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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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Related concepts (2)
Ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity).