Picard–Lindelöf theorem

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the study of differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s, 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

Charles Émile Picard

Charles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...

, Ernst Lindelöf, Rudolf Lipschitz

Rudolf Lipschitz

Rudolf Otto Sigismund Lipschitz was a German mathematician and professor at the University of Bonn from 1864. Peter Gustav Dirichlet was his teacher. He supervised the early work of Felix Klein....

and Augustin Cauchy.

Consider the initial value problem

Initial value problem

In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...

Suppose

is Lipschitz continuous in

and continuous

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

. Then, for some value

, there exists a unique solution

to the initial value problem within the range

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

Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

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

Banach fixed point theorem

In mathematics, the Banach fixed-point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points...

, that the sequence of "Picard iterates"

is convergent

Limit of a sequence

The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...

and that the limit

Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

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

are two solutions,
shows that

, thus proving the global uniqueness (the local uniqueness is a consequence of the uniqueness of the Banach fixed point).

Detailed proof

Let

be the compact cylinder where

is defined, this is

and

.

Let

,

this is, the maximum slope of the function in modulus. Finally, let

be the Lipschitz constant of

with respect to the second variable.

We will proceed to apply Banach fixed point theorem

Banach fixed point theorem

using the metric on

induced by the uniform norm

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

, or equivalently, that the norm of

is less than

,
which can be restated as

The last step is the imposition, so we are impose the requirement

Let us impose now the Picard's operator to be contractive under certain hypothesis over

that later on we will be able to omit.

Given two functions

, in order to appy the Banach fixed point theorem

Banach fixed point theorem

we want

So let

be such that

then using the definition of

Then since

is Lipschitz with respect to the second variable, we have that:

This is contractive if

.

We have established that the Picard's operator is a contraction on the Banach spaces with the metric induced by the uniform norm. This allows us to apply the Banach fixed point theorem to conclude that the operator has a unique fixed point. In particular, there is a unique function

such that

This function is the unique solution of the initial value problem, valid on the interval

where

satisfies the condition

Optimization of the solution's interval

Nevertheless, there is a corollary of the Banach fixed point theorem that states that if an operator

is contractive for some

then

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

we have already seen it, let us suppose it is true for

and let us check it for

.

Therefore, taking into account this inequality we can assure that for some

large enough, the quantity

and hence

will be contractive. So by the previous corollary

will have a unique fixed point. So, finally, we have been able to optimize the interval of the solution by taking

.

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.

Other existence theorems

The Picard–Lindelöf theorem shows that the solution exists and that it is unique. The Peano existence theorem

Peano existence theorem

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...

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 with initial condition is continuous but not Lipschitz continuous. Indeed, the solution of this equation is not unique; two different solutions are given besides the trivial one

Even more general is Carathéodory's existence theorem

Carathéodory's existence theorem

In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem...

, which proves existence (in a more general sense) under weaker conditions on ƒ.

External links

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