Peano existence theorem

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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.

History[edit]

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

Theorem[edit]

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 .

Proof[edit]

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 theorems[edit]

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.

Notes[edit]

  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.
  3. ^ (Coddington & Levinson 1955, p. 6)

References[edit]