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.
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