Picard–Lindelöf local existence and uniqueness theorem

Statement

Let $F$ be continuous on an open set in $\mathbb R\times\mathbb R^m$ and locally Lipschitz in the state, locally uniformly in time. Through every point $(t_0,y_0)$ in that set there is a unique local solution of $y'=F(t,y)$ , $y(t_0)=y_0$ . Uniqueness means that two solutions with this data agree wherever their intervals overlap.

Quantitative construction

On a closed rectangle $|t-t_0|\le a$ , $|y-y_0|\le r$ inside the domain, choose $|F|\le M$ and a state Lipschitz constant $L$ . For $0<h\le a$ with $hM\le r$ and $hL<1$ , the Picard map preserves the radius- $r$ ball in the complete space of continuous curves on $[t_0-h,t_0+h]$ and contracts it. Its fixed point solves the equation by the fundamental theorem of calculus.

References

Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, Chapters 2–4.