is a contraction operator, i.e. the following condition is true:

$$\label{eqn206}
\|\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}\cdot\vec{u}_{\mbox{\tiny V}}\; -\;\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}\cdot\vec{u}_{\mbox{\tiny V}}^{'}\|_{p}\;\leq\;\alpha\cdot\|\vec{u}_{\mbox{\tiny V}}\;-\;\vec{u}_{\mbox{\tiny V}}^{'} \|_{p}\;$$

where $\alpha\;<\;1$ and is independent from $\vec{u}_{\mbox{\tiny V}},\;\vec{u}_{\mbox{\tiny V}}^{'} \in \overrightarrow{TS}$ for any t $\in$ \[0,$\infty$).\
Proof by contradiction.

Let us assume that the opposite is true. Then there exist such $\vec{u}_{\mbox{\tiny V}n},\;\vec{u}^{'}_{\mbox{\tiny V}n} \in \overrightarrow{TS}$ (n=1,2,$\ldots$) and $$\lim_{ n\rightarrow \infty}\vec{u}_{\mbox{\tiny V}n}, \lim_{ n\rightarrow \infty}\vec{u}^{'}_{\mbox{\tiny V}n} \in \overrightarrow{TS}\;$$ that

$$\label{eqn207}
\|\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}\cdot\vec{u}_{\mbox{\tiny V}n}\; -\;\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}\cdot\vec{u}^{'}_{\mbox{\tiny V}n}\|_{p}\;=\;\alpha_{n}\cdot\|\vec{u}_{\mbox{\tiny V}n}\;-\;\vec{u}^{'}_{\mbox{\tiny V}n} \|_{p}\;\;\;\;\;\;\;\;(n=1,2,\ldots;\;\;\alpha_{n}\;\rightarrow\;1 )\;\;$$

Then the limiting result in $(\ref{eqn207})$ would lead to equality\

$\|\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}\cdot\vec{u}_{\mbox{\tiny V}}\; - \;\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}\cdot\vec{u}_{\mbox{\tiny V}}^{'}\|_{p}\;=\;\|\vec{u}_{\mbox{\tiny V}}\;-\;\vec{u}_{\mbox{\tiny V}}^{'} \|_{p}\;,$\

which is contradicting condition 4. Hence, $\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}$ is a contraction operator.\
<span style="font-variant:small-caps;">Theorem 2.</span> **Existence and uniqueness of solution.**$\;\;\;\textsc{\cite{KA64}}\;$

Let us consider a contraction operator $\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}$. Then there exists the unique solution $\vec{u}_{\mbox{\tiny V}}^{*}$ of equation $(\ref{eqn203ab})$

in space $\overrightarrow{TS}$ for any t $\in$ \[0,$\infty$). Also in this case it is possible to obtain $\vec{u}_{\mbox{\tiny V}}^{*}$ as a limit of sequence $\{\vec{u}_{\mbox{\tiny V}n}\}$ ,

where

$\vec{u}_{\mbox{\tiny V},n+1}\; =\;\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}(\vec{u}_{\mbox{\tiny V}n}) \;\;\;\;\;\;\;\;(n=0,1,\ldots),$\

and $\vec{u}_{\mbox{\tiny V}0}\; = \; 0$.

The rate of conversion of the sequence $\{\vec{u}_{\mbox{\tiny V}n}\}$ to the solution can be defined from the following inequality:

$$\label{eqn208}
\|\vec{u}_{\mbox{\tiny 