Page 480 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 32: The Fredholm Theorem

n Notes
R ( , ) v ( ) ( )
s
t
t
s
v
v 1
is also written in the form
n 1
R ( , ) ( ) (1) ( )
t
s
t
s
v v
v 1
(1) (1) (1) (1)
t
t
t
t
If 1 ( ), 2 ( ), ..., n 1 ( ) are linearly independent, then, by setting v ( ) v ( ) , the number
t
(1)
t
n is diminished. If otherwise, say, n 1 ( ) is written as a linear combination of
(1) (2) (1)
t
t
t
1 ( ), 2 ( ), ..., n 2 ( ) then R(s, t) is also written as
n 2
t
R ( , ) (1) ( ) (1) ( )
s
s
t
v
v
v 1
Repeating this argument alternatively for and , we finally obtain two sets of linearly
independent functions
t
t
s
s
s
( ), ( ), ..., ( ) and ( ), ( ),..., ( )
t
1 2 m 1 2 m
in terms of which R(s, t) is written as
m
R ( , ) v ( ) ( )
s
t
t
s
v
v 1
provided that ( , ) 0K s t and ( , ) 0R s t . Then by setting (s) = (s), and (t) = (t), the proof
v v v v
is completed
we now set
m
t
t
K 1 ( , ) K ( , ) v ( ) ( )
t
s
s
s
v
v 1
and denote the resolvent Kernel of K (s, t) by
1
n
s
s
t
( , ) K ( ) ( , )
t
1 1
n 1
Then, the equation (1) is written as
b
s
s
( ) K 1 ( , ) ( )dt
t
t
a
m
b
s
t
t
s
f ( ) v ( ) ( ) ( )dt ...(17)
v
a
v 1
and we can prove in the same way as in last that (s) is determined by
b m b
t
r
( ) v ( ) 1 ( , ) v ( )dr v ( ) ( )dt
r
s
s
t
s
a a
v 1
...(18)
b
f ( ) 1 ( , ) ( )dr
r
s
r
s
f
a
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