Continuous dependence on initial data

Statement

Let $J$ be an interval, let $U$ be an open subset of $\mathbb{R}^n$, let $K \ge 0$, and let $f : J \times U \to \mathbb{R}^n$ be continuous. Assume that $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ for all $t \in J$ and $x, y \in U$. Let $a < b$ with $[a, b] \subseteq J$, and let $\gamma_1, \gamma_2 : [a, b] \to U$ be solutions of $x' = f(t, x)$. Then $$\|\gamma_1(t) - \gamma_2(t)\| \le \|\gamma_1(a) - \gamma_2(a)\|\, e^{K (t - a)} \quad \text{for each } t \in [a, b].$$

Remarks

So the solution at time $t$ is a Lipschitz function of the initial value, with constant $e^{K(t - a)}$. The proof applies Grönwall's inequality to $u(t) = \|\gamma_1(t) - \gamma_2(t)\|$. Existence, uniqueness and continuous dependence together make the initial value problem well posed, in the sense of Jacques Hadamard. For equal initial values, the bound gives uniqueness again.

Explanations

L1 human

If two solutions start close together, they stay close for a while. Their distance can grow, but at most exponentially, with rate given by the Lipschitz constant. So a small error in the initial value gives a small error in the solution over a bounded time.