Page 436 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 25: Volterra Equations and L Kernels and Functions
2

where the Kernel K(x, u) and the function f(x) belong to the class L . In the last sections we had Notes
2
seen that the solution is given by the formula
x
u
x
f
x
u
y ( ) f ( ) H ( , , ) ( )du ...(2)
x
0
where the resolvent Kernel H(x, u, ) is given by the series of iterated Kernels
u
x
H ( , , ) v K v 1 ( , ) ...(3)
x
y
v 0
The series (3) converges almost everywhere. H(x, u, ) satisfies the integral equation
x
z
x
x
u
u
x
z
K ( , ) H ( , , ) K ( , ) H ( , , )dz
u
y
x ..(4)
K
z
z
u
H ( , , ) ( , )dz
x
y
Proof: with the help of the Schwarz inequality, we first find
2
x
K 2 2 ( , ) K ( , u 1 ) (u 1 , )du 1
K
x
u
u
x
y
x 2 x 2
)
x
y
K ( , u du 1 k (u 1 , )du 1
1
y y
h h
x
y
x
u
K 2 ( , u 1 )du 1 K 2 (u 1 , ) du 1 A 2 ( ) B 2 ( ),
0 0
and successively
x x
u
K 2 ( , ) K 2 ( , u )du K 2 (u , )du
x
x
u
3 1 1 2 1 1
y y
h x
)
x
u
K 2 ( , u 1 )du 1 A 2 (u B 2 ( )du 1
1
0 y
x
A 2 ( )B 2 ( ) A 2 (u 1 )du 1
u
x
y
x x
x
u
K 2 4 ( , ) K 2 ( , u 1 )du 1 K 3 2 (u 1 , )du 1
x
u
y y
h x x
)
x
u
K 2 ( , u 1 )du 1 A 2 (u B 2 ( )du 1 A 2 (u 2 )du 2
1
0 y u
x x
x
A 2 ( )B 2 ( ) A 2 (u 1 )du 1 A 2 (u 2 )du 2
u
y u
         
In general, we can write
u
u
x
u
x
K 2 ( , ) A 2 ( )B 2 ( ) F ( , ) (n = 1, 2, 3, ...) ...(5)
x
n 2 n
where
x x
F
u
dz
F ( , ) A 2 (u )du ,F ( , ) A 2 (u ) (u , ) ,...
u
u
x
x
1 1 1 2 1 1 1
y u
or generally
x
z
z
u
x
F n ( , ) A 2 ( ) F n 1 ( , ) , (n = 2, 3, ...) ...(6)
u
dz
y
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