|
|
|
|
Picard's theorem
|
(Theorem)
|
|
|
The above theorem is also named the Picard-Lindelöf theorem and can be generalized to a system of first order ordinary differential equations
Theorem 2 (generalization of Picard's theorem [ KF]) Let $E$ be an open subset of $\mathbb{R}^{n+1}$ and a continuous function $f(x,y_1,\ldots,y_n)$ defined as $f=(f_1,\ldots,f_n)\colon E \to \mathbb{R}^n$ . If $(t_0,y_{10},\ldots,y_{n0})\in E$ and $f$ satisfies the Lipschitz condition in the variable $y_1,\ldots,y_n$ in $E$ : $$|f_i(x,y_1,\dots,y_n)-f_i(x,y_1'\ldots,y_n')| \leq M \max_{1\leq j\leq n}| y_j-y_j'|$$ where $M$ is a constant. Then the system of ordinary differential equation defined as
with the initial condition $$y_1(x_0) = y_{10},\ldots, y_n(x_0) = y_{n0}$$ has a unique solution $$y_1(x) ,\ldots, y_n(x)$$ on some interval $|x- x_0| \leq\delta$ .
see also:
- KF
- Kolmogorov, A.N. & Fomin, S.V.: Introductory Real Analysis, Translated & Edited by Richard A. Silverman. Dover Publications, Inc. New York, 1970.
|
"Picard's theorem" is owned by Daume.
|
|
(view preamble | get metadata)
Cross-references: existence and uniqueness of solution of ordinary differential equations, first order, interval, solution, initial condition, ordinary differential equation, variable, Lipschitz condition, continuous function, open subset, theorem
There are 2 references to this entry.
This is version 3 of Picard's theorem, born on 2005-02-03, modified 2008-01-09.
Object id is 6706, canonical name is PicardsTheorem2.
Accessed 5739 times total.
Classification:
| AMS MSC: | 34A12 (Ordinary differential equations :: General theory :: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
