The aim of this method is to solve equations of type:
Let
be the solution of (E).
The idea is to bring back to equation of type:
where
is a fixed point of
.
We introduce a convergent sequence
to the fixed point
of
, which is the solution of equation (E).
{{{Fixed Point Theorem}}}
{{Existence.}}
If
and
, then g has a fixed point
in [a, b].
{{Unicity.}}
If
and if exists a constant
in ]0,1[ such that
over [a, b] then:
 the fixed point
is unique
 the iteration
will converge to the unique fixed point
of
.
{{Proof of the existence.}}
We define
over [a,b], like follows:
Clearly:
and
because
. Now we apply the intermediate value theorem to
:
thus:
{{Proof of the unicity.}}
We suppose now that there exists two fixed points
with
. We apply the mean value theorem : there exists
with:
we get
This is a contradiction! Finally
The sequence
is well defined because
and consequently
provided that
. As previously , we apply the mean value theorem, we show the existence of
for all
with
Finally:
since
est dans ]0,1[.
{{Corollary}}


{{Proof of the Corollary.}}
The first inequality is obvious. To prove the second inequality,we use
this one:
Let
:
when m goes to infinity:
{{{Order and Rate of convergence of a sequence}}}
We suppose that
converges to
:
where
is the error between
and
.
If exists two constants
and
such that:
We say that the sequence
converges with order p
with a rate of convergence
.
{{In Particular...}}
If p=1 and C<1 we say that the sequence converges linearly .
If p=2, convergence is called quadratic convergence.
If p=3, convergence is called cubic convergence.
{{{Order of convergence of the fixed point method}}}
Obviously several cases are possible, we can build several functions
and that also depends on the nature of
.
If
since
, the rate of convergence
and the convergence is linear(order 1) and
since
on [a, b].
If
, we must introduce a Taylor Developement of
around
. Don't forget that
converges to 0. For example,
a Taylor developement of order 3 gives:
with
, thus
and
The rate of convergence
and the convergence is quadratic(order 2). We can generalize this result with the following therorem.
{{Theorem.}}
if
then the order of convergence of the fixed point method is k.
{{Proof.}}
Taylor developement of order k around
of
gives:
with
, d'oà¹
and
since