Is q-Lipschitz continuity necessary condition for fixed point iteration.?

Question

Is q-Lipschitz continuity necessary condition for fixed point iteration? Meaning does it have to be so that $d(f(x),f(y))\leq qd(x,y)$ where $0\leq{q}<1$.

Answer 1

It's necessary in the sense that if you omit that condition, or just weaken it to $0\le q\le 1$ then the theorem no longer holds.

I's not necessary in the sense that there are certainly examples not satisfying the condition where the iterates converge to a fixed point. An example with a unique fixed point: Define $f:[-1/2,1/2]\to[-1/2,1/2]$ by $f(x)=x^2\sin(cx)$. Iterates converge to a fixed point, but if $c$ is large enough there exist points where $|f'(x)|>1$, hence it's not $q$-Lipschitz for any $q\le 1$.

Answer 2

No. Consider $f(x)=x$. It is not a contraction, but for any $x_0$, the sequence of iterates $x_{n+1}=f(x_n)$ converges to the fixed point $x_0$. However uniqueness is lost in this example.