Describe Picard Iteration.
Theorem- Let be continuous for
and satisfy a Lipschitz condition with constant in this region. Let there and let . Then the initial value problem
has a unique solution for .
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:
The first approximation is made by neglecting the integral. The next is made by using 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 . It will be shown that the iterates are defined for and satisfy the inequalities
The first step of induction is easy:
The induction hypothesis is to assume that
The completes the induction.
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
Take the norm of both sides and use the Lipschitz condition.
This produces a sequence of inequalities.
From part (i), .
The induction hypothesis is
Now it follows that
Now can be written as the partial sum
But the series on the left is domainated term-by-term by the series
As , this series is uniformly and asymptotically convergent to
So by the comparison test, comverges uniformly for to a limit function .
Again from the general term of the iteration, it can be seen that each iterate is a continuous function of for . Hence, since converges uniformly to , it follows that is a continuous function of for .
It follows from that as ,
It remains to show that satisfies the IVP .
Let in the general iteration term.
Finally using the Lipschitz condition,
and the right hand side tends to zero as .