For $T<\infty$， if $\phi: [0,T] \rightarrow R$ solves \begin{align} &\frac{d\phi}{dt} =F(\phi(t),t) \\ &\phi(0) =\alpha \end{align} where $F(x,t)$ is smooth. Then, how to show there is $\epsilon_0>0$ such that , for any $0<\epsilon\le \epsilon_0$, \begin{align} &\frac{d\phi_\epsilon}{dt} =F(\phi_\epsilon(t),t) +\epsilon \\ &\phi_\epsilon(0) =\alpha +\epsilon \end{align} has solution $\phi_\epsilon$ on $[0,T]$ ? Besides, how to show $\phi_\epsilon \rightarrow \phi$ uniformly ?
PS: This problem is from the proof of Theorem 3.1.1 (Weak maximum principle for scalars) of Topping's Lectures on the Ricci flow. I have a little knowledge of ODE, just know some first order equation and its local existence.