Picard--Lindelöf theorem

From Wikipedia.
Theorem (Existence and Uniqueness Theorem):
Let $D\subseteq \mathbb{R}\times {\mathbb{R}}^n$ be a closed rectangle with $(t_0,y_0)\in \mathrm{int}D$, the interior of $D$. Let $f:D\to {\mathbb{R}}^n$ be a function that is continuous in $t$ and Lipschitz continuous in $y$ (with Lipschitz constant independent from $t$). Then there exists some $\epsilon >0$ such that the initial value problem

has a unique solution $y(t)$ on the interval $[t_0-\epsilon ,t_0+\epsilon ]$.

Remarks

• Proof: the IVP is equivalent to the Volterra integral equation $y(x)=y_0+{\int }_{x_0}^{x}f(t,y(t))dt$, and the theorem is the contraction argument on the associated Picard iteration: see integral equation for an initial value problem.
• If we assume only continuity, the Peano existence theorem assure only existence, not uniqueness.
• The theorem can be weakened to:
  Theorem (Existence and Uniqueness Theorem):
  Let $R=[a,b]\times [c,d]\subset {\mathbb{R}}^2$ be a rectangle such that the point $(x_0,y_0)$ is in the interior of $(a,b)\times (c,d)$.
  If:
  1. $f(x,y)$ is continuous on $R$.
  2. The partial derivative $\frac{\partial f}{\partial y}(x,y)$ exists and is continuous on $R$.
     Then, there exists an open interval $I\subset (a,b)$ such that $x_0\in I$, and a unique function $y(x)$ defined on $I$ that satisfies the initial value problem:

Continuity of $\frac{\partial f}{\partial y}$ is a sufficient condition for Lipschitz continuity.