I have a TR (research work) concerning the equation: $$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$$ f and g are defined and continuous in $\mathbb R$ and with values in $\mathbb R$. Furthermore f is assumed to be T periodic (There is no initial condition)
First question:
Assume that there is a periodic solution. Using the mean formula, to show that $\frac 1T \int_0 ^ T f (t) dt \in g (\mathbb R) $
Second question
what to add as hypotheses for the existence of a periodic solution
For first question:
$ g \circ x $ being continuous, there is a $ c $ between $ 0 $ and $ T $ such as $ \frac 1T\int_0 ^ T g (x (s)) ds = g (x (c)) $. So if $ x $ is a solution $ T $ periodic, because $ x (T) = x (0) $, we will have:
$$\frac1T \int_0 ^ T f (s) ds = \frac 1T \int_0 ^ T (g (x (s)) + x '(s)) ds = \frac1T \int_0 ^ T g (x (s)) ds = g (x (c)) \in g (\mathbb R). $$
For second question:
I need help what to add as hypotheses?