Some Approximation in Cone Metric Space and Variational Iterative Method ()
1. Introduction
Fixed point theory has fascinated many researchers since 1922 with the celebrated Banach’s fixed point theorem. There exist a fast literature on the topic and this is a very active field of research at present (See [1-7]).
By using same definition and meaning in stating is also looking in [2] and [3] etc. we introducing the following results for needing. For convenience, the authors give the following definition and lemma (see the proof of theorem 3 [2])
Let
always be a real Banach space, and
be the subset of
is called a cone, if and only if:
(i)
is closed, nonempty, and 
(ii) 
(iii) 
Given a cone
we define a partial ordering
with respect to
by
if and only if 
We shall write
to indicate
but
while
implies
int
denotes the interior of
.
Definition 1.1 Let
be a non-empty set in
, and suppose the mapping
satisfies:
(i)
for any
and 
(ii)
for all 
(iii)
for all 
Then
is cone distance on
is called a cone metric space.
It is obvious that cone metric space generalize the metric spaces.
Definition 1.2 Let
is said to be a complete cone metric space, if every Cauchy sequence is convergent in 
Let
be a metric space. We denote by
the family of non-empty closed bounded subset of
Let
be the Hausdorff metric on
. That is for

where
is the distance from point
to the sub-set
An element
is said to be a fixed point of a multi-valued mapping 
Lemma 1.3 Suppose that
be a cone metric space and the mapping
hold the sequence
in
satisfying the following conditions:

and that is

Then the sequence
is a Cauchy sequence in
.
2. Several Common Fixed Point Theorems
In recent years, the fixed point theory and application has rapidly development. Huang and Zhang [2] introduced the concept of a cone metric space that a element in Banach space equipped with a cone which induces a nature order partial order. In the same work, they investigated the convergence and that inequality and they extend the contractive principle to in partial order set s with some applications to matrix equations and common solution of integral equations.
Such theorems are very important tools for proving the existence and eventually the uniqueness of the solutions to various mathematical models (integral and partial differential equations, variations inequalities etc.).
First, we state following some extend conclusion ([3, 4]). Next, authors consider the variation iterative method to some integral and differential equations, and effective method ([5-8]) for examples and numerical test as some Fig case.
Now we first give common fixed point Theorem in similar method for two operators to extend Theorem 2.1 [2] with one operator case. Assume that
be a complete cone metric space.
Theorem 2.1 Let
be a complete cone metric space,
a normal cone with normal constant
Suppose that mappings
satisfies the Contractive condition
(2.1)
for each
where
is a constant
Then
has a unique common fixed point in
And so for any
the iterative sequence
converges to the fixed point.
Corollary 2.2 Let
and
then we obtain that theorem 2.1 in [2].
In the same way, authors can extend theorem 4 [2], and omit again these stating.
3. Some Notes of Common Fixed Point
Theorem A (See Theorem 1 [3]) Assume that
be a complete cone metric space. Let mappings
satisfying following Lipchitz conditions for any 
(3.0)
where
are nonnegative real value functions on
such that

and that

Then there is a unique common fixed point in
for
and
and for any
the iterative sequence
convergent to the common fixed point of 

Remark in [3], the example 2 illustrate this effect of meanings with this Theorem (non-expansion mapping, not contractive case that have uniqueness common fixed point). Look for multiple-value mappings in some case [5].
We can easy note Theorem A. Now we give complete this fixed point problem below.
Theorem 3.1 Same as the assume of theorem 1 [3]. Let
be a complete cone metric space, and there exists positive integer
and mappings
satisfying following Lipchitz conditions for any
that is satisfy following inequality such that
(3.1)
where
are non-negative real functions on
If it holds, 
Then there is a unique common fixed point in
for
.
Proof By the proof of theorem 1 [3] that we known that
and
has a unique common fixed point
and
from

then
is also common fixed point of
and
Hence, we have
by the uniqueness of them. In the same way, we know that
that is,
This
is a common fixed point of
, and

We have 
On the other hand, if
then clearly,
in the same way, we know
which a contradiction. So, we complete the proof of this theorem.
Theorem 3.2 Let
be a complete cone metric space, and mappings
satisfying following Lipchitz conditions for any
that is satisfy following inequality (
nonnegative real constant)
(3.2)
and

If
have not common fixed point each other, then the
exist at least
number fixed points in 
Proof By the theorem 1 [3], we known that
and
has a unique common fixed point
that is
and that in the same way,

Then S have at least
number fixed points.
Corollary 3.3 Let
more positive integer cases in theorem 3.1.
Remark 3.4 (see corollary 5 [3]) Assume that
be complete cone metric space, and mappings
satisfy following condition (
non-negative real constant):
(3.3)
And

then
must have uniqueness common fixed point, and for any
iterative sequence
convergent to the common fixed point

Here example 2 in [5] for non-expansion mappings, also have common fixed point case with important meanings.
4. Common Fixed Point of Four Mappings
Many authors have extended the contraction mapping principle in difference direction. Some extension of Banach’s fixed point theorem through the rational expression form with it’s inequality. The purpose of this section, is to establish some common fixed point theorem for four mappings in this space.
Theorem 4.1 Assume that
be a complete cone metric space, and mappings
continuous and satisfying following conditions for any

and that inequality
(4.1)
Then
have a unique common fixed point.
Proof Let
be an arbitrary point of
and from
we can choose a point
such that
Also
we can choose a point
such that
In general, we can Choose
and
define a sequence
in
as follows,

Now, by (4.1) we have that

Similarly, we have

which implies that

Hence, it is well know that
is Cauchy sequence. From the
is complete, then there exists
such that the
convergences to
in
Since
and
are subsequences of
then it will convergence to same point u.
Next, from these
are continuous maps, we can obtain following results

It follows from
then we have
and we get
In the same reason,

we again obtain
By (4.1), if
then

this is a contradiction. Hence, we have that

Let
then we obtain that

and

By (4.1), if
then

we get 
It is a contradiction. Thus, 
Similarly,

we get 
It is also a contradiction. Therefore, 
By (4.1), if
then

also a contradiction. Thus, we have

then this implies
for common fixed point of
.
(Uniqueness) Let two points
be the difference common fixed point of
.
By (4.1), we have

A contradiction in the above same reason, then this implies the uniqueness of common fixed point of
.
Remarks
(i) As
we obtain special case.
(ii)
(Identity map),
we get some special case.
Theorem 4.2 Assume that
be a complete cone metric space. Same as theorem 4.1, and satisfies these conditions below


for any
(4.2)
Then
have a unique common fixed point.
Proof By theorem 4.1, we known that there is a unique fixed point
in
and 
Obviously,
This completes the proof of theorem 4.2.
5. Some Notes for Multi-Valued Mappings
According the direction of [5], we give out some coincidence point theorem of maps to extend theorem 2.1 and theorem 2.3 [5].
Let
be a strictly increasing function such that
(i) 
(ii)
for each 
(iii)
for each 
Now, we can easy obtain following theorems.
Theorem 5.1 Let
be a complete metric space and
be multi-valued maps
satisfying for each 

where
(5.1)
If
have not common fixed point each other, then there exist at least
number fixed points in 
Proof From theorem 2.1 [5], we known there exist
in
such that
again for
in 
The same way,

Since
not equality each other, then S have at least
number fixed point. This completes the proof.
Theorem 5.2 Let
be a complete metric space and
be multi-valued maps
and
be a map satisfying
(i) 
(ii)
is complete(iii) there exists a function
such that
for every
(5.2)
And for each 
(5.3)
If
and
have not coincidence point each other, then the
and
at least exist
number coincidence points in 
Proof From theorem 2.3 [5], we known there exist coincidence point
in
such that
in the same way, the coincidence point
in
with

Since

not equality each other, therefore
and
have at least
number coincidence point. That is,

Then this completes the proof.
Corollary 5.3 Let
be a complete metric space and
be multi-valued maps
and
be a map satisfying
(i) 
(ii)
is complete(iii)
for each
where
such that
for every 
If
have not coincidence point each other, then 
If we take
we easy get these conclusion below, where
is the identity map on 
Corollary 5.4 Let
be a complete metric space and
be multi-valued maps
and satisfying for each

where
such that
for every 
If
have not common fixed point each other, then the
and
exist at least
number points in
That is case

Corollary 5.5 When
, for each
Then we get also similar conclusion case: 
6. Solution of Integral Equation by VIM
Recently, the variation iteration method (VIM) has been favorably applied to some various kinds of nonlinear problems, for example, fractional differential equations, nonlinear differential equations, nonlinear thermo elasticity elasticity, nonlinear wave equations.
In this section, we apply the variation iteration method (simple writing VIM) to Integral-differential equations below (see [6,7]). To illustrate the basic idea of the method, we consider:

The basic character of the method is to construct functional for the system, which reads:

which can be identified optimally via variation theory,
is the nth approximate solution, and
denotes a restricted variation, i.e.
There is a iterative formula:

of this equation
(*)
Theorem 6.1 (see theorem 3.1 [6]) Consider the iteration scheme
and
. (6.1)
Now, for
to construct a sequence of Successive iterations that for the
for solution of integral Equation
.
In addition, we assume that

and
then if
the above iteration converges in the norm of
to the solution of integral equation
.
Corollary 6.2 If
and

then assume
if
the above iteration converges in the norm of
to the solution of integral equation
.
Corollary 6.3 If
and

then assume
if
the above iteration converges in the norm of
to the solution of integral Equation
.
Example 6.4 Consider that integral equation
(6.2)
where

then if
(See Figure 1(a)) then the iterative
that convergent the solution of Equation (6.2) by corollary 6.2 of theorem 6.1. Therefore, we check that

From (6.1), we have that

Let

and

(See Figure 1(b), when n = 4).
The exact solution
Then we obtain exact solution below

Remark The exact solution
and approximate solution
of Example 6.4 (See Figure 1(c)).
Example 6.5 We consider that integral equation
(6.3)
From (6.1), we have that

We take

The exact solution
then we obtain exact solution below

By corollary 6.2 of Theorem 6.1, where that

Then the iterative sequence is convergent to the exact solution of the Equation (6.3).
In fact,

then if
the iterative sequence is convergent the solution of Equation (6.3).
Example 6.6 Consider that integral equation (
positive integer),
(6.4)
where
and we have that

We can take that
(k-positive integer), and

Inductively, we have

Then by Theorem 6.1 and simple computation, we obtain that

then if
the iterative is convergent the solution of integral equation (6.4) (Similar as examples case in [6, 7]).
7. Some Effective Modification and Numerical Test for [8]
In this section, we apply the effective modification method of He’s VIM to solve some integral-differential equations. In [6-8] by the variation iteration method (VIM ) simulate the system of this form

To illustrate its basic idea of the method, we consider the following general nonlinear system

the highest derivative and is assumed easily invertible,
is a linear differential operator of order less than,
represents the nonlinear terms, and
is the source term. Applying the inverse operator
to both sides of above equation, we obtain

The variational iteration method (VIM) proposed by Ji-Huan He (see [6-8] recently has been intensively studied by scientists and engineers. the references cited therein) is one of the methods which have received much concern. It is based on the Lagrange multiplier and it merits of simplicity and easy execution. Unlike the traditional numerical methods. Along the direction and technique in [4,8] we may get more examples below.
Example 7.1 (similar as example in [8]) Consider the following nonlinear Fredholm integral equation
(7.1)
Applying the inverse operator
to both side of Equation (7.1), yields:

from 
So,

Inductively,

Then
is exact solution of (7.1). The numerical results are shown in Figure 2.
Example 7.2 Consider the following Volterra-Fredholm integral-differential equation
(7.2)
Similar as example1 in this way, we easy get this solution.
According to the method, we divide
into two parts defined
and

By calculating this

So, we have

In fact, in this way

and

Writing

therefore, we have

Inductively,

And
Hence the
is the exact solution of (7.2).
Example 7.3 Consider the following integral-differential equation
(7.3)
where
In similar example1, we easy have it .
According to the method, we divide
into two parts defined by

Figure 2. Figures of exact solution u(x) for Example 7.1.

Taking
then we have

where 
And the processes,

Thus,
then
is the exact solution of (7.3) by only one iteration leads to a solution. The numerical results are shown in Figure 3.
Example 7.4 (See example 2 in [8]) Consider the following partial differential equation
(7.4)
Let that integer
The modified methods:
Applying the inverse operator to both sides of (7.4) yields

where 
Here, we divide
into two parts defined by

Using the relation
we obtain

and so on

Hence,
is the exact solution of (7.4) and by only one iteration leads to that exact solution. Taking
that is example 2 in [8].
The numerical results are shown in Figure 4.
Remark 7.5 Some solving integral-differential equations by VIM may see [9], and that some random Altman type inequality for fixed point results see [10].
The fixed point results of Multi-value mapping are also discussed in [11].
Remark 7.6 By [12], the authors consider the mixed problem for non-linear Burgers equation:
(7.5)
The authors point out the problem describes physic phenomenon of motive quality and conservation of law in dynamic problem, it is important model in flow mechanics. Where
express the velocity of flow body,

Figure 3. Figures of exact solution u(x) for Example 7.3.
and
express the constant of motive flow body,
-initial function.
Burger’s equation has attracted much attention. The approximation solution for this Burger’s equation is also interesting tasks.
8. Concluding Remarks
In this Letter, we give out new fixed point theorems in cone metric space and apply the variation iteration method to integral-differential equation, and extend some results in [3,6-8]. The obtained solution shows the method is also a very convenient and effective for some various non-linear integral and differential equations, only one iteration leads to exact solutions.
Recently, the impulsive differential delay equation and stochastic schrodinger equation is also a very interesting topic, and may look [11] etc.
9. Acknowledgements
This work is supported by the Natural Science Foundation (No.07ZC053) of Sichuan Education Bureau and the key program of Science and Technology Foundation (No.07zx2110) of Southwest University of Science and Technology.
The authors would like to thank the reviewers for the useful comments and some more better results.