Page 551 - Handbook Of Integral Equations

11.3-3. Construction of the Resolvent

The resolvent of the integral equation (1) is defined via the iterated kernels by the formula
∞

R(x, t; λ)= λ n–1 K n (x, t), (5)
n=1

where the series on the right-hand side is called the Neumann series of the kernel K(x, t). It
converges to a unique square integrable solution of Eq. (1) provided that

b b
1
|λ| < , B = K (x, t) dx dt. (6)
2
B a a
If, in addition, we have
b
2
K (x, t) dt ≤ A, a ≤ x ≤ b,
a
where A is a constant, then the Neumann series converges absolutely and uniformly on [a, b].
A solution of a Fredholm equation of the second kind of the form (1) is expressed by the formula

b
y(x)= f(x)+ λ R(x, t; λ)f(t) dt, a ≤ x ≤ b. (7)
a
Inequality (6) is essential for the convergence of the series (5). However, a solution of Eq. (1)
can exist for values |λ| >1/B as well.

Remark 1. A solution of the equation
b

y(x) – λ K(x, t)y(t) dt = f(x), a ≤ x ≤ b,
a
with weak singularity, where the kernel K(x, t) has the form

L(x, t)
K(x, t)= , 0 < α <1,
|x – t| α
and L(x, t) is a function continuous on the square S = {a ≤ x ≤ b, a ≤ t ≤ b}, can be obtained by the
successive approximation method provided that
1 – α
∗
|λ| < , B = sup |L(x, t)|.
2B (b – a) 1–α
∗
The equation itself can be reduced to a Fredholm equation of the form

b
y(x) – λ n K n (x, t)y(t) dt = F(x), a ≤ x ≤ b,
a
n–1 b

F(x)= f(x)+ λ p K p (x, t)f(t) dt,
a
p=1
where K p (x, t)(p =1, ... , n)isthe pth iterated kernel, with K n (x, t) being a Fredholm kernel for
–1
1
n > (1 – α) –1 and bounded for n >(1 – α) .
2

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 534