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Existence of periodic solutions of a Liénard equation with a singularity of repulsive type
 Shiping Lu^{1},
 Yajiao Wang^{1}Email author and
 Yuanzhi Guo^{1}
 Received: 5 April 2017
 Accepted: 13 June 2017
 Published: 27 June 2017
Abstract
Keywords
 Liénard equation
 ManásevichMawhin’s continuation theorem
 singularity
 periodic solution
1 Introduction
2 Preliminary lemmas
Let \(C_{T}=x\in C(\mathbb{R},\mathbb{R}):x(t+T)=x(t)\) for all \(t\in\mathbb{R}\) with the norm defined by \(x_{\infty}=\max_{t\in [0,T]}x(t)\). For any Tperiodic solution \(h(t)\) with \(h \in C_{T}\), \(h_{+}(t)\) and \(h_{}(t)\) is denoted by \(\max\{(h(t),0)\}\) and \(\min\{ (h(t),0)\}\), respectively, and \(\overline{h}=\frac{1}{T}\int ^{T}_{0}h(s)\,ds\). Clearly, \(h(t)=h_{+}(t)h_{}(t)\) for all \(t\in \mathbb{R}\), and \(\overline{h}=\overline{h}_{+}\overline{h}_{}\). Furthermore, \(\\varphi\_{p}:=(\int_{0}^{T}\varphi(t)^{p}\,dt)^{\frac{1}{p}}\), \(p\in[1,+\infty)\), \(\varphi\in C_{T}\).
The following lemma is a corollary of Theorem 3.1 in [36].
Lemma 2.1
 1.for each \(\lambda\in(0,1]\), each possible positive Tperiodic solution x to the equationsatisfies the inequalities \(M_{0}< x(t)< M_{1}\) and \(x'(t)< M_{2}\) for all \(t\in[0,T]\);$$ u''+\lambda f(u)u'\lambda \frac{\alpha(t)}{u^{\mu}}=\lambda h(t), $$
 2.each possible solution \(x\in(0,+\infty)\) to the equationsatisfies the inequality \(M_{0} < x < M_{1}\);$$ \frac{\overline{\alpha}}{x^{\mu}}+\overline{h}=0 $$
 3.the inequality$$ \biggl(\frac{\overline{\alpha}}{M_{0}^{\mu}}+\overline{h}\biggr) \biggl(\frac{\overline {\alpha}}{M_{1}^{\mu}}+ \overline{h}\biggr) < 0 $$
Lemma 2.2
[29]
 (H_{1}):

\(\lim_{x\rightarrow0^{+}}\int_{x}^{1}f(s)\,ds=+\infty\).
 (H_{2}):

\((\frac{\overline{\alpha}}{\overline{h}})^{\frac{1}{\mu }}>T^{\frac{1}{2}}[\frac{T}{\pi}\h\_{2} +(T^{\frac{1}{2}}(\frac{\overline{\alpha}}{\overline{h}})^{\frac{1}{\mu }}\h\_{2})^{\frac{1}{2}}]\).
Remark 2.1
3 Main results
Theorem 3.1
If (H_{1}) holds, then equation (1.6) has a positive Tperiodic solution if and only if \(\overline{h}<0\).
Proof
Theorem 3.2
Suppose that \(\overline {h}<0\) and (H_{2}) holds. Then equation (1.6) has at least one positive Tperiodic solution.
Proof
Example 3.1
Example 3.2
Remark
The above two examples can neither be studied by using the results in [31, 32, 34] and [35], since \(f(x)\) in (3.19) and in (3.20) are all singular at \(x=0\), nor be studied by using the results in [33], since the restoring force terms of \(\frac{\sin^{2}t}{x^{\frac{2}{3}}}\) in (3.19) and \(\frac{\sin^{2}8t}{x^{\frac{3}{4}}}\) in (3.20) have weak singularities at \(x=0\).
Declarations
Acknowledgements
The authors thank the referees for valuable comments. This research is supported by the NSF of China (No. 11271197).
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
 Lei, J, Zhang, M: Twist property of periodic motion of an atom near a charged wire. Lett. Math. Phys. 60(1), 917 (2002) MathSciNetView ArticleMATHGoogle Scholar
 Bevc, V, Palmer, JL, Süsskind, C: On the design of the transition region of axisymmetric magnetically focused beam valves. J. Br. Inst. Radio Eng. 18, 696708 (1958) Google Scholar
 Ye, Y, Wang, X: Nonlinear differential equations in electron beam focusing theory. Acta Math. Appl. Sin. 1, 1341 (1978) (in Chinese) Google Scholar
 Huang, J, Ruan, S, Song, J: Bifurcations in a predatorprey system of Leslie type with generalized Holling type III functional response. J. Differ. Equ. 257(6), 17211752 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Plesset, MS: The dynamics of cavitation bubbles. J. Appl. Mech. 16, 228231 (1949) Google Scholar
 Habets, P, Sanchez, L: Periodic solutions of some Liénard equations with singularities. Proc. Am. Math. Soc. 109, 10351044 (1990) MATHGoogle Scholar
 Tanaka, K: A note on generalized solutions of singular Hamiltonian systems. Proc. Am. Math. Soc. 122, 275284 (1994) MathSciNetView ArticleMATHGoogle Scholar
 Terracini, S: Remarks on periodic orbits of dynamical systems with repulsive singularities. J. Funct. Anal. 111, 213238 (1993) MathSciNetView ArticleMATHGoogle Scholar
 Gaeta, S, Manásevich, R: Existence of a pair of periodic solutions of an ode generalizing a problem in nonlinear elasticity via variational methods. J. Math. Anal. Appl. 123, 257271 (1988) View ArticleMATHGoogle Scholar
 Fonda, A: Periodic solutions for a conservative system of differential equations with a singularity of repulsive type. Nonlinear Anal. 24, 667676 (1995) MathSciNetView ArticleMATHGoogle Scholar
 Jebelean, P, Mawhin, J: Periodic solutions of singular nonlinear differential perturbations of the ordinary pLaplacian. Adv. Nonlinear Stud. 2(3), 299312 (2002) MathSciNetView ArticleMATHGoogle Scholar
 Torres, PJ: Mathematical Models with Singularities  A Zoo of Singular Creatures. Atlantis Press, Amsterdam (2015). ISBN:9789462391055 View ArticleMATHGoogle Scholar
 Nagumo, M: On the periodic solution of an ordinary differential equation of second order. In: Zenkoku Shijou Suugaku Danwakai, pp. 5461 (1944) (in Japanese). English translation in Mitio Nagumo Collected Papers, Sringer, Berlin (1993) Google Scholar
 Derwidué, L: Systemes différentiels non linéaires ayant solutions périodiques. Acad. R. Belg., Cl. Lett. Sci., V Ser. 49, 1132 (1963) MathSciNetMATHGoogle Scholar
 Fauré, R: Solutions périodiques d’équations différentielles et méthode de LeraySchauder. Ann. Inst. Fourier 14(1), 195204 (1964) MathSciNetView ArticleMATHGoogle Scholar
 Gordon, WB: Conservative dynamical systems involving strong forces. Trans. Am. Math. Soc. 204, 113135 (1975) MathSciNetView ArticleMATHGoogle Scholar
 Lazer, AC, Solimini, S: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 99, 109114 (1987) MathSciNetView ArticleMATHGoogle Scholar
 Torres, PJ: Existence of onesigned periodic solutions of some second order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 190, 643662 (2003) MathSciNetView ArticleMATHGoogle Scholar
 Torres, PJ, Zhang, M: Twist periodic solutions of repulsive singular equations. Nonlinear Anal. 56, 591599 (2004) MathSciNetView ArticleMATHGoogle Scholar
 Torres, PJ: Bounded solutions in singular equations of repulsive type. Nonlinear Anal. 32, 117125 (1998) MathSciNetView ArticleMATHGoogle Scholar
 Jiang, D, Chu, J, Zhang, M: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ. 211, 282302 (2005) MathSciNetView ArticleMATHGoogle Scholar
 del Pino, M, Manásevich, R: Infinitely many Tperiodic solutions for a problem arising in nonlinear elasticity. J. Differ. Equ. 103, 260277 (1993) MathSciNetView ArticleMATHGoogle Scholar
 del Pino, M, Manásevich, R, Montero, A: TPeriodic solutions for some second order differential equations with singularities. Proc. R. Soc. Edinb., Sect. A 120(34), 231243 (1992) MathSciNetView ArticleMATHGoogle Scholar
 Fonda, A, Manasevich, R, Zanolin, F: Subharmonics solutions for some second order differential equations with singularities. SIAM J. Math. Anal. 24, 12941311 (1993) MathSciNetView ArticleMATHGoogle Scholar
 Torres, PJ: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 232, 277284 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Chu, J, Torres, PJ, Zhang, M: Periodic solutions of second order nonautonomous singular dynamical systems. J. Differ. Equ. 239, 196212 (2007) MathSciNetView ArticleMATHGoogle Scholar
 Li, X, Zhang, Z: Periodic solutions for second order differential equations with a singular nonlinearity. Nonlinear Anal. 69, 38663876 (2008) MathSciNetView ArticleMATHGoogle Scholar
 Hakl, R, Torres, PJ, Zamora, M: Periodic solutions of singular second order differential equations: upper and lower functions. Nonlinear Anal. 74, 70787093 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Hakl, R, Torres, PJ: On periodic solutions of second order differential equations with attractiverepulsive singularities. J. Differ. Equ. 248, 111126 (2010) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, M: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203(1), 254269 (1996) MathSciNetView ArticleMATHGoogle Scholar
 Martins, R: Existence of periodic solutions for secondorder differential equations with singularities and the strong force condition. J. Math. Anal. Appl. 317, 113 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Wang, Z: Periodic solutions of Liénard equation with a singularity and a deviating argument. Nonlinear Anal., Real World Appl. 16(1), 227234 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Hakl, R, Torres, PJ, Zamora, M: Periodic solutions of singular second order differential equations: the repulsive case. Topol. Methods Nonlinear Anal. 39, 199220 (2012) MathSciNetMATHGoogle Scholar
 Lu, S: A new result on the existence of periodic solutions for Liénard equations with a singularity of repulsive type. J. Inequal. Appl. 2017, 37 (2017). doi:10.1186/s1366001612858 View ArticleMATHGoogle Scholar
 Lu, S, Zhong, T, Gao, Y: Periodic solutions of pLaplacian equations with singularities. Adv. Differ. Equ. 2016, 146 (2016). doi:10.1186/s1366201608756 MathSciNetView ArticleGoogle Scholar
 Manásevich, R, Mawhin, J: Periodic solutions for nonlinear systems with pLaplacianlike operators. J. Differ. Equ. 145, 367393 (1998) MathSciNetView ArticleMATHGoogle Scholar