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.