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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 760918, 8 pages
Periodic Solutions for Duffing Type -Laplacian Equation with Multiple Constant Delays
1College of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, Hunan 415000, China
2College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China
Received 19 November 2011; Accepted 17 January 2012
Academic Editor: Elena Braverman
Copyright © 2012 Hong Zhang and Junxia Meng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Using inequality techniques and coincidence degree theory, new results are provided concerning the existence and uniqueness of periodic solutions for the Duffing type -Laplacian equation with multiple constant delays of the form Moreover, an example is provided to illustrate the effectiveness of the results in this paper.
Referring to the work of Esmailzadeh and Nakhaie-Jazar , Duffing type equation is the simplest case of a vibrating system with nonlinear restoring force generator element. This is equivalent to a mechanical vibrating system with either a hard or soft spring. Thus, this equation and its modifications have been extensively and intensively studied. In particular, the existence of periodic solutions for Duffing type equations with and without delays have been discussed by various researchers (see, e.g., [2–8] and the references given therein). However, to the best of our knowledge, the existence and uniqueness of periodic solutions of Duffing type -Laplacian equation whose delays more than two have not been sufficiently researched. Motivated by this, we shall consider the Duffing type -Laplacian equations with multiple constant delays of the form where and is given by for and , and are constants, and , are continuous functions, and are -periodic, and are -periodic in the first argument, and . The main purpose of this paper is to establish sufficient conditions for the existence and uniqueness of -periodic solutions of (1.1). The results of this paper are new and complement previously known results. Moreover, we give an example to illustrate the results.
2. Preliminary Results
Throughout this paper, we will denote
For the periodic boundary value problem where is -periodic in the first variable, we have the following lemma.
Lemma 2.1 (see ). Let be an open bounded set in , if the following conditions hold. (i)For each the problem has no solution on .(ii)The equation has no solution on .(iii)The Brouwer degree of Then, the periodic boundary value problem (2.2) has at least one -periodic solution on .
We can easily obtain the homotopic equation of (1.1) as follows:
Lemma 2.2. Assume that the following conditions are satisfied.
() There exists a constant such that(1), (2)Moveover, if is a -periodic solution of (2.6), then
Proof. Let be a -periodic solution of (2.6). Then, integrating (2.6) over , we have
Using the integral mean-value theorem, it follows that there exists such that
We now prove that there exists a constant such that
Indeed, suppose otherwise. Then,
Let . From , (2.9), and (2.11), we see that there exist such that
which, together with (2.11), implies
Without loss of generality, we may assume that (the situation is analogous for ). Then, we have
According to (2.14) and , we obtain
this contradicts the fact (2.9); thus, (2.10) is true.
Let where and is an integer. Then, by the same approach used in the proof of inequality (3.3) of , we have This completes the proof of Lemma 2.2.
Lemma 2.3. Let holds. Assume that the following condition is satisfied.
() There exist nonnegative constants such that for all .
Then, (1.1) has at most one -periodic solution.
Proof. Suppose that and are two -periodic solutions of (1.1). Set . Then, we obtain Multiplying and (2.18) and then integrating it from 0 to T, from and Schwarz inequality, we get Since , we have Thus, for all . Therefore, (1.1) has at most one -periodic solution. The proof of Lemma 2.3 is now complete.
3. Main Results
Theorem 3.1. Let () and () hold. Then, (1.1) has a unique -periodic solution in .
Proof. By Lemma 2.3, it is easy to see that (1.1) has at most one -periodic solution in . Thus, in order to prove Theorem 3.1, it suffices to show that (1.1) has at least one -periodic solution in . To do this, we are going to apply Lemma 2.1. Firstly, we claim that the set of all possible -periodic solutions of (2.6) in is bounded.
Let be a -periodic solution of (2.6). Multiplying and (2.6) and then integrating it from 0 to T, we have Since , then there exists such that . And since , we have where .
In view of (3.1), (), and Schwarz inequality, we get It follows that
Again from () and Schwarz inequality, (3.2) and (3.4) yield which, together with (2.7), implies that there exists a positive constant such that, for all , Set then we know that (2.6) has no -periodic solution on as and when or , from , we can see that so condition (ii) of Lemma 2.1 is also satisfied. Set and when , we have Thus, is a homotopic transformation and so condition (iii) of Lemma 2.1 is satisfied. In view of the previous Lemma 2.1, (1.1) has at least one solution with period . This completes the proof.
4. Example and Remark
Example 4.1. Let , and for all . Then, the following Liénard type -Laplacian equation with two constant delays has a unique -periodic solution.
Remark 4.2. Obviously, the results in [2–5] obtained on Duffing type -Laplacian equation with single delay and without multiple delays cannot be applicable to (4.1). This implies that the results of this paper are essentially new.
The authors would like to express their sincere appreciation to the editor and anonymous referee for their valuable comments which have led to an improvement in the presentation of the paper. This work was supported by the construct program of the key discipline in Hunan province (Mechanical Design and Theory), the Scientific Research Fund of Hunan Provincial Natural Science Foundation of PR China (Grant no. 11JJ6006), the Natural Scientific Research Fund of Hunan Provincial Education Department of PR China (Grants no. 11C0916, 11C0915, 11C1186), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (Grant no. Y6110436), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R. China (Grant no. Z201122436).
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