Convergence of iterations of interval maps

Find: $\lim \limits_{n\to \infty }\underbrace{\sin \sin \cdots \sin }_{n\uparrow }x$.

In order to find an approximate solution of Kepler's equation (Уравнение Кеплера)

suppose

Prove that $\xi =\lim \limits_{n\to \infty }x_{n}$ exists, and that the number $\xi$ is the unique root of equation (1).

The function $f$ is monotone, continuous on $[0 ; 1]$, and $0 \leqslant f(x) \leqslant 1$ for any $x \in [0 ; 1]$. Prove that for any $a_{1} \in [0 ; 1]$ the sequence $a_{1}, a_{n+1} = f\left(a_{n}\right), n \in \mathbb {N}$, converges to one of the solutions of the equation $f(x) = x$.

Let the function $f$ be continuous on $[a ; b]$, and let the sequence $\left\{ x_{n}\right\} : x_{0} \in [a ; b], x_{n} = f\left(x_{n-1}\right), n \in \mathbb {N}$ (i.e., for any $n \in \mathbb {N} \quad f\left(x_{n-1} \in [a ; b]\right)$) be defined.

Prove that:

The function $f$ is strictly increasing, while the function $g$ is strictly decreasing on $[a ; b]$, and $\mathbb {E}\left[f\right] \cap \mathbb {E}\left[g\right] \neq \varnothing$.

For the sequence $\left\{ x_{n}\right\}$, given recursively: $x_{1} = 1 / 2, x_{n+1} = f\left(x_{n}\right), n \in \mathbb {N}$, where

find the limits of the subsequences $\left\{ x_{2 k}\right\}$ and $\left\{ x_{2 k-1}\right\}$. Plot the graph of the function $f$ and show, in the figure, the construction of the first five terms of the sequence.

The range of the function $f$, defined on $[a ; b]$, is contained in $[a ; b]$. For any $x, y \in [a ; b], x \neq y$, the inequality holds

For a sequence given recursively, prove that a limit exists and find it:

Determine for which values:

Prove that the sequence $\left\{ x_{n}\right\}$, such that

In solving the equation $x=\frac{1}{x}+a$ ($a$ a positive parameter) by the iteration method, a recurrent sequence $x_{n}=\frac{1}{x_{n-1}}+a, x_{0}>0$ arises. Prove that the limit $l=\lim x_{n}$ exists, and find the asymptotics of the differences $x_{n}-l$.

Let $f \in C(\left[a, b\right])$, with $f(x) \in [a, b]$ for all $x \in [a, b]$. Consider a sequence $\left\{ x_{n}\right\}$ such that $x_{1} \in [a, b]$ and $x_{n+1}= \left(1-c_{n}\right) x_{n}+c_{n} f\left(x_{n}\right), c_{n} \in (0,1)$ for all $n \in \mathbb {N}$.

A fixed point of a map $f$ is a point $x$ in its domain of definition such that $f(x)=x$. Let a function $f \in C(\left[a, b\right])$ be increasing, $f(\left[a, b\right]) \subset [a, b]$. Prove that for every $x_{0} \in [a, b]$ the sequence $x_{n}=f\left(x_{n-1}\right)$ ($n \in \mathbb {N}$) converges to a fixed point. Does this hold if $f$ is a decreasing function?

A fixed point of a map $f$ is a point $x$ in its domain of definition such that $f(x)=x$. Let $f \in C(\mathbb {R})$. For $x_{0} \in \mathbb {R}$, set $x_{1}=f\left(x_{0}\right), x_{k}=f\left(x_{k-1}\right) (k \in \mathbb {N})$. Prove that if for some $x_{0} \in \mathbb {R}$ the sequence $\left\{ \frac{1}{n} \sum_{0 \leqslant k<n} f\left(x_{k}\right)\right\}$ is bounded, then $f$ has a fixed point. Does this hold for continuous maps $f: \mathbb {C} \rightarrow \mathbb {C}$?

Let $f$ be a map of an interval $\Delta \subset \mathbb {R}$ into itself, $f^{0}(x)=x$, $f^{1}(x)=f(x)$, $f^{n}(x)=f\left(f^{n-1}(x)\right)$ for $n>1$. We say that the method of successive approximations with initial point $x_{0} \in \Delta$ converges if the limit $\lim x_{n}$ exists, where $x_{n}=f^{n}\left(x_{0}\right)$.

Let the function $f \in C(\left[a, b\right])$ be increasing. Prove that the method of successive approximations converges for any initial point $x_{0} \in [a, b]$, if

Let $f$ be a map of an interval $\Delta \subset \mathbb {R}$ into itself, $f^{0}(x)=x$, $f^{1}(x)=f(x)$, $f^{n}(x)=f\left(f^{n-1}(x)\right)$ for $n>1$. We say that the method of successive approximations with initial point $x_{0} \in \Delta$ converges if the limit $\lim x_{n}$ exists, where $x_{n}=f^{n}\left(x_{0}\right)$.

Prove the convergence of the method of successive approximations for any starting point $x_{0} \in \Delta$ and find $\lim x_{n}$ for the following maps $f$:

Let $f$ be a map of an interval $\Delta \subset \mathbb {R}$ into itself, $f^{0}(x)=x$, $f^{1}(x)=f(x)$, $f^{n}(x)=f\left(f^{n-1}(x)\right)$ for $n>1$. We say that the method of successive approximations with initial point $x_{0} \in \Delta$ converges if the limit $\lim x_{n}$ exists, where $x_{n}=f^{n}\left(x_{0}\right)$.

Determine for which starting points $x_{0}$ in the domain of definition of the map $f$ the method of successive approximations converges, and find $\lim x_{n}$, if

The function $f$ defined by

has three fixed points, say $\alpha , \beta , \gamma$, where

For arbitrarily chosen $x_{1}$, define $\left\{ x_{n}\right\}$ by setting $x_{n+1} = f\left(x_{n}\right)$.