Integral Equations

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

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

CHAPTER I. INTEGRAL EQUATIONS

SECTION 2. THE FREDHOLM ALTERNATIVE THEOREMS

A first understanding of the problem of solving an integral equation

y = Ky + f

can be made by reviewing the Fredholm Alternative Theorems in this context.

(Review the alternative theorem for matrices.)

I. Exactly one of the following holds:

(a)(First Alternative) if f is in L2{0,1}, then

has one and only one solution.

(b)(Second Alternative)

as a nontrivial solution.

II. (a) If the first alternative holds for the equation

then it also holds for the equation

z(x) = I(0,1, ) K(t,x) z(t) dt + g(x).

(b) In either alternative, the equation

have the same number of linearly independent solutions.

III. Suppose the second alternative holds. Then

has a solution if and only if

for each solution z of the adjoint equation

Comparing this context for the Fredholm Alternative Theorems with an understanding of matrix examples seems irresistible. Since these ideas will re-occur in each section, the student should pause to make these comparisons.

EXAMPLE: Suppose that E is the linear space of continuous functions on the interval [-1,1]. with

and that

The equation y = K(y) has a non-trivial solution: the constant function 1. To see this, one computes

One implication of these computations is that the problem y = Ky + f is a second alternative problem. It may be verified that y(x) = 1 is also a nontrivial solution for y = K*y. It follows from the third of the Fredholm alternative theorems that a necessary condition for y = Ky + f to have a solution is that

Note that one such f is f(x) = x + x3.

EXERCISE 1.2

(1) Suppose that E is the linear space of continuous functions on [0,1] with

and that

(2) Show that y = Ky has non-trivial solution the constant function 1.

(3) Show that y = K*y has non-trivial solution the function [[pi]] + 2 cos([[pi]]x).

(4) What conditions must hold on f in order that

y = Ky + f

should have a solution?