Let $f:\mathbb R\times \mathbb R\longrightarrow\mathbb R$ be a $C^1$ function, periodic in the first variable, and such that $f(t,1)\leq > 0\leq f(t,0)$ for all $t$.
Consider the differential equation $$u'(t)=f(t,u(t))$$ and prove that there exists a periodic solution $u$ such that $0\leq u\leq 1$.
It seems that we need the condition $f(t,1)\leq 0\leq f(t,0)$ for periodicity too, since, for example, $f(t,x)=\sin t+2$ provides only non-periodic solutions (correct?).
Moreover, I can't understand how the $C^1$ condition is involved: it assures us for example that $f$ is globally Lipschitz, and that we have $C^1$-dependence on the initial values, but I can't see how this can help us.