The Solution of the Optimality Equations:

Operator T is called contracting if there exists $0 \leq\lambda<1$ such that

$$\begin{eqnarray*}\Vert T\vec{u}-T\vec{v}\Vert \leq \lambda \Vert\vec{u}-\vec{v}\Vert & \forall \vec{u}, \vec{v} \in R^{n} \end{eqnarray*}$$

Theorem 5.6   Let $T: R^n\rightarrow R^n$ a contracting operator, then

1.

there exists a unique $\vec{v^*}$ such that $T\vec{v}^*=\vec{v}^*$

2.

For each starting point $\vec{v}_0$ the series $\vec{v}_{n+1}=T\vec{v}_n$ converges to $\vec{v^*}$

Proof:We define $\vec{v}_{n+1}=T\vec{v}_n$