© Hui-Sheng Ding et al. 2010
Received: 1 February 2010
Accepted: 19 March 2010
Published: 30 March 2010
We investigate the following nonlinear equations with -Laplacian-like operators : some criteria to guarantee the existence and uniqueness of periodic solutions of the above equation are given by using Mawhin's continuation theorem. Our results are new and extend some recent results due to Liu (B. Liu, Existence and uniqueness of periodic solutions for a kind of Lienard type -Laplacian equation, Nonlinear Analysis TMA, 69, 724–729, 2008).
Recall that -Laplacian equations have been of great interest for many mathematicians. Especially, there is a large literature (see, e.g., [1–7] and references therein) about the existence of periodic solutions to the following -Laplacian equation:
However, there are seldom results about the existence of periodic solutions to (1.1). The main difficulty lies in the -Laplacian-like operator of (1.1), which is more complicated than in (1.3). Since there is no concrete form for the -Laplacian-like operator of (1.1), it is more difficult to prove the existence of periodic solutions to (1.1).
Therefore, in this paper, we will devote ourselves to investigate the existence of periodic solutions to (1.1). As one will see, our theorem generalizes some recent results even for the case of (see Remark 2.2).
Next, let us recall some notations and basic results. For convenience, we denote
In the proof of our main results, we will need the following classical Mawhin's continuation theorem.
Lemma 1.1 ().
2. Main Results
In this section, we prove an existence and uniqueness theorem for (1.1).
Suppose the following assumptions hold:
On the other hand, it follows from
If this is not true, we consider the following two cases.
Now, we have proved that
In addition, we have the following interesting corollary.
Suppose (A1) and
At last, we give two examples to illustrate our results.
The authors are grateful to the referee for valuable suggestions and comments, which improved the quality of this paper. The work was supported by the NSF of China, the NSF of Jiangxi Province of China (2008GQS0057), the Youth Foundation of Jiangxi Provincial Education Department (GJJ09456), and the Youth Foundation of Jiangxi Normal University.
- Lu S: New results on the existence of periodic solutions to a -Laplacian differential equation with a deviating argument. Journal of Mathematical Analysis and Applications 2007,336(2):1107-1123. 10.1016/j.jmaa.2007.03.032MATHMathSciNetView ArticleGoogle Scholar
- 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.2006.12.041MATHMathSciNetView ArticleGoogle Scholar
- Liu B:Periodic solutions for Liénard type -Laplacian equation with a deviating argument. Journal of Computational and Applied Mathematics 2008,214(1):13-18. 10.1016/j.cam.2007.02.004MATHMathSciNetView ArticleGoogle 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.MATHMathSciNetGoogle Scholar
- Cheung W-S, Ren J:Periodic solutions for -Laplacian differential equation with multiple deviating arguments. Nonlinear Analysis: Theory, Methods & Applications 2005,62(4):727-742. 10.1016/j.na.2005.03.096MATHMathSciNetView ArticleGoogle 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.2005.11.002MATHMathSciNetView ArticleGoogle 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.2007.06.007MATHMathSciNetView ArticleGoogle 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.1998.3425MATHMathSciNetView ArticleGoogle Scholar
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