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Viewing Version
9
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'existence and uniqueness of solution of ordinary differential equations'
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| Title of object: |
existence and uniqueness of solution of ordinary differential equations |
| Canonical Name: |
ExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations |
| Type: |
Theorem |
| Created on: |
2003-05-07 09:33:12 |
| Modified on: |
2005-02-12 14:03:23 |
| Classification: |
msc:34-00, msc:35-00, msc:34A12 |
Revision comment (for changes between this and next version):
| Changes for correction #7662 ('typo (fonction)'). |
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Content:
Let $E\subset W$ where $E$ is an open subset of $W$ which is a normed vector space, and let $f$ be a continuous differentiable map
$$f: E \to W.$$ Then the ordinary differential equation defined as
$$\dot{x} = f(x)$$
with the initial condition
$$x(0) = x_0$$
where $x_0 \in E$ has a unique solution on some interval containing zero. More specifically there exists $\alpha>0$ such that the following is a unique fonction
$$x:(-\alpha,\alpha) \to E$$
such that $\dot{x}=f\circ x$ and $x(0)=x_0$.\cite{HS}
\begin{thebibliography}{1}
\bibitem[HS]{HS} Hirsch, W. Morris, Smale, Stephen.: Differential Equations, Dynamical Systems, And Linear Algebra. Academic Press, Inc. New York, 1974.
\end{thebibliography} |
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