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  .


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.


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  .