Page 678 - Handbook Of Integral Equations

An Urysohn equation of the second kind can be rewritten in the canonical form
b

u(x)= K x, t, u(t) dt. (12)
a
Remark 3. Conditions for existence and uniqueness of the solution of an Urysohn equation are
discussed below in Subsections 14.3-4 and 14.3-5.

If in Eq. (9) the kernel is K x, t, y(t) = Q(x, t)Φ t, y(t) , and Q(x, t) and Φ(t, y) are given
functions, then we obtain an integral equation of the Hammerstein type:
b

Q(x, t)Φ t, y(t) dt = F x, y(x) , (13)
a
where, as usual, all functions in the equation are assumed to be continuous.
If Eq. (13) can be rewritten in the form
b

Q(x, t)Φ t, y(t) dt = f(x), (14)
a
then (14) is called a Hammerstein equation of the first kind. Similarly, an equation of the form
b

y(x) – Q(x, t)Φ t, y(t) dt = f(x), (15)
a
is called a Hammerstein equation of the second kind.
A Hammerstein equation of the second kind can be rewritten in the canonical form
b

u(x)= Q(x, t)Φ ∗ t, u(t) dt. (16)
a
The existence of the canonical forms (4), (8), (12) and (16) means that the distinction between
the inhomogeneous and homogeneous nonlinear integral equations is unessential, unlike the case of
linear equations. Another specific feature of a nonlinear equation is that it frequently has several
solutions.
Remark 4. Since a Hammerstein equation is a special case of an Urysohn equation, the methods
discussed below for the latter are certainly applicable to the former.
Remark 5. Some other types of nonlinear integral equations with constant limits of integration
are considered in Chapter 6.
•
References for Section 14.1: N. S. Smirnov (1951), M. A. Krasnosel’skii (1964), M. L. Krasnov, A. I. Kiselev,
and G. I. Makarenko (1971), P. P. Zabreyko, A. I. Koshelev, et al. (1975), F. G. Tricomi (1985), A. F. Verlan’ and
V. S. Sizikov (1986).
14.2. Nonlinear Volterra Integral Equations

14.2-1. The Method of Integral Transforms
Consider a Volterra integral equation with quadratic nonlinearity
x

µy(x)– λ y(x – t)y(t) dt = f(x). (1)
0
To solve this equation, the Laplace transform can be applied, which, with regard to the convolution
theorem (see Section 7.2), leads to a quadratic equation for the transform ˜y(p)= L{y(x)}:
2
˜
µ ˜y(p)– λ ˜y (p)= f(p).
This implies
2
˜
µ ± µ –4λf(p)
˜ y(p)= . (2)
2λ
–1
The inverse Laplace transform y(x)= L { ˜y(p)} (if it exists) is a solution to Eq. (1). Note that for
the two different signs in formula (2), there are two corresponding solutions of the original equation.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 661