1.3 Fundamental Theorem for the Flow

Main Idea

The local existence and uniqueness of flow with a continuous dependence on the initial condition. Known as Picard-Lineröf-Cauchy-Lipschitz theorem.

Lipschitz Continuity

The Lipschitz condition permits one to construct the so-called Picard-Linderöf (LS) iteration, and to show that it is a contraction. Then existence and uniqueness then come from there.

Fundamental Theorem

Consider the system

Suppose $F(x)$ is Lipschitz continuous on $(a,b)$ with Lipschitz constant $K$.

1. Existence:
   For $a<x_0<b$, there exists a solution $x(t)$ to

defined for some time interval $-\tau <t<\tau$.
In other words, ${\varphi }^t(x_0)$ exists locally.

2. Uniqueness:
   If $x(t)$ and $y(t)$ are two solutions with $x(0)=x_0=y(0)$, then

on the largest interval of time around $t=0$ where both are defined.

3. Continuous Dependence on Initial Condition:
   Let $T>0$ such that ${\varphi }^t(x_0)$ is defined for $-T\le t\le T$.
   Then for any $\epsilon >0$, there exists $\delta >0$ such that if $|y_0-x_0|<\delta$, then ${\varphi }^t(y_0)$ is defined for $-T\le t\le T$ and

Example:

On $[x_0-c,x_0+c]$,

$f^{\prime }(x)=2kx\Rightarrow |f^{\prime }(x)|\le 2k(|x_0|+c)$, so

→ $f$ is locally Lipschitz ⇒ unique local solution exists.
Exact solution:

Integrate:

Use $x(0)=x_0$:

Solve for $x(t)$:

As $t\to t^{\ast -}$, $x(t)\to \mathrm{\infty }$ ⇒ not globally Lipschitz

Only global case: $x_0=0\Rightarrow x(t)\equiv 0$.