Operator Equation and Application of Variation Iterative Method ()
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 
If
is 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
(2.2)
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
(2.3)
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
(2.4)
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
(2.5)
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
(3.1)
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
(3.4)
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
(4.1)
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
(4.2)
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
(5.1)
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
(5.2)
where from that 
by iterative method:
Clearly, 
is evident exact solution of (5.2).
6. Some Notes for Schrodinger Equations
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
(6.1)
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:
(6.3)
. (6.4)
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
(6.5)
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
(6.61)
(6.62)
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.
Figure 1. The φ(x) is the space form of wave function φ(x.t) for (6.3) under action of shake V(x) = 0.5x2, by φ0, φ1, ···, φ4 express for 0-level, 1-level,···, 4-level wave function respectively.
Figure 2. The ϕ(x) is the space form of wave function ϕ(x,t) for (6.3) under action of shake power V(x) = 0.5x2, by φ0, φ1, ···, φ4 express for 0-level, 1-level,···, 4-level wave function respectively.
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
Figure 3. The φ(x) is numerical solution by action of (6.3) under the shake power V(x) = 0.5x2 and in boundary value condition φ(–2) = φ(2) = 0.7, the φ3(x) express 3-level (here step length = 0.04, the energy En = ((nπ)2, n = 3).
Figure 4. The φ(x) is numerical solution by action of (6.3) under the shake power V(x) = 0.5x2 and in boundary value condition φ(–2) = φ(2) = 0.7, the φ3(x) express 3-level (here step length=0.04, the energy En = ((nπ)2, n = 3).
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.