10.11.2. Theorem 10.7.
We now apply the fixed point theorem to establish an existence theorem for the
integral equation
b
  x     x     dt K  x, t    t 
(10.31)
a
where the givens are the function   C a, b , the constant , and the kernel K.
Furthermore, we assume K to be bounded on the square
S    x, y  : a  x  b, a  y  b 
in such a way that the operator T defined by
b
T   x     x     dt K  x, t    t 
(10.31a)
a
maps C  a, b into C  a, b , i.e., T    C a, b whenever   C  a, b .
Theorem 10.7:
Equations.
An Existence-Uniqueness Theorem For Integral
If, under the foregoing conditions, we have
  x, y   S
0  K  x, y   M
then for each  such that
1
 
M ba
(10.32)
(10.33)
there is one and only one function   C  a, b that satisfies (10.31).
Proof
The theorem is proved if we can show that T is a contraction operator since, by
(10.31a), the fixed point  of T is the unique solution to (10.31).
Consider then the difference
b
T 1  x   T 2  x     dt K  x, t  1  t    2  t  
a
Using the upper bound for K in (10.32), we have
T 1  x   T  2  x    M b  a 1   2   1  2
where    M b  a . By (10.33), we see that 0    1 , so that T is a contraction
operator. QED.