Page 477 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Theorem 1: Under the assumption
b b 2
t
K
| ( , )| ds dt 1 ...(3)
s
a a
the equation (1) [(2)] admits one and only one solution (s)[ (t)] for any f(s)[g(t)]; in particular
(s) 0[ (t) 0] for the homogeneous equation
b
( ) K ( , ) ( )dt 0 ...(4)
s
s
t
t
a
b
( ) K ( , ) ( )ds 0 ...(5)
s
t
s
t
a
(1)
(2)
Proof: Starting with the Kernel K(s, t), we define the iterated Kernels K (s, t), K (s, t), ...., K (n)
(s, t), ... as follows:
K (1) ( , ) K ( , )
s
s
t
t
b
t
r
s
K
r
K (2) ( , ) K ( , ) ( , )dr
s
t
a
   ...(6)
b
n
K ( ) ( , ) K ( , )K (n 1) ( , )dr
t
r
t
r
s
s
a
The following relation clearly holds for the iterated Kernels
b ( ) ( )
(n
m
m
n
)
t
s
K ( , ) K ( , )K ( , )dr ...(7)
t
r
s
r
a
By (6) and the Schwarz inequality, we have
b b (n 1)
n
( )
2
2
r
s
|K ( , )| 2 | ( , )| dr |K ( , )| dr
t
r
K
t
s
a a
hence
b b ( )
n
2
s
|K ( , )| ds dt
t
a a
b b b b (n 1)
2
2
K
r
| ( , )| ds dt |K ( , )| dr dt
t
r
s
a a a a
Repeating this procedure, we finally obtain
n
b b ( ) b b
n
2
2
t
t
|K ( , )| ds dt | ( , )| ds dt ...(8)
K
s
s
a a a a
On the other hand, according to (6) and (7), we see that for n 3.
b b
n
r
r
K
t
r
K ( ) ( , ) K ( , )K (n 2) ( , ) ( , )dr dr 1
s
s
t
r
1
1
a a
Hence by the Schwarz inequality we have
b b b b
2
n
2
t
s
K
r
r
K
r
|K ( ) ( , )| 2 |K (n 2) ( , )| dr dr 1 | ( , ) ( , )| dr dr 1
s
t
r
1
1
a a a a
Accordingly, by making use of (8), we obtain
n 2
b b b b
2
2
n
2
r
t
K
t
s
K
|K ( ) ( , )| 2 | ( , )| ds dt | ( , )| dr | ( , )| dr 1
t
s
r
s
K
1
a a a a
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