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Concept
Peano existence theorem
Summary
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems.
Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations.
Let be an open subset of with
a continuous function and
a continuous, explicit first-order differential equation defined on D, then every initial value problem
for f with
has a local solution
where is a neighbourhood of in ,
such that for all .
The solution need not be unique: one and the same initial value may give rise to many different solutions .
By replacing with , with , we may assume . As is open there is a rectangle .
Because is compact and is continuous, we have and by the Stone–Weierstrass theorem there exists a sequence of Lipschitz functions converging uniformly to in . Without loss of generality, we assume for all .
We define Picard iterations as follows, where . , and . They are well-defined by induction: as
is within the domain of .
We have
where is the Lipschitz constant of . Thus for maximal difference , we have a bound , and
By induction, this implies the bound which tends to zero as for all .
The functions are equicontinuous as for we have
so by the Arzelà–Ascoli theorem they are relatively compact. In particular, for each there is a subsequence
converging uniformly to a continuous function . Taking limit
in
we conclude that . The functions are in the closure of a relatively compact set, so they are themselves relatively compact. Thus there is a subsequence converging uniformly to a continuous function . Taking limit in we conclude that , using the fact that are equicontinuous by the Arzelà–Ascoli theorem. By the fundamental theorem of calculus, in .
The Peano theorem can be compared with another existence result in the same context, the Picard–Lindelöf theorem.
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