Picard's Existence Theorem

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

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

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 $$\left\{{\left({x, y}\right) \in \reals^2: \left|{x - a}\right| \le h, \left|{y - b}\right| \le k}\right\}$$ such that $$M h \le k$$.

Let $$R \subseteq D$$.

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


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

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