Let $n$ and $p$ be two integers greater than or equal to 2. We fix throughout this part a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which we assume to be $p$-periodic, that is such that $\forall k \in \mathbb { N } , A _ { k + p } = A _ { k }$. We denote by $\operatorname { Sol }$ (III.1) the set of sequences $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of vectors of $\mathbb { C } ^ { n }$ satisfying the recurrence relation $$\forall k \in \mathbb { N } , \quad Y _ { k + 1 } = A _ { k } Y _ { k }$$ Justify that we define a sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ by setting $\left\{ \begin{array} { l } \Phi _ { 0 } = I _ { n } \\ \Phi _ { k + 1 } = A _ { k } \Phi _ { k } \quad \forall k \in \mathbb { N } \end{array} \right.$ and that $\left( Y _ { k } \right) _ { k \in \mathbb { N } } \in \operatorname { Sol }$ (III.1) if and only if $\forall k \in \mathbb { N } , Y _ { k } = \Phi _ { k } Y _ { 0 }$.

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix belonging to $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$. Prove that there exists a unique sequence $\left( P _ { k } \right) _ { k \in \mathbb { N } } \in \left( \mathrm { GL } _ { n } ( \mathbb { C } ) \right) ^ { \mathbb { N } }$, periodic of period $p$, such that $$\forall k \in \mathbb { N } , \quad \Phi _ { k } = P _ { k } B ^ { k }$$

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$, and $\left( P _ { k } \right) _ { k \in \mathbb { N } }$ the unique $p$-periodic sequence in $\left( \mathrm { GL } _ { n } ( \mathbb { C } ) \right) ^ { \mathbb { N } }$ such that $\Phi _ { k } = P _ { k } B ^ { k }$ for all $k \in \mathbb{N}$. Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be a solution of (III.1). Justify the existence of $M = \max _ { k \in \mathbb { N } } \left\| P _ { k } \right\| _ { 0 }$. Show that for all $k \in \mathbb { N } , \left\| \Phi _ { k } \right\| _ { 0 } \leqslant n M \left\| B ^ { k } \right\| _ { 0 }$.

Let $n$ and $p$ be two integers greater than or equal to 2. We fix a sequence $\left( A _ { k } \right) _ { k \in \mathbb { N } }$ of matrices of $\mathrm { GL } _ { n } ( \mathbb { C } )$ which is $p$-periodic. The sequence $\left( \Phi _ { k } \right) _ { k \in \mathbb { N } }$ is defined by $\Phi _ { 0 } = I _ { n }$ and $\Phi _ { k + 1 } = A _ { k } \Phi _ { k }$. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$, and $\left( P _ { k } \right) _ { k \in \mathbb { N } }$ the unique $p$-periodic sequence such that $\Phi _ { k } = P _ { k } B ^ { k }$. Let $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ be a solution of (III.1).
a) Prove that if $\lim _ { k \rightarrow + \infty } \left\| B ^ { k } \right\| _ { 0 } = 0$, then $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.
b) Prove that if the sequence $\left( \left\| B ^ { k } \right\| _ { 0 } \right) _ { k \in \mathbb { N } }$ is bounded, then the sequence $\left( \left\| Y _ { k } \right\| _ { \infty } \right) _ { k \in \mathbb { N } }$ is also bounded.

We still assume that $p$ is an integer greater than or equal to 2. Let $R \in \mathbb { C } [ X ]$ be a polynomial of degree greater than or equal to 1 with simple roots. Prove that the polynomial $R \left( X ^ { p } \right)$ has simple roots if and only if $R ( 0 ) \neq 0$.

We still assume that $p$ is an integer greater than or equal to 2. We denote by $B$ a matrix in $\mathrm { GL } _ { n } ( \mathbb { C } )$ satisfying $B ^ { p } = \Phi _ { p }$. Suppose that $B$ is diagonalizable and that all its eigenvalues have modulus strictly less than 1. Prove that for every solution $\left( Y _ { k } \right) _ { k \in \mathbb { N } }$ of (III.1), $\lim _ { k \rightarrow + \infty } \left\| Y _ { k } \right\| _ { \infty } = 0$.

Let $A$ be a continuous function, periodic of period $T > 0$ and $X$ a function of class $\mathcal { C } ^ { 1 }$ $$A : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathcal { M } _ { 2 } ( \mathbb { C } ) \\ & t \mapsto A ( t ) \end{aligned} \quad X \right. : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto \binom { x _ { 1 } ( t ) } { x _ { 2 } ( t ) } \end{aligned}$$ We are interested in the homogeneous differential system with unknown $X$ $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$. We denote by $U$ and $V$ the two solutions of the differential system (IV.1) satisfying $U \left( t _ { 0 } \right) = \binom { 1 } { 0 }$ and $V \left( t _ { 0 } \right) = \binom { 0 } { 1 }$.
We consider the linear differential system (IV.2) whose solutions are functions of class $\mathcal { C } ^ { 1 }$ with values in $\mathcal { M } _ { 2 } ( \mathbb { C } )$ $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ For all $t \in \mathbb { R }$, we set $E ( t ) = [ U ( t ) , V ( t ) ]$. Verify that $E$ is the solution of (IV.2) satisfying $E \left( t _ { 0 } \right) = I _ { 2 }$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We consider the linear differential system $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ If $M : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathcal { M } _ { 2 } ( \mathbb { C } ) \\ & t \mapsto [ F ( t ) , G ( t ) ] \end{aligned} \right.$ is a solution of (IV.2) and $W = \binom { w _ { 1 } } { w _ { 2 } } \in \mathbb { C } ^ { 2 }$, prove that the function $Y : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto M ( t ) W = w _ { 1 } F ( t ) + w _ { 2 } G ( t ) \end{aligned}$ is a solution of (IV.1).

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$. We denote by $U$ and $V$ the two solutions of (IV.1) satisfying $U \left( t _ { 0 } \right) = \binom { 1 } { 0 }$ and $V \left( t _ { 0 } \right) = \binom { 0 } { 1 }$, and set $E ( t ) = [ U ( t ) , V ( t ) ]$.
Let $t _ { 1 } \in \mathbb { R }$ and $W = \binom { w _ { 1 } } { w _ { 2 } } \in \mathbb { C } ^ { 2 }$. Assume that $E \left( t _ { 1 } \right) W = \binom { 0 } { 0 }$. Show that the function $Y : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto E ( t ) W = w _ { 1 } U ( t ) + w _ { 2 } V ( t ) \end{aligned} \right.$ is zero. Deduce that for all real $t , E ( t )$ is invertible.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We consider the linear differential system $$\forall t \in \mathbb { R } , \quad M ^ { \prime } ( t ) = A ( t ) M ( t ) \tag{IV.2}$$ Let $M \in \mathcal { C } ^ { 1 } \left( \mathbb { R } , \mathcal { M } _ { 2 } ( \mathbb { C } ) \right)$ be a solution of system (IV.2). Show that for all real $t , M ( t ) = E ( t ) M \left( t _ { 0 } \right)$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. Deduce from the previous question that there exists a unique matrix $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ independent of $t$ such that for all real $t , E ( t + T ) = E ( t ) B$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the unique matrix such that $E(t+T) = E(t)B$ for all $t$. The Floquet multipliers of (IV.1) are the eigenvalues of $B$.
Let $\rho \in \mathbb { C }$ be a Floquet multiplier of (IV.1) and $Z \in \mathbb { C } ^ { 2 }$ be an eigenvector of $B$ associated with this eigenvalue. We denote $Y : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto E ( t ) Z \end{aligned}$.
a) Prove that $\forall t \in \mathbb { R } , Y ( t + T ) = \rho Y ( t )$.
b) Prove that there exists a complex number $\mu$ and a function $S : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto S ( t ) \end{aligned} \right.$ non-zero and $T$-periodic such that $\forall t \in \mathbb { R } , Y ( t ) = \mathrm { e } ^ { \mu t } S ( t )$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the Floquet matrix of (IV.1) and its eigenvalues are the Floquet multipliers. Give a necessary and sufficient condition on the Floquet multipliers for the differential system (IV.1) to admit a non-zero periodic solution of period $T$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the Floquet matrix of (IV.1) and its eigenvalues are the Floquet multipliers. Assume that the matrix $B$ is diagonalizable. Give a necessary and sufficient condition on the Floquet multipliers for the differential system (IV.1) to admit an unbounded solution on $\mathbb { R }$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set for all $t \in \mathbb { R } , W ( t ) = \operatorname { det } ( E ( t ) )$. Show that for all real $t , W ^ { \prime } ( t ) = \operatorname { tr } ( A ( t ) ) W ( t )$.

Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set $W ( t ) = \operatorname { det } ( E ( t ) )$ and denote $\rho _ { 1 }$ and $\rho _ { 2 }$ the Floquet multipliers of (IV.1). Deduce that $\rho _ { 1 } \rho _ { 2 } = \exp \left( \int _ { 0 } ^ { T } \operatorname { tr } ( A ( s ) ) \mathrm { d } s \right)$.

Let $\lambda \in \mathbb { R }$. Show that the problem
$$\left\{ \begin{array} { l } - v _ { \lambda } ^ { \prime \prime } ( x ) + c ( x ) v _ { \lambda } ( x ) = f ( x ) , x \in [ 0,1 ] \\ v _ { \lambda } ( 0 ) = 0 \\ v _ { \lambda } ^ { \prime } ( 0 ) = \lambda \end{array} \right.$$
admits a unique solution $v _ { \lambda } \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$.

Show that for all $\lambda \in \mathbb { R } , v _ { \lambda }$ can be expressed in the form:
$$v _ { \lambda } = \lambda w _ { 1 } + w _ { 2 }$$
with $w _ { 1 } \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$ the unique solution of the system
$$\left\{ \begin{array} { l } - w _ { 1 } ^ { \prime \prime } ( x ) + c ( x ) w _ { 1 } ( x ) = 0 , x \in [ 0,1 ] \\ w _ { 1 } ( 0 ) = 0 \\ w _ { 1 } ^ { \prime } ( 0 ) = 1 \end{array} \right.$$
and $w _ { 2 }$ a function independent of $\lambda$ to be characterized.

Show that $w _ { 1 } ( 1 ) \neq 0$.

Deduce that there exists a solution $u \in \mathcal { C } ^ { 2 } ( [ 0,1 ] , \mathbb { R } )$ to problem (1): $$\left\{ \begin{array} { l } - u ^ { \prime \prime } ( x ) + c ( x ) u ( x ) = f ( x ) , x \in [ 0,1 ] \\ u ( 0 ) = u ( 1 ) = 0 \end{array} \right.$$ Show that this solution is unique.

Show that if $f$ is positive, then $u$ is also positive.

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. What are the $2\pi$-periodic solutions of (II.2): $$z''(\theta) + \lambda z(\theta) = 0$$

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We assume here that $\lambda = 0$. Solve (II.1) on $\mathbb{R}^{+*}$: $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We now assume $\lambda \neq 0$. Give a necessary and sufficient condition for (II.2) $$z''(\theta) + \lambda z(\theta) = 0$$ to admit non-zero $2\pi$-periodic solutions. Give these solutions.

In this subsection II.C, we consider two functions of class $\mathcal{C}^2$, $u : \mathbb{R}^{*+} \rightarrow \mathbb{R}$ and $v : \mathbb{R} \rightarrow \mathbb{R}$ and we set $$\forall (r,\theta) \in \mathbb{R}^{*+} \times \mathbb{R} \quad f(r\cos(\theta), r\sin(\theta)) = u(r)v(\theta)$$ We now assume $\lambda \neq 0$. Solve (II.1) on $\mathbb{R}^{+*}$: $$r^2 z''(r) + r z'(r) - \lambda z(r) = 0$$ One may consider, justifying its existence, a function $Z$ of class $\mathcal{C}^2$ on $\mathbb{R}$ such that, for all $r > 0$, $z(r) = Z(\ln(r))$.