V}n}\; -\;\vec{u}_{\mbox{\tiny V}}^{*}\|_{p}\;\leq\;\frac{\alpha^{n}}{(1 - \alpha)}\|\vec{u}_{\mbox{\tiny V}1}\;-\;\vec{u}_{\mbox{\tiny V}0} \|_{p}\;\;\;\;\;\;\;\;\;(n=0,1,\ldots)$$

Proof:

It is clear that

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

It follows from $(\ref{eqn206})$ that\

$\|\vec{u}_{\mbox{\tiny V},n+1}\; -\;\vec{u}_{\mbox{\tiny V}n}\|_{p}\;\leq\;\alpha\cdot\|\vec{u}_{\mbox{\tiny V}n}\;-\;\vec{u}_{\mbox{\tiny V},n-1} \|_{p}$.\

Using similar inequalities one after another while decreasing n we will receive:\

$\|\vec{u}_{\mbox{\tiny V},n+1}\; -\;\vec{u}_{\mbox{\tiny V}n}\|_{p}\;\leq\;\alpha^{n}\cdot\|\vec{u}_{\mbox{\tiny V}1}\;-\;\vec{u}_{\mbox{\tiny V}0} \|_{p}$.\

From this result it follows that

$$\begin{aligned}
\label{eqn209}
\|\vec{u}_{\mbox{\tiny V},n+l}\; -\;\vec{u}_{\mbox{\tiny V}n}\|_{p}\;\leq\;\|\vec{u}_{\mbox{\tiny V},n+l}\; -\;\vec{u}_{\mbox{\tiny V},n+l-1}\|_{p}+\cdots+\|\vec{u}_{\mbox{\tiny V},n+1}\; -\;\vec{u}_{\mbox{\tiny V}n}\|_{p}\;\leq\;
\nonumber\\
\nonumber\\
\leq(\alpha^{n+l-1}\;+\cdots+\;\alpha^{n})\|\vec{u}_{\mbox{\tiny V}1}\;-\;\vec{u}_{\mbox{\tiny V}0} \|_{p}\leq\frac{\alpha^{n}}{(1 - \alpha)}\|\vec{u}_{\mbox{\tiny V}1}\;-\;\vec{u}_{\mbox{\tiny V}0} \|_{p}.
\quad\quad\end{aligned}$$

Because of $\alpha^{n} \rightarrow 0$ for n$\;\;\rightarrow\; \infty$ the obtained estimation $(\ref{eqn209})$ shows that sequence $\{\vec{u}_{\mbox{\tiny V}n}\}$ is a Cauchy

sequence. Since the space $\overrightarrow{TS}$ is a perfect space, this sequence converges to an element $\vec{u}_{\mbox{\tiny V}}^{*}\in \overrightarrow{TS}$,

such that $\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}(\vec{u}_{\mbox{\tiny V}}^{*})$ has sense. We use inequality $(\ref{eqn206})$ again and have:\

$\|\vec{u}_{\mbox{\tiny V},n+1}\; -\;\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}(\vec{u}_{\mbox{\tiny V}}^{*})\|_{p}\;=\;\|\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}(\vec{u}_{\mbox{\tiny V}n})\; -\;\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}(\vec{u}_{\mbox{\tiny V}}^{*})\|_{p}\;\leq\;\alpha\cdot\|\vec{u}_{\mbox{\tiny V}n}\;-\;\vec{u}_{\mbox{\tiny V}}^{*} \|_{p}\;\;\;\;\;\;\;\;\;(n=0,1,2,\ldots)$\

The right part of the above inequality tends to 0 for n$\;\;\rightarrow\; \infty$ and it means that $\;\;\;\vec{u}_{\mbox{\tiny V},n+1}\; \rightarrow\;\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}(\vec{u}_{\mbox{\tiny V}}^{*})$

and $\;\;\;\vec{u}_{\mbox{\tiny V}}^{*}\; =\;\bar{\bar{S}_{\mbox{\tiny V}}}^{\nabla}(\vec{u}_{\mbox{\tiny V}}^{*})$. In other words, $\;\;\vec{u}_{\mbox{\tiny V}}^{*}\;\;$ is the solution of equation $(\ref{eqn203ab})$.

Uniqueness of the solution also follows from $(\ref{eqn206})$. In fact, if there 