In mathematics, the Picard–Lindelöf theorem or Picard's existence theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem

$y\text{'}(t)=f(t,y(t)),\backslash quad\; y(t\_0)=y\_0$

has exactly one solution if f is Lipschitz continuous in $y$, continuous in $t$ as long as $y(t)$ stays bounded.

A simple proof of existence of the solution is successive approximation: (also called Picard iteration)

Set

$\backslash varphi\_0(t)=y\_0\; \backslash ,\backslash !$

and

$\backslash varphi\_i(t)=y\_0+\backslash int\_\{t\_0\}^\{t\}f(s,\backslash varphi\_\{i-1\}(s))\backslash ,ds.$

It can then be shown rather easily, by using the Banach fixed point theorem, that the sequence of the $\backslash varphi\_i\; \backslash ,\backslash !$ (called the Picard iterates) is convergent and that the limit is a solution to the problem.

An application of Grönwall's lemma to $|\backslash varphi(t)-\backslash psi(t)|$, where $\backslash varphi$ and $\backslash psi$ are two solutions, shows that $\backslash varphi(t)\backslash equiv\backslash psi(t)$, thus proving the uniqueness.

See also

• Frobenius theorem (differential topology)
• Integrability conditions for differential systems

References

• M. E. Lindelöf, Sur l'application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre; Comptes rendus hebdomadaires des séances de l'Académie des sciences. Vol. 114, 1894, pp. 454-457. Digitized version online via http://gallica.bnf.fr/ark:/12148/bpt6k3074r . (In that article Lindelöf discusses a generalization of an earlier approach by Picard.)

Mathematical analysis | Mathematical theorems

مبرهنة بيكار ليندلوف | Satz von Picard-Lindelöf