Ordinary Differential Equations/The Picard–Lindelöf theorem

In this section, our aim is to prove several closely related results, all of which are occasionally called "Picard-Lindelöf theorem". This type of result is often used when it comes to arguing for the existence and uniqueness of a certain ordinary differential equation, given that some boundary conditions are satisfied.

Local resultsEdit

Picard–Lindelöf Theorem (Banach fixed-point theorem version):

Let   be an interval, let   be a function, and let

 

be the associated ordinary differential equation. If   is Lipschitz continuous in the second argument, then this ODE possesses a unique solution on   for each possible initial value  , where  ,   being the Lipschitz constant of the second argument of  .

Proof:

We first rewrite the problem as a fixed-point problem. Indeed, using the fundamental theorem of calculus, one can show that the simultaneous equations

 

are equivalent to the single equation

 ,

where   is to be determined at a later stage. This means that the function   is a fixed point of the function

 .

Now   satisfies a Lipschitz condition as follows:

 

where we took the norm on   to be the supremum norm. If now  , then   is a contraction, and hence the Banach fixed-point theorem is applicable, giving us both existence and uniqueness. 

Replacing the fixed-point principle by summation techniques, we get a slightly better result in the sense that the domain of definition of the function   does not have to be all of  .

Picard–Lindelöf theorem (telescopic series version):

Let   be a function which is continuous and Lipschitz continuous in the second argument, where  , and let   with the property that   for some  . If in this case  , where  , then the initial value problem

 

possesses a unique solution.

Proof:

We first prove uniqueness. To do so, we use Gronwall's inequalities. Suppose   are both solutions to the problem. Then

 ,

and hence by Gronwall's inequalities

 

for both   (right Gronwall's inequality) and   (left Gronwall's inequality).

Now on to existence. Once again, we inductively define

  (the constant function),
 .

Since   is not necessarily defined on any larger set than  , we have to prove that this definition always makes sense, i.e. that   is defined for all   and  , that is,   for  . We prove this by induction.

For  , this is trivial.

Assume now that   for  . Then

 

For   we obtain an analogous bound.

By the telescopic sum, we have

 .

Furthermore, for   and  ,

 

Hence, by induction,

 .

Again, by the very same argument, an analogous bound holds for  .

Thus, by the Weierstraß M-test, the telescopic sum

 

converges uniformly; in particular,   converges.

It is now possible to interchange differentiation and summation in the latter sum; for, on the one hand, we are uniformly convergent, and on the other hand,

 ,

which converges to   for   due to theorem 2.5 and the convergence of  ; note that the image of each   is contained within the compact set  , the closure of  . Hence indeed

 

on  .