Picard's Existence Theorem

Theorem
Let $\map f {x, y} : \R^2 \to \R$ be continuous in a region $D \subseteq \R^2$.

Let $\exists M \in \R: \forall x, y \in D: \size {\map f {x, y} } < M$.

Let $\map f {x, y}$ satisfy in $D$ the Lipschitz condition in $y$:


 * $\size{\map f {x, y_1} - \map f {x, y_2} } \le A \size {y_1 - y_2}$

where $A$ is independent of $x, y_1, y_2$.

Let the rectangle $R$ be defined as $\set {\tuple {x, y} \in \R^2: \size {x - a} \le h, \size {y - b} \le k}$ such that $M h \le k$.

Let $R \subseteq D$.

Then $\forall x \in \R: \size {x - a} \le h$, the first order ordinary differential equation:


 * $y' = \map f {x, y}$

has one and only one solution $y = \map y x$ for which $b = \map y a$.