Page 460 - Handbook Of Integral Equations

1 .If K(a, a) ≠ 0, then f(x) must be constrained by f(a)=0.
◦

◦
2 .If K(a, a)= K (a, a)= ··· = K x (n–1) (a, a)=0, 0< K x (n) (a, a) < ∞, then the right-hand side
x

of the equation must satisfy the conditions
f(a)= f (a)= ··· = f x (n) (a)=0.

x

◦
3 .If K(a, a)= K (a, a)= ··· = K x (n–1) (a, a)=0, K (n) (a, a)= ∞, then the right-hand side of the
x
x
equation must satisfy the conditions

f(a)= f (a)= ··· = f x (n–1) (a)=0.
x
For polar kernels of the form (4) and continuous f(x), no additional conditions are imposed on
the right-hand side of the integral equation.
Remark 1. Generally, the case in which the integration limit a is infinite is not excluded.
8.1-2. Existence and Uniqueness of a Solution
Assume that in Eq. (1) the functions f(x) and K(x, t) are continuous together with their first
derivatives on [a, b] and on S, respectively. If K(x, x) ≠ 0(x ∈ [a, b]) and f(a) = 0, then there exists
a unique continuous solution y(x) of Eq. (1).

Remark 2. The problem of existence and uniqueness of a solution to a Volterra equation of
the first kind is closely related to conditions under which this equation can be reduced to Volterra
equations of the second kind (see Section 8.3).

Remark 3. A Volterra equation of the first kind can be treated as a Fredholm equation of the
first kind whose kernel K(x, t) vanishes for t > x (see Chapter 10).
•
References for Section 8.1: E. Goursat (1923), H. M. M¨ untz (1934), F. G. Tricomi (1957), V. Volterra (1959),
S. G. Mikhlin (1960), M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), C. Corduneanu (1973),
V. I. Smirnov (1974), P. P. Zabreyko, A. I. Koshelev, et al. (1975), A. J. Jerry (1985), A. F. Verlan’ and V. S. Sizikov (1986).

8.2. Equations With Degenerate Kernel:
K(x, t)= g (x)h (t)+ ··· + g (x)h (t)
1
n
1
n
8.2-1. Equations With Kernel of the Form K(x, t)= g 1 (x)h 1 (t)+ g 2 (x)h 2 (t)
Any equation of this type can be rewritten in the form
x x

g 1 (x) h 1 (t)y(t) dt + g 2 (x) h 2 (t)y(t) dt = f(x). (1)
a a
2
2
It is assumed that g 1 (x) ≠ const g 2 (x), h 1 (t) ≠ const h 2 (t), 0 < g (a)+ g (a)< ∞, and f(a)=0.
1 2
The change of variables
x

u(x)= h 1 (t)y(t) dt (2)
a
followed by the integration by parts in the second integral in (1) with regard to the relation u(a)=0
yields the following Volterra equation of the second kind:
x
h 2 (t)
[g 1 (x)h 1 (x)+ g 2 (x)h 2 (x)]u(x) – g 2 (x)h 1 (x) u(t) dt = h 1 (x)f(x). (3)
a h 1 (t) t

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
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