Picard–Lindelöf theorem

Statement

Let $J = [t_{\min}, t_{\max}]$ and $t_0 \in J$, let $x_0 \in \mathbb{R}^n$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ satisfy the Picard–Lindelöf hypotheses at $(t_0, x_0)$ with constants $a, L, K$. Let $B$ be the closed ball of radius $a$ around $x_0$. Then there is exactly one continuous curve $\alpha : J \to B$ with $\alpha(t_0) = x_0$ and with this property. For each $t \in J$, $\alpha$ has the derivative $f(t, \alpha(t))$ at $t$ within $J$.

Remarks

The proof uses the integral equation form and the fact that an iterate of the Picard operator is a contraction. The limit of Picard iteration is the solution. A common form: let $f$ be continuous on a neighbourhood of $(t_0, x_0)$ and locally Lipschitz in $x$. Then the initial value problem has a unique solution on $[t_0 - \varepsilon, t_0 + \varepsilon]$ for some $\varepsilon > 0$. The theorem is local. For $x' = 1 + x^2$, $x(0) = 0$, the solution $\tan t$ exists only for $|t| < \pi/2$. Without the Lipschitz condition, solutions exist by the Peano existence theorem, but they need not be unique. See the non-uniqueness example.

History

The theorem is named after Émile Picard and Ernst Lindelöf. It is also called the Cauchy–Lipschitz theorem, after Augustin-Louis Cauchy and Rudolf Lipschitz.

Explanations

L1 human

If the vector field is regular enough, the present state determines the whole motion near the start time. Regular enough means continuous in time and Lipschitz in the state. Then there is one and only one solution through the initial point, at least for a short time.