Estimate for approximate solutions

Statement

Let $J$ be an interval, let $U$ be an open subset of $\mathbb{R}^n$ and let $K \ge 0$. Let $f : J \times U \to \mathbb{R}^n$ satisfy $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ for all $t \in J$ and $x, y \in U$. Let $a < b$ with $[a, b] \subseteq J$, and let $\delta, \varepsilon_1, \varepsilon_2 \ge 0$. Let $\gamma_1, \gamma_2 : [a, b] \to U$ be continuous on $[a, b]$ and differentiable on $(a, b)$, with $\|\gamma_i'(t) - f(t, \gamma_i(t))\| \le \varepsilon_i$ for $i = 1, 2$ and $t \in (a, b)$. If $\|\gamma_1(a) - \gamma_2(a)\| \le \delta$, then for each $t \in [a, b]$, $$\|\gamma_1(t) - \gamma_2(t)\| \le G_{\delta, K, \varepsilon_1 + \varepsilon_2}(t - a),$$ where $G$ is the Grönwall bound.

Remarks

A curve with $\|\gamma'(t) - f(t, \gamma(t))\| \le \varepsilon$ is an $\varepsilon$-approximate solution. For $\varepsilon_1 = \varepsilon_2 = 0$, the result is continuous dependence on initial data. The proof applies the derivative form of Grönwall's inequality with the additional term $\varepsilon_1 + \varepsilon_2$ to $\|\gamma_1 - \gamma_2\|$.

Explanations

L1 human

A curve that almost solves the equation stays close to a true solution. The error has two sources: the difference of initial values and the defects $\varepsilon_i$. Both can grow at most exponentially in time.