Page 677 - Handbook Of Integral Equations

If in Eq. (1) the kernel is K x, t, y(t) = Q(x, t)Φ t, y(t) , where Q(x, t) and Φ(t, y) are known
functions, then we obtain the Volterra integral equation in the Hammerstein form:
x

Q(x, t)Φ t, y(t) dt = F x, y(x) . (5)
a
In some cases Eq. (5) can be rewritten in the form
x

Q(x, t)Φ t, y(t) dt = f(x). (6)
a
Equation (6) is called a Volterra equation of the first kind in the Hammerstein form. Similarly, an
equation of the form
x

y(x) – Q(x, t)Φ t, y(t) dt = f(x), (7)
a
is called a Volterra equation of the second kind in the Hammerstein form.
It is possible to reduce Eq. (7) to the canonical form

x

u(x)= Q(x, t)Φ ∗ t, u(t) dt, (8)
a
where u(x)= y(x) – f(x).
Remark 1. Since a Volterra equation in the Hammerstein form is a special case of a Volterra
equation in the Urysohn form, the methods discussed below for the latter are certainly applicable to
the former.
Remark 2. Some other types of nonlinear integral equations with variable limits of integration
are considered in Chapter 5.

14.1-2. Nonlinear Equations With Constant Integration Limits
Nonlinear integral equations with constant integration limits can be represented in the form

b

K x, t, y(t) dt = F(x, y(x)), α ≤ x ≤ β, (9)
a

where K x, t, y(t) is the kernel of the integral equation and y(x) is the unknown function. Usually,
all functions in (9) are assumed to be continuous and the case of α = a and β = b is considered.
The form (9) does not cover all possible forms of nonlinear integral equations with constant
integration limits; however, just as the form (1) for the Volterra equations, it includes the most
frequently used and most studied types of these equations. A nonlinear integral equation (1) with
constant limits of integration is called an integral equation of the Urysohn type.
If Eq. (9) can be rewritten in the form

b

K x, t, y(t) dt = f(x), (10)
a
then (10) is called an Urysohn equation of the first kind. Similarly, the equation
b

y(x) – K x, t, y(t) dt = f(x), (11)
a
is called an Urysohn equation of the second kind.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 660