We denote by $E$ the vector space of functions with real values continuous on $\mathbb { R } _ { + }$. For every element $f$ of $E$ and all $x \in \mathbb { R } _ { + }$ we set $F ( x ) = \int _ { 0 } ^ { x } f ( u ) \mathrm { d } u$.

\begin{enumerate}
  \item Justify that $F$ is of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$ and give for all $x \in \mathbb { R } _ { + }$ the expression of $F ^ { \prime } ( x )$.

Let $\Psi : f \in E \mapsto \Psi ( f )$ defined by: $\forall x \in \mathbb { R } _ { + } , \Psi ( f ) ( x ) = \int _ { 0 } ^ { 1 } f ( x t ) \mathrm { d } t$.
  \item Express, for all strictly positive real $x$, $\Psi ( f ) ( x )$ using $F ( x )$.
  \item Justify that the function $\Psi ( f )$ is continuous on $\mathbb { R } _ { + }$ and give the value of $\Psi ( f ) ( 0 )$.
  \item Show that $\Psi$ is an endomorphism of $E$.
  \item Surjectivity of $\Psi$

Let $h : x \in \mathbb { R } _ { + } \longmapsto h ( x ) = \left\{ \begin{array} { l l } x \sin \left( \frac { 1 } { x } \right) & \text { for } x > 0 \\ 0 & \text { for } x = 0 \end{array} \right.$.
  \begin{enumerate}
    \item[5.1.] Show that the function $h$ is continuous on $\mathbb { R } _ { + }$.
    \item[5.2.] Is the function $h$ of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$?
    \item[5.3.] Let $g \in \operatorname { Im } ( \Psi )$. Show that the function $x \mapsto x g ( x )$ is of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$.
    \item[5.4.] Do we have $h \in \operatorname { Im } ( \Psi )$?
    \item[5.5.] Conclude.
  \end{enumerate}
  \item Show that $\Psi$ is injective.
  \item Search for the eigenvectors of $\Psi$
  \begin{enumerate}
    \item[7.1.] Justify that 0 is not an eigenvalue of $\Psi$.

Let $\mu \in \mathbb { R }$. We consider the differential equation $(L)$ on $\mathbb { R } _ { + } ^ { * }$:
$$y ^ { \prime } + \frac { \mu } { x } y = 0$$
    \item[7.2.] Solve $(L)$ on $\mathbb { R } _ { + } ^ { * }$.
    \item[7.3.] Determine the solutions of $(L)$ that can be extended by continuity on $\mathbb { R } _ { + }$.
    \item[7.4.] Then determine the eigenvalues of $\Psi$ and the associated eigenspaces.
  \end{enumerate}
  \item Let $n \in \mathbb { N } , n > 1$. For $i \in \llbracket 1 , n \rrbracket$, we set:
$$f _ { i } : x \in \mathbb { R } _ { + } \longmapsto f _ { i } ( x ) = x ^ { i } \text { and } g _ { i } : x \in \mathbb { R } _ { + } \longmapsto g _ { i } ( x ) = \begin{cases} x ^ { i } \ln ( x ) & \text { for } x > 0 \\ 0 & \text { for } x = 0 \end{cases}$$
We denote $\mathscr { B } = \left( f _ { 1 } , \ldots , f _ { n } , g _ { 1 } , \ldots , g _ { n } \right)$ and $F _ { n }$ the vector subspace of $E$ generated by $\mathscr { B }$.
  \begin{enumerate}
    \item[8.1.] We want to show that the family $\mathcal { B } = \left( f _ { 1 } , \ldots , f _ { n } , g _ { 1 } , \ldots , g _ { n } \right)$ is a basis of $F _ { n }$.

Let $\left( \alpha _ { i } \right) _ { i \in \llbracket 1 , n \rrbracket }$ and $\left( \beta _ { j } \right) _ { j \in \llbracket 1 , n \rrbracket }$ be scalars such that $\sum _ { i = 1 } ^ { n } \alpha _ { i } f _ { i } + \sum _ { j = 1 } ^ { n } \beta _ { j } g _ { j } = 0$.
    \begin{enumerate}
      \item[8.1.1.] Show that $\alpha _ { 1 } = \beta _ { 1 } = 0$. One may simplify the expression (*) by $x$ when $x$ is non-zero.
      \item[8.1.2.] Let $p \in \llbracket 1 , n - 1 \rrbracket$. Suppose that $\alpha _ { 1 } = \cdots = \alpha _ { p } = \beta _ { 1 } = \cdots = \beta _ { p } = 0$. Prove that $\alpha _ { p + 1 } = \beta _ { p + 1 } = 0$.
      \item[8.1.3.] Conclude and determine the dimension of the vector space $F _ { n }$.
    \end{enumerate}
    \item[8.2.] Where we prove that $\Psi$ induces an endomorphism on $F _ { n }$
    \begin{enumerate}
      \item[8.2.1.] Let $x > 0$ and $p \in \mathbb { N } ^ { * }$. Show that the integral $\int _ { 0 } ^ { x } t ^ { p } \ln ( t ) \mathrm { d } t$ is convergent and calculate it.
      \item[8.2.2.] Deduce that $\Psi$ induces an endomorphism $\Psi _ { n }$ on $F _ { n }$.
    \end{enumerate}
    \item[8.3.] Give the matrix of the application $\Psi _ { n }$ in the basis $\mathcal { B }$.
    \item[8.4.] Prove that $\Psi _ { n }$ is an automorphism of $F _ { n }$.
    \item[8.5.] Let $z : x \in \mathbb { R } _ { + } \longmapsto z ( x ) = \left\{ \begin{array} { l l } \left( x + x ^ { 2 } \right) \ln ( x ) & \text { for } x > 0 \\ 0 & \text { for } x = 0 \end{array} \right.$. After verifying that $z \in F _ { n }$, determine $\Psi _ { n } ^ { - 1 } ( z )$.
  \end{enumerate}
\end{enumerate}