Picards_Existence_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 $b = \map y a$.