Suppose $a \in \mathbb{R}$ and $g\in C(\mathbb{R})$ is a nonnegative periodic function $g(t+1)=g(t)$. Find conditions on $a,g$ such that the linear inhomogeneous equation $$\dot{x}=ax+g(t)$$ has a periodic solution. When is this solution unique?
I can also use Poincare map
I should use the fact that the solution of the inhomogeneous equation $$\dot{x} = a(t)x +g(t)$$ is given by $$x(t) = x_{0}A(t,t_{0})+\int_{t_{0}}^{t} A(t,s)g(s)ds,$$ where $$A(t,s) = e^{\int_{s}^{t}a(s)ds}$$.+
So my idea was to show that $A(t,s) \neq 0$ but I am not sure if that´s right path at all. I would appreciate if someone could give me a suggestion or help to formulate the solution mathematically correct?
I´ve seen this solution already Find periodic solution of differential equation but there is no mention of $a$ being a constant and $g$ a periodic function or how does that relate to uniqness.