*3.3. Continuous Dependence*

**Theorem 7.** *The solution (16) of the problem (1)–(2) and (4) depends continuously on the two random data* (*x*0, *y*0).

**Proof.** Let (*x*ˆ, *y*ˆ) be the solution of the coupled system

$$\begin{aligned} \mathfrak{X}(t) &= \; \mathfrak{X}\_0 - \int\_0^\tau h\_1(s, \mathfrak{X}(s)) dW(s) + \int\_0^t f\_1(s, \mathfrak{Y}(\phi\_1(s))) ds \\ \mathfrak{Y}(t) &= \; \mathfrak{Y}\_0 - \int\_0^\eta h\_2(s, \mathfrak{Y}(s)) ds + \int\_0^t f\_2(s, \mathfrak{X}(\phi\_2(s))) dW(s), \end{aligned}$$

such that (*x*0, *y*0) − (*x*ˆ0, *y*ˆ0)*<sup>X</sup>* < *δ*3. Then we have

$$\begin{aligned} \mathbf{x}(t) - \mathbf{\hat{x}}(t) &= \mathbf{x}\_0 - \mathbf{\hat{x}}\_0 - \int\_0^\tau [h\_1(s, \mathbf{\hat{x}}(s)) - h\_1(s, \mathbf{x}(s))] dW(s) \\ &+ \int\_0^t [f\_1(s, \mathbf{y}(\phi\_1(s))) - f\_1(s, \mathbf{\hat{y}}(\phi\_1(s)))] ds \end{aligned}$$

and

$$\begin{split} \|\mathbf{x}(t) - \mathbf{\hat{x}}(t)\|\_{2} &\leq \quad \|\mathbf{x}\_{0} - \mathbf{\hat{x}}\_{0}\|\_{\mathbb{C}} + c\sqrt{T}\|\mathbf{x} - \mathbf{\hat{x}}\|\_{\mathbb{C}} + bT\|\mathbf{y} - \mathbf{\hat{y}}\|\_{\mathbb{C}} \\ &\leq \quad \|\mathbf{x}\_{0} - \mathbf{\hat{x}}\_{0}\|\_{\mathbb{C}} + cT\|\mathbf{x} - \mathbf{\hat{x}}\|\_{\mathbb{C}} + bT\|\mathbf{y} - \mathbf{\hat{y}}\|\_{\mathbb{C}} \\ &\leq \quad \|\mathbf{x}\_{0} - \mathbf{\hat{x}}\_{0}\|\_{2} + cT\max\{\|\mathbf{x} - \mathbf{\hat{x}}\|\_{\mathbb{C}}, \|\mathbf{y} - \mathbf{\hat{y}}\|\_{\mathbb{C}}\} + bT\max\{\|\mathbf{x} - \mathbf{\hat{x}}\|\_{\mathbb{C}}, \|\mathbf{y} - \mathbf{\hat{y}}\|\_{\mathbb{C}}\} \\ &\leq \quad \max\{\|\mathbf{x}\_{0} - \mathbf{\hat{x}}\_{0}\|\_{2}, \|\mathbf{y}\_{0} - \mathbf{y}\_{0}\|\_{2}\} + (b + c)T\max\{\|\mathbf{x} - \mathbf{\hat{x}}\|\_{\mathbb{C}}, \|\mathbf{y} - \mathbf{\hat{y}}\|\_{\mathbb{C}}\}. \end{split}$$

By the same way we can obtain

$$\|\|y(t) - \hat{y}(t)\|\|\_{2} \le \max\{\|\|x\_0 - \hat{x}\_0\|\|\_{2'}, \|\|y\_0 - \hat{y}\_0\|\|\_{2}\} + (b + c)T \max\{\|\|x - \hat{x}\|\|\_{\mathcal{L}'}, \|\|y - \hat{y}\|\|\_{\mathcal{L}}\}$$

and

$$\begin{array}{rcl} \| (\mathfrak{x}, \mathfrak{y}) - (\mathfrak{x}, \mathfrak{y}) \|\_{X} &=& \max \{ \| (\mathfrak{x} - \mathfrak{x} \|\_{\mathbb{C}^{\prime}} \| (\mathfrak{y} - \mathfrak{y} \|\_{\mathbb{C}}) \|\_{\mathbb{C}} \} \\ &\leq& \max \{ \| \| \mathfrak{x}\_{0} - \mathfrak{x}\_{0} \|\_{2}, \| \| y\_{0} - \mathfrak{y}\_{0} \|\_{2} \} + (\mathfrak{b} + c) T \max \{ \| \mathfrak{x} - \mathfrak{x} \|\_{\mathbb{C}^{\prime}}, \| y - \mathfrak{y} \|\_{\mathbb{C}} \} \\ &\leq& \delta\_{3} + (\mathfrak{b} + c) T \max \{ \| \mathfrak{x} - \mathfrak{x} \|\_{\mathbb{C}^{\prime}}, \| y - \mathfrak{y} \|\_{\mathbb{C}} \} \end{array}$$

which gives our result

$$\|(x, y) - (\pounds, \mathfrak{j})\|\_{X} \le \frac{\delta\_3}{1 - T(b + c)} = \epsilon\_3$$

and completes the proof.

**Theorem 8.** *The solution (16) of the problem (1), (2) and (4) depends continuously on the two random functions h*<sup>1</sup> *and h*2*.*

**Proof.** Let (*x*ˆ, *y*ˆ) be the solutions of the coupled system of stochastic integral Equations (1), (2) and (4) such that

$$\begin{aligned} \mathfrak{k}(t) &= \ x\_0 - \int\_0^\tau h\_1^\*(s, \mathfrak{k}(s)) dW(s) + \int\_0^t f\_1(s, \mathfrak{j}(\phi\_1(s))) ds \\ \mathfrak{j}(t) &= \ y\_0 - \int\_0^\eta h\_2^\*(s, \mathfrak{j}(s)) ds + \int\_0^t f\_2(s, \mathfrak{k}(\phi\_2(s))) dW(s) .\end{aligned}$$

Let *h*<sup>∗</sup> *<sup>i</sup>* (*t*, *u*(*t*)) − *h*(*t*, *u*(*t*))<sup>2</sup> ≤ *δ*4, *i* = 1, 2 then

$$\begin{split} \|\|\mathbf{x}(t) - \hat{\mathbf{x}}(t)\|\|\_{2} &= \|\int\_{0}^{\tau} [h\_{1}^{\*}(s, \mathbf{\hat{x}}(s)) - h\_{1}(s, \mathbf{x}(s))] dW(s) + \int\_{0}^{t} [f\_{1}(s, y(\boldsymbol{\phi}\_{1}(s))) - f\_{1}(s, \boldsymbol{\hat{y}}(\boldsymbol{\phi}\_{1}(s)))] ds\|\_{2} \\ &\leq \sqrt{\int\_{0}^{\tau} \|h\_{1}^{\*}(s, \mathbf{\hat{x}}(s)) - h\_{1}(s, \mathbf{x}(s))\|\_{2}^{2} ds} + \int\_{0}^{t} \|f\_{1}(s, y(\boldsymbol{\phi}\_{1}(s))) - f\_{1}(s, \boldsymbol{\hat{y}}(\boldsymbol{\phi}\_{1}(s)))\|\_{2} ds} \\ &\leq \sqrt{\int\_{0}^{\tau} \|[h\_{1}^{\*}(s, \mathbf{\hat{x}}(s)) - h\_{1}^{\*}(s, \mathbf{x}(s))]\|\_{2} + \|h\_{1}^{\*}(s, \mathbf{x}(s)) - h\_{1}(s, \mathbf{x}(s))\|\_{2}]^{2} ds} \\ &+ \int\_{0}^{t} \|f\_{1}(s, y(\boldsymbol{\phi}\_{1}(s))) - f\_{1}(s, \boldsymbol{\hat{y}}(\boldsymbol{\phi}\_{1}(s)))\|\_{2} ds \\ &\leq \sqrt{\int\_{0}^{\tau} (c\|\boldsymbol{x}(s) - \hat{\mathbf{x}}(s)\|\_{2} + \delta\_{4})^{2} ds} + \int\_{0}^{t} b \|y(s) - \hat{y}(s)\|\_{2} ds} \end{split}$$


Similarly we can obtain

$$||y - \hat{y}||\_{\mathcal{C}} \le (b + c)T \max\{||\mathfrak{x} - \hat{\mathfrak{x}}||\_{\mathcal{C}}, ||y - \hat{y}||\_{\mathcal{C}}\} + \delta\_4 T$$

and

$$\|(\mathbf{x}, \mathbf{y}) - (\mathbf{\hat{x}}, \mathbf{\hat{y}})\|\_{X} = \max\{\|\mathbf{x} - \mathbf{\hat{x}}\|\_{\mathbb{C}^{\prime}} \|\mathbf{y} - \mathbf{\hat{y}}\|\_{\mathbb{C}}\} \le (b + c)T \max\{\|\mathbf{x} - \mathbf{\hat{x}}\|\_{\mathbb{C}^{\prime}} \|\mathbf{y} - \mathbf{\hat{y}}\|\_{\mathbb{C}}\} + \delta\_{4}T.$$
 This implies that

(*x*, *y*) − (*x*ˆ, *y*ˆ)*<sup>X</sup>* ≤ *δ*4*T* <sup>1</sup> <sup>−</sup> *<sup>T</sup>*(*<sup>b</sup>* <sup>+</sup> *<sup>c</sup>*) <sup>=</sup>

which completes the proof.

**Example 1.** *Consider the coupled system*

$$\frac{d\mathbf{x}}{dt}(t) = \begin{array}{rcl} \frac{a(t) + y(t)}{5(1 + \|y(t)\|\_2)}, & t \in (0, 1] \\\\ d y(t) = \begin{array}{rcl} t\mathbf{x}(t) \\ \frac{\mathbf{z}(1 + \|\mathbf{x}\|\_2)}{2(1 + \|\mathbf{x}\|\_2)} dW(t), & t \in (0, 1] \end{array} \tag{24}$$

*subject to*

$$\mathbf{x}\_0 = \int\_0^\tau \frac{e^{-s} \mathbf{y}(s)}{120 + s^2} d\mathcal{W}(s), \quad \mathbf{y}\_0 = \int\_0^\eta \frac{\mathbf{x}(s)}{\sqrt{s + 36}} ds \tag{25}$$

*where*

$$\|\|f\_1(t, y(t))\|\|\_2 \le \frac{1}{5} [|a(t)| + \|\|y(t)\|\|\_2], \quad \|f\_2(t, x(t))\|\|\_2 \le \frac{1}{2\|\|\chi(t)\|\|\_2}$$

*and*

$$\|h\_1(t, y(t))\|\_2 \le \frac{\|y(t)\|\_2}{120}, \quad \|h\_2(t, x(t))\|\_2 \le \frac{\|x(t)\|\_2}{6}.$$

*Easily, the coupled system (24) with nonlocal integral conditions (25) satisfies all the Assumptions 1–5 of Theorem 1. with b* = <sup>1</sup> <sup>2</sup> , *<sup>c</sup>* <sup>=</sup> <sup>1</sup> <sup>6</sup> *, then there exists at least one solution of the system (24) on* [0, 1]*.*
