Theorem 19 (unique solution)

Take a set and assume is continuous on . If satisfies a Lipschitz Condition on in the variable then the initial value problem

,

has a unique solution for .

>

For example, take the initial value problem

,

then is a continuous function and we can, while holding fixed, apply the Mean Value Theorem on any interval in :

.

Hence, for , we have that

,

so that satisfies a Lipschitz condition in with Lipschitz constant . Hence, this initial value problem has a unique solution.

>