We recently introduced the following theorem in my current lecture:
Theorem (Stability): Let $f(t,u)$ and $g(t,u)$ be two continuous functions on a cylinder $D = I \times \Omega$ where the interval $I$ contains $t_0$ and $\Omega$ is a convex set in $\mathbb{R}^d$. Furthermore, let $f$ admit a Lipschitz condition with constant $L$ on $D$. Let $u$ and $v$ be solutions to the IVP
$$u' = f(t,u)\,\text{ for all }t \in I,\hspace{30pt}v(t_0) = u_0$$ $$v' = g(t,v)\,\text{ for all }t \in I,\hspace{30pt}v(t_0) = v_0$$
Then, there holds:
$$|u(t) - v(t)| \leq e^{L|t-t_0|}\cdot(|u_0 - v_0| + \int_{t_0}^t \max_{x \in \Omega} |f(s,x) - g(s,x)|\,ds)$$
I do understand the theorem per se, but have no clue where to apply this theorems, since $u$ and $v$ are solutions to completely different IVPs. Where is this theorem useful? I can't imagine where this is going..
An intuition why this theorem is useful would be amazing.