Page 510 - Handbook Of Integral Equations

or logarithmic, i.e.,
K(x, t)= L(x, t)ln |x – t| + M(x, t), (4)

where L(x, t) and M(x, t) are continuous on S and L(x, x) /≡ 0, then K(x, t) is called a kernel with
weak singularity, and the equation itself is called an equation with weak singularity.
Remark 2. Kernels with logarithmic singularity and polar kernels with 0 < α < 1 are Fredholm
2
kernels.
Remark 3. In general, the case in which the limits of integration a and/or b can be infinite is not
excluded, but in this case the validity of condition (2) must be verified with special care.

10.1-3. Integral Equations of Convolution Type
The integral equation of the first kind with difference kernel on the entire axis (this equation is
sometimes called an equation of convolution type of the first kind with a single kernel) has the form

∞
K(x – t)y(t) dt = f(x), –∞ < x < ∞, (5)
–∞
where f(x) and K(x) are the right-hand side and the kernel of the integral equation and y(x)isthe
unknown function (in what follows we use the above notation).
An integral equation of the first kind with difference kernel on the semiaxis has the form

∞

K(x – t)y(t) dt = f(x), 0 < x < ∞. (6)
0
Equation (6) is also called a one-sided equation of the first kind or a Wiener–Hopf integral equation
of the first kind.
An integral equation of convolution type with two kernels of the first kind has the form

0
∞
K 1 (x – t)y(t) dt + K 2 (x – t)y(t) dt = f(x), –∞ < x < ∞, (7)
0 –∞
where K 1 (x) and K 2 (x) are the kernels of the integral equation (7).
Recall that a function g(x) satisfies the H¨ older condition on the real axis if for any real x 1 and x 2
we have the inequality

λ
|g(x 2 ) – g(x 1 )|≤ A|x 2 – x 1 | , 0 < λ ≤ 1,
and for any x 1 and x 2 sufficiently large in absolute value we have

λ
1 1
|g(x 2 ) – g(x 1 )|≤ A – , 0 < λ ≤ 1,
x 2 x 1

where A and λ are positive (the latter inequality is the H¨ older condition in the vicinity of the point
at infinity).
Assume that the functions y(x) and f(x) and the kernels K(x), K 1 (x), and K 2 (x) are such that
their Fourier transforms belong to L 2 (–∞, ∞) and, moreover, satisfy the H¨ older condition.
For a function y(x) to belong to the above function class it suffices to require y(x) to belong to
L 2 (–∞, ∞) and xy(x) to be absolutely integrable on (–∞, ∞).

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 492