Small Integral Kernels

## Integral Equations and the Method of Green's Functions James V. Herod*

Page maintained by Evans M. Harrell, II, harrell@math.gatech.edu.

SECTION 4: SOLVING y = Ky + f, where K IS SMALL.

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