ODE with periodic initial/end condition

ODE with periodic initial/end condition

Are there any existence/uniqueness results for solutions to the ODE
$$y'(t) = f(y(t),t)$$ $$y(0) = y(T) = y_0$$ on the time interval $[0,T]$
where $f$ is Caretheodory and $y_0$ is given.
I am looking for $y \in C^0$ say and $y' \in L^2$.