James V. Herod*
Page maintained by Evans M. Harrell, II, harrell@math.gatech.edu.
If the kernel is not separable, an alternate hypothesis that will enable one to solve the equation y = Ky + f is to suppose that the kernel for K is small. Of course this does not mean that K is of the form K(x,t) = .007 x t . Rather, we ask that K should be small in a sense developed below. The technique for getting a solution in this case is to iterate.
Take [[phi]]0(x) to be f(x) and [[phi]]1 to be defined by
It is reasonable to ask: does this generated sequence converge to a limit and in what sense does it converge? The answer to both questions can be found under appropriate hypothesis on K.
THEOREM If K satisfies the condition that
then limp [[phi]]p(x) exists and the convergence is uniform on [0,1] - in the sense that if u = limp[[phi]]p then
limp maxx | u(x) - [[phi]]p(x) | = 0.
SUGGESTION FOR PROOF: Note that
Furthermore, if p is a positive integer, the distance between successive iterates can be computed:
Inductively, this does not exceed
Thus, if
and n > m then
Hence, the sequence {[[phi]]p} of functions converges uniformly on [0,1] to a limit function and this limit provides a solution to the equation
COROLLARY. If
and
u = limp [[phi]]p
then
EXERCISE 1.4:
1. Let
(a) Show that if 0 <= x <= 1,
In fact,
(b) Toward solving y(x) = K[y](x) + x , compute [[phi]]0, [[phi]]1, and [[phi]]2.
(c) Give a bound on the error between the solution y and [[phi]]2.
ans: |y - [[phi]]2| <= 1/4
(d) Solve y(x) = Ky(x) + x in closed form for this K. (Reference Exercise 1.3 IV.)
2. Repeat 1 (a)-(c) with
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