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.
Let be an interval, let be a continuous 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