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 Guiseppe Peano and Augustin Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems.

The Peano theorem can be compared with another existence result in the same context, the Picard–Lindelöf theorem. The Picard–Lindelöf theorem assumes the stronger condition of Lipschitz continuity. The Peano theorem requires only continuity. Consequently, the Picard–Lindelöf ensures an unique solution while the example

x' = x^{\frac{1}{2}}

shows there may be infinitely many solutions if the right-hand side of an ODE is not Lipschitz continuous.

[edit] History

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

[edit] Theorem

Let D be an open and simply connected subset of R × R with

f: D \to \mathbb{R}
f(x,y(x)) = y'(x)

a continuous, explicit first order differential equation defined on D, then an initial value problem

y(x0) = y0

for f with (x_0, y_0) \in D has a local solution

z: I \to \mathbb{R}

with I a neighbourhood of x0.

[edit] Reference

  • G. Peano, Sull’integrabilità delle equazioni differenziali del primo ordine, Atti Accad. Sci. Torino, 21 (1886) 677–685.
  • 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.
  • E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955.
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