# ODE0.2

Describe Picard Iteration.

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

$|t-t_{0}|\leq \alpha ,|x-x_{0}|\leq \beta \,$

and satisfy a Lipschitz condition with constant $L\,$ in this region. Let $|f(x,t)|\leq M\,$ 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 $|t-t_{0}|\leq \delta \,$.

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 $|t-t_{0}|\leq \delta \,$ and satisfy the inequalities

$|x^{{N+1}}(t)-x_{0}|\leq \beta ({\mathrm {for}}|t-t_{0}|\leq \delta )\,$

The first step of induction is easy:

$|x^{{0}}(t)-x_{0}|=0\leq \beta ({\mathrm {for}}|t-t_{0}|\leq \delta )\,$

The induction hypothesis is to assume that

$|x^{r}(t)-x_{0}|\leq \left|\int _{{t_{0}}}^{t}|f(x^{N}(s),s)|ds\right|\,$

$\leq M|t-t_{0}|\,$

$\leq \beta ({\mathrm {for}}|t-t_{0}|\leq \delta )\,$

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.

$|x^{{N+1}}(t)-x^{N}(t)|\leq L{\Bigg |}\int _{{t_{0}}}^{t}{\Big |}x^{N}(s)-x^{{N-1}}Big|ds{\Bigg |}\,$

This produces a sequence of inequalities.

From part (i), $|x^{1}(t)-x^{0}(t)|\leq M|t-t_{0}|\,$.

The induction hypothesis is

$|x^{r}(t)-x^{{r-1}}(t)|\leq ML^{{r-1}}{\frac {|t-t_{0}|^{r}}{r!}}(r=1,...,N)\,$

Now it follows that

$|x^{{N+1}}(t)-x^{N}(t)|\leq {\frac {ML^{N}}{N!}}{\Bigg |}\int _{{t_{0}}}^{t}|s-t_{0}|^{N}ds{\Bigg |}\,$

$=ML^{N}{\frac {|t-t_{0}|^{{N+1}}}{(N+1)!}}\,$

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 $|t-t_{0}|\leq \delta \,$ 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 $|t-t_{0}|\leq \delta \,$. Hence, since $x^{N}(t)\,$ converges uniformly to $x(t)\,$, it follows that $x(t)\,$ is a continuous function of $t\,$ for $|t-t_{0}|\leq \delta \,$.

It follows from $|x^{{N+1}}(t)-x_{0}|\leq \beta ({\mathrm {for}}|t-t_{0}|\leq \delta )\,$ that as $N\to \infty \,$,

$|x(t)-x_{0}|\leq \beta ({\mathrm {for}}|t-t_{0}|\leq \delta )\,$

It remains to show that $x(t)\,$ satisfies the IVP $x(t)=x_{0}+\int _{{t_{0}}}^{t}f(x(s),s)ds\,$.

Let $N\to \infty \,$ in the general iteration term.

$x^{{N+1}}(t)\to x(t)\,$

and

$\int _{{t_{0}}}^{t}f(x^{N}(s),s)ds\to \int _{{t_{0}}}^{t}f(x(s),s)ds\,$

Finally using the Lipschitz condition,

${\Bigg |}\int _{{t_{0}}}^{t}\left\{f(x(s),s)-f(x^{N}(s),s)\right\}ds{\Bigg |}\leq {\Bigg |}\int _{{t_{0}}}^{t}L|x(s)-x^{N}(s)|ds{\Bigg |}\,$

$\leq L\delta \max _{{|t-t_{0}|\leq \delta }}|x(t)-x^{N}(t)|\,$

and the right hand side tends to zero as $N\to \infty \,$.