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x

R(x, t)= K(x, t)+ K(s, t)R(x, s) ds, (4)
t
in which the integration is performed with respect to different pairs of variables of the kernel and
the resolvent.

9.1-2. A Relationship Between Solutions of Some Integral Equations
Let us present two useful formulas that express the solution of one integral equation via the solutions
of other integral equations.
◦
1 . Assume that the Volterra equation of the second kind with kernel K(x, t) has a resolvent R(x, t).
∗
Then the Volterra equation of the second kind with kernel K (x, t)= –K(t, x) has the resolvent
R (x, t)= –R(t, x).
∗
2 . Assume that two Volterra equations of the second kind with kernels K 1 (x, t) and K 2 (x, t) are
◦
given and that resolvents R 1 (x, t) and R 2 (x, t) correspond to these equations. In this case the Volterra
equation with kernel
x

K(x, t)= K 1 (x, t)+ K 2 (x, t) – K 1 (x, s)K 2 (s, t) ds (5)
t
has the resolvent
x
R(x, t)= R 1 (x, t)+ R 2 (x, t)+ R 1 (s, t)R 2 (x, s) ds. (6)
t
Note that in formulas (5) and (6), the integration is performed with respect to different pairs of
variables.
•
References for Section 9.1: E. Goursat (1923), H. M. M¨ untz (1934), V. Volterra (1959), S. G. Mikhlin (1960),
M. L. Krasnov, A. I. Kiselev, and G. I. Makarenko (1971), J. A. Cochran (1972), V. I. Smirnov (1974), P. P. Zabreyko,
A. I. Koshelev, et al. (1975), A. J. Jerry (1985), F. G. Tricomi (1985), A. F. Verlan’ and V. S. Sizikov (1986), G. Gripenberg,
S.-O. Londen, and O. Staffans (1990), C. Corduneanu (1991), R. Gorenflo and S. Vessella (1991), A. C. Pipkin (1991).

9.2. Equations With Degenerate Kernel:

K(x, t)= g (x)h (t)+ ··· + g (x)h (t)
1
n
1
n
9.2-1. Equations With Kernel of the Form K(x, t)= ϕ(x)+ ψ(x)(x – t)
The solution of a Volterra equation (see Subsection 9.1-1) with kernel of this type can be expressed
by the formula

y = w , (1)
xx
where w = w(x) is the solution of the second-order linear nonhomogeneous ordinary differential
equation
w – ϕ(x)w – ψ(x)w = f(x), (2)

xx x
with the initial conditions

w(a)= w (a) = 0. (3)
x
Let w 1 = w 1 (x) be a nontrivial particular solution of the corresponding homogeneous linear differ-
ential equation (2) for f(x) ≡ 0. Assume that w 1 (a) ≠ 0. In this case, the other nontrivial particular
solution w 2 = w 2 (x) of this homogeneous linear differential equation has the form

x Φ(t) x
w 2 (x)= w 1 (x) 2 dt, Φ(x)=exp ϕ(s) ds .
a [w 1 (t)] a

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 460