Operator Equation and Application of Variation Iterative Method

In this paper, we study some semi-closed 1-set-contractive operators A and investigate the boundary conditions under which the topological degrees of 1-set contractive fields, deg (I-A, Ω, p) are equal to 1. Correspondingly, we can obtain some new fixed point theorems for 1-set-contractive operators which extend and improve many famous theorems such as the Leray-Schauder theorem, and operator equation, etc. Lemma 2.1 generalizes the famous theorem. The calculation of topological degrees and index are important things, which combine the existence of solution of for integration and differential equation and or approximation by iteration technique. So, we apply the effective modification of He’s variation iteration method to solve some nonlinear and linear equations are proceed to examine some a class of integral-differential equations, to illustrate the effectiveness and convenience of this method.

• Topology Degrees and Index; 1-Set-Contract Operators; Modified Variation Iteration Method; Integral-Differential Equation

1. Introduction

In recent years, the fixed point theory and application has rapidly development.

That topological degree theory and fixed point index theory play an important role in the study of fixed points for various classes of nonlinear operators in Banach spaces (see [1-6]). We begin recall theorem A and lemma 1.1 [3]. Then, several new fixed point theorems are obtained in Section 2, and the common solutions of the system of operator equations in Section 3. We also extend some examples for search solution of integral equation and integral-differential equation in Section 4 and Section 5 by variation iterative method. In last part, we compare some figures, by numerical test and note that simple case of Schrodinger equation. The main results are Theorem 2.2, Theorem 3.4-3.5, Example 3, Example 6, etc.

2. Several Fixed Point Theorems

Let be a real Banach space, a bounded open subset of and the zero element of

If is a completely continuous operator, we have some well known theorems as follows (see [3,4]).

First, we need following some definitions and conclusion (see [3]). For convenience, we first recall theorem A.

Theorem A (see Theorem 1.1 in [3]) Suppose that has no fixed point on and one of the following conditions is satisfied1) (Leray-Schauder), for and

2) (Rothe), for all

3) (Petryshyn) Let, for all

4) (Altman), for all then, and hence has at least one fixed point in.

Lemma 2.1 (see Corollary 2.1 [3]) Let be a real Banach Space, is a bounded open subset of and

Ifis a semi-closed 1-set-contractive operator such that satisfies the L-S boundary condition

for all and      (2.1)

then, and so has a fixed point in

Remark This lemma 2.1 generalizes the famous L-S theorem to the case of semi-closed 1-set-contractive operators.

First, we state following some extend conclusion (see theorem [5]).

Theorem 2.2 Let be the same as in lemma 2.1. Moreover, if there exists, - positive integer such that

Then if has no fixed points on and so has a fixed point in.

Proof. By lemma 2.1, we can prove theorem 2.2. Suppose that has no fixed point on.

Then assume it is not true, there exists such that. It is easy to see that

Now, consider the function defined by

for any

Since

and by formal differential, is a strictly increasing function in and so for. Thus

Consequently, noting that , we have

which contradicts (2.2), and so the condition is satisfied. Therefore, it follows from lemma 2.1 that the conclusions of theorem 2.2 hold.

Theorem 2.3 Let be the same as in lemma 2.1. Moreover, if there exists, positive integer such that

Then if has no fixed points on and so has a fixed point in.

Proof. Similar proof of that theorem 2.2.

Now, we consider the function defined by

for any and.

So,

is a strictly increasing function in and for. We have

for any.

Consequently, noting that , we have

which contradicts (2.3). Therefore, it follows from lemma 2.1 that the conclusions of theorem 2.3 hold.

Corollary 2.4 If

then (2.3) holds. By theorem 2.3, has a fixed point in.

We get easy theorem 2.5 in bellow. So, extend (vi) of theorem 2.6 in [3], omit the similar proof.

Theorem 2.5 Let be the same as in lemma 2.1. Moreover, if there exists and - positive integer such that

Then, if has no fixed points on and so has at least one fixed point in. (Let that is theorem 2.4 in [5]).

3. Operator Equations

We will extend Lemma 2 and Theorem 2, adopt same notation and method in [7] in following form.

Let be a real Banach space, and -positive integer.

Lemma 3.1 When the following holds:

Proof. Let, similar the proof of lemma 2 in [7], we easy get In fact, by derivative of it, we have

Since

We obtain that

that is,

Thus, Therefore, is a strictly monotone increasing function in.When we have and that is.

Hence,

where We complete the proof of this lemma 3.1.

Theorem 3.2 Let be a bounded open convex subset in and Suppose that is a semi-closed 1-set-contradictive operator, and m, n-positive integer such that

for every

Then the operator equation has solution in.

Proof. By (3.1), we know that has no solution in, that is, for every We shall prove

for every for every   (3.2)

In fact, suppose that (3.2) is not true that is there exists a and an such that that is.

By (3.1), we obtain

for every

This is because hence then we have

Let as we have

That is then this is a contradiction to Lemma 3.1.

Thus,

for every for every  (3.3)

From (3.2) and (3.3), we know that By Ref [6], we obtain that Then this operator equation has a solution in

Theorem 3.4 Let be a bounded open convex subset in and Suppose that are semi-closed.

1-set-contradictive operator, and m, n-positive integer such that

Then the operator equation has common solution in (omit the proof of this theorem).

Theorem 3.5 Let Same as assume theorem 3.1. Suppose that are semi-closed 1-set-contradictive operator, and m, n-positive integer, substitute (3.5) for inequality bellow

Then the operator equation has common solution in (omit this proof).

4. Solution of Integral Equation

Recently, the variational iteration method (VIM) has been favorably applied to some various kinds of nonlinear problems, for example, fractional differential equations, nonlinear differential equations, nonlinear thermoelasticity, nonlinear wave equations.

In this section, we apply the variation iteration method (simple writing VIM) to Integral equations bellow (see [8,9]). 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 4.1 (see theorem 3.1 [8]) Consider the iteration scheme and

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 4.2 If and

then assume if the above iteration converges in the norm of to the solution of integral equation (*).

Example 1 Consider that integral equation

where, and

From that

We have

From theorem 4.1 and simple computation, we obtain again that

and by theorem 4.1 if then iterative

is convergent.

Then inductively, we have

The solution of integral Equation (4.1) by calculating as follows.

Example 2 We consider that integral equation

From (*), we have that

In fact,

and by Corollary 4.2, then if iterative sequence is convergent the solution of Equation (4.2).

5. Some Effective Modification

In this section, we apply the effective modification method of He’s VIM to solve some integral-differential equations.

In [10] 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 Equation (1), and we obtain

The variation iteration method (VIM) proposed by Ji-Huan He (see [5,10] has recently 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 [5], we may get more examples bellow.

Example 3 Consider the following integral-differential equation

where In similar example1, we easy have it.

According to the method, we divide into two parts defined by

Taking, then we have

where and the processes:

Thus, then is the exact solution of (5.1) by only one iteration leads to a solution.

Example 4 (similar example 3 in [5]) Consider the following nonlinear Fredholm integral equation

where from that

by iterative method:

Clearly, is evident exact solution of (5.2).

6. Some Notes for Schrodinger Equations

The quantum mechanics theory and application in more field are widely important meaning.

Along the direction and technique in [11] and [12], we may get more examples.

As we all know the solution of initial problem for Schrodinger equation bellow

Assume that real part and imaginary part of

are real analytical function for then this solution of the problem may express in form:

Now, the authors consider again one-dimension Schrodinger equation as application form:

where look in (6.3), that be the part in space for wave function, the in (6.4) be the potential function be arrange plank constant, be the practical mass, express energy.

The Equation (6.3) for with extensive equation, by calculating and search the general solution that

So, by (6.3) and with power of (6.4), we consider that two case:

1) (see [13,14]) The infinite deep power trap

2) The shake Power

We take parameters

Then

Furthermore, from (6.5), we obtain analytic solution for and So, we have that

See Figures 1 and 2 below.

Therefore, by using of mathematical software with Matlab (see [14]), we may proceed numerical imitate, to get approximate solution, see Figures 3 and 4.

In fact, according to the finite difference principle, a one-dimensional Schrodinger equation can be converted into a set of nodal liner equations expressed in a matrix equation after the space is divided into a series of discrete nodes with an equal interval. The matrix left division command offered in the MATLAB software can be used to derive the function approximation of each unknown nodal function.

7. Concluding Remarks

In this Letter, we consider operator equations and apply

the variation iteration method to integral-differential equations, and extend some results in [3,8,10]. The obtained solution shows the method is also a very convenient and effective for various integral-differential equations, only one iteration leads to exact solutions. Recently, the impulsive differential delay equations is also a very interesting topic, and we may see [10] etc.

In our future work, we may try to do some research in this field and may be could obtain some better results.

8. Acknowledgements

This work is supported by the Natural Science Foundation (No. 11ZB192) of Sichuan Education Bureau and the key program of Science and Technology Foundation (No. 11ZD1007) of Southwest University of Science and Technology.

The author thanks the Editor kindest suggestions, and thanks the referee for his comments.

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