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Peano existence theorem

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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.

Contents

History

Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations.

Theorem

Let D be an open subset of R × R with

f : D R

a continuous function and

y ( x ) = f ( x , y ( x ) )

a continuous, explicit first-order differential equation defined on D, then every initial value problem

y ( x 0 ) = y 0

for f with ( x 0 , y 0 ) D has a local solution

z : I R

where I is a neighbourhood of x 0 in R , such that z ( x ) = f ( x , z ( x ) ) for all x I .

The solution need not be unique: one and the same initial value (x0,y0) may give rise to many different solutions z.

The Peano theorem can be compared with another existence result in the same context, the Picard–Lindelöf theorem. The Picard–Lindelöf theorem both assumes more and concludes more. It requires Lipschitz continuity, while the Peano theorem requires only continuity; but it proves both existence and uniqueness where the Peano theorem proves only the existence of solutions. To illustrate, consider the ordinary differential equation

y = | y | 1 2 on the domain [ 0 , 1 ] .

According to the Peano theorem, this equation has solutions, but the Picard–Lindelöf theorem does not apply since the right hand side is not Lipschitz continuous in any neighbourhood containing 0. Thus we can conclude existence but not uniqueness. It turns out that this ordinary differential equation has two kinds of solutions when starting at y ( 0 ) = 0 , either y ( x ) = 0 or y ( x ) = x 2 / 4 . The transition between y = 0 and y = ( x C ) 2 / 4 can happen at any C.

The Carathéodory existence theorem is a generalization of the Peano existence theorem with weaker conditions than continuity.

References

Peano existence theorem Wikipedia