- Research Article
- Open Access
© 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).
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Periodic Solution
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 ().
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.
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