Picard's Existence Theorem

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

Let $\exists M \in \R: \forall x, y \in D: \left|{f \left({x, y}\right)}\right| < M$.

Let $f \left({x, y}\right)$ satisfy in $D$ the Lipschitz condition in $y$:


 * $\left|{f \left({x, y_1}\right) - f \left({x, y_2}\right)}\right| \le A \left|{y_1 - y_2}\right|$

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

Let the rectangle $R$ be defined as $\left\{{\left({x, y}\right) \in \R^2: \left|{x - a}\right| \le h, \left|{y - b}\right| \le k}\right\}$ such that $M h \le k$.

Let $R \subseteq D$.

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


 * $y' = f \left({x, y}\right)$

has one and only one solution $y = y \left({x}\right)$ for which $b = y \left({a}\right)$.