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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}

Theorem 5.6   Let $T: R^n\rightarrow R^n$ a contracting operator, then
there exists a unique $ \vec{v^*}$ such that $T\vec{v}^*=\vec{v}^*$
For each starting point $\vec{v}_0$ the series $\vec{v}_{n+1}=T\vec{v}_n$ converges to $ \vec{v^*}$

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


Yishay Mansour