ODE0.2

From Exampleproblems

Theorem- Let $f(x,t)\,$ be continuous for

$|t-t_0|\le \alpha, |x-x_0|\le \beta\,$

and satisfy a Lipschitz condition with constant $L\,$ in this region. Let $|f(x,t)|\le M\,$ there and let $\delta = \min\left\{\alpha,\beta/M\right\}\,$ . Then the initial value problem

$x(t)=x_0+\int_{t_0}^t f(x(s),s) ds\,$

has a unique solution for $|t-t_0|\le\delta\,$ .

Proof- The uniqueness of the solution has already been established in ODE0.3. The existence of the solution is proved in three steps.

(i) Define the following sequence of iterates:

$x^0(t)=x_0 \,$

$x^1(t)=x_0+\int_{t_0}^t f(x^0(s),s) ds \,$

...

$x^{N+1}(t) = x_0 + \int_{t_0}^t f(x^N(s),s)ds \,$

The first approximation $x^0\,$ is made by neglecting the integral. The next $x^1\,$ is made by using $x^0\,$ in the intgrand as a correction term.

In order for the iterates to be defined, it is required to prove that each iterate stays within its domain of definition of $f(x,t)\,$ . It will be shown that the iterates are defined for $|t-t_0|\le \delta\,$ and satisfy the inequalities

$|x^{N+1}(t)-x_0|\le\beta (\mathrm{for} |t-t_0|\le\delta)\,$

The first step of induction is easy:

$|x^{0}(t)-x_0|=0\le\beta (\mathrm{for} |t-t_0|\le\delta)\,$

The induction hypothesis is to assume that

$|x^r(t)-x_0| \le \left|\int_{t_0}^t |f(x^N(s),s)| ds\right| \,$

$\le M |t-t_0| \,$

$\le \beta (\mathrm{for} |t-t_0|\le\delta)\,$

The completes the induction.

(ii)

Next it is shown that the sequence of iterates is uniformly and absolutely convergent. From the general term of the iteration it is clear that

$x^{N+1}(t)-x^N(t) = \int_{t_0}^t \left\{f(x^N(s),s) - f(x^{N-1}(s),s)\right\} ds\,$

Take the norm of both sides and use the Lipschitz condition.

$|x^{N+1}(t)-x^N(t)| \le L \Bigg| \int_{t_0}^t \Big| x^N(s) - x^{N-1} Big| ds \Bigg|\,$

This produces a sequence of inequalities.

From part (i), $|x^1(t)-x^0(t)| \le M |t-t_0| \,$ .

The induction hypothesis is

$|x^r(t)-x^{r-1}(t)| \le ML^{r-1} \frac{|t-t_0|^r}{r!} (r=1,...,N)\,$

Now it follows that

$|x^{N+1}(t) - x^N(t)| \le \frac{ML^N}{N!} \Bigg| \int_{t_0}^t |s-t_0|^N ds \Bigg| \,$

$=ML^N \frac{|t-t_0|^{N+1}}{(N+1)!}\,$

Now $x^N(t)\,$ can be written as the partial sum

$x^0(t) + \sum_{r=1}^N \left\{ x^r(t) - x^{r-1}(t) \right\} = x^N(t)\,$

But the series on the left is domainated term-by-term by the series

$x_0 + \sum_{r=1}^N ML^{r-1} \frac{|t-t_0|^r}{r!} \,$

As $m\to\infty\,$ , this series is uniformly and asymptotically convergent to

$x_0 + \frac{M}{L} \left[ \exp\{L|t-t_0|\}-1\right]\,$

So by the comparison test, $x^N(t)\,$ comverges uniformly for $|t-t_0|\le \delta\,$ to a limit function $x(t)\,$ .

(iii)

Again from the general term of the iteration, it can be seen that each iterate is a continuous function of $t\,$ for $|t-t_0|\le\delta\,$ . Hence, since $x^N(t)\,$ converges uniformly to $x(t)\,$ , it follows that $x(t)\,$ is a continuous function of $t\,$ for $|t-t_0|\le\delta\,$ .