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Uniqueness of Periodic Solution for a Class of Liénard Laplacian Equations
Advances in Difference Equations volume 2010, Article number: 235749 (2010)
Abstract
By topological degree theory and some analysis skills, we consider a class of generalized Liénard type Laplacian equations. Upon some suitable assumptions, the existence and uniqueness of periodic solutions for the generalized Liénard type Laplacian differential equations are obtained. It is significant that the nonlinear term contains two variables.
1. Introduction
As it is well known, the existence of periodic and almost periodic solutions is the most attracting topics in the qualitative theory of differential equations due to their vast applications in physics, mathematical biology, control theory, and others. More general equations and systems involving periodic boundary conditions have also been considered. Especially, the existence of periodic solutions for the Duffing equation, Rayleigh equation, and Liénard type equation, which are derived from many fields, such as fluid mechanics and nonlinear elastic mechanics, has received a lot of attention.
Many experts and scholars, such as Manásevich, Mawhin, Gaines, Cheung, Ren, Ge, Lu, and Yu, have contributed a series of existence results to the periodicity theory of differential equations. Fixed point theory, Mawhin's continuation theorem, upper and lower solutions method, and coincidence degree theory are the common tools to study the periodicity theory of differential equations. Among these approaches, the Mawhin's continuation theorem seems to be a very powerful tool to deal with these problems.
Some contributions on periodic solutions to differential equations have been made in [1–13]. Recently, periodic problems involving the scalar Laplacian were studied by many authors. We mention the works by Manásevich and Mawhin [3] and Cheung and Ren [4, 8, 10].
In [3], Manásevich and Mawhin investigated the existence of periodic solutions to the boundary value problem
where the function is quite general and satisfies some monotonicity conditions which ensure that is homeomorphism onto Applying LeraySchauder degree theory, the authors brought us the widely used ManásevichMawhin continuation theorem. When is the socalled onedimensional pLaplacian operator given by
Recently, by Mawhin's continuation theorem, Cheung and Ren studied the existence of periodic solutions for a Laplacian Liénard equation with a deviating argument in [4] as follows:
and two results (Theorems 3.1 and 3.2 ) on the existence of periodic solutions were obtained.
Ge and Ren [5] promoted Mawhin's continuation theorem to the case which involved the quasilinear operator successfully; this also prepared conditions for using Mawhin's continuation theorem to solve nonlinear boundary value problem.
Liu [7] has dealt with the existence and uniqueness of periodic solutions of the Liénard type Laplacian differential equation of the form
by using topological degree theory, and one sufficient condition for the existence and uniqueness of periodic solutions of this equation was established.
The aim of this paper is to study the existence of periodic solutions to a class of Laplacian Liénard equations as follows:
where , is given by for , , , and periodic in the first variable, where is a given constant, and
The paper is organized as follows. In Section 2, we give the definition of norm in Banach space and the main lemma. In Section 3, combining Lemma 2.1 with some analysis skills, two sufficient conditions about the existence of solutions for (1.4) are obtained. The nonlinear terms and contain two variables in this paper, which is seldom considered in the other papers, and the results are new.
2. Preliminary Results
For convenience, we define
and the norm is defined by , for all
Clearly, is a Banach space endowed with such norm.
For the periodic boundary value problem
where is a continuous function and periodic in the first variable, we have the following result.
Lemma 2.1 (see [3]).
Let be an open bounded set in . If the following conditions hold:

(i)
for each the problem
(2.4)has no solution on ,

(ii)
the equation
(2.5)has no solution on ,

(iii)
the Brouwer degree of is ,
then the periodic boundary value problem (2.3) has at least one periodic solution on
Set
We can rewrite (1.4) in the following form:
where and
Lemma 2.2.
Suppose the following condition holds:
(A_{1}) , for all .
Then (1.4) has at most one periodic solution.
Proof.
Let and be two periodic solutions of (1.4). Then, from (2.7), we obtain
Set
Then it follows from (2.8) that
We claim that for all By way of contradiction, in view of and for all , we obtain Then there must exist ; for convenience, we can choose such that
which implies that
Set
Then, Since , if from the first equation of (2.12), we have
which contradicts assumption , so ; it implies that that is, Hence we have
Substituting (2.15) into the second equation of (2.12), we have
Noticing () and that and , from (2.16), we know that
this contradicts the second equation of (2.12). So we have , for all
By using a similar argument, we can also show that
Then, from (2.10) we obtain
For every if then it contradicts (2.19), so ; it implies that then , for all
Hence, (1.4) has at most one periodic solution. The proof of Lemma 2.2 is completed now.
3. Main Results
Theorem 3.1.
Let () hold. Suppose that there exists a positive constant d such that
(A_{2}) , for all
(A_{3})
Then (1.4) has one unique periodic solution if
Proof.
Consider the homotopic equation of (1.4) as follows:
By Lemma 2.2, combining it is easy to see that (1.4) has at least one periodic solution. For the remainder, we will apply Lemma 2.1 to study (3.1). Firstly, we will verify that all the possible periodic solutions of (3.1) are bounded.
Let be an arbitrary solution of (3.1) with period . By integrating the two sides of (3.1) from to and we obtain
Consider and , there exists such that while for we see that
where Let be the global maximum point of on Then as we claim that
Otherwise, we have there must exist a constant such that , for ; therefore, is strictly increasing for which implies that is strictly increasing for Thus, (3.4) is true. Then
In view of (), (3.5) implies that ; similar to the global minimum point of on Since it follows that there exists a constant such that Then we have
Combining the above two inequalities, we obtain
Considering () there exist constants and the sufficiently small such that
Set
From (3.7), we have
where Combining the classical inequality when where is a constant, since
then we consider the following two cases.
Case 1.
If then Combining (3.7), we know that
when then we have
Case 2.
When then from the above classical inequality, we obtain
Substituting the above inequality into (3.10) we get
Since is periodic, multiplying by (3.1) and then integrating from to , in view of () we have
Substituting (3.14) into (3.15) and since
we obtain
where
Since and from (3.17), we know that there exists a constant such that Then,
So, there exists a constant such that
Set
Then (3.1) has no solution on as and when or ; from we can see that
so condition (ii) holds.
Set
Then, when , we have
thus is a homotopic transformation and
So condition (iii) holds. In view of Lemma 2.1, there exists at least one solution with period . This completes the proof.
Theorem 3.2.
Let () hold. Suppose that there exist positive constants and satisfying the following conditions:
(A_{4}) , when
(A_{5}) ,
(A_{6}) , when
Then for (1.4) there exists one unique periodic solution when
Proof.
We can rewrite (3.1) in the following from:
Let be a periodic solution of (3.26), then must be a periodic solution of (3.1). First we claim that there is a constant such that
Take as the global maximum point and global minimum point of on , respectively, then
From the first equation of (3.26) we have so
We claim that
By way of contradiction, (3.29) does not hold, then So there exists such that , for ; therefore, , for so , that is, , for This contradicts the definition of so we have
Substituting into the second equation of (3.26), we obtain
By condition () we have Similarly, we get
Case 1.
If define Obviously
Case 2.
If from the fact that is a continuous function in there exists a constant between and such that
So we have that (3.27) holds. Next, in view of there are integer and constant such that hence
So
We claim that all the periodic solutions of (3.1) are bounded and
Let
Multiplying both sides of (3.1) by and integrating from to together with () and (), we have
where and That is,
Using Hölder's inequality and we have
so
there must be a positive constant such that
hence together with (3.31), we have
This proves the claim and that the rest of the proof of the theorem is identical to that of Theorem 3.1.
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Acknowledgments
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation Funded Project (20080441126, 200902564), Shandong Postdoctoral Funded Project (200802018), and the Natural Scientific Foundation of Shandong Province (Y2008A28, ZR2009AL003) and is also supported by University of Jinan Research Funds for Doctors (XBS0843).
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Cao, F., Han, Z., Zhao, P. et al. Uniqueness of Periodic Solution for a Class of Liénard Laplacian Equations. Adv Differ Equ 2010, 235749 (2010). https://doi.org/10.1155/2010/235749
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Keywords
 Periodic Solution
 Periodic Problem
 Homotopic Equation
 Rayleigh Equation
 Classical Inequality