Peano theorem

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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]).


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