\_{0}\|\_{2} + (K + cr\_{2})T + (M + br\_{2})T = r\_{2} \end{split}$$

where

$$r\_2 = \frac{||y\_0||\_2 + KT + MT}{1 - (b + c)T} > 0.1$$

Let *r* = max{*r*1,*r*2}, (*x*, *y*) ∈ *Q*, then

$$\begin{aligned} \|F(\mathfrak{x}, \mathfrak{y})\|\_{X} &= \| (F\_{1}\mathfrak{y}, F\_{2}\mathfrak{x}) \|\_{X} \\ &= \max\{ \|(F\_{1}\mathfrak{y})\|\_{C'} \|(F\_{2}\mathfrak{x})\|\_{C} \} < r. \end{aligned}$$

This proves that *F* : *Q* → *Q* and the class of functions {*F*(*x*, *y*)} is uniformly bounded on *Q*.

**Lemma 3.** *The class of functions* {*F*(*x*, *y*)} *is equicontinuous on Q.*

**Proof.** Let *x*, *y* ∈ *Q*, *t*1, *t*<sup>2</sup> ∈ [0, *T*] such that |*t*<sup>2</sup> − *t*1| < *δ*, then

$$\begin{aligned} \|(F\_1 y)(t\_2) - (F\_1 y)(t\_1)\|\_2 &= \|\int\_0^{t\_2} f\_1(s, y(\phi\_1(s))) ds - \int\_0^{t\_1} f\_1(s, y(\phi\_1(s))) ds\|\_2 \\ &\le \int\_{t\_1}^{t\_2} \|f\_1(s, y(\phi\_1(s)))\|\_2 \\ &\le \quad (M + b \|y\|\_C)(t\_2 - t\_1) \end{aligned} \tag{14}$$

This proves the equicontinuity of the class {*F*1*y*} and

$$\begin{split} \|(\mathsf{F}\_{2}\mathsf{x})(t\_{2}) - (\mathsf{F}\_{2}\mathsf{x})(t\_{1})\|\_{2} &= \|\int\_{0}^{t\_{2}} f\_{2}(s, \mathsf{x}(\phi\_{2}(s)))d\mathsf{W}(s) - \int\_{0}^{t\_{1}} f\_{2}(s, \mathsf{x}(\phi\_{2}(s)))d\mathsf{W}(s)\|\_{2} \\ &\leq \sqrt{\int\_{t\_{1}}^{t\_{2}} \|f\_{2}(s, \mathsf{x}(\phi\_{2}(s)))\|\_{2}^{2}ds} \\ &\leq \quad (M+b\|\mathsf{x}\|\_{C})\sqrt{(t\_{2}-t\_{1})}. \end{split} \tag{15}$$

This proves the equicontinuity of the class {*F*1*x*}. Now

$$\begin{aligned} \left( (F(\mathbf{x},\mathbf{y}))(t\_2) - F(\mathbf{x},\mathbf{y}) \right)(t\_1) &= \left( (F\_1 \mathbf{y})(t\_2), (F\_2 \mathbf{x})(t\_2) \right) - \left( (F\_1 \mathbf{y})(t\_1), (F\_2 \mathbf{x})(t\_1) \right) \\ &= \left( (F\_1 \mathbf{y})(t\_2) - (F\_1 \mathbf{y})(t\_1) \right) \left( (F\_2 \mathbf{x})(t\_2) - (F\_2 \mathbf{x})(t\_1) \right), \end{aligned}$$

then from (14) and (15), we can deduce the equicontinuity of the class {*F*(*x*, *y*)} on *Q*.
