Page 476 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 32: The Fredholm Theorem

b Notes
t
t
s
s
s
g ( ) ( ) K ( , ) ( )dt ...(3)
a
also admits a unique continuous solution (s) for any continuous function g(s). In the second
case the associated homogeneous equation
b
s
( ) K ( , ) ( )dt ...(4)
s
t
t
a
s
s
s
s
admits a number r of linearly independent continuous solutions 1 ( ), 2 ( ), 3 ( )... r ( ). In
the second case, the equation (1) admits a solution if and only if
b
s
s
t ( ) ( )ds 0 (i = 1, 2, ...r) ...(5)
i
a
If condition (5) is satisfied, the general solution of (1) is written as
r
s
s
s
Q ( ) Q (1) ( ) C Q j ( ) ...(6)
j
j 1
(1)
by means of a particular solution Q (s) of (1) and r arbitrary constants C , C , ..., C . Similarly, the
1 2 r
conjugate equation (3) admits a solution if and only if
b
s
g ( )Q j ( ) 0 ( j 1, 2, 3.... ) ...(7)
s
r
a
If condition (7) is satisfied, the general solution of (3) is written as
r
s
s
( ) (1) ( ) C ;( )
s
j
j 1
by means of a particular solution (1) (s) of (3) and r arbitrary constants C , C , ...C .
1 2 r
The theorem also shows that the unique solution of (1) exists for any continuous function f(x) if
and only if is not an eigenvalue.
The proof of the Fredholm’s alternative theorem is given in two parts for continuous Kernel
K(s, t) on the domain a s b, a t b. We shall start proving the theorem by Schmidt’s method
instead of L -class method. Of course both the methods had to the same conclusion.
2
32.2 Proof of Fredholm Theorem
b b
2
t
K
s
The case when | ( , )| ds dt 1
a a
For the sake of simplicity, we take = 1 and consider the equation
b
s
s
( ) K ( , ) ( )dt f ( ) ...(1)
t
s
t
a
An equation in the unknown (t), of the form
b
t
( ) K ( , ) ( )ds g ( ) ...(2)
s
t
t
s
a
g(t) being a given continuous function on the interval a t b, is called the conjugate equation
of (1).
LOVELY PROFESSIONAL UNIVERSITY 469