Page 653 - Handbook Of Integral Equations

Consider the Fredholm integral equation

ϕ(t)+ λ K(t, τ)ϕ(τ) dτ = f(t), (16)
L

where L is a smooth contour, t and τ are complex coordinates of its points, ϕ(t) is the desired
function, f(t) is the right-hand side of the equation, and K(t, τ) is the kernel.
If for some λ, the homogeneous Fredholm equation has a nontrivial solution (or nontrivial
solutions), then λ is called a characteristic value, and the nontrivial solutions themselves are called
eigenfunctions of the kernel K(t, τ) or of Eq. (16).
The set of characteristic values of Eq. (16) is at most countable. If this set is infinite, then its
only limit point is the point at infinity. To each characteristic value, there are corresponding finitely
many linearly independent eigenfunctions. The set of characteristic values of an integral equation
is called its spectrum. The spectrum of a Fredholm integral equation is a discrete set.
If λ does not coincide with any characteristic value (in this case the value λ is said to be regular),
i.e., the homogeneous equation has only the trivial solution, then the nonhomogeneous equation (16)
is solvable for any right-hand side f(t).
The general solution is given by the formula

ϕ(t)= f(t) – R(t, τ; λ)f(τ) dτ, (17)
L
where the function R(t, τ; λ) is called the resolvent of the equation or the resolvent of the kernel
K(t, τ) and can be expressed via K(t, τ).
If a value of the parameter λ is characteristic for Eq. (16), then the homogeneous integral
equation

ϕ(t)+ λ K(t, τ)ϕ(τ) dτ = 0, (18)
L
as well as the transposed homogeneous equation

ψ(t)+ λ K(τ, t)ϕ(τ) dτ = 0, (19)
L

has nontrivial solutions, and the number of solutions of Eq. (18) is finite and is equal to the number
of linearly independent solutions of Eq. (19).
The general solution of the homogeneous equation can be represented in the form
n

ϕ(t)= C k ϕ k (t), (20)
k=1

where ϕ 1 (t), ... , ϕ n (t) is a (complete) finite set of linearly independent eigenfunctions that corre-
spond to the characteristic value λ, and C k are arbitrary constants.
If the homogeneous equation (18) is solvable, then the nonhomogeneous equation (16) is, in
general, unsolvable. This equation is solvable if and only if the following conditions hold:

f(t)ψ k (t) dt = 0, (21)
L

where {ψ k (t)} (k =1, ... , n) is a (complete) finite set of linearly independent eigenfunctions of the
transposed equation that correspond to the characteristic value λ.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC

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