- Open Access
Existence and uniqueness of a positive periodic solution for Rayleigh type ϕ-Laplacian equation
© Xin and Cheng; licensee Springer. 2014
Received: 31 March 2014
Accepted: 30 June 2014
Published: 15 August 2014
By using the Manásevich-Mawhin continuation theorem and some analysis skills, we establish some sufficient condition for the existence and uniqueness of positive T-periodic solutions for a generalized Rayleigh type ϕ-Laplacian operator equation. The results of this paper are new and they complement previous known results.
where the function is continuous and . is an -Carathéodory function and , , which means it is measurable in the first variable and continuous in the second variable. For every , there exists such that for all and a.e. ; and f, g is a T-periodic function about t and . and is T-periodic.
Here is a continuous function and , which satisfies
(A1) for , ;
(A2) there exists a function , as , such that for .
It is easy to see that ϕ represents a large class of nonlinear operators, including is a p-Laplacian, i.e., for .
We know that the study on ϕ-Laplacian is relatively infrequent, the main difficulty lies in the fact that the ϕ-Laplacian operator typically possesses more uncertainty than the p-Laplacian operator. For example, the key step for to get a priori solutions, , is no longer available for general ϕ-Laplacian. So, we need to find a new method to get over it.
By using the Manásevich-Mawhin continuation theorem and some analysis skills, we establish some sufficient condition for the existence of positive T-periodic solutions of (1.1). The results of this paper are new and they complement previous known results.
2 Main results
here is assumed to be Carathéodory.
Lemma 2.1 (Manásevich-Mawhin )
- (i)for each , the problem
- (ii)the equation
- (iii)the Brouwer degree of F
Then the periodic boundary value problem (2.1) has at least one periodic solution on .
Lemma 2.2 If is bounded, then x is also bounded.
Proof Since is bounded, then there exists a positive constant N such that . From (A2), we have . Hence, we can get for all . If x is not bounded, then from the definition of α, we get for some , which is a contradiction. So x is also bounded. □
Lemma 2.3 Suppose that the following condition holds:
(A3) for all t, , .
Then (1.1) has at most one T-periodic solution in .
Hence, (1.1) has at most one T-periodic solution in . The proof of Lemma 2.3 is now complete. □
For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:
(H1) there exists a positive constant D such that for and , for and ;
(H2) there exist constants and such that for ;
(H3) there exist positive constants ρ and γ such that for ;
By using Lemmas 2.1-2.3, we obtain our main results.
Theorem 2.1 Assume that conditions (H1)-(H4) and (A3) hold. Then (1.1) has a unique positive T-periodic solution if .
Case (1): If , define , obviously, .
Case (2): If , from , we know . Define , we have . This proves (2.5).
where , .
So condition (iii) is satisfied. In view of Lemma 2.1, there exists at least one solution with period T.
which implies that (1.1) has a unique positive solution with period T. This completes the proof. □
We illustrate our results with some examples.
Comparing (2.12) to (1.1), we see that , , , . Obviously, we know that is a homeomorphism from ℝ to ℝ satisfying (A1) and (A2). Consider for , then (A3) holds. Moreover, it is easily seen that there exists a constant such that (H1) holds. Consider , here , , and , here , . So, we can get that conditions (H2) and (H3) hold. Choose , we have , here , , then (H4) holds and . So, by Theorem 2.1, we can get that (2.12) has a unique positive periodic solution.
So, we know that (A1) and (A2) hold. Consider for , then (A3) holds. Moreover, it is easily seen that there exists a constant such that (H1) holds. Consider , here , , and , here , . So, we can get that conditions (H2) and (H3) hold. Choose , we have , here , , then (H4) holds and . Therefore, by Theorem 2.1, we know that (2.13) has a unique positive periodic solution.
Research is supported by the National Natural Science Foundation of China (Nos. 11326124, 11271339).
- Cheung WS, Ren JL: Periodic solutions for p -Laplacian Rayleigh equations. Nonlinear Anal. TMA 2006, 65: 2003-2012. 10.1016/j.na.2005.11.002MathSciNetView ArticleMATHGoogle Scholar
- Cheng ZB, Ren JL: Periodic solutions for a fourth-order Rayleigh type p -Laplacian delay equation. Nonlinear Anal. TMA 2009, 70: 516-523. 10.1016/j.na.2007.12.023MathSciNetView ArticleMATHGoogle Scholar
- Feng L, Guo LX, Lu SP: New results of periodic solutions for Rayleigh type p -Laplacian equation with a variable coefficient ahead of the nonlinear term. Nonlinear Anal. TMA 2009, 70: 2072-2077. 10.1016/j.na.2008.02.107View ArticleMathSciNetMATHGoogle Scholar
- Ma TT: Periodic solutions of Rayleigh equations via time-maps. Nonlinear Anal. TMA 2012, 75: 4137-4144. 10.1016/j.na.2012.03.004View ArticleMathSciNetMATHGoogle Scholar
- Lu SP, Gui ZJ: On the existence of periodic solutions to p -Laplacian Rayleigh differential equation with a delay. J. Math. Anal. Appl. 2007, 325: 685-702. 10.1016/j.jmaa.2006.02.005MathSciNetView ArticleMATHGoogle Scholar
- Liang RX: Existence and uniqueness of periodic solution for forced Rayleigh type equations. J. Appl. Math. Comput. 2012, 40: 415-425. 10.1007/s12190-012-0568-6MathSciNetView ArticleMATHGoogle Scholar
- Xiao B, Liu W: Periodic solutions for Rayleigh type p -Laplacian equation with a deviating argument. Nonlinear Anal., Real World Appl. 2009, 10: 16-22. 10.1016/j.nonrwa.2007.08.010MathSciNetView ArticleMATHGoogle Scholar
- Wang LJ, Shao JY: New results of periodic solutions for a kind of forced Rayleigh-type equation. Nonlinear Anal., Real World Appl. 2010, 11: 99-105. 10.1016/j.nonrwa.2008.10.018MathSciNetView ArticleMATHGoogle Scholar
- Xiong WM, Shao JY: Existence and uniqueness of positive periodic solutions for Rayleigh type p -Laplacian equation. Nonlinear Anal., Real World Appl. 2009, 10: 275-280.MathSciNetView ArticleMATHGoogle Scholar
- Zong MG, Liang HZ: Periodic solutions for Rayleigh type p -Laplacian equation with deviating arguments. Appl. Math. Lett. 2007, 20: 43-47. 10.1016/j.aml.2006.02.021MathSciNetView ArticleMATHGoogle Scholar
- Manásevich R, Mawhin J: Periodic solutions for nonlinear systems with p -Laplacian-like operator. J. Differ. Equ. 1998, 145: 367-393. 10.1006/jdeq.1998.3425View ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.