# Peano theorem

From Encyclopedia of Mathematics

One of the existence theorems for solutions of an ordinary differential equation (cf. Differential equation, ordinary), established by G. Peano [1], and consisting in the following. Suppose one is given the differential equation

(*) |

If the function is bounded and continuous in a region , then through each interior point of this region there passes at least one integral curve for (*). It may be that more than one integral curve passes through a certain point, e.g. for the equation there exists an infinite set of integral curves passing through :

where and are arbitrary constants.

There are generalizations (including multi-dimensional ones) of Peano's theorem (see [2], [3]).

#### References

[1] | G. Peano, "Démonstration de l'intégrabilité des équations différentielles ordinaires" Math. Ann. , 37 (1890) pp. 182–228 |

[2] | I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian) |

[3] | P. Hartman, "Ordinary differential equations" , Birkhäuser (1982) |

**How to Cite This Entry:**

Peano theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Peano_theorem&oldid=14971

This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article