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.
Define
as:
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.
Now,
.
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.