Existence and Uniqueness Theorem for 1st Order IVPs

Theorem
Let $x' = f(t,x)$, $x(t_0) = x_0$ be an explicit ODE of dimension $n$.

Suppose that there exists a neighborhood $V = [t_0 - \ell_0,t_0 + \ell_0] \times \overline{B}(x_0,\epsilon)$ of $(t_0,x_0)$ in phase space $\R \times \R^n$ such that $f$ is Lipschitz continuous on $V$.

Then there exists $\ell < \ell_0$ such that there exists a unique solution $x(t)$ defined for $t \in [t_0 - \ell,t_0 + \ell]$.

Proof
For $0 < \ell < \ell_0$, let $\mathcal X = \mathcal C([t_0 - \ell,t_0 + \ell]; \R^n)$.

By Fixed Point Formulation of Explicit ODEs it is sufficient to find a fixed point of the map $T : \mathcal X \to \mathcal X$ defined by:


 * $\displaystyle (Tx)(t) = x_0 + \int_{t_0}^t f(s,x(s))\ ds$

Now we have that continuous functions on an interval form a Banach space, i.e. that $\mathcal X$ is a Banach space.

We also have that a closed subset of a complete metric space is complete.

Therefore the Banach Fixed-Point Theorem it is sufficient to find a non-empty subset $\mathcal Y \subseteq \mathcal X$ such that:
 * $\mathcal Y$ is closed in $\mathcal X$
 * $T\mathcal Y \subseteq \mathcal Y$
 * $T$ is a contraction on $\mathcal Y$