Some Approximation in Cone Metric Space and Variational Iterative Method ()

Ning Chen, Jiqian Chen

School of Science, Southwest University of Science and Technology, Mianyang, China.

**DOI: **10.4236/am.2012.312276
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School of Science, Southwest University of Science and Technology, Mianyang, China.

In this paper, we give some new results of common fixed point theorems and coincidence point case for some iterative method. By using of variation iteration method and an effective modification of He’s variation iteration method discusses some integral and differential equations, we give out some new conclusion and more new examples.

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Chen, N. and Chen, J. (2012) Some Approximation in Cone Metric Space and Variational Iterative Method. *Applied Mathematics*, **3**, 2007-2018. doi: 10.4236/am.2012.312276.

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 conewe define a partial ordering with respect to by if and only if

We shall write to indicate but while implies intdenotes 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 pointsuch 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

(a)(b)(c)

Figure 1. By Matlab in numerable test. (a) The figure expresses exact solution u(x) for Example 6.4; (b) The figure expresses approximate solution u_{4}(x) for the solution u(x) of Example 6.4; (c) The figure expresses absolute error for Example 6.4.

(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 divideinto two parts definedand

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 divideinto 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.

Figure 4. Figures of exact solution u(x) for example 7.4. (where parameters taking as).

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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