The solution need not be unique: one and the same initial value may give rise to many different solutions .
Proof
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 .
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 .
Related theorems
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
on the domain
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 , either or . The transition between and can happen at any .
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 of , the continuity of alone is insufficient for guaranteeing the existence of solutions for the associated initial value problem.[4]