Existence and uniqueness of solutions
There are several theorems that establish existence and uniqueness of solutions to initial value
problems involving ODEs both locally and globally. See Picard–Lindelöf theorem for a brief
discussion of this issue.
Picard–Lindelöf theorem
From Wikipedia, the free encyclopedia
Jump to: navigation, search
In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's
existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and
uniqueness of solutions to first-order equations with given initial conditions.
The theorem is named after Charles Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin
Cauchy.

edit] Picard–Lindelöf theorem
Consider the initial value problem
Suppose f is Lipschitz continuous in y and continuous in t. Then, for some value ε > 0, there
exists a unique solution y(t) to the initial value problem within the range [t0 − ε,t0 + ε].[1]
[edit] Proof sketch
The proof relies on transforming the differential equation, and applying fixed-point theory. By
integrating both sides, any function satisfying the differential equation must also satisfy the
integral equation
A simple proof of existence of the solution is obtained by successive approximations. In this
context, the method is known as Picard iteration.
Set
and
It can then be shown, by using the Banach fixed point theorem, that the sequence of "Picard
iterates"
is convergent and that the limit is a solution to the problem. Exploiting the fact that
the width of the interval where the local solution is defined is entirely determined by the
Lipschitz constant of the function,one can assure global existence of the solution, i.e. the solution
exists and is unique until it leaves the domain of definition of the ODE. An application of
Grönwall's lemma to
where and ψ are two solutions, shows that
, thus proving the global uniqueness (the local uniqueness is a consequence of the
uniqueness of the Banach fixed point).
[edit] Detailed proof
be the compact cylinder where f is defined, this is
Let
and
.
Let
,
this is, the maximum slope of the function in modulus. Finally, let L be the Lipschitz constant of
f with respect to the second variable.
We define an operator between two functional spaces of continuous functions, Picard's operator,
as follows:
defined by:
.
We impose that it is well-defined, in other words, that its image must be a function taking values
on Bb(x0), or equivalently, that the norm of
is less than b.
The last step is the imposition, so we are required to demand α < b / M. Let us impose now the
Picard's operator to be contractive under certain hypothesis over α that later on we will be able to
omit.
Given two functions
we want:
.
Then since f is Lipschitz with respect to the second variable, we have that:
.
This is contractive if α <
1 / L or equivalently, in order to have equality, if
.
Therefore, since the Picard's operator is an operator between Banach spaces (in particular, metric
spaces induced by the norm) and contractive, by means of the Banach fixed point theorem there
exists a unique function
such that
this is, solution of the initial value problem defined on Iα where α must satisfy the condition
given above, α = min{a,b / M,1 / (2L)}.
[edit] Optimization of the solution's interval
Nevertheless, there is a corollary of the Banach fixed point theorem that states that if an operator
Tn is contractive for some
then T has a unique fixed point. We will try to apply this
theorem to the Picard's operator. But before doing that, let us recall a lemma that will be very
useful to apply the aforementioned corollary.
Lemma:
We will check this by induction:
For m
= 1 we have already seen it, let us suppose it is true for m − 1 and let us check it for m:
.
Therefore, taking into account this inequality we can assure that for some m large enough, the
quantity
and hence Γm will be contractive. So by the previous corollary Γ will
have a unique fixed point. So, finally, we have been able to optimize the interval of the solution
by taking α = min{a,b / M}.
The importance of this result is that the interval of definition of the solution does eventually not
depend on the Lipschitz constant of the field, but essentially depends on the interval of definition
of the field and its maximum absolute value of it.
[edit] Other existence theorems
The Picard–Lindelöf theorem shows that the solution exists and that it is unique. The Peano
existence theorem shows only existence, not uniqueness, but it assumes only that ƒ is continuous
in y, instead of Lipschitz continuous. For example, the right-hand side of the equation y ′ = y1/3
with initial condition y(0) = 0 is continuous but not Lipschitz continuous. Indeed, the solution of
this equation is not unique; two different solutions are given besides the trivial one y(t) = 0
[2]
Even more general is Carathéodory's existence theorem, which proves existence (in a more
general sense) under weaker conditions on ƒ.