Peano existence theorem

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 D be an open subset of R × R 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 (x0,y0) may give rise to many different solutions z.

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 C.

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

ReferencesEdit