Let $f,g:\mathbb{R}^2\to\mathbb{R}$ continuous functions such as $f(t,x)\geq g(t,x)$ for all $(t,x)\in\mathbb{R}^2$. Consider $\varphi:I\to \mathbb{R}$ and $\psi:J\to\mathbb{R}$ solutions of the following Cauchy's problems. $x'(t)=f(t,x), x(0)=0$ and $y'(t)=g(t,y),y(0)=0$, respctively. Prove that $\varphi(t) \geq \psi(t)$, for all $t\in I\cap J$
My attempt: I've tried to create a system as follow:$z'(t)=f(t,x)-g(t,y), z(0)=0$. And now I don't know how to conclude. Any sugestions?
Put $\zeta(t) := \psi(t) - \varphi(t)$ for $t \in I \cap J \cap [0, \infty)$. We want to prove that $\zeta(t) \le 0$ for all $t \in I \cap J \cap [0, \infty)$.
Suppose to the contrary that there exists $c \in I \cap J \cap [0, \infty)$ such that $\zeta(c) > 0$. Let $b := \sup\{t \in I \cap J \cap [0, c): \zeta(t) \le 0\}$. By construction, $b \in [0, c)$ and $\zeta(t) > 0$ for all $t \in (b, c]$.
We have $$ \zeta'(t) = \psi'(t) - \varphi'(t) = g(t, \psi(t)) - f(t, \varphi(t)) \\ = [g(t, \psi(t)) - g(t,\varphi(t))] + [g(t, \varphi(t)) - f(t, \varphi(t))] \le g(t, \psi(t)) - g(t,\varphi(t)) $$ for all $t \in I \cap J \cap [0, \infty)$.
By the (local) Lipschitz property of $g$ there exists $L > 0$ such that $$ \lvert g(t, \psi(t)) - g(t,\varphi(t)) \rvert \le L \lvert \psi(t) - \varphi(t) \rvert, \quad t \in [b, c], $$ consequently, $$ g(t, \psi(t)) - g(t,\varphi(t)) \le L (\psi(t) - \varphi(t)), \quad t \in [b, c]. $$ We have thus obtained that $$ \zeta'(t) - L \zeta(t) \le 0, \quad t \in [b, c], $$ which gives $$ \frac{d}{dt} (e^{-Lt} \zeta(t)) \le 0, \quad t \in [b, c]. $$ Therefore $$ \zeta(t) \le e^{L(t - b)} \zeta(b) = 0 $$ for all $t \in [b, c]$, a contradiction.
(The above proof is modeled on Theorem 9.IX on p. 96 of Wolfgang Walter's book Ordinary Differential Equations.)