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 ![](https://www.scirp.org/html/6-7400891\e7bfd237-7525-4850-8d33-3a798281f336.jpg)
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 ![](https://www.scirp.org/html/6-7400891\793043cc-1101-4a13-b9c3-f02adc2cc83a.jpg)
2) (Rothe)
, for all ![](https://www.scirp.org/html/6-7400891\b4991809-1b55-4a4f-b8cf-bf92b2f713a2.jpg)
3) (Petryshyn) Let
, for all ![](https://www.scirp.org/html/6-7400891\00956450-5156-4dfa-a068-ec83fafc0d05.jpg)
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 ![](https://www.scirp.org/html/6-7400891\be494bfd-56f9-48f9-bf57-72cbcd64f102.jpg)
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 ![](https://www.scirp.org/html/6-7400891\956cd7c3-05c0-46ea-b595-69ceef6fc9f3.jpg)
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 ![](https://www.scirp.org/html/6-7400891\c3af0784-21b6-4b7e-990b-66d1e5c5a431.jpg)
Now, consider the function defined by
![](https://www.scirp.org/html/6-7400891\5a18b506-f1c7-45ae-be55-131931ff4b57.jpg)
for any ![](https://www.scirp.org/html/6-7400891\0f8be297-c6b1-4cbf-ae37-63cb34a0c9ee.jpg)
Since
![](https://www.scirp.org/html/6-7400891\89be14e1-2aa6-421b-958c-f1b80ee87496.jpg)
and by formal differential,
is a strictly increasing function in
and so
for
. Thus
![](https://www.scirp.org/html/6-7400891\6b0d39d5-e772-4209-a6c5-4b26552cba6b.jpg)
Consequently, noting that
, we have
![](https://www.scirp.org/html/6-7400891\0688dc1e-bb96-4290-bdbc-e139eea5c767.jpg)
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
![](https://www.scirp.org/html/6-7400891\d23db67c-0a01-4d46-a1a9-40cd49d5926a.jpg)
for any
and
.
So,
is a strictly increasing function in
and
for
. We have
![](https://www.scirp.org/html/6-7400891\26f04ad8-ab3f-4122-a12e-d022dfc38230.jpg)
for any
.
Consequently, noting that
, we have
![](https://www.scirp.org/html/6-7400891\42284525-698b-4315-87bb-d442c9de39fe.jpg)
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:
![](https://www.scirp.org/html/6-7400891\8080a98e-5634-4e06-b64a-6104be58670c.jpg)
Proof. Let
, similar the proof of lemma 2 in [7], we easy get
In fact, by derivative of it, we have
![](https://www.scirp.org/html/6-7400891\02ee9c2b-d9fb-4f72-83fe-044cb1366756.jpg)
Since
![](https://www.scirp.org/html/6-7400891\d15e28c8-60a5-4ef7-b490-998ff7376097.jpg)
We obtain that
![](https://www.scirp.org/html/6-7400891\2a62d76a-f613-4ff3-9208-340ebd835035.jpg)
that is,
![](https://www.scirp.org/html/6-7400891\a602b4a1-8137-4391-9e61-fe9c566fa869.jpg)
Thus,
Therefore,
is a strictly monotone increasing function in
.When
we have
and
that is
.
Hence,
![](https://www.scirp.org/html/6-7400891\68ce3274-3327-4a3f-99a7-3bbc3dbf93c5.jpg)
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
![](https://www.scirp.org/html/6-7400891\110846c2-8a79-473c-a839-bf322d8762bb.jpg)
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
![](https://www.scirp.org/html/6-7400891\9d374bd5-6ec8-4247-99df-2f71e477d141.jpg)
for every ![](https://www.scirp.org/html/6-7400891\91d2e150-74bc-45a7-8136-69014c589de1.jpg)
This is because
hence
then we have ![](https://www.scirp.org/html/6-7400891\1d4c037a-b62c-442d-ace5-ad0576c93404.jpg)
Let
as
we have ![](https://www.scirp.org/html/6-7400891\11d99ef9-b6c9-48c0-9f59-958a1b0e6756.jpg)
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 ![](https://www.scirp.org/html/6-7400891\57d66d59-65fa-4a46-b5bb-2bed3c116eb4.jpg)
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
![](https://www.scirp.org/html/6-7400891\8581a2aa-edcc-4b70-8214-4f5191157843.jpg)
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:
![](https://www.scirp.org/html/6-7400891\66d6f05e-810a-4997-9d01-9b2971b18a2d.jpg)
The basic character of the method is to construct functional for the system, which reads:
![](https://www.scirp.org/html/6-7400891\e12aa785-c280-4495-a2ca-facc5be961ba.jpg)
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:
![](https://www.scirp.org/html/6-7400891\d1affa2f-f788-469d-8a1e-84f6bc6ab132.jpg)
of this equation
(*)
Theorem 4.1 (see theorem 3.1 [8]) Consider the iteration scheme
and
![](https://www.scirp.org/html/6-7400891\b8e35409-2900-46eb-826e-4831f432e19c.jpg)
Now, for
to construct a sequence of successive iterations that for the
for solution of integral equation (*).
In addition, we assume that
![](https://www.scirp.org/html/6-7400891\9014e1a8-e3e3-42d9-a078-5f34eb19494a.jpg)
and
then if
the above iteration converges in the norm of
to the solution of integral equation (*).
Corollary 4.2 If
and
![](https://www.scirp.org/html/6-7400891\6c2d1b2f-ad22-4423-8e87-b4c3afd874ca.jpg)
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
![](https://www.scirp.org/html/6-7400891\b7833bdb-0889-4590-ae7d-61c06afc5148.jpg)
From that
![](https://www.scirp.org/html/6-7400891\5c064050-0689-473a-a964-7d626f162f97.jpg)
We have ![](https://www.scirp.org/html/6-7400891\b3840083-4263-41e9-867c-aa2125d204d0.jpg)
![](https://www.scirp.org/html/6-7400891\31208e6b-b04a-4da1-889c-1d76029d83df.jpg)
![](https://www.scirp.org/html/6-7400891\0c42ba32-3b93-4f9d-bc92-37ce882be935.jpg)
From theorem 4.1 and simple computation, we obtain again that
![](https://www.scirp.org/html/6-7400891\9f65757d-d8cb-475e-8f20-1f8a5583da44.jpg)
and by theorem 4.1 if
then iterative
![](https://www.scirp.org/html/6-7400891\a3bfbba8-db97-4e5e-ac5a-0fc0e1d0b6d2.jpg)
is convergent.
Then inductively, we have
![](https://www.scirp.org/html/6-7400891\9913aa58-9619-44cf-9edd-72fca13ed612.jpg)
The solution of integral Equation (4.1) by calculating as follows.
![](https://www.scirp.org/html/6-7400891\9355001a-5bd5-43f1-8581-8b84c2841cdc.jpg)
Example 2 We consider that integral equation
(4.2)
![](https://www.scirp.org/html/6-7400891\5ceda64c-0db7-4c1f-9dca-30be73612e14.jpg)
From (*), we have that
![](https://www.scirp.org/html/6-7400891\2e242b32-18f6-42cd-b8cf-97bbe754a88f.jpg)
In fact,
![](https://www.scirp.org/html/6-7400891\e9b0ccc1-b0cb-451f-a9e5-4048a21414cf.jpg)
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
![](https://www.scirp.org/html/6-7400891\dbd29eaf-08ed-4798-a1d4-c4b79ab217c5.jpg)
To illustrate its basic idea of the method .we consider the following general nonlinear system
![](https://www.scirp.org/html/6-7400891\f4f32543-af14-479f-b7f3-d2c72d01121f.jpg)
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
![](https://www.scirp.org/html/6-7400891\60708533-8071-4e60-8461-cfc547f1c851.jpg)
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
![](https://www.scirp.org/html/6-7400891\cae696ef-bd17-4f5c-9fea-d5b682d68cd2.jpg)
Taking
, then we have
![](https://www.scirp.org/html/6-7400891\9c0bd7c9-d041-4e2d-9f5f-df03811a0546.jpg)
where
and the processes:
![](https://www.scirp.org/html/6-7400891\d4081240-0b7a-4232-87fb-5ad4f8954b98.jpg)
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
![](https://www.scirp.org/html/6-7400891\14aaa4d3-6e7b-4196-9dfb-2b00990640d6.jpg)
![](https://www.scirp.org/html/6-7400891\6832fb5e-ec4f-4631-b9bb-eebec7ccf911.jpg)
by iterative method:
![](https://www.scirp.org/html/6-7400891\37430076-e3aa-4847-b611-e105ff54a651.jpg)
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
![](https://www.scirp.org/html/6-7400891\55252534-e7bb-4ab8-b270-47704497216c.jpg)
2) The shake Power
![](https://www.scirp.org/html/6-7400891\1b631bd3-fd21-42ec-af07-4c4e5e692ad9.jpg)
We take parameters ![](https://www.scirp.org/html/6-7400891\8568d0e5-5fe2-416d-8b9c-5aafe3788521.jpg)
Then
![](https://www.scirp.org/html/6-7400891\63738af3-2b20-4c74-8cd3-7298669c75fc.jpg)
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.
![](https://www.scirp.org/html/6-7400891\6d86dd3a-a425-4b3d-be3f-b95f8785f06f.jpg)
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.
![](https://www.scirp.org/html/6-7400891\95b5ed66-becc-4bcd-9180-cbc7f371ce66.jpg)
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
![](https://www.scirp.org/html/6-7400891\d6af1573-78a0-4880-92af-f38ee30f31d1.jpg)
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).
![](https://www.scirp.org/html/6-7400891\f95f23de-9019-43e3-a351-f451aa933f53.jpg)
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.