Page 482 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 32: The Fredholm Theorem

where the are the solution of (23). Notes
v
Let be the matrix of the equations (22), in the unknowns , and that of the equations (23), in
the unknowns . Then
is the transposed matrix of ...(25)
Hence det 0 if and only if det 0.
We first consider the case when det 0, and hence, det 0. In this case, the equation (22)[(23)]
for any function f(s)[g(t)], admits a unique solution
= ( , , ..., ) [ = ( , , ..., )]
1 2 m 1 2 m
Therefore, for the given function f(s)[g(t)], the equation (1) [2] admits a unique solution (s)[ (t)].
In particular, if f(s) 0 [g(t) 0], then

( 1 , 2 , ..., m ) (0, 0, ..., 0)
[ ( 1 , 2 , ..., m ) (0, 0, ..., 0)]

hence, (s) 0[ (t) 0]
We next consider the case when det = 0, and hence det = 0. For the sake of simplicity we write
(22), (23) as
m
c v v f ( = 1, 2, ..., m) ...(23 )
v 1
m
c v v g ( = 1, 2, ..., m) ...(24 )
v 1
respectively. The matrices , are of course written as
= ( c ), = ( c )
v v
where = 0 for , and = 1 for = . For the case when det = det = 0, the following facts
are known:
The associated systems of linear homogeneous equations
m
c 0
v v ( = 1, 2, ..., m) ...(22 )
v 1
and
m
c v v 0 ( = 1, 2, ..., m) ...(23 )
v 1
admit a number r(r 1) of linearly independent solutions

(1) ( 11 , 12 , ..., 1m ), ...,
( ) ( 1 r , r 2 , ..., rm )
r
and
(1) ( 11 , 12 , ..., 1m ), ...,
( ) ( 1 r , r 2 , ..., rm )
r
respectively. The inhomogeneous system (22 ) admits a solution for given f , f , ..., f if and only
1 2 m
if
m
f j 0 (j = 1, 2, ..., m)
1

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