Picard–Lindelöf theorem

In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to certain initial value problems.

The theorem is named after Charles Émile Picard, Ernst Lindelöf, Rudolph Lipschitz and Augustin Cauchy.

Picard–Lindelöf theorem

Consider the initial value problem:$y\text{'}(t)=f(t,y(t)),quad\; y(t\_0)=y\_0,\; quad\; t\; in\; [t\_0-alpha,\; t\_0+alpha]\; .$Suppose $f$ is bounded, Lipschitz continuous in $y$, and continuous in $t$. Then, for some value $epsilon>0$, there exists a unique solution $y(t)$ to the initial value problem within the range $[t\_0-epsilon,\; t\_0+epsilon]$.

Proof sketch

A simple proof of existence of the solution is obtained by successive approximations. In this context, the method is known as Picard iteration.

Set

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

and

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

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

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

See also

* Frobenius theorem (differential topology)
* Integrability conditions for differential systems
* Peano existence theorem

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/f454.table . (In that article Lindelöf discusses a generalization of an earlier approach by Picard.)

External links

* [http://www.krellinst.org/UCES/archive/classes/CNA/dir2.6/uces2.6.html Fixed Points and the Picard Algorithm]
* [http://math.fullerton.edu/mathews/n2003/PicardIterationMod.html Picard Iteration]
* [http://www.math.byu.edu/~grant/courses/m634/f99/lec4.pdf Proof of the Picard–Lindelöf theorem]