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F(x) is then called an iteration function, and it can be used for the generation
of the sequence:

x = F(x ) (5.2)
k
k+1
To guarantee that this method gives accurate results in a specific case, the
function should be continuous and it should satisfy the contraction condition:

Fx( ) − Fx( ) ≤ s x − x (5.3)
n m n m

where 0 ≤ s < 1; that is, the changes in the value of the function are smaller
than the changes in the value of the arguments. To prove that under these
conditions, the iterative function possesses a fixed point (i.e., that ultimately
the difference between two successive iterations can be arbitrarily small) that
can be immediately obtained from the above contraction condition [Eq. (5.3)].
PROOF Let the x guess be the first term in the iteration, then:

Fx() − Fx( ) ≤ s x − x (5.4)
1 guess 1 guess

but since

Fx( ) = x and F x( ) = x (5.5)
guess 1 1 2
then

x − x ≤ s x − x (5.6)
2 1 1 guess

Similarly,

Fx() − Fx( ) ≤ s x − x (5.7)
2 1 2 1

translates into

x − x ≤ s x − x ≤ s x − x (5.8)
2
3 2 2 1 1 guess
The argument can be extended to the (m + 1)-iteration, where we can assert
that:

m
x − x ≤ s x − x (5.9)
m+1 m 1 guess
© 2001 by CRC Press LLC