Picard–Lindelöf theorem
From Wikipedia, the free encyclopedia
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.
Contents |
[edit] Picard–Lindelöf theorem
Consider the initial value problem
![y'(t)=f(t,y(t)),\quad y(t_0)=y_0, \quad t \in [t_0-\alpha, t_0+\alpha].](http://upload.wikimedia.org/math/f/f/b/ffb916f7df6cf7484505e5860b715d0e.png)
Suppose f is Lipschitz continuous in y and continuous in t. Then, for some value ε > 0, there exists a unique solution y(t) to the initial value problem within the range [t0 − ε,t0 + ε].[1]
[edit] Other existence theorems
The Picard–Lindelöf theorem shows that the solution exists and that it is unique. The Peano existence theorem shows only existence, not uniqueness, but it assumes only that ƒ is continuous in y. For example, the right-hand side of the equation y ′ = y1/3 with initial condition y(0) = 0 is continuous but not Lipschitz continuous. Indeed, the solution of this equation is not unique; two different solutions are given by
Even more general is Carathéodory's existence theorem, which proves existence (in a more general sense) under weaker conditions on ƒ.
[edit] 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
and
It can then be shown, by using the Banach fixed point theorem, that the sequence of "Picard iterates" 
is convergent and that the limit is a solution to the problem. Exploiting the fact that the width of the interval where the local solution is defined is entirely determined by the Lipschitz constant of the function,one can assure global existence of the solution, i.e. the solution exists and is unique until it leaves the domain of definition of the ODE. An application of Grönwall's lemma to 
where 
and ψ are two solutions, shows that 
, thus proving the global uniqueness (the local uniqueness is a consequence of the uniqueness of the Banach fixed point).
[edit] See also
[edit] Notes
- ^ Coddington & Levinson (1955), Theorem I.3.1
- ^ Coddington & Levinson (1955), page 7
[edit] References
- Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential Equations, New York: McGraw-Hill.
- 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.)
