Page 610 - Handbook Of Integral Equations

Now system (1) can be rewritten as the single Fredholm equation

nb–(n–1)a
Y (x) – λ K(x, t)Y (t) dt = F(x), a ≤ x ≤ nb – (n – 1)a.
a

If the kernels K ij (x, t) are square integrable on the square S = {a ≤ x ≤ b, a ≤ t ≤ b} and the
right-hand sides f i (x) are square integrable on [a, b], then the kernel K(x, t) is square integrable on
the new square
S n = {a < x < nb – (n – 1)a, a < t < nb – (n – 1)a},

and the right-hand side F(x) is square integrable on [a, nb – (n – 1)a].
If condition (4) is satisfied, then the kernel K(x, t) satisfies the inequality

b
2
K (x, t) dt ≤ A ∗ , a < x < nb – (n – 1)a,
a
where A ∗ is a constant.
•
Reference for Section 11.19: S. G. Mikhlin (1960).

11.20. Regularization Method for Equations With Infinite
Limits of Integration

11.20-1. Basic Equation and Fredholm Theorems
Consider an integral equation of the second kind in the form

1 ∞ 1 0 ∞
y(x)+ √ K 1 (x – t)y(t) dt + √ K 2 (x – t)y(t) dt + M(x, t)y(t) dt = f(x), (1)
2π 0 2π –∞ –∞

where –∞ < x < ∞. We assume that the functions y(x) and f(x) and the kernels K 1 (x) and K 2 (x)
are such that their Fourier transforms belong to L 2 (–∞, ∞) and satisfy the H¨ older condition. We
also assume that the Fourier transforms of the kernel M(x, t) with respect to each variable belong
to L 2 (–∞, ∞) and satisfy the H¨ older condition and, in addition,

∞ ∞
2
|M(x, t)| dx dt < ∞.
–∞ –∞
It should be noted that Eq. (1) with M(x, t) ≡ 0 is the convolution-type integral equation with two
kernels which was discussed in Subsection 11.9-2.
The transposed homogeneous equation has the form

1 ∞ 1 0 ∞
ϕ(x)+ √ K 1 (t – x)ϕ(t) dt + √ K 2 (t – x)ϕ(t) dt + M(t, x)ϕ(t) dt = 0, (2)
2π 0 2π –∞ –∞

where –∞ < x < ∞.
Assume that the normality conditions (see Subsection 11.9-2) hold, that is,

1+ K 1 (u) ≠ 0, 1 + K 2 (u) ≠ 0, –∞ < u < ∞. (3)

THEOREM 1. The number of linearly independent solutions of the homogeneous (f(x) ≡ 0)
equation (1) and that of the transposed homogeneous (g(x) ≡ 0) equation (2) are finite.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 593