Uniqueness of Periodic Solution for a Class of Liénard -Laplacian Equations
© Fengjuan Cao et al. 2010
Received: 31 December 2009
Accepted: 23 February 2010
Published: 28 February 2010
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.
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  and Cheung and Ren [4, 8, 10].
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
and two results (Theorems 3.1 and 3.2 ) on the existence of periodic solutions were obtained.
Ge and Ren  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.
by using topological degree theory, and one sufficient condition for the existence and uniqueness of -periodic solutions of this equation was established.
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
Clearly, is a Banach space endowed with such norm.
where is a continuous function and -periodic in the first variable, we have the following result.
Lemma 2.1 (see ).
- (i)for each the problem(2.4)
has no solution on ,
- (ii)the equation(2.5)
has no solution on ,
the Brouwer degree of is ,
then the periodic boundary value problem (2.3) has at least one -periodic solution on
Suppose the following condition holds:
(A1) , for all .
Then (1.4) has at most one -periodic solution.
this contradicts the second equation of (2.12). So we have , for all
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
Let ( ) hold. Suppose that there exists a positive constant d such that
(A2) , for all
Then (1.4) has one unique -periodic solution if
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.
then we consider the following two cases.
when then we have
so condition (ii) holds.
So condition (iii) holds. In view of Lemma 2.1, there exists at least one solution with period . This completes the proof.
Let ( ) hold. Suppose that there exist positive constants and satisfying the following conditions:
(A4) , when
(A6) , when
Then for (1.4) there exists one unique -periodic solution when
From the first equation of (3.26) we have so
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
By condition ( ) we have Similarly, we get
If define Obviously
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
We claim that all the periodic solutions of (3.1) are bounded and
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.
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).
- 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.2006.12.041MathSciNetView ArticleMATHGoogle Scholar
- 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.2007.08.010MathSciNetView ArticleMATHGoogle Scholar
- 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.1998.3425MathSciNetView ArticleMATHGoogle Scholar
- 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.MathSciNetMATHGoogle Scholar
- 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.2004.01.007MathSciNetView ArticleMATHGoogle Scholar
- 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.2007.01.013MathSciNetView ArticleMATHGoogle Scholar
- 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.2007.06.007MathSciNetView ArticleMATHGoogle Scholar
- 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.MathSciNetMATHGoogle Scholar
- Gaines RE, Mawhin JL: Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Mathematics. Volume 568. Springer, Berlin, Germany; 1977:i+262.Google Scholar
- 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.2005.11.002MathSciNetView ArticleMATHGoogle Scholar
- 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.2006.05.007MathSciNetView ArticleMATHGoogle Scholar
- Cao F, Han Z, Sun S:Existence of periodic solutions for -Laplacian equations on time scales. Advances in Difference Equations 2010, 2010:-13.Google Scholar
- 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.MathSciNetGoogle Scholar
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