Peano theorem
From Encyclopedia of Mathematics
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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]).
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
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