Page 478 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 32: The Fredholm Theorem

The term in braces on the right side is continuous on the domain a s b, a t b; hence Notes
bounded. Therefore, according to the assumption (3), the series.

n
t
t
s
( , ) K ( ) ( , )
s
...(9)
n 1
converge uniformly on the domain a s b, a t b. Hence by term-by-term integration and
by using (7) we obtain
b
s
r
r
t
s
t
( , ) K ( , ) K ( , ) ( , )dr ...(10)
t
s
a
b
r
t
t
r
s
t
K
s
( , ) K ( , ) ( , ) ( , )dr ...(11)
s
a
The series (9) is known as the Neumann series for the Kernel K(s, t).
Now, by making use of (10), we can prove that
b
s
t
s
( ) f ( ) ( , ) ( )dt ...(12)
f
t
s
a
satisfies the equation (1). In fact, substituting (12) in (1) and using (10), we have
b
t
s
s
t
( ) K ( , ) ( )dt
a
b b b
s
t
t
f ( ) ( , ) ( )dt K ( , ) f ( ) ( , ) ( )dr dt
t
r
r
t
t
s
f
f
s
a a a
b b
t
r
t
s
s
r
s
f ( ) ( , ) K ( , ) K ( , ) ( , )dr f ( )dt
s
t
t
a a

s
f ( )
Conversely, we can prove that if (s) satisfies the equation (1), then (s) satisfies (12). In fact,
b
s
t
t
substituting f(s) = (s) K ( , ) ( )dt in (12) and using (11), we see that
a
b
( ) K ( , ) ( )dt
t
t
s
s
a
b b
t
t
s
( , ) ( ) K ( , ) ( )dr dt
r
t
r
a a
b b
r
s
K
s
r
s
( ) ( , ) K ( , ) ( , ) ( , )dr ( )dt
t
s
t
t
t
a a
( )
s
Accordingly, we see that the equation (1) is equivalent to the equation (12). Similarly, we can
prove that conjugate equation (2) is equivalent to the equation
b
t
g
s
( ) g ( ) ( , ) ( )ds ...(13)
s
t
t
a
Example: Under the assumption (3), every solution (s) of the equation (1) is given by
(12) by means of the Kernel (s, t) and every solution (t) of the conjugate equation (2) is given
by (13) by means of the conjugate Kernel (s, t) of (s, t), defined by
(s, t) = (t, s) ...(14)
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