*3.2. Uniqueness Theorem*

**Theorem 6.** *Let the assumptions* (*A*∗1)*–*(*A*∗2) *and* (*A*3)*–*(*A*5) *be satisfied then the solution of problem (1), (2) and (4) is unique.*

**Proof.** Let (*x*1, *y*1) and (*x*2, *y*2) be two solutions of the problem (1), (2) and (4) on the form

$$\begin{aligned} \left(\mathbf{x}(t), \mathbf{y}(t)\right) &= \left(\mathbf{x}\_0 \quad - \int\_0^t h\_1(s, \mathbf{x}(s))dW(s) + \int\_0^t f\_1(s, \mathbf{y}(\phi\_1(s)))ds, \\ \mathbf{y}\_0 & \quad - \int\_0^\eta h\_2(s, \mathbf{y}(s))ds + \int\_0^t f\_2(s, \mathbf{x}(\phi\_2(s)))dW(s)), \end{aligned} \tag{21}$$

then we can obtain

$$\begin{split} \|\|\mathbf{x}\_{1}(t) - \mathbf{x}\_{2}(t)\|\|\_{2} &\quad \leq \ c\sqrt{T} \|\|\mathbf{x}\_{1} - \mathbf{x}\_{2}\|\|\_{\mathcal{C}} + bT \|\|\mathbf{y}\_{1} - \mathbf{y}\_{2}\|\|\_{\mathcal{C}} \\ &\leq \ (b+c)T \|\|\mathbf{x}\_{1} - \mathbf{x}\_{2}\|\|\_{\mathcal{C}} + (b+c)T \|\|\mathbf{y}\_{1} - \mathbf{y}\_{2}\|\|\_{\mathcal{C}} \\ &\leq \ (b+c)T \max\{\|\mathbf{x}\_{1} - \mathbf{x}\_{2}\|\_{\mathcal{C}}, \ \|\mathbf{y}\_{1} - \mathbf{y}\_{2}\|\|\_{\mathcal{C}}\}. \end{split} \tag{22}$$

Similarly, we can obtain

$$\|\|y\_1(t) - y\_2(t)\|\|\_2 \le \varepsilon \quad (b+c)T \max\{\|\mathbf{x}\_1 - \mathbf{x}\_2\|\_{\mathcal{C}}, \|\|y\_1 - y\_2\|\_{\mathcal{C}}\}.\tag{23}$$

Hence from (22) and (23)

$$\begin{array}{rcl} \| (\mathfrak{x}\_{1}, y\_{1}) - (\mathfrak{x}\_{2}, y\_{2}) \|\_{X} &=& \| (\mathfrak{x}\_{1} - \mathfrak{x}\_{2}), (y\_{1} - y\_{2}) \|\_{X} \\ &\leq& \max \{ \| \mathfrak{x}\_{1} - \mathfrak{x}\_{2} \|\_{\mathcal{C}'}, \| y\_{1} - y\_{2} \|\_{\mathcal{C}} \} \\ &\leq& (b + c)T \max \{ \| \mathfrak{x}\_{1} - \mathfrak{x}\_{2} \|\_{\mathcal{C}'}, \| y\_{1} - y\_{2} \|\_{\mathcal{C}} \}. \end{array}$$

This implies that

$$\|\left(1 - (b + c)T\right)\|\left(\mathfrak{x}\_1, y\_1\right) - \left(\mathfrak{x}\_2, y\_2\right)\|\_X \le 0.5$$

Then

$$\|\left(\mathfrak{x}\_{1\prime}y\_1\right) - \left(\mathfrak{x}\_{2\prime}y\_2\right)\|\_{X} = 0$$

and (*x*1, *y*1)=(*x*2, *y*2) which proves that the solution of the problem (1), (2) and (4) is unique.
