| Version 7 |
Version 6 |
| Let $E\subset W$ where $E$ is an open subset of $W$ which is a normed vector space, $f\in C^1(E)$ is a continuous differentiable map such that |
Let $E\subset W$ where $E$ is an open subset of $W$ which is a normed vector space, $f\in C^1(E)$ is a continuous differentiable map such that |
| $$f: E \to W.$$ Then the ordinary differential equation defined as |
$$f: E \to W.$$ Then the ordinary differential equation defined as |
| $$\dot{x} = f(x)$$ |
$$\dot{x} = f(x)$$ |
| with the initial condition |
with the initial condition |
| $$x(0) = x_0$$ |
$$x(0) = x_0$$ |
| where $x_0 \in E$ has a unique solution on some interval $I$. More specifically there exists $\alpha>0$ such that the following is a unique solution |
where $x_0 \in E$ |
|
has a unique solution on some interval $I$. More specifically there exists $\alpha>0$ such that the following is a unique solution |
| $$x:(-\alpha,\alpha) \to E$$ |
$$x:(-\alpha,\alpha) \to E$$ |
|
which also satify the initial condition of the initial value problem.\cite{HS}
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which also satify the initial condition of the initial value problem.\cite{1}
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\begin{thebibliography}{1}
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\begin{thebibliography}{HS}
|
| \bibitem[HS]{HS} Hirsch, W. Morris, Smale, Stephen.: Differential Equations, Dynamical Systems, And Linear Algebra. Academic Press, Inc. New York, 1974. |
\bibitem[HS]{HS} Hirsch, W. Morris, Smale, Stephen.: Differential Equations, Dynamical Systems, And Linear Algebra. Academic Press, Inc. New York, 1974. |
| \end{thebibliography} |
\end{thebibliography} |
