I need to prove: If $F$ is contractive from $[a, b]$ to $[a, b]$ and $x_{n+1} = F(x_n)$ with $x_0\in[a,b]$ then $|x_n -s| \leq C\lambda^n$ for an appropriate $C$ where $s$ is a fixed point of $F(x)$. I also would like to find an upper bound for $C$.
Here is what I have so far:
\begin{align} |x_n-s| =& |F(x_{n-1})- F(s)| \\ \leq&\lambda|x_{n-1}-s| \\ =& \lambda|F(x_{n-2})-F(s)| \\ \leq& \lambda^2|x_{n-2}-s| \\ \vdots& \\ \leq&\lambda^n|x_0 -s| \end{align} $\implies C=|x_0-s|$. Is this sufficient? I don't like having an $s$ on both sides of the equation. But I could say $(b-a) \geq C$ and thus: \begin{equation} |x_n-s| \leq (b-a)\lambda^n \end{equation}
I could use this as my upper bound.
- Is this correct?
- What improvements would you suggest?
All help is greatly appreciated!