Positive periodic solution for p-Laplacian neutral Rayleigh equation with singularity of attractive type
- Yun Xin1,
- Hongmin Liu1Email author and
- Zhibo Cheng2, 3
https://doi.org/10.1186/s13660-018-1654-6
© The Author(s) 2018
Received: 4 January 2018
Accepted: 7 March 2018
Published: 14 March 2018
Abstract
Keywords
1 Introduction
Gaines and Mawhin’s work has attracted the attention of many scholars in the field of the Rayleigh equations. More recently, the existence of periodic solutions for Rayleigh equation was extensively studied (see [2–11] and the references therein). Some classical tools have been used to study Rayleigh equation in the literature, including the method of upper and lower solutions [6], the time map continuation theorem [7, 9], fixed point theory [4], the Manásevich–Mawhin continuation theorem [10, 11], and coincidence degree theory [2, 3, 5, 8].
In this paper, by applications of an extension of Mawhin’s continuation theorem in [17] and some analysis techniques, we see the existence of a positive periodic solution for (1.5). Our results improve and extend the results in [12, 15, 16].
2 Preliminary lemmas
Lemma 2.1
(see [18])
Lemma 2.2
Proof
Lemma 2.3
(see [19])
The following lemma involves the consequences of Theorem 3.1 of [17].
Lemma 2.4
- (i)for each \(\lambda \in (0,1)\) the equationhas no solution on ∂Ω;$$ \bigl(\phi_{p}(Au)'(t) \bigr)'+\lambda f \bigl(t,u'(t) \bigr)+\lambda g \bigl(t,u(t) \bigr)=\lambda e(t) $$(2.1)
- (ii)the equationhas no solution on \(\partial \Omega \cap \mathbb{R}\);$$F(a):=\frac{1}{\omega } \int^{\omega }_{0}g(t,a)\,dt=0 $$
- (iii)the Brouwer degreethen Eq. (2.1) has at least one periodic solution on Ω̄.$$\deg \{F,\Omega \cap \mathbb{R},0\}\neq 0, $$
3 Main results: positive periodic solution for (1.5)
In this section, we will consider the existence of a positive periodic solution for (1.5) with singularity.
Theorem 3.1
- \((H_{1})\) :
-
there exists a positive constant K such that \(\vert f(t,v)\vert \leq K\), for \((t,v)\in \mathbb{R}\times \mathbb{R}\);
- \((H_{2})\) :
-
there exist positive constants \(D_{1}\) and \(D_{2}\) with \(D_{1}>D_{2}>0\) such that \(g(t,u)<-K\) for \((t,u)\in \mathbb{R}\times (D_{1},+\infty)\) and \(g(t,u)>K\) for \((t,u)\in \mathbb{R}\times (0,D _{2})\);
- \((H_{3})\) :
-
there exist positive constants a, b such that$$-g(t,u)\leq a u^{p-1}+b,\quad \textit{for all } u>0. $$
Proof
Firstly, we will claim that the set of all possible ω-periodic solutions of (2.1) is bounded. Let \(u(t)\in C^{1}_{\omega }\) be an arbitrary solution of (2.1) with period ω.
4 Example
Example 4.1
5 Conclusions
In this article we introduce the existence of a periodic solution for a p-Laplacian neutral Rayleigh equation with singularity of attractive type. Due to the attractive condition being in contradiction with the repulsive condition, the methods of [12, 15, 16] are no long applicable to the proof of a periodic solution for equation (1.5) with singularity of attractive singularity. In this paper, we give attractive conditions (1.6) and \((H_{3})\), and we see the existence of a periodic solution for (1.5) by applications of the extension of Mawhin’s continuation theorem [17]. Moreover, in view of the mathematical points, the results satisfying the conditions of an attractive singularity are valuable to understand the periodic solution for Rayleigh equations.
Declarations
Acknowledgements
YX, HML and ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by National Natural Science Foundation of China (No. 11501170), Education Department of Henan Province project (No. 16B110006) and Henan Polytechnic University Outstanding Youth Fund (J2016-03).
Authors’ contributions
YX, HML and ZBC worked together on the derivation of the mathematical results. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Gaines, R.E., Mawhin, J.L.: Coincidence Degree, and Nonlinear Differential Equations. Lecture Notes in Mathematics, vol. 568. Springer, Berlin (1977) View ArticleMATHGoogle Scholar
- Cheng, Z.B., Ren, J.L.: Periodic solutions for a fourth-order Rayleigh type p-Laplacian delay equation. Nonlinear Anal. 70, 516–523 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Cheung, W., Ren, J.L.: Periodic solutions for p-Laplacian Rayleigh equations. Nonlinear Anal. 65, 2003–2012 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Cheung, W., Ren, J.L., Han, W.: Positive periodic solution of second-order neutral functional differential equations. Nonlinear Anal. 71, 3948–3955 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Du, B., Lu, S.P.: On the existence of periodic solutions to a p-Laplacian Rayleigh equation. Indian J. Pure Appl. Math. 40, 253–266 (2009) MathSciNetMATHGoogle Scholar
- Habets, P., Torres, P.: P: Some multiplicity results for periodic solutions of a Rayleigh differential equation. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 8, 335–351 (2001) MathSciNetMATHGoogle Scholar
- Ma, T.: Periodic solutions of Rayleigh equations via time-maps. Nonlinear Anal. 75, 4137–4144 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Wang, L., Shao, J.: New results of periodic solutions for a kind of forced Rayleigh-type equations. Nonlinear Anal., Real World Appl. 11, 99–105 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Wang, Z.H.: On the existence of periodic solutions of Rayleigh equations. Z. Angew. Math. Phys. 56, 592–608 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Wang, Y., Dai, X.: Existence and stability of periodic solutions of a Rayleigh type equation. Bull. Aust. Math. Soc. 79, 377–390 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Xin, Y., Cheng, Z.B.: Existence and uniqueness of a positive periodic solution for Rayleigh type ϕ-Laplacian equation. Adv. Differ. Equ. 2014, 225 (2014) MathSciNetView ArticleGoogle Scholar
- Chen, L.J., Lu, S.P.: A new result on the existence of periodic solutions for Rayleigh equations with a singularity of repulsive type. Adv. Differ. Equ. 2017, 106 (2017) MathSciNetView ArticleGoogle Scholar
- Lu, S.P., Zhang, T., Chen, L.: Periodic solutions for p-Laplacian Rayleigh equations with singularities. Bound. Value Probl. 2016, 96 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Sun, X., Yu, P., Qin, B.: Global existence and uniqueness of periodic waves in a population model with density-dependent migrations and Allee effect. Int. J. Bifurc. Chaos Appl. Sci. Eng. 27, 1750192 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Wang, Z.H., Ma, T.: Periodic solutions of Rayleigh equations with singularities. Bound. Value Probl. 2015, 154 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Xin, Y., Cheng, Z.B.: Study on a kind of neutral Rayleigh equation with singularity. Bound. Value Probl. 2017, 92 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Lu, S.P.: Periodic solutions to a second order p-Laplacian neutral functional differential system. Nonlinear Anal. TMA 69, 4215–4229 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, M.R.: Periodic solutions of linear and quasilinear neutral functional differential equations. J. Math. Anal. Appl. 189, 378–392 (1995) MathSciNetView ArticleMATHGoogle Scholar
- Xin, Y., Cheng, Z.B.: Positive periodic solution of p-Laplacian Liénard type differential equation with singularity and deviating argument. Adv. Differ. Equ. 2016, 41 (2016) View ArticleGoogle Scholar