Ordinary differential equations Existence, uniqueness and dependence on data Theorem

Continuous dependence on initial data

Builds on Lipschitz condition Solution of an ordinary differential equation Grönwall's inequality

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.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Lipschitz condition definition
Grönwall's inequality statement
Solution of an ordinary differential equation definition
Continuous dependence on initial data statement

Tag	`0G3DBXRRQJ`
URI	`m2://0G3DBXRRQJ`
Kind	statement, theorem
Writing	W0 stub
MSC	34A12
Status	This copy of the site has no build data. Run `m2 status 0G3DBXRRQJ`.
Uses	Lipschitz condition, Solution of an ordinary differential equation, Grönwall's inequality

Continuous dependence on initial data

Statement

Remarks

Reading path

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