Periodic solutions for p-Laplacian Rayleigh equations with singularities
- Shipin Lu^{1}Email author,
- Tao Zhong^{1} and
- Lijuan Chen^{1}
Received: 23 January 2016
Accepted: 6 May 2016
Published: 11 May 2016
Abstract
In this paper, the problem of existence of periodic solutions is studied for p-Laplacian Rayleigh equations with a singularity at \(x=0\). By using the topological degree theory, some new results are obtained.
Keywords
Rayleigh equation topological degree singularity periodic solution1 Introduction
2 Preliminary lemmas
The following two lemmas (Lemma 2.1 and Lemma 2.2) are all consequences of Theorem 3.1 in [19].
Lemma 2.1
- (1)For each \(\lambda\in(0,1]\), each possible positive T-periodic solution x to the equationsatisfies the inequalities \(\eta_{0}< x(t)<\eta_{1}\) and \(|x'(t)|< M_{2}\) for all \(t\in[0,T]\).$$ \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'+ \lambda f \bigl(u' \bigr)-\lambda g_{1}(u)+\lambda g_{2}(t,u)=\lambda h(t) $$
- (2)Each possible solution c to the equationsatisfies the inequality \(\eta_{0}< c<\eta_{1}\).$$ g_{1}(c)-g_{2}(c)+\bar{h}=0 $$
- (3)It holds$$ \bigl(g_{1}(\eta_{0})-g_{2}( \eta_{0})+\bar{h} \bigr) \bigl(g_{1}(\eta _{1})-g_{2}(\eta_{1})+\bar{h} \bigr)< 0. $$
Lemma 2.2
- (1)For each \(\lambda\in(0,1]\), each possible positive T-periodic solution x to the equationsatisfies the inequalities \(\eta_{0}< x(t)<\eta_{1}\) and \(|x'(t)|< M_{2}\) for all \(t\in[0,T]\).$$ \bigl(\bigl|u'\bigr|^{p-2}u' \bigr)'+ \lambda f \bigl(u' \bigr)+\lambda g_{1}(u)-\lambda g_{2}(u)=\lambda h(t) $$
- (2)Each possible solution c to the equationsatisfies the inequality \(\eta_{0}< c<\eta_{1}\).$$ g_{1}(c)-g_{2}(c)-\bar{h}=0 $$
- (3)It holds$$ \bigl(g_{1}(\eta_{0})-g_{2}( \eta_{0})-\bar{h} \bigr) \bigl(g_{1}(\eta _{1})-g_{2}(\eta_{1})-\bar{h} \bigr)< 0. $$
In order to study the existence of positive periodic solutions to equation (1.5) and equation (1.6), we list the following assumptions:
(H_{2}) \(g_{1}(x)\ge0\) for all \(x\in(0,+\infty)\), and \(\int _{0}^{1}g_{1}(s)\,ds=+\infty\);
(H_{3}) \(f(0)=0\);
Lemma 2.3
- (1)for each possible positive T-periodic solution \(u(t)\) of equation (2.3) there exists \(\tau\in[0,T]\) such that$$ m_{0}< u(\tau)< m_{1}; $$(2.5)
- (2)each possible solution c to the equationsatisfies the inequality \(m_{0}< c< m_{1}\);$$ g_{1}(c)-g_{2}(c)+\bar{h}=0 $$
- (3)
\(g_{1}(u)-g_{2}(u)+\bar{h}>0\) for all \(u\in(0,m_{0}]\), and \(g_{1}(u)-g_{2}(u)+\bar{h}<0\) for all \(u\in[m_{1},+\infty)\).
Proof
(2) Conclusion (2), as well as conclusion (3), follows directly from assumption (H_{1}). □
Similar to the proof of Lemma 2.3, we obtain the following result.
Lemma 2.4
- (1)each possible positive T-periodic solution \(u(t)\) to equation (2.4) satisfies$$ m_{0}< u(t)< m_{1} \quad\textit{for all } t\in[0,T]; $$
- (2)each possible solution c to the equationsatisfies the inequality \(m_{0}< c< m_{1}\);$$ g_{1}(c)-g_{2}(c)-\bar{h}=0 $$
- (3)
\(g_{1}(u)-g_{2}(u)-\bar{h}>0\) for all \(u\in(0,m_{0}]\), and \(g_{1}(u)-g_{2}(u)-\bar{h}<0\) for all \(u\in[m_{1},+\infty)\).
3 Main results
Theorem 3.1
Assume that assumptions (H_{1}), (H_{2}), and (H_{4}) hold, then equation (1.5) has at least one positive T-periodic solution.
Proof
Theorem 3.2
Assume that assumptions (H_{1}) and (H_{3}) hold, then equation (1.6) has at least one positive T-periodic solution.
Proof
Corollary 3.1
Assume that (3.20), (3.21), and assumptions (H_{2}) and (H_{4}) hold, then equation (1.5) has at least one positive T-periodic solution.
Corollary 3.2
Assume that (3.20), (3.21), and assumption (H_{3}) hold, then equation (1.6) has at least one positive T-periodic solution.
Example 3.1
Example 3.2
Corresponding to equation (1.6), \(f(u)=u^{4}\), \(g_{1}(u)=\frac {1}{u^{\frac{1}{2}}}-\frac{1}{u^{\frac{1}{4}}}\), \(g_{2}(u)=u^{3}\), and \(h(t)=\cos t\). It is easy to see that conditions (3.20) and (3.21), and assumption (H_{3}) are all satisfied. By using Corollary 3.2, we see that equation (3.23) has at least one positive 2π-periodic solution.
Declarations
Acknowledgements
The work is sponsored by the National Natural Science Foundation of China (11271197). The authors are grateful to anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.
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
- Jebelean, P, Mawhin, J: Periodic solutions of singular nonlinear perturbations of the ordinary p-Laplacian. Adv. Nonlinear Stud. 2(3), 299-312 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Forbat, F, Huaux, A: Détermination approchée et stabilité locale de la solution périodique d’une équation différentielle non lineáire. Mém. Publ. Soc. Sci. Arts Lett. Hainaut 76, 3-13 (1962) MathSciNetMATHGoogle Scholar
- Gordon, WB: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc. 204, 113-135 (1975) MathSciNetView ArticleMATHGoogle Scholar
- Gordon, WB: A minimizing property of Keplerian orbits. Am. J. Math. 99, 961-971 (1977) MathSciNetView ArticleMATHGoogle Scholar
- Solimini, S: On forced dynamical systems with a singularity of repulsive type. Nonlinear Anal. 14, 489-500 (1990) MathSciNetView ArticleMATHGoogle Scholar
- Terracini, S: Remarks on periodic orbits of dynamical systems with repulsive singularities. J. Funct. Anal. 111, 213-238 (1993) MathSciNetView ArticleMATHGoogle Scholar
- Torres, PJ: Existence of one-signed periodic solutions of some second order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 190, 643-662 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Hakl, R, Torres, PJ: On periodic solutions of second-order differential equations with attractive-repulsive singularities. J. Differ. Equ. 248, 111-126 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Jiang, D, Chu, J, Zhang, M: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ. 211, 282-302 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Chu, J, Torres, PJ, Zhang, M: Periodic solutions of second order non-autonomous singular dynamical systems. J. Differ. Equ. 239, 196-212 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Chu, J, Fan, N, Torres, PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 388, 665-675 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Martins, R: Existence of periodic solutions for second-order differential equations with singularities and the strong force condition. J. Math. Anal. Appl. 317, 1-13 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Lei, J, Zhang, MR: Twist property of periodic motion of an atom near a charged wire. Lett. Math. Phys. 60(1), 9-17 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Torres, PJ: Existence and stability of periodic solutions for second order semilinear differential equations with a singular nonlinearity. Proc. R. Soc. Edinb., Sect. A 137, 195-201 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Fonda, A, Toader, R: Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach. J. Differ. Equ. 244, 3235-3264 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Li, X, Zhang, Z: Periodic solutions for second-order differential equations with a singular nonlinearity. Nonlinear Anal. 69, 3866-3876 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Zhang, M: Nonuniform nonresonance at the first eigenvalue of the p-Laplacian. Nonlinear Anal. 29(1), 41-51 (1997) MathSciNetView ArticleMATHGoogle Scholar
- Wang, Z: Periodic solutions of Lienard equations with a singularity and a deviating argument. Nonlinear Anal. 16, 227-234 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Manaśevich, R, Mawhin, J: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differ. Equ. 145, 367-393 (1998) MathSciNetView ArticleMATHGoogle Scholar