Says I have two (scalar) ODE: u' = f(u,t) and v' = g(v,t) where
- Both $f$ and $g$ are piecewise-continuous and locally Lipschitz, for existence & uniqueness of solutions $u(t)$ and $v(t)$.
- $f(x,t) \leq g(x,t)$ for all $x$ and $t$.
I believe that if $u(0) \leq v(0)$ then $u(t) \leq v(t)$ for all $t \geq 0$. But I don't know if there is such theorem, or if not, how to prove it.