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* {{citation | last1=Coddington | first1=Earl A. | last2=Levinson | first2=Norman | title=Theory of Ordinary Differential Equations | publisher=[[McGraw-Hill]] | location=New York | year=1955}}
* {{citation | last1=Coddington | first1=Earl A. | last2=Levinson | first2=Norman | title=Theory of Ordinary Differential Equations | publisher=[[McGraw-Hill]] | location=New York | year=1955}}
* {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
* {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
* {{citation | last1=Murray | first1=Francis J. | last2=Miller | first2=Kenneth S. | title=Existence Theorems for Ordinary Differential Equations | publisher=[[Krieger]] | location=New York || descriptive=Reprinted | year=1976 | descriptive=Original Edition by New York University Press | year=1954 |

[[Category:Theorems in analysis]]
[[Category:Theorems in analysis]]
[[Category:Ordinary differential equations]]
[[Category:Ordinary differential equations]]

Revision as of 06:19, 30 March 2013

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

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

Theorem

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 .[1]

Note that the solution need not be unique: one and the same initial value (x0,y0) may give rise to many different solutions z.

Extensions

The theorem holds as stated for explicit first order systems also in higher dimensions, when D is an open subset of R × Rn. It is however in general wrong for differential equations in infinite-dimensional Banach spaces.

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.

Notes

References

  • G. Peano, Sull’integrabilità delle equazioni differenziali del primo ordine, Atti Accad. Sci. Torino, 21 (1886) 437–445.[1]
  • G. Peano, Demonstration de l’intégrabilité des équations différentielles ordinaires, Mathematische Annalen, 37 (1890) 182–228.
  • W. F. Osgood, Beweis der Existenz einer Lösung der Differentialgleichung dy/dx = f(x, y) ohne Hinzunahme der Cauchy-Lipschitzchen Bedingung, Monatsheft Mathematik,9 (1898) 331–345.
  • Coddington, Earl A.; Levinson, Norman (1955), Theory of Ordinary Differential Equations, New York: McGraw-Hill
  • Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
  • {{citation | last1=Murray | first1=Francis J. | last2=Miller | first2=Kenneth S. | title=Existence Theorems for Ordinary Differential Equations | publisher=Krieger | location=New York || descriptive=Reprinted | year=1976 | descriptive=Original Edition by New York University Press | year=1954 |