Periodic solution of differential equation let be the ODE $ -y''(x)+f(x)y(x)=0 $
if the function $ f(x+T)=f(x) $ is PERIODIC does it mean that the ODE has only periodic solutions ? 
if all the solutions are periodic , then can all be determined by Fourier series ??

No, it doesn't mean that. For instance, $f(x)=0$ is periodic with any period, but $y''(x)=0$ has non-periodic solutions $y(x)=ax+b$.
