has a solution satisfying the initial condition , then it must satisfy the following integral equation:
Now we will solve this equation by the method of successive approximations.
And define as
We will now prove that:
- If is bounded and the Lipschitz condition is satisfied, then the sequence of functions converges to a continuous function
- This function satisfies the differential equation
- This is the unique solution to this differential equation with the given initial condition.
First, we prove that lies in the box, meaning that . We prove this by induction. First, it is obvious that . Now suppose that . Then so that
. This proves the case when , and the case when is proven similarily.
We will now prove by induction that . First, it is obvious that . Now suppose that it is true up to n-1. Then
due to the Lipschitz condition.
Therefore, the series of series is absolutely and uniformly convergent for because it is less than the exponential function.
Therefore, the limit function exists and is a continuous function for .
Now we will prove that this limit function satisfies the differential equation.