We are given $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ continuous and 1-periodic in each of its arguments, of class $\mathscr{C}^1$. We assume $x \neq 0$ (but close to 0), and that a solution $h$ of equation (4) exists (1-periodic, $\mathscr{C}^1$, zero average). We set $\tilde{\alpha}(t) = \alpha(t) + x h(\alpha(t))$ where $\alpha$ satisfies $\alpha'(t) = \omega + x g(\alpha(t))$.
Show that there exists a function $\varepsilon : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ such that $$\tilde { \alpha } ^ { \prime } ( t ) = \omega + x \nu + x \varepsilon ( x , t )$$ and $\sup _ { t \in \mathbb { R } } \| \varepsilon ( x , t ) \| \rightarrow 0$ when $x \rightarrow 0$.

We are given $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ continuous and 1-periodic in each of its arguments, of class $\mathscr{C}^1$. We assume $x \neq 0$ (but close to 0), and that a solution $h$ of equation (4) exists (1-periodic, $\mathscr{C}^1$, zero average). We set $\tilde{\alpha}(t) = \alpha(t) + x h(\alpha(t))$ where $\alpha$ satisfies $\alpha'(t) = \omega + x g(\alpha(t))$, $\alpha(0)=(0,0)$. From question 15b, $\tilde{\alpha}'(t) = \omega + x\nu + x\varepsilon(x,t)$.
Let $T > 0$ be fixed. Show that there exists a function $\eta : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ such that $$\alpha ( t ) = ( \omega + x \nu ) t + x ( h ( 0,0 ) - h ( \omega t ) ) + x \eta ( x , t )$$ and $\sup _ { t \in [ 0 , T ] } \| \eta ( x , t ) \| \rightarrow 0$ when $x \rightarrow 0$.

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).

Show that $S _ { N }$ converges uniformly to $S$ on $\mathbf { R } _ { + }$ when $N \rightarrow + \infty$ and that
$$\left\| S _ { N } - S \right\| _ { \infty , \mathbf { R } _ { + } } \leq \frac { M \mathrm { e } ^ { 2 M } } { 2 ^ { N + 1 } }$$
where $S _ { N } ( x ) = S _ { 0 } \mathrm { e } ^ { y _ { N } ( x ) } = \frac { 1 } { 2 } \exp \left( \sum _ { n = 0 } ^ { N } a _ { n } \mathrm { e } ^ { - \lambda _ { n } x } \right)$, $S(x) = S_0 e^{y(x)} = \frac{1}{2} e^{y(x)}$, and $\left\| y_N - y \right\|_{\infty, \mathbf{R}_+} \leq \frac{M}{2^N}$.