# 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 results[edit | edit source]

**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 .