Let $f:\mathbb{R}\longrightarrow [0,+\infty)$ is smooth, 1-periodic with \begin{equation*} \min_{x\in [0,1]} f(x) = 0. \end{equation*} I consider the following ODE: \begin{equation*} \begin{cases} \dot{\eta}(s) &= \Phi(f(\eta(s))), \qquad\text{for}\qquad s>0,\\ \eta(0) &= 0, \end{cases} \end{equation*} where $\Phi: [0,\infty)\longrightarrow [0,\infty)$ is strictly increasing, smooth on $(0,+\infty)$ with $\Phi(0) = 0$.
Is it true that if $f(0) = 0$ then the unique solution is $\eta(s)\equiv 0$? I am thinking about some sort of nonlinear Gronwall's inequality to show this but haven't be successful yet.