lautet:

Seien a, b > 0, (x₀, y₀) ∈ ℝⁿ⁺¹und

\begin{eqnarray}G:=\{(x,{\bf{\text{y}}})\in {{\mathbb{R}}}^{n+1}||x-{x}_{0}|\lt a,\Vert {\bf{\text{y}}}-{{\bf{\text{y}}}}_{0}\Vert \lt b\}.\end{eqnarray}

\begin{eqnarray}|x-{x}_{0}|\cdot \Vert f(x,{{\bf{\text{y}}}}_{1})-f(x,{{\bf{\text{y}}}}_{2})\Vert \le \Vert {{\bf{\text{y}}}}_{1}-{{\bf{\text{y}}}}_{2}\Vert .\end{eqnarray}

Dann besitzt das Anfangswertproblem

\begin{eqnarray}{{\bf{\text{y}}}}^{\text{'}}=f(x,{\bf{\text{y}}}),\quad\quad\quad {\bf{\text{y}}}({x}_{0})={{\bf{\text{y}}}}_{0}\end{eqnarray}