Ordinary Differential Equations/Successive Approximations

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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:

  1. If is bounded and the Lipschitz condition is satisfied, then the sequence of functions converges to a continuous function
  2. This function satisfies the differential equation
  3. This is the unique solution to this differential equation with the given initial condition.

Proof[edit]

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.