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.
[1] In 1890 he published a new correct proof using successive approximations.
a continuous function and
a continuous, explicit first-order differential equation defined on D, then every initial value problem
has a local solution
[3] The solution need not be unique: one and the same initial value
may give rise to many different solutions
and by the Stone–Weierstrass theorem there exists a sequence of Lipschitz functions
Without loss of generality, we assume
We define Picard iterations
is the Lipschitz constant of
, and By induction, this implies the bound
we have so by the Arzelà–Ascoli theorem they are relatively compact.
converging uniformly to a continuous function
Taking limit
converging uniformly to a continuous function
Taking limit
are equicontinuous by the Arzelà–Ascoli theorem.
By the fundamental theorem of calculus,
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 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
The Carathéodory existence theorem is a generalization of the Peano existence theorem with weaker conditions than continuity.
The Peano existence theorem cannot be straightforwardly extended to a general Hilbert space
: for an open subset
alone is insufficient for guaranteeing the existence of solutions for the associated initial value problem.