- Research Article
- Open Access
© 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.
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
Lemma 2.1 (see ).
Suppose the following condition holds:
3. Main Results
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.
so condition (ii) holds.
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).
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