prove existence and uniqueness of the solution of integral equation $(\ref{eqn203ab})$.

For this purpose we will operate with the following properties of matrix integral operator $\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}$:

1\. Matrix integral operator $\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}$ depends continuously on its parameter t $\in$ \[0,$\infty$) (based on formulas $(\ref{eqn160})$ - $(\ref{eqn162})$). $\\$

2\. Matrix integral operator $\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}$ maps vector-functions $\vec{u}_{\mbox{\tiny V}}$ from perfect space $\overrightarrow{TS}$ onto perfect space $\overrightarrow{TS}$. This property directly follows from the properties of Fourier transform $\cite{GC68},\;$ and the form of integrands of integral operators $S_{ij},\; B$ (based on formulas $(\ref{eqn160})$ - $(\ref{eqn162})$). $\\$

3\. Matrix integral operator $\bar{\bar{S}_{\mbox{\tiny V}}}$ is “quadratic”. $\\$

4\. $\|\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}\cdot\vec{u}_{\mbox{\tiny V}}\; - \;\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}\cdot\vec{u}_{\mbox{\tiny V}}^{'}\|_{p}\;<\;\|\vec{u}_{\mbox{\tiny V}}\;-\;\vec{u}_{\mbox{\tiny V}}^{'} \|_{p}\;$ for any $\vec{u}_{\mbox{\tiny V}},\;\vec{u}_{\mbox{\tiny V}}^{'} \in \overrightarrow{TS}\;\;$ (${\vec{u}_{\mbox{\tiny V}}\;\neq \;\vec{u}_{\mbox{\tiny V}}^{'}})$ and any t $\in$ \[0,$\infty$) (based on properties 1, 2, 3 and formulas $(\ref{eqn160})$ - $(\ref{eqn162}),(\ref{eqn203f} )$). $\\$

Properties mentioned above allow us to prove that matrix integral operator $\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}$ is a contraction operator.\
<span style="font-variant:small-caps;">Theorem 1.</span> **Contraction operator.**$\;\;\;\textsc{\cite{KA64}}\;$

Matrix integral operator $\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}$ maps perfect space $\overrightarrow{TS}$ onto perfect space $\overrightarrow{TS}$, and for any $\vec{u}_{\mbox{\tiny V}},\;\vec{u}_{\mbox{\tiny V}}^{'}\in \overrightarrow{TS}$

(${\vec{u}_{\mbox{\tiny V}}\;\neq \;\vec{u}_{\mbox{\tiny V}}^{'}}$) the condition 4 is valid.

Then matrix integral operator $\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}$ 