Page 435 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes exist and are finite in the Lebesgue sense while N is finite. Such a Kernel as well as the function
will be called L Kernel and L2-function, respectively.
2
The consequences of the Kernel being L -Kernel are many. One of them is as follows:
2
The functions

h 1 2 h 1 2
A ( ) K 2 ( , )du , ( ) K 2 ( , )dx ...(4)
u
x
x
B
u
x
u
0 0
exist almost everywhere for 0 x h an 0 u h respectively. Also A(x), B(u) belong to L class.
2
and finally that
h h
2 2 2
K A ( )dx B ( )du ...(5)
x
u
0 0
Secondly, if (x) is any L -function in (0, h) then the two functions
2
h h
x
x
u
u
x
u
x
( ) K ( , ) ( )du , ( ) K ( , ) ( )dx ...(6)
u
0 0
are also L -functions. This is an immediate consequence of the Schwarz inequality
2
b 2 b b
g
x
x
x
f ( ) ( )dx f 2 ( )dx g 2 ( ) .
dx
x
a a a
From (6) it follows that
K , K ...(7)
In the same way, it is easy to show that the composition of two L Kernels K(x, u) and H(u, t) i.e.
2
the formation of two new Kernels
h
x
G 1 ( , ) K ( , u H (u u )du 1
x
u
)
1
1
0
h ...(8)
K
x
G 2 ( , ) H ( , u 1 ) (u u )du 1
x
u
1
0
yields two new L -Kernels, such that
2
G 1 K H , G 2 H K ...(9)
and so on. In fact this last formula give us useful bounds for the norms of the iterated Kernels
K n K n ...(10)
Self Assessment
1. Show that the nth iterated Kernel K (x, u) satisfies the bound
n
K n K n
25.4 Solution of Volterra Integral Equation of Second Kind
In the section we want to prove the existence and uniqueness of the solution of the Volterra
integral equation of the second kind
x
u
x
y ( ) K ( , ) ( )du f ( ) (0 x h) ...(1)
y
x
u
x
0
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