Q: Give an evaluation of error between two consecutive terms for methods of type $p_{n+1}=g(p_{n})$.
I tried solving it, but I think that my solution is correct only when the method converges to a fixed point, or when the function that we consider fulfills the fixed point theorem, which gives the necessary conditions for existence and uniqueness of a fixed point. Let me give you an idea about what theorem states:
Theorem. If $g \in C[a,b]$ and $g(x) \in [a,b]$ for all $x \in [a,b]$, then $g$ has at least one fixed point in $[a,b]$. If, in addition, $g'(x)$ exists on $(a,b)$ and a positive constant $k<1$ exists with $|g'(x)| \leq k$, for all $x \in (a,b)$ then there is exactly one fixed point in $[a,b]$.
First I tried something like this, considering that the questions states that the fixed point theorem stands, and using the Mean Value Theorem:
$|p_{n+1}-p_{n}|=|g(p_{n})-g(p_{n-1})|=|g'(\epsilon)||p_{n}-p_{n-1}| \leq k|p_{n}-p_{n-1}| \leq ... \leq k^n|p_1-p_0|.$
But then, what if we don't know that every derivative of our function at points between $(p_i,p_{i-1})$ for $i \in {1,2,...,n}$ is bounded by $k$? So my dilemma is here, does the question imply that the fixed point theorem stands, and if not necessary, can we consider that the solution is a two-part solution, and the first part is the one that I stated, and the second part is this one?(or is only this one, because the scope of this solution goes beyond the first part, or what I'm doing is totally wrong):
If we take $M_1, M_2, ... , M_n$ as the bounds of our derivative functions at $\epsilon_1, \epsilon_2, ..., \epsilon_n$, $\epsilon_i \in (p_i,p_{i-1}), i \in {1,2,...,n}$ i.e $|g'(\epsilon_1)| \leq M_1,|g'(\epsilon_2)| \leq M_2,...,|g'(\epsilon_n)| \leq M_n$ where $M_i \in \mathbb{R}, i\in{1,2,3,...,n}$ and $M= M_1\cdot M_2 \cdot ... \cdot M_n$.
$|p_{n+1}-p_{n}|=|g(p_{n})-g(p_{n-1})|=|g'(\epsilon_1)||p_{n}-p_{n-1}| \leq M_1|p_{n}-p_{n-1}| = M_1|g(p_{n-1})-g(p_{n-2})|=M_1|g'(\epsilon_2)||p_{n-1}-p_{n-2}|\leq M_1M_2|p_{n-1}-p_{n-2}|\leq ... \leq M_1M_2\cdot...\cdot M_n|p_1-p_0|=M|p_1-p_0|.$
What if our function does not converge? Can we give an evaluation(approximation)? If my tries are not correct, please give me an idea how to answer that question. Thank you!
