Page 555 - Handbook Of Integral Equations

11.5. Fredholm Theorems and the Fredholm Alternative

11.5-1. Fredholm Theorems
THEOREM 1. If λ is a regular value, then both the Fredholm integral equation of the second kind
and the transposed equation are solvable for any right-hand side, and both the equations have unique
solutions. The corresponding homogeneous equations have only the trivial solutions.
THEOREM 2. For the nonhomogeneous integral equation to be solvable, it is necessary and
sufficient that the right-hand side f(x) satisfies the conditions

b
f(x)ψ k (x) dx =0, k =1, ... , n,
a
where ψ k (x) is a complete set of linearly independent solutions of the corresponding transposed
homogeneous equation.
THEOREM 3. If λ is a characteristic value, then both the homogeneous integral equation and the
transposed homogeneous equation have nontrivial solutions. The number of linearly independent
solutions of the homogeneous integral equation is finite and is equal to the number of linearly
independent solutions of the transposed homogeneous equation.
THEOREM 4. A Fredholm equation of the second kind has at most countably many characteristic
values, whose only possible accumulation point is the point at infinity.

11.5-2. The Fredholm Alternative
The Fredholm theorems imply the so-called Fredholm alternative, which is most frequently used in
the investigation of integral equations.
THE FREDHOLM ALTERNATIVE. Either the nonhomogeneous equation is solvable for any right-
hand side or the corresponding homogeneous equation has nontrivial solutions.
The first part of the alternative holds if the given value of the parameter is regular and the second
if it is characteristic.
Remark. The Fredholm theory is also valid for integral equations of the second kind with weak
singularity.
•
References for Section 11.5: S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971),
J. A. Cochran (1972), V. I. Smirnov (1974), A. J. Jerry (1985), D. Porter and D. S. G. Stirling (1990), C. Corduneanu (1991),
J. Kondo (1991), W. Hackbusch (1995), R. P. Kanwal (1997).

11.6. Fredholm Integral Equations of the Second Kind
With Symmetric Kernel

11.6-1. Characteristic Values and Eigenfunctions

Integral equations whose kernels are symmetric, that is, satisfy the condition K(x, t)= K(t, x), are
called symmetric integral equations.
Each symmetric kernel that is not identically zero has at least one characteristic value.
For any n, the set of characteristic values of the nth iterated kernel coincides with the set of nth
powers of the characteristic values of the first kernel.
The eigenfunctions of a symmetric kernel corresponding to distinct characteristic values are
orthogonal, i.e., if

b b

ϕ 1 (x)= λ 1 K(x, t)ϕ 1 (t) dt, ϕ 2 (x)= λ 2 K(x, t)ϕ 2 (t) dt, λ 1 ≠ λ 2 ,
a a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 538