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I define a recurrence relation as follow:

$$ω_{n+1}=f(ω_{n})$$ for some function $f$.

Assume that $f:ℝ⁺→ℝ⁺$ is a derivable, strictly increasing (so it is bijective) and a contraction function. Then $(ω_{n})$ converges to the fixed point of $f$, i.e., $ω_{n}$→$x$ with $f(x)=x$ and this is the only type of solutions for the above recurrence equation. This is the Banach fixed point theorem.

My question is: Find sufficient and necessary conditions on which the sequence $(ω_{n})$ converges to the fixed point of $f$ via:

(a) finite number of iterations

(b) infinite number of iterations

I know that this is depends on the choice of initial conditions. However, I cannot find the good idea to start.

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**Hint**

Let $C$ be a closed subset of $\mathbb R$ such that

  • $f$ is continuous at $C$

  • The unique fixed point $L\in C$

  • $\exists N\in\mathbb N\;\; \forall n\geq N\;\; \omega_n\in C$

then

if $(\omega_n)_n$ converges to $\omega,\;\;\;\omega \in C$ and

$$\lim_{n\to+\infty}\omega_{n+1}=$$

$$\lim_{n\to+\infty}f(\omega_n)=$$

$$f(\lim_{n\to+\infty}\omega_n)=$$

$$\implies \omega=f(\omega)$$

$$\implies \omega=L.$$

The number of iterations, depends on initial value $\omega_0$ and how $f$ is contractive.

we have $|f(\omega_n)-L|\leq K^n|\omega_1-\omega_0|<\epsilon$

this gives the number $n$ of iterations.

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