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###

Uniqueness of

If

then

Hence
in contradiction to the premises.

Thus
is unique.

Next we will show that the operator *L* is a contracting
operator.

**Claim 5.7**
,

*L* is a contracting operator.

**Proof:**For all ,
,
we choose ,
and assume
.

We define *a*_{s}^{*} such that:

We have shown that

(The same proof holds for
)

Thus, for *all **,
*

*
Hence **L** is a contracting operator.
*

**Proof:**

- 1.
- As
*L* has been shown to be a contracting operator there is a
unique solution for the equation *Lv* = *v* by theorem
. This fixed point is
.
- 2.
- Is true by the same argument.

** Next:** Example:
** Up:** The Solution of the
** Previous:** is a fixed point
*Yishay Mansour*

*1999-11-24*