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 $$\forall x \in \mathbb { R } , \left| \phi ^ { \prime } ( x ) \right| < 1$$
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 $$\forall x \in F , \forall y \in F , \quad \| \phi ( y ) - \phi ( x ) \| \leqslant k \| y - x \| .$$ (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 $$E = \{ x \in [ 0,1 ] ; x \leqslant g ( x ) \} .$$
Let $A \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ 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 positive 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 positive integer $n$ and all $x \in \mathbb { C } ^ { 2 }$ we have $$\left\| A ^ { n } x \right\| \leqslant \beta ( \rho ( A ) + \epsilon ) ^ { n } \| x \|$$
Let $A \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ be a matrix and let $\eta$ be a strictly positive real number. (a) For $x \in \mathbb { C } ^ { 2 }$, show that the series $$\sum _ { n } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ is convergent. We denote $$N ( x ) = \sum _ { n = 0 } ^ { \infty } ( \rho ( A ) + \eta ) ^ { - n } \left\| A ^ { n } x \right\|$$ the sum of this series. (b) Show that $x \mapsto N ( x )$ is a norm on $\mathbb { C } ^ { 2 }$, which satisfies the following inequality $$\forall x \in \mathbb { C } ^ { 2 } , \quad N ( A x ) \leqslant ( \rho ( A ) + \eta ) N ( x ) .$$ (c) Show that there exists a real $C > 0$ such that for all $x \in \mathbb { C } ^ { 2 }$ we have $$\| x \| \leqslant N ( x ) \leqslant C \| x \|$$
(a) If $B \in \mathrm { M } _ { \ell } ( \mathbb { C } )$ is diagonalizable, show that there exists a norm $\| \cdot \| _ { B }$ on $\mathbb { C } ^ { \ell }$ such that $\| B x \| _ { B } \leqslant \rho ( B ) \| x \| _ { B }$ for all $x \in \mathbb { C } ^ { \ell }$. Hint: one may verify that if $P \in \mathrm { GL } _ { \ell } ( \mathbb { C } )$, then $x \mapsto \| P x \|$ is a norm on $\mathbb { C } ^ { \ell }$. (b) Show that there exists a matrix $C \in \mathrm { M } _ { 2 } ( \mathbb { C } )$ such that, for every norm $N$ on $\mathbb { C } ^ { 2 }$ there exists $y \in \mathbb { C } ^ { 2 }$ such that $N ( C y ) > \rho ( C ) N ( y )$.
Let $\phi : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ be a map and let $x ^ { * }$ be a fixed point of $\phi$. Let $A \in \mathrm { M } _ { 2 } ( \mathbb { R } )$ be a matrix satisfying $\rho ( A ) < 1$, and let $M > 0$ be a real number. We assume that $\phi$ satisfies $$\forall x \in \mathbb { R } ^ { 2 } , \quad \left\| \phi ( x ) - \phi \left( x ^ { * } \right) - A \left( x - x ^ { * } \right) \right\| \leqslant M \left\| x - x ^ { * } \right\| ^ { 2 }$$ Show that there exists $\varepsilon > 0$ such that for all $x _ { 0 } \in \mathbb { R } ^ { 2 }$ satisfying $\left\| x _ { 0 } - x ^ { * } \right\| < \varepsilon$, the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ defined by $x _ { n + 1 } = \phi \left( x _ { n } \right)$ (for $n \geqslant 0$) converges to $x ^ { * }$ when $n \rightarrow + \infty$.
Let $I$ be an open interval of $\mathbb { R }$. We are given a function $f : I \rightarrow \mathbb { R }$ of class $\mathcal { C } ^ { 3 }$, such that $f ^ { \prime } ( x ) > 0$ for all $x \in I$. Show that $f$ is bijective from $I$ onto the open interval $f ( I )$. We denote by $g : f ( I ) \rightarrow I$ its inverse function. Recall the value of $g ^ { \prime } ( f ( x ) )$. Express $g ^ { \prime \prime } ( f ( x ) )$ as a function of the successive derivatives of $f$ at $x$.
We keep the hypotheses and notation of questions 3.2 and 3.3. We now assume $0 \in f ( I )$ and we denote $x ^ { * } = g ( 0 )$. For $x \in I$ we denote by $I _ { x }$ the closed interval with endpoints $x$ and $x ^ { * }$. (a) Let $x , y \in I$. Show that there exists $( \bar { x } , \bar { y } ) \in I _ { x } \times I _ { y }$, such that $$H _ { f } ( x , y ) - x ^ { * } = \left( x - x ^ { * } \right) \left( y - x ^ { * } \right) \frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } ( \bar { x } , \bar { y } )$$ (b) Compute $$\frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } \left( x ^ { * } , x ^ { * } \right)$$ as a function of the derivatives of $f$.
Illustrate the construction of the secant method by means of a figure. When $f ^ { \prime } > 0$ on $I$, express $x _ { n + 1 }$ as a function of $x _ { n - 1 } , x _ { n }$ by means of the function $H _ { f }$ defined in question 3 of the third part. (Recall: the secant method initializes with $x_0, x_1 \in I$, and at each step considers the line $L_n$ passing through $(x_{n-1}, f(x_{n-1}))$ and $(x_n, f(x_n))$, defining $x_{n+1}$ as the $x$-intercept of $L_n$.)
In this question, we examine the special case of a polynomial function of degree two $f$ defined by the formula $f ( x ) = ( x - \alpha ) ( x - \beta )$ where $\alpha$ and $\beta$ are real and $\alpha > \beta$. We take $I = ] ( \alpha + \beta ) / 2 , + \infty [$. For $x \in \mathbb { R }$ we define $h ( x ) = \frac { x - \alpha } { x - \beta }$, with the convention $h ( \beta ) = \infty$. (a) For $x \in \mathbb { R }$ show that we have $| h ( x ) | < 1$ if and only if $x \in I$. (b) Explicitly state the recurrence relation satisfied by the sequence $u _ { n } : = h \left( x _ { n } \right)$ and deduce that the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined for any $x _ { 0 }$ and $x _ { 1 }$ in $I$. (c) Show that the sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ tends to 0 and deduce that $\left( x _ { n } \right) _ { n \geqslant 0 }$ tends to $\alpha$. (d) Let $\phi = \frac { 1 + \sqrt { 5 } } { 2 }$. Show that there exists a strictly negative real number $s$ such that $$x _ { n } - \alpha = O \left( e ^ { s \phi ^ { n } } \right) .$$
We return to the general case, $f$ being any function of class $\mathcal { C } ^ { 3 }$. We assume that $f$ vanishes at a point $x ^ { * } \in I$, for which $f ^ { \prime } \left( x ^ { * } \right) > 0$. (a) Show that there exists $\epsilon > 0$ such that $\left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right] \subset I$ and $f ^ { \prime } > 0$ on the interval $\left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right]$. We fix such an $\epsilon$ for the rest and we define $$M = \sup _ { ( x , y ) \in \left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right] ^ { 2 } } \left| \frac { \partial ^ { 2 } H _ { f } } { \partial x \partial y } ( x , y ) \right| .$$ (b) We assume that $x _ { n - 1 } , x _ { n } \in \left[ x ^ { * } - \epsilon , x ^ { * } + \epsilon \right]$. Show that $$\left| x _ { n + 1 } - x ^ { * } \right| \leqslant M \left| x _ { n - 1 } - x ^ { * } \right| \cdot \left| x _ { n } - x ^ { * } \right| .$$ (c) We fix $\left. \epsilon ^ { \prime } \in \right] 0 , \epsilon ]$ such that $M \epsilon ^ { \prime } < 1$. Show that if $x _ { 0 } , x _ { 1 }$ belong to $\left[ x ^ { * } - \epsilon ^ { \prime } , x ^ { * } + \epsilon ^ { \prime } \right]$ then the sequence $\left( x _ { n } \right) _ { n \geqslant 0 }$ is well defined and converges to $x ^ { * }$.