Let us consider the follwing cauchy problem: $$\eqalign{ & y'(t) = f(t,y(t)) \cr & y(0) = 0 \cr} $$ Assume that $f$ is strictely positive function and it is globally lipschitz and it is $\tau $ periodic with respect to the first variable. In addition, assume that $y(t)\in[0,1]$ for all positive $t$. Let $T$ be the unique positive constant such that $y(T)=1$. I have two questions: Are the solution is $\tau$-periodic?. Do we have $\tau$=$T$?.
Thank you in advance.