For all $n \in \mathbb{N}^\star$, justify that there exists a unique function $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfying $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$.

To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $$U ( f ) ( x ) = \int _ { 0 } ^ { x } \left( 1 - \mathrm { e } ^ { - t } \right) \frac { f ( t ) } { t } \mathrm {~d} t + \left( \mathrm { e } ^ { x } - 1 \right) \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Let $f \in E$. Show that $U ( f )$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$ and satisfies, for all $x > 0$, $$( U ( f ) ) ^ { \prime } ( x ) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$

To each function $f \in E$, we associate the endomorphism $U$ of $E$ defined for all $x > 0$ by $$U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Is the real number $0$ an eigenvalue of $U$?

To each function $f \in E$, we associate the endomorphism $U$ of $E$ defined for all $x > 0$ by $$U ( f ) ( x ) = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ It has been shown that $U(f)$ satisfies $y'' - y' = -f(x)/x$ on $\mathbb{R}_+^*$. Let $\lambda \in \mathbb { R } ^ { * }$. We assume that $\lambda$ is an eigenvalue of $U$. Let $f$ be an eigenvector associated with it. Show that $f$ is a solution of the differential equation $(E_{1/\lambda}) : x(y'' - y') + \frac{1}{\lambda} y = 0$.

For $p \in \mathbb{N}^*$, let $P$ be a non-zero polynomial solution of $(E_p) : x(y'' - y') + py = 0$. It has been shown that $U(f)$ satisfies $y'' - y' = -f(x)/x$ and that $U$ is self-adjoint ($\langle f | U(g) \rangle = \langle U(f) | g \rangle$). Prove that the function $pU(P) - P$ satisfies on $\mathbb { R } _ { + } ^ { * }$ the differential equation $y ^ { \prime \prime } - y ^ { \prime } = 0$.

For $p \in \mathbb{N}^*$, let $P$ be a non-zero polynomial solution of $(E_p) : x(y'' - y') + py = 0$. It has been shown that $pU(P) - P$ satisfies $y'' - y' = 0$ on $\mathbb{R}_+^*$. Show that $P$ is an eigenvector of $U$ for the eigenvalue $1/p$.

We consider the Cauchy problem associated with $F_0$ defined by: $$\forall y \in ]0, +\infty[, \quad F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$$ with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$. We denote $\phi_0$ the solution of this problem on $[0, +\infty[$.
(a) Show that there exists $\varepsilon > 0$ such that for all $t \in ]0, \varepsilon]$ we have $y_{\text{init}} < \phi_0(t) < \theta$.
(b) By considering the function $z_0(t) = \ln\left(\phi_0(t)/\theta\right)$ find the expression of $\phi_0$.
(c) Deduce that $\phi_0$ satisfies $y_{\text{init}} < \phi_0(t) < \theta$ for all $t \in ]0, +\infty[$ and that moreover $\phi_0$ is strictly increasing.

For $0 < \mu \leqslant 1$, we consider $F_\mu$ defined by: $$\forall y \in ]0, +\infty[, \quad F_\mu(y) = \frac{a}{\mu} y\left(1 - \left(\frac{y}{\theta}\right)^\mu\right)$$ with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$. By considering the function $z_\mu(t) = \phi_\mu(t)^{-\mu}$ find the expression of the solution $\phi_\mu$ on $[0, +\infty[$ associated with $F_\mu$.

For $0 < \mu \leqslant 1$, we consider $F_\mu$ defined by: $$\forall y \in ]0, +\infty[, \quad F_\mu(y) = \frac{a}{\mu} y\left(1 - \left(\frac{y}{\theta}\right)^\mu\right)$$ and $F_0$ defined by $F_0(y) = ay\ln\left(\frac{\theta}{y}\right)$, with $a, \theta > 0$ and $0 < y_{\text{init}} < \theta$.
(a) Show that $F_\mu$ converges pointwise to $F_0$ as $\mu$ tends to 0.
(b) Show that $\phi_\mu$ converges pointwise to $\phi_0$ as $\mu$ tends to 0.

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. For all $r \geqslant 0$, we denote $B_r$ the closed ball with center $y_{\text{init}}$ and radius $r$.
Show that we can choose $r > 0$ and $T > 0$ such that $B_r \subset \Omega$ and such that for all $N \in \mathbb{N}^*$, we can define by recursion, by setting $\Delta t = \frac{T}{N}$, a sequence $(y_n)_{0 \leqslant n \leqslant N}$ taking values in $B_r$ such that: $$y_0 = y_{\text{init}}, \quad y_{n+1} = y_n + \Delta t F(y_n), \forall n \in \{0, \cdots, N-1\}$$

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With $r > 0$, $T > 0$, $\Delta t = \frac{T}{N}$ and the sequence $(y_n)_{0 \leqslant n \leqslant N}$ as defined in question III.1, show that we can construct a unique function $\phi_N$ continuous on $[0, T]$, affine on each interval $[n\Delta t, (n+1)\Delta t]$ for all $n \in \{0, \cdots, N-1\}$ and such that $\phi_N(n\Delta t) = y_n$ for all $n \in \{0, \cdots, N\}$.

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. Show that we can define a sequence of step functions $\psi_N : [0, T] \rightarrow \mathbb{R}^d$ such that $\psi_N(t) = \phi_N(t)$ for $t \in \{n\Delta t \mid n \in \{0, \cdots, N\}\}$ and such that: $$\forall N \in \mathbb{N}^*, \phi_N(t) = y_{\text{init}} + \int_0^t F(\psi_N(s))\, ds \text{ for all } t \in [0, T]$$ We will specify the expression of the functions $\psi_N$.

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With the step functions $\psi_N$ defined in question III.4, deduce that there exists a subsequence of $\psi_N$ that converges uniformly on $[0, T]$ and specify its limit.

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. Show that $(\phi, T)$ is a solution of the Cauchy problem $$\left\{\begin{array}{l} y'(t) = F(y(t)) \\ y(0) = y_{\text{init}} \end{array}\right.$$ and deduce the following theorem:
Theorem 2: If $F$ is a continuous function, then there exists at least one solution of the Cauchy problem (1).

We consider the particular case for $d = 1$ given for all $y \in \mathbb{R}$ by $F(y) = 3|y|^{2/3}$ and $y_{\text{init}} = 0$. Show that this Cauchy problem admits infinitely many global solutions.

We consider $\mathcal{F} : \mathbb{R}^d \rightarrow \mathcal{P}_c(\mathbb{R}^d)$ taking values in the set $\mathcal{P}_c(\mathbb{R}^d)$ of compact subsets of $\mathbb{R}^d$, and the differential inclusion problem: $$\left\{\begin{array}{l} y'(t) \in \mathcal{F}(y(t)) \\ y(0) = y_{\text{init}} \end{array}\right.$$
Show that if for every compact $K \subset \mathbb{R}^d$, there exists $C_K > 0$ such that $\mathcal{F}$ satisfies: $$\forall x, y \in K, \forall v_x \in \mathcal{F}(x), \forall v_y \in \mathcal{F}(y), \quad \langle v_x - v_y, x - y \rangle \leqslant C_K \|x - y\|^2$$ then problem (2) admits at most one maximal solution. (Hint: You may look at $\|X(t) - Y(t)\|^2$ for $X$ and $Y$ two solutions.)

We consider the differential inclusion problem given by $d = 2$ and $\mathcal{F} : \mathbb{R}^2 \rightarrow \mathcal{P}_c(\mathbb{R}^2)$ defined for all $x = (x_1, x_2) \in \mathbb{R}^2$ by: $$\mathcal{F}(x) = \begin{cases} \{v^-\} & \text{if } x_1 < 0 \\ \{v^+\} & \text{if } x_1 > 0 \\ [v_1^+, v_1^-] \times [v_2^+, v_2^-] & \text{if } x_1 = 0 \end{cases}$$ where $v^- = (v_1^-, v_2^-) \in \mathbb{R}^2$ and $v^+ = (v_1^+, v_2^+) \in \mathbb{R}^2$ with $v_1^- \geqslant v_1^+$ and $v_2^- \geqslant v_2^+$.
We set $v^- = (1, 2)$ and $v^+ = (-1, 2)$.
(a) Show that $\mathcal{F}$ satisfies condition (3).
(b) We choose $y_{\text{init}} = (0, 0)$. Find all maximal solutions of problem (2).
(c) We choose $y_{\text{init}} = (1, 0)$. Find all maximal solutions of problem (2).

We consider the differential inclusion problem given by $d = 2$ and $\mathcal{F} : \mathbb{R}^2 \rightarrow \mathcal{P}_c(\mathbb{R}^2)$ defined for all $x = (x_1, x_2) \in \mathbb{R}^2$ by: $$\mathcal{F}(x) = \begin{cases} \{v^-\} & \text{if } x_1 < 0 \\ \{v^+\} & \text{if } x_1 > 0 \\ [v_1^+, v_1^-] \times [v_2^+, v_2^-] & \text{if } x_1 = 0 \end{cases}$$ where $v^- = (v_1^-, v_2^-) \in \mathbb{R}^2$ and $v^+ = (v_1^+, v_2^+) \in \mathbb{R}^2$ with $v_1^- \geqslant v_1^+$ and $v_2^- \geqslant v_2^+$.
We set $v^- = (0, 1)$ and $v^+ = (1, 1)$.
(a) Show that $\mathcal{F}$ does not satisfy condition (3).
(b) We choose $y_{\text{init}} = (1, 0)$. Find all maximal solutions of problem (2).
(c) We choose $y_{\text{init}} = (0, 0)$. Find all maximal solutions of problem (2).

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 $F ( x ) = \int _ { 0 } ^ { x } f ( u ) \mathrm { d } u$.

1. 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: $\forall x \in \mathbb { R } _ { + } , \Psi ( f ) ( x ) = \int _ { 0 } ^ { 1 } f ( x t ) \mathrm { d } t$.
2. Express, for all strictly positive real $x$, $\Psi ( f ) ( x )$ using $F ( x )$.
3. Justify that the function $\Psi ( f )$ is continuous on $\mathbb { R } _ { + }$ and give the value of $\Psi ( f ) ( 0 )$.
4. Show that $\Psi$ is an endomorphism of $E$.
5. Surjectivity of $\Psi$
   Let $h : x \in \mathbb { R } _ { + } \longmapsto h ( x ) = \left\{ \begin{array} { l l } x \sin \left( \frac { 1 } { x } \right) & \text { for } x > 0 \\ 0 & \text { for } x = 0 \end{array} \right.$.
   1. [5.1.] Show that the function $h$ is continuous on $\mathbb { R } _ { + }$.
   2. [5.2.] Is the function $h$ of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$?
   3. [5.3.] Let $g \in \operatorname { Im } ( \Psi )$. Show that the function $x \mapsto x g ( x )$ is of class $C ^ { 1 }$ on $\mathbb { R } _ { + }$.
   4. [5.4.] Do we have $h \in \operatorname { Im } ( \Psi )$?
   5. [5.5.] Conclude.
6. Show that $\Psi$ is injective.
7. Search for the eigenvectors of $\Psi$
   1. [7.1.] Justify that 0 is not an eigenvalue of $\Psi$.
      Let $\mu \in \mathbb { R }$. We consider the differential equation $(L)$ on $\mathbb { R } _ { + } ^ { * }$: $$y ^ { \prime } + \frac { \mu } { x } y = 0$$
   2. [7.2.] Solve $(L)$ on $\mathbb { R } _ { + } ^ { * }$.
   3. [7.3.] Determine the solutions of $(L)$ that can be extended by continuity on $\mathbb { R } _ { + }$.
   4. [7.4.] Then determine the eigenvalues of $\Psi$ and the associated eigenspaces.
8. 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 $\mathscr { B }$.
   1. [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 $\sum _ { i = 1 } ^ { n } \alpha _ { i } f _ { i } + \sum _ { j = 1 } ^ { n } \beta _ { j } g _ { j } = 0$.
      1. [8.1.1.] Show that $\alpha _ { 1 } = \beta _ { 1 } = 0$. One may simplify the expression (*) by $x$ when $x$ is non-zero.
      2. [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 $\alpha _ { p + 1 } = \beta _ { p + 1 } = 0$.
      3. [8.1.3.] Conclude and determine the dimension of the vector space $F _ { n }$.
   2. [8.2.] Where we prove that $\Psi$ induces an endomorphism on $F _ { n }$
      1. [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.
      2. [8.2.2.] Deduce that $\Psi$ induces an endomorphism $\Psi _ { n }$ on $F _ { n }$.
   3. [8.3.] Give the matrix of the application $\Psi _ { n }$ in the basis $\mathcal { B }$.
   4. [8.4.] Prove that $\Psi _ { n }$ is an automorphism of $F _ { n }$.
   5. [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 $\Psi _ { n } ^ { - 1 } ( z )$.

Mathematics Specialty - EXERCISE II (20 points)

The questions in Part I can be treated independently. In this exercise, $K$ and $a$ are strictly positive real constants.

Part I - Preliminary Studies

Consider the differential equation $\left( E _ { 1 } \right) : z ^ { \prime } ( t ) + z ( t ) = \frac { 1 } { K }$, where $z$ is a function defined and differentiable on $[ 0 ; + \infty [$. II-1- Give the general solution of $( E _ { 1 } )$ on the interval $[ 0 ; + \infty [$. Consider the function $f$ defined for every positive real $t$ by: $f ( t ) = \frac { 10 } { 1 + a e ^ { - t } }$. II-2- Complete the table of variations of $f$ on the interval $[ 0 ; + \infty [$. Specify the value of $f$ at $0$ and the limit of $f$ at $+ \infty$. II-3- Determine, as a function of $a$, the set of solutions of the equation $f ( t ) = 5$.

Part II - Evolution of a Marmot Population

Let $y _ { 0 }$ be a strictly positive real number. We study the evolution of a marmot population, which initially numbers $y _ { 0 }$ thousand individuals. We admit that the size of the population, expressed in thousands of individuals, after $t$ years (with $t \geq 0$) is a function $y$ differentiable on $[ 0 ; + \infty [$, solution of the differential equation: $$\left( E _ { 2 } \right) : y ^ { \prime } ( t ) = y ( t ) \left( 1 - \frac { y ( t ) } { K } \right)$$ The constant $K$ is called the carrying capacity of the environment, expressed in thousands of individuals. We admit that there exists a unique function $y$ solution of $\left( E _ { 2 } \right)$ that satisfies $y ( 0 ) = y _ { 0 }$. We admit that this function takes strictly positive values on $[ 0 ; + \infty [$. We set $z ( t ) = \frac { 1 } { y ( t ) }$ for every positive real $t$. II-4-a- Express $z ^ { \prime } ( t )$ as a function of $y ^ { \prime } ( t )$ and $y ( t )$. II-4-b- We wish to show that $z$ is a solution of $\left( E _ { 1 } \right)$ if, and only if, $y$ is a solution of $\left( E _ { 2 } \right)$. Complete:

• Line 1 using an expression involving $z ^ { \prime } ( t )$ and $z ( t )$;
• Line 2 and Line 3 using an expression involving $y ^ { \prime } ( t )$ and $y ( t )$.

II-5-a- Deduce the general solution of $( E _ { 2 } )$. II-5-b- We admit that the unique solution $y$ of $\left( E _ { 2 } \right)$ satisfying $y ( 0 ) = y _ { 0 }$ is written in the form $y ( t ) = \frac { K } { 1 + a e ^ { - t } }$. Express $a$ as a function of $y _ { 0 }$ and $K$. In a certain valley with carrying capacity $K = 10$, the marmots have disappeared. Scientists wish to reintroduce $y _ { 0 }$ thousand marmots, with $0 < y _ { 0 } < 10$. In the remainder of the exercise, we will take $K = 10$. II-6- Justify that the value of $a$ obtained in question II-5-b- is indeed strictly positive. II-7-a- Using the result from question II-3-, give the value of $a$ such that $y ( 5 ) = 5$. II-7-b- Deduce the exact value of $y _ { 0 }$ such that $y ( 5 ) = 5$. Justify your answer. II-7-c- The calculator gives $0.0669285092$ as the result of the calculation of the value of $y _ { 0 }$ from the previous question. What is the minimum number of marmots to reintroduce so that at least $5$ thousand marmots are present after $5$ years following their reintroduction?

Deduce that for $f \in C^{2}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime}$ and $f^{\prime\prime}$ have slow growth, we have
$$\forall t \in \mathbf{R}_{+}^{*}, \forall x \in \mathbf{R}, \quad \frac{\partial P_{t}(f)(x)}{\partial t} = L\left(P_{t}(f)\right)(x)$$
where $\forall x \in \mathbf{R}, \quad L(f)(x) = f^{\prime\prime}(x) - x f^{\prime}(x).$

Deduce that for $f \in C^2(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f'$ and $f''$ have slow growth, we have $$\forall t \in \mathbf{R}_+^*, \forall x \in \mathbf{R}, \quad \frac{\partial P_t(f)(x)}{\partial t} = L\!\left(P_t(f)\right)(x),$$ where $L(f)(x) = f''(x) - xf'(x)$.

We admit that $S$ is of class $C^{1}$ on $\mathbf{R}_{+}^{*}$ and that
$$\forall t \in \mathbf{R}_{+}^{*}, \quad S^{\prime}(t) = \int_{-\infty}^{+\infty} \frac{\partial P_{t}(f)(x)}{\partial t} \left(1 + \ln\left(P_{t}(f)(x)\right)\right) \varphi(x) \mathrm{d}x$$
Show that
$$\forall t \in \mathbf{R}_{+}^{*}, \quad S^{\prime}(t) = \int_{-\infty}^{+\infty} L\left(P_{t}(f)\right)(x) \left(1 + \ln\left(P_{t}(f)(x)\right)\right) \varphi(x) \mathrm{d}x$$

We admit that $S$ is of class $C^1$ on $\mathbf{R}_+^*$ and that $$\forall t \in \mathbf{R}_+^*, \quad S'(t) = \int_{-\infty}^{+\infty} \frac{\partial P_t(f)(x)}{\partial t}\left(1 + \ln\!\left(P_t(f)(x)\right)\right)\varphi(x)\,\mathrm{d}x.$$ Show that $$\forall t \in \mathbf{R}_+^*, \quad S'(t) = \int_{-\infty}^{+\infty} L\!\left(P_t(f)\right)(x)\left(1 + \ln\!\left(P_t(f)(x)\right)\right)\varphi(x)\,\mathrm{d}x.$$

Show that a power series $f(x) = \sum_{n=0}^{\infty} \frac{c_n}{n!} x^n \in \mathbf{Q}\llbracket x \rrbracket$ is a solution of a differential equation if and only if its Laplace transform $$\widehat{f}(x) \stackrel{\text{def}}{=} \sum_{n=0}^{\infty} c_n x^n$$ is a solution of a differential equation.