James V. Herod*
Page maintained by Evans M. Harrell, II, harrell@math.gatech.edu.
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
and its adjoint 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?
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