Question

prove that FPI( fixed point iteration) for the smooth function g(x)=(1/4)sin^2(x)-1 converges locally

Answer 1

Let us call your function g
\[
\left| g(x)-g(y)\right| = \left| \frac{\sin ^2(x)}{4}-\frac{\sin ^2(y)}{4}\right|=\\
\frac{1}{4} \left| \sin (x)-\sin (y)\right| \left| \sin (x)+\sin (y)\right|
\leq \frac{2 \left| x-y\right| }{4}=\frac{\left| x-y\right| }{2}
\]
Your function is a Lipschitz function.

Answer 2

Apply Banach Theorem and you are done

Answer 3

Your function is a contraction. since
\[
|g(x) - g(y)| \le \frac 12 |x-y|
\]

Answer 4

can u explain ,is it converge locally?
i do not understand where x-y in second line comes from?