Theorem 2 (Uniqueness of the fixed point)

Theorem 2: Under the condition of Theorem 1, and if exists on as well as a positive constant such that for all in ( ) then has a unique fixed point in .

Proof:

Assume there are twc different fixed points (there is at least one!).

Then the Mean Value Theorem says that there exists a value in and hence in such that .

So,

> expr:=abs(p-q);

> expr=abs(g(p)-g(q));

> expr=abs(D(g)(zeta))*abs(p-q);

> expr<K*abs(p-q);

> expr<abs(p-q);

This is a contradiction. Therefore, there can only be one fixed point in the interval .