Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$. Let the differential system $$X'(t) = A(t) X(t) + b(t) \tag{3}$$ where $b : \mathbb{R} \rightarrow \mathbb{C}^n$ is a continuous function on $\mathbb{R}$ and $T$-periodic. We assume that $1$ is not an eigenvalue of $\exp(TB)$. Show that (3) possesses a unique $T$-periodic solution.
Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$. Let the differential system
$$X'(t) = A(t) X(t) + b(t) \tag{3}$$
where $b : \mathbb{R} \rightarrow \mathbb{C}^n$ is a continuous function on $\mathbb{R}$ and $T$-periodic.
We assume that $1$ is not an eigenvalue of $\exp(TB)$. Show that (3) possesses a unique $T$-periodic solution.