Peano existence theorem

Summary

In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems.

HistoryEdit

Peano first published the theorem in 1886 with an incorrect proof.[1] In 1890 he published a new correct proof using successive approximations.[2]

TheoremEdit

Let   be an open subset of   with   a continuous function and   a continuous, explicit first-order differential equation defined on D, then every initial value problem   for f with   has a local solution   where   is a neighbourhood of   in  , such that   for all  .[3]

The solution need not be unique: one and the same initial value   may give rise to many different solutions  .

ProofEdit

By replacing   with  ,   with  , we may assume  . As   is open there is a rectangle  .

Because   is compact and   is continuous, we have   and by the Stone–Weierstrass theorem a sequence of Lipschitz functions   converging uniformly to   in  . Without loss of generality, we assume   for all  .

We define Picard iterations   as follows, where  .  , and  . They are well-defined by induction: as

 

  is within the domain of  .

We have

 

where   is the Lipschitz constant of  . Thus for maximal difference  , we have a bound  , and

 

By induction, this implies the bound   which tends to zero as   for all  .

The functions   are equicontinuous as for   we have

 

so by the Arzelà–Ascoli theorem they are relatively compact. In particular, for each   there is a subsequence   converging uniformly to a continuous function  . Taking limit   in

 

we conclude that  . The functions   are in the closure of a relatively compact set, so they are themselves relatively compact. Thus there is a subsequence   converging uniformly to a continuous function  . Taking limit   in   we conclude that  , using the fact that   are equicontinuous by the Arzelà–Ascoli theorem. By the fundamental theorem of calculus,   in  .

Related theoremsEdit

The Peano theorem can be compared with another existence result in the same context, the Picard–Lindelöf theorem. The Picard–Lindelöf theorem both assumes more and concludes more. It requires Lipschitz continuity, while the Peano theorem requires only continuity; but it proves both existence and uniqueness where the Peano theorem proves only the existence of solutions. To illustrate, consider the ordinary differential equation

  on the domain  

According to the Peano theorem, this equation has solutions, but the Picard–Lindelöf theorem does not apply since the right hand side is not Lipschitz continuous in any neighbourhood containing 0. Thus we can conclude existence but not uniqueness. It turns out that this ordinary differential equation has two kinds of solutions when starting at  , either   or  . The transition between   and   can happen at any  .

The Carathéodory existence theorem is a generalization of the Peano existence theorem with weaker conditions than continuity.

NotesEdit

  1. ^ Peano, G. (1886). "Sull'integrabilità delle equazioni differenziali del primo ordine". Atti Accad. Sci. Torino. 21: 437–445.
  2. ^ Peano, G. (1890). "Demonstration de l'intégrabilité des équations différentielles ordinaires". Mathematische Annalen. 37 (2): 182–228. doi:10.1007/BF01200235. S2CID 120698124.
  3. ^ (Coddington & Levinson 1955, p. 6)

ReferencesEdit

  • Osgood, W. F. (1898). "Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x, y) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung". Monatshefte für Mathematik. 9: 331–345. doi:10.1007/BF01707876. S2CID 122312261.
  • Coddington, Earl A.; Levinson, Norman (1955). Theory of Ordinary Differential Equations. New York: McGraw-Hill.
  • Murray, Francis J.; Miller, Kenneth S. (1976) [1954]. Existence Theorems for Ordinary Differential Equations (Reprint ed.). New York: Krieger.
  • Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.