PlanetMath: dependence on initial conditions of solutions of ordinary differential equations

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dependence on initial conditions of solutions of ordinary differential equations

(Theorem)

Let $E\subset W$ where $W$ is a normed vector space, $f\in C^1(E)$ is a continuous differentiable map $f: E \to W$ . Furthermore consider the ordinary differential equation $$\dot{x} = f(x)$$ with the initial condition

$x(0) = x_0$ .

Let $x(t)$ be the solution of the above initial value problem defined as $$x:I \to E$$ where $I = [-a,a]$ . Then there exist $\delta >0$ such that for all $y_0 \in N_\delta (x_0)$ ($y_0$ in the $\delta$ neighborhood of $x_0$ ) has a unique solution $y(t)$ to the initial value problem above except for the initial value changed to $x(0)=y_0$ . In addition $y(t)$ is twice continouously differentialble function of $t$ over the interval $I$ .

"dependence on initial conditions of solutions of ordinary differential equations" is owned by Daume.

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Cross-references: interval, function, addition, neighborhood, initial value problem, solution, initial condition, ordinary differential equation, differentiable map, continuous, normed vector space

This is version 4 of dependence on initial conditions of solutions of ordinary differential equations, born on 2003-05-09, modified 2006-02-07.
Object id is 4257, canonical name is DependenceOnInitialConditionsOfSolutionsOfOrdinaryDifferentialEquations.
Accessed 2610 times total.

Classification:

AMS MSC:	34-00 (Ordinary differential equations :: General reference works )
	35-00 (Partial differential equations :: General reference works )

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