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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 Leray-Schauder degree theory, the authors brought us the widely used Manásevich-Mawhin continuation theorem. When
is the so-called one-dimensional p-Laplacian 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:
(A1) , 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
(A2) , for all
(A3)
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:
(A4) , when
(A5) ,
(A6) , 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.
References
Lu S:Existence of periodic solutions to a
-Laplacian Liénard differential equation with a deviating argument. Nonlinear Analysis: Theory, Methods & Applications 2008,68(6):1453-1461. 10.1016/j.na.2006.12.041
Xiao B, Liu B:Periodic solutions for Rayleigh type
-Laplacian equation with a deviating argument. Nonlinear Analysis: Real World Applications 2009,10(1):16-22. 10.1016/j.nonrwa.2007.08.010
Manásevich R, Mawhin J:Periodic solutions for nonlinear systems with
-Laplacian-like operators. Journal of Differential Equations 1998,145(2):367-393. 10.1006/jdeq.1998.3425
Cheung W-S, Ren J:On the existence of periodic solutions for
-Laplacian generalized Liénard equation. Nonlinear Analysis: Theory, Methods & Applications 2005,60(1):65-75.
Ge W, Ren J:An extension of Mawhin's continuation theorem and its application to boundary value problems with a
-Laplacian. Nonlinear Analysis: Theory, Methods & Applications 2004,58(3-4):477-488. 10.1016/j.na.2004.01.007
Zhang F, Li Y:Existence and uniqueness of periodic solutions for a kind of Duffing type
-Laplacian equation. Nonlinear Analysis: Real World Applications 2008,9(3):985-989. 10.1016/j.nonrwa.2007.01.013
Liu B:Existence and uniqueness of periodic solutions for a kind of Liénard type
-Laplacian equation. Nonlinear Analysis: Theory, Methods & Applications 2008,69(2):724-729. 10.1016/j.na.2007.06.007
Cheung W-S, Ren J:Periodic solutions for
-Laplacian Liénard equation with a deviating argument. Nonlinear Analysis: Theory, Methods & Applications 2004,59(1-2):107-120.
Gaines RE, Mawhin JL: Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematics. Volume 568. Springer, Berlin, Germany; 1977:i+262.
Cheung W-S, Ren J:Periodic solutions for
-Laplacian Rayleigh equations. Nonlinear Analysis: Theory, Methods & Applications 2006,65(10):2003-2012. 10.1016/j.na.2005.11.002
Peng S, Zhu S:Periodic solutions for
-Laplacian Rayleigh equations with a deviating argument. Nonlinear Analysis: Theory, Methods & Applications 2007,67(1):138-146. 10.1016/j.na.2006.05.007
Cao F, Han Z, Sun S:Existence of periodic solutions for
-Laplacian equations on time scales. Advances in Difference Equations 2010, 2010:-13.
Cao F, Han Z:Existence of periodic solutions for
-Laplacian differential equation with a deviating arguments. Journal of University of Jinan (Sci. Tech.) 2010,24(1):95-98.
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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DOI: https://doi.org/10.1155/2010/235749
Keywords
- Periodic Solution
- Periodic Problem
- Homotopic Equation
- Rayleigh Equation
- Classical Inequality