Page 560 - Handbook Of Integral Equations

Remark 2. On the basis of the bilinear expansion (8) and the Hilbert–Schmidt theorem, the
solution of the symmetric Fredholm integral equation of the first kind

b

K(x, t)y(t) dt = f(x), a ≤ x ≤ b,
a
can be constructed in a similar way in the form

∞

y(x)= f k λ k ϕ k (x),
k=1
and the necessary and sufficient condition for the existence and uniqueness of such a solution
in L 2 (a, b) is the completeness of the system of the eigenfunctions ϕ k (x) of the kernel K(x, t) together
2 2
with the convergence of the series ∞ f λ , where the λ k are the corresponding characteristic values.
k k
k=1
It should be noted that the verification of the last condition for specific equations is quite
complicated. In the solution of Fredholm equations of the first kind, the methods presented in
Chapter 10 are usually applied.

11.6-6. The Fredholm Alternative for Symmetric Equations
The above results can be unified in the following alternative form.
A symmetric integral equation
b

y(x) – λ K(x, t)y(t) dt = f(x), a ≤ x ≤ b, (23)
a
for a given λ, either has a unique square integrable solution for an arbitrarily given function
f(x) ∈ L 2 (a, b), in particular, y = 0 for f = 0, or the corresponding homogeneous equation has
finitely many linearly independent solutions Y 1 (x), ... , Y r (x), r >0.
For the second case, the nonhomogeneous equation has a solution if and only if the right-hand
side f(x) is orthogonal to all the functions Y 1 (x), ... , Y r (x) on the interval [a, b]. Here the solution
is defined only up to an arbitrary additive linear combination A 1 Y 1 (x)+ ··· + A r Y r (x).

11.6-7. The Resolvent of a Symmetric Kernel

The solution of a Fredholm equation of the second kind (23) can be written in the form
b
y(x)= f(x)+ λ R(x, t; λ)f(t) dt, (24)
a
where the resolvent R(x, t; λ) is given by the series

∞
ϕ k (x)ϕ k (t)
R(x, t; λ)= . (25)
λ k – λ
k=1
Here the collections ϕ k (x) and λ k form the system of eigenfunctions and characteristic values of
Eqs. (23). It follows from formula (25) that the resolvent of a symmetric kernel has only simple
poles.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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