Ordinary Differential Equations/Successive Approximations

$y^{\prime }=f(x,y)$ has a solution $y$ satisfying the initial condition $y(x_0)=y_0$, then it must satisfy the following integral equation:

$y=y_0+{\int }_{x_0}^{x}f(t,y(t))dt$

Now we will solve this equation by the method of successive approximations.

Define $y_1$ as:

$y_1=y_0+{\int }_{x_0}^{x}f(t,y_0)dt$

And define $y_n$ as

$y_n=y_0+{\int }_{x_0}^{x}f(t,y_{n-1})dt$

We will now prove that:

1. If $f(x,y)$ 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

First, we prove that $y_n$ lies in the box, meaning that $|y_n(x)-y_0|<\frac{1}{2}h$. We prove this by induction. First, it is obvious that $|y_1(x)-y_0|\le \frac{1}{2}h$. Now suppose that $|y_{n-1}(x)-y_0|\le \frac{1}{2}h$. Then $|f(t,y_{n-1}(t))|\le M$ so that

$|y_n(x)-y_0|\le {\int }_{x_0}^x|f(t,y_{n-1}(t))|dt\le M(x-x_0)\le \frac{1}{2}Mw\le \frac{1}{2}h$. This proves the case when $x_0<x$, and the case when $x<x_0$ is proven similarily.

We will now prove by induction that $|y_n(x)-y_{n-1}(x)|<\frac{M{K}^{n-1}}{n!}(x-x_0{)}^n$. First, it is obvious that $|y_1(x)-y_0|<M(x-x_0)$. Now suppose that it is true up to n-1. Then

$|y_n(x)-y_{n-1}(x)|\le {\int }_{x_0}^x|f(t,y_{n-1}(t))-f(t,y_{n-2}(t))|dt<{\int }_{x_0}^{x}K|y_{n-1}(t)-y_{n-2}(t)|dt$ due to the Lipschitz condition.

Now,

$|y_n(x)-y_{n-1}(x)|<\frac{M{K}^{n-1}}{(n-1)!}{\int }_{x_0}^x||u-x_0{|}^{n-1}du=\frac{M{K}^{n-1}}{n!}|x-x_0{|}^n$.

Therefore, the series of series $y_0+\sum _{n=1}^{\mathrm{\infty }}(y_n(x)-y_{n-1}(x))$ is absolutely and uniformly convergent for $|x-x_0|\le \frac{1}{2}w$ because it is less than the exponential function.

Therefore, the limit function $y(x)=y_0+\sum _{n=1}^{\mathrm{\infty }}(y_n(x)-y_{n-1}(x))=\underset{n\to \mathrm{\infty }}{lim}{y}_n(x)$ exists and is a continuous function for $|x-x_0|\le \frac{1}{2}w$.

Now we will prove that this limit function satisfies the differential equation.