Let $[ a , b ]$ be a closed bounded interval of $\mathbb { R }$. If $\phi : [ a , b ] \rightarrow [ a , b ]$ is continuous, show that $\phi$ has at least one fixed point.
If $\phi : \mathbb { R } \rightarrow \mathbb { R }$ is of class $\mathcal { C } ^ { 1 }$ and satisfies $$\sup \left\{ \left| \phi ^ { \prime } ( x ) \right| ; x \in \mathbb { R } \right\} < 1$$ show that $\phi$ has at least one fixed point (one may study the sign of $x - \phi ( x )$ for $| x |$ sufficiently large). Show that this fixed point is unique.
By means of the function $\psi ( x ) = \sqrt { 1 + x ^ { 2 } }$, show that in the previous question hypothesis (1) cannot be replaced by $$\forall x \in \mathbb { R } , \quad \left| \phi ^ { \prime } ( x ) \right| < 1 .$$
Let $\ell$ be a strictly positive integer. Let $F$ be a closed subset of $\mathbb { R } ^ { \ell }$ and let $\phi : F \rightarrow F$ be a map. We assume that there exists $k \in [ 0,1 [$ such that $$\forall x \in F , \quad \forall y \in F , \quad \| \phi ( y ) - \phi ( x ) \| \leqslant k \| y - x \| .$$ (a) We choose a point $x _ { 0 } \in F$. Show that the formula $x _ { n + 1 } = \phi \left( x _ { n } \right)$ defines a sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ of elements of $F$, and that this sequence is convergent in $F$. (b) Deduce that $\phi$ has a unique fixed point in $F$. (c) This fixed point being denoted $x ^ { * }$, bound $\left\| x _ { n } - x ^ { * } \right\|$ as a function of $\left\| x _ { 0 } - x ^ { * } \right\|$. (d) In what precedes, we assume that $$\phi = \underbrace { \theta \circ \cdots \circ \theta } _ { m \text { times } } ,$$ where $\theta : F \rightarrow F$ is a map and $m \geqslant 2$ is an integer. Show that $\theta$ has a fixed point, and a unique one, in $F$.
Let $g : [ 0,1 ] \rightarrow [ 0,1 ]$ be an increasing function (but not necessarily continuous). Show that $g$ has at least one fixed point. Hint: one may consider the set $$E = \{ x \in [ 0,1 ] ; x \leqslant g ( x ) \} .$$
For a triangular matrix $T = \left( \begin{array} { l l } \lambda & a \\ 0 & \mu \end{array} \right) \in \mathbf { M } _ { 2 } ( \mathbb { C } )$, explicitly compute the successive powers $T ^ { n }$ for $n$ a strictly positive integer.
Let $A \in \mathbf { M } _ { 2 } ( \mathbf { C } )$ be a matrix and let $\epsilon > 0$ be a real number. (a) Show the existence of a real number $\alpha > 0$ such that for every non-negative integer $n$ the absolute values of the coefficients of $A ^ { n }$ are bounded by $\alpha ( \rho ( A ) + \epsilon ) ^ { n }$. (b) Deduce the existence of a real number $\beta > 0$ such that for every non-negative integer $n$ and every $x \in \mathbb { C } ^ { 2 }$ we have $$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$