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Difference between revisions of "Peano theorem"

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One of the existence theorems for solutions of an ordinary differential equation (cf. [[Differential equation, ordinary|Differential equation, ordinary]]), established by G. Peano [[#References|[1]]], and consisting in the following. Suppose one is given the differential equation
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p0719101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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{{MSC|34A12}}
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[[Category:Ordinary differential equations]]
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{{TEX|done}}
  
If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p0719102.png" /> is bounded and continuous in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p0719103.png" />, then through each interior point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p0719104.png" /> of this region there passes at least one [[Integral curve|integral curve]] for (*). It may be that more than one integral curve passes through a certain point, e.g. for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p0719105.png" /> there exists an infinite set of integral curves passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p0719106.png" />:
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One of the existence theorems for solutions of an ordinary differential equation (cf. [[Differential equation, ordinary|Differential equation, ordinary]]), established by G. Peano on {{Cite|Pe}}. More precisely
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p0719107.png" /></td> </tr></table>
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'''Theorem'''
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Let $U\subset \mathbb R^n$ be an open set and $f: U\times [0,T] \to \mathbb R^n$ a continuous function. Then, for every $x_0\in U$ there is a positive $\delta$ and a solution $x: [0,\delta]\to U$ of the ordinary differential equation $\dot{x} (t) = f (x(t), t)$ satisfying the initial condition $x(0)=x_0$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p0719108.png" /></td> </tr></table>
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Strictly speaking, the theorem above is the $n$-dimensional generalization of the original Peano's result, which he established in the case $n=1$. The solution $x$ of the theorem is called the ''[[Integral curve|integral curve]] through $x_0$''. Peano's theorem guarantees the existence of at least one solution, but the continuity hypothesis is far from guaranteeing its uniqueness. For the latter one usually assumes a [[Lipschitz condition]] on $f$, namely $|f(x_1, t)- f (x_2, t)|\leq M |x_1-x_2|$, as in the classical [[Cauchy-Lipschitz theorem]] (see also [[Osgood criterion]] for a refinement of this statement).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p0719109.png" /></td> </tr></table>
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===References===
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p07191010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071910/p07191011.png" /> are arbitrary constants.
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There are generalizations (including multi-dimensional ones) of Peano's theorem (see [[#References|[2]]], [[#References|[3]]]).
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|valign="top"|{{Ref|Am}}|| H. Amann, "Ordinary differential equations. An introduction to nonlinear analysis." de Gruyter Studies in Mathematics, 13. Walter de Gruyter & Co., Berlin,  1990.
 
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====References====
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|valign="top"|{{Ref|Ha}}|| P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)
<table><TR><TD valign="top">[1]</TD> <TD valign="top"G. Peano,  "Démonstration de l'intégrabilité des équations différentielles ordinaires"  ''Math. Ann.'' , '''37'''  (1890)  pp. 182–228</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.G. Petrovskii,  "Ordinary differential equations" , Prentice-Hall  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P. Hartman,  "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR></table>
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|valign="top"|{{Ref|Pe}}|| G. Peano,  "Démonstration de l'intégrabilité des équations différentielles ordinaires"  ''Math. Ann.'' , '''37'''  (1890)  pp. 182–228
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|valign="top"|{{Ref|Pet}}|| I.G. Petrovskii,  "Ordinary differential equations" , Prentice-Hall  (1966)  (Translated from Russian)
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Latest revision as of 10:00, 29 November 2013

2020 Mathematics Subject Classification: Primary: 34A12 [MSN][ZBL]

One of the existence theorems for solutions of an ordinary differential equation (cf. Differential equation, ordinary), established by G. Peano on [Pe]. More precisely

Theorem Let $U\subset \mathbb R^n$ be an open set and $f: U\times [0,T] \to \mathbb R^n$ a continuous function. Then, for every $x_0\in U$ there is a positive $\delta$ and a solution $x: [0,\delta]\to U$ of the ordinary differential equation $\dot{x} (t) = f (x(t), t)$ satisfying the initial condition $x(0)=x_0$.

Strictly speaking, the theorem above is the $n$-dimensional generalization of the original Peano's result, which he established in the case $n=1$. The solution $x$ of the theorem is called the integral curve through $x_0$. Peano's theorem guarantees the existence of at least one solution, but the continuity hypothesis is far from guaranteeing its uniqueness. For the latter one usually assumes a Lipschitz condition on $f$, namely $|f(x_1, t)- f (x_2, t)|\leq M |x_1-x_2|$, as in the classical Cauchy-Lipschitz theorem (see also Osgood criterion for a refinement of this statement).


References

[Am] H. Amann, "Ordinary differential equations. An introduction to nonlinear analysis." de Gruyter Studies in Mathematics, 13. Walter de Gruyter & Co., Berlin, 1990.
[Ha] P. Hartman, "Ordinary differential equations" , Birkhäuser (1982)
[Pe] G. Peano, "Démonstration de l'intégrabilité des équations différentielles ordinaires" Math. Ann. , 37 (1890) pp. 182–228
[Pet] I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian)
How to Cite This Entry:
Peano theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peano_theorem&oldid=30802
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article