Fixed Point Iteration

Theorem (First Fixed Point Theorem). Assume that , i. e.is continuous on.
Then we have the following conclusions.
(i).If the range of the mapping satisfies for all , then has a fixed point in .
(ii).Furthermore, suppose that is defined over and that a positive constant exists with
for all,then has a unique fixed point in .

Theorem (Second Fixed Point Theorem). Assume that the following hypothesis hold true.
(a) is a fixed point of a function ,
(b),
(c) is a positive constant,
(d), and
(e)for all.
Then we have the following conclusions.
(i).If for all,then the iterationwill converge to the
unique fixed point .In this case, is said to be an attractive fixed point.
(ii).If for all,then the iterationwill not converge to .
In this case, is said to be a repelling fixed point and the iteration exhibits local divergence.

Remark 1.It is assumed that in statement (ii).

Remark 2.Because is continuous on an interval containing , it is permissible to use the simpler criterion and in (i) and (ii), respectively.

Corollary. Assume that satisfies hypothesis (a)-(e)of the previous theorem.Bounds for the error involved when usingto approximateare given by
for,
and
for.

Graphical Interpretation of Fixed-point Iteration

Since we seek a fixed point to ,it is necessary that the graph of the curveand the lineintersect at the point .
The following animations illustrate two types iteration: monotone and oscillating.