James V. Herod*

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

There is an alternate, and independent, concept of K being small which leads
to convergence of the iteration process in the norm of L^{2}[0,1] This
alternate hypothesis asks that

**THEOREM** If K satisfies the condition that

then lim_{p} [[phi]]p(x) exists and the convergence is in norm- meaning that if u
= lim_{p}[[phi]]p then

lim_{p} || u(x) - [[phi]]p(x) || = 0.

INDICATION OF PROOF. The analysis of the nature of the convergence will go like this:

|| [[phi]]1 - [[phi]]0 || ^{2}

is defined to be

As before,

and

u = lim_{p} [[phi]]p

then

||u - [[phi]]m || __<__ F(r^{m+1},1 - r) ||f||.

**THE RESOLVENT.**

Before addressing the final case - where

** K** does not have a separable kernel,

we generate "resolvents" for the integral equations.

Re-examining the iteration process:

[[phi]]0(x) = f(x),

[[phi]]1(x) = **K**[[phi]]0(x) + f(x)

[[phi]]2(x) = **K**(**K**([[phi]]0))x +
**K**(f)(x) + f(x)

.

.

One writes [[phi]]0=f, [[phi]]1=**K**f+f, [[phi]]2 =
**K**[**K**f+f] + f = **K**^{2}f+**K**f+f, .....

In fact, with

Hence, the kernel K2 associated with **K**^{2} is

Inductively,

and

We have, in this section, conditions which imply that

[[Sigma]]p=1 **K**^{p}f

converges and that its limit y satisfies y = **K**y + f. Many authors call
this series of operators the "resolvent" and denote

** R** = [[Sigma]]p=1 **K**^{p}.

Note that **R** is a function which operates on elements of
L^{2}[0,1]. One writes that y = **K**y + f

has solution

y(x) = [( 1 + **R** ) f](x) = f(x) + I(0,1, ) R(x,t) f(t) dt.

Suggestive algebra can be made by identifying (1 + **R** ) as

(1 - **K** ) ^{-1} = 1 + **K**( 1 - **K** ) ^{-1},
so that **R** = **K** ( 1 - **K** ) ^{-1}.

Please refer to the accompanying notebook for the solution by iteration of a typical integral equation, including error estimates.

**EXERCISE 1.5**.

1. Suppose that

Give a formula for

2. Compute

for each K in the previous exercise set. ans: 1/12 and 1/6.

3. Let

For this K, find y such that y(x) = **K**y(x) + x. Note that

What is the significance of this observation?

ans: x +1/8

4. Let

For this K, find y such that y(x) = **K**y(x) + x. Note that

What is the significance of this observation?

ans: exp(x) - 1

5. Suppose that

so that the kernel of **K** is cos(x+t) and the kernel of **H** is
sin(x+t). What is the kernel of K[H]?

6. Find the kernel for the resolvent of the **K** whose kernel is K(x,t) =
x t.

Ans: R(x,t) = K(x,t) + K2(x,t) + K3(x,t) + . = 3xt/2.

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