existence and uniqueness of solution of ordinary differential equations

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 function

x:(-\alpha,\alpha)\to E

such that $\dot{x}=f\circ x$ and $x(0)=x_{0}$ .[HS]

References

HS Hirsch, W. Morris, Smale, Stephen.: Differential Equations, Dynamical Systems, And Linear Algebra. Academic Press, Inc. New York, 1974.

Title	existence and uniqueness of solution of ordinary differential equations
Canonical name	ExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations
Date of creation	2013-03-22 13:36:50
Last modified on	2013-03-22 13:36:50
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	13
Author	Daume (40)
Entry type	Theorem
Classification	msc 35-00
Classification	msc 34-00
Classification	msc 34A12
Related topic	PicardsTheorem2
Related topic	CauchyKowalewskiTheorem