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Periodic solutions for a singular damped differential equation
- Jing Li^{1},
- Shengjun Li^{2} and
- Ziheng Zhang^{3}Email author
https://doi.org/10.1186/s13661-014-0269-1
© Li et al.; licensee Springer 2015
Received: 20 October 2014
Accepted: 17 December 2014
Published: 10 January 2015
Abstract
Based on a variational approach, we prove that a second-order singular damped differential equation has at least one periodic solution when some reasonable assumptions are satisfied.
Keywords
MSC
1 Introduction
Recently, Eq. (1.1) has also been investigated by several authors; see, for instance, [12, 13] (application of Leray-Schauder alternative principle) and [14] (using Schauder’s fixed point theorem). In general cases, it is very difficult or impossible to apply variational methods to Eq. (1.1) when \(\int_{0}^{T}q(t)\,dt>0\). In this paper, we consider the case \(\int_{0}^{T}q(t)\,dt=0\) and under some reasonable assumptions, we establish the corresponding variational framework of T-periodic solutions for Eq. (1.1) on an appropriate Sobolev space and give a new criterion to guarantee the existence of at least one nontrivial T-periodic solution of Eq. (1.1) using a variant of the mountain pass theorem. We refer the reader to [15–17] for the details about variational methods.
- (H1)
\(q, g\in C(\mathbb{R}/{T\mathbb{Z}})\) with \(\int_{0}^{T} q(t)\,dt=0\);
- (H2)\(f\in C((0,\infty),\mathbb{R})\) has a repulsive singularity at \(u=0\), i.e.,$$\lim_{u\rightarrow0^{+}}f(u)=-\infty; $$
- (H3)
\(\lim_{u\rightarrow0^{+}}F(u)=+\infty\), where \(F(u)= \int_{1}^{u} f(s)\,ds\);
- (H4)
\(M=\sup\{f(s): 0< s<+\infty\}\) is bounded;
- (H5)\(\lim_{u\rightarrow+\infty} (F(u)- \bar{g} u) =+\infty\), where \(\bar{g}\) is defined by$$\bar{g}\stackrel{\mathrm{def}}{=} \frac{1}{\int_{0}^{T} \exp (\int_{0}^{t}q(s)\,ds )\,dt}\int_{0}^{T} g(t)\exp \biggl(\int_{0}^{T}q(s)\,ds \biggr)\,dt. $$
Theorem 1.1
Assume that (H1)-(H5) are satisfied. Then Eq. (1.1) has at least one nontrivial T-periodic solution.
Remark 1
The remaining part of this paper is organized as follows. Some preliminaries are presented in Section 2. In Section 3, the proof of Theorem 1.1 is given.
2 Preliminary results
Lemma 2.1
[15, Proposition 1.3] (Wirtinger’s inequality)
In order to obtain the existence of T-periodic solutions of Eq. (2.1), the following version of the mountain pass theorem will be used in our argument.
Lemma 2.2
[15, Theorem 4.10]
3 Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1.
Proof
The proof will be divided into four steps.
Step 1. \(\Phi_{\lambda}\) satisfies the (PS)-condition.
(ii) Assume that the first possibility occurs, i.e., \(m_{n}\rightarrow-\infty\) as \(n\rightarrow+\infty\). We replace \(M_{n}\) by \(-m_{n}\) in the preceding arguments, and we also get a contradiction.
Therefore, \(\Phi_{\lambda}\) satisfies the (PS) condition. This completes the proof of the claim.
Step 3. We show that there exists \(\lambda_{0}\in(0,1)\) with the property that, for every \(\lambda\in(0,\lambda_{0})\), any solution u of Eq. (2.1) satisfying \(\Phi_{\lambda}(u)\geq-m\) is such that \(\min u\geq\lambda_{0}\), and hence u is a solution of Eq. (1.1).
- (i)
\(\lambda_{n}\leq\frac{1}{n}\);
- (ii)
\(u_{n}\) is a solution of Eq. (2.1) with \(\lambda =\lambda_{n}\);
- (iii)
\(\Phi_{\lambda_{n}}(u_{n})\geq-m\);
- (iv)
\(\min u_{n}<\frac{1}{n}\).
Step 4. We prove that \(\Phi_{\lambda}\) has a mountain pass geometry for \(\lambda\leq\lambda_{0}\).
Since \(\inf_{u\in\partial\Omega}\Phi_{\lambda }(u)\geq -m\), it follows from Step 3 that \(u_{\lambda}\) is a solution of Eq. (1.1). Now the proof is finished. □
Declarations
Acknowledgements
The authors would like to thank the anonymous referee for his/her valuable comments, which have improved the correctness and presentation of the manuscript. This work is supported by the National Natural Science Foundation of China (Grant No. 11101304).
Open Access 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.
Authors’ Affiliations
References
- Mawhin, J: Topological degree and boundary value problems for nonlinear differential equations. In: Furi, M, Zecca, P (eds.) Topological Methods for Ordinary Differential Equations. Lecture Notes in Mathematics, vol. 1537, pp. 74-142. Springer, New York (1993) View ArticleGoogle Scholar
- Bonheure, D, De Coster, C: Forced singular oscillators and the method of lower and upper solutions. Topol. Methods Nonlinear Anal. 22, 297-317 (2003) MATHMathSciNetGoogle Scholar
- Rachunková, I, Tvrdý, M, Vrkoč, I: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differ. Equ. 176, 445-469 (2001) View ArticleMATHGoogle Scholar
- Zhang, MR: Periodic solutions of damped differential systems with repulsive singular forces. Proc. Am. Math. Soc. 127, 401-407 (1999) View ArticleMATHGoogle Scholar
- Franco, D, Webb, JRL: Collisionless orbits of singular and nonsingular dynamical systems. Discrete Contin. Dyn. Syst. 15, 747-757 (2006) View ArticleMATHMathSciNetGoogle 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) View ArticleMATHGoogle Scholar
- Chu, JF, Torres, PJ: Applications of Schauder’s fixed point theorem to singular differential equations. Bull. Lond. Math. Soc. 39, 653-660 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Jiang, DQ, Chu, JF, Zhang, MR: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ. 211, 282-302 (2005) View ArticleMATHMathSciNetGoogle Scholar
- Boucherif, A, Daoudi-Merzagui, N: Periodic solutions of singular nonautonomous second differential equations. Nonlinear Differ. Equ. Appl. 15, 147-158 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Daoudi-Merzagui, N, Derrab, F, Boucherif, A: Subharmonic solutions of nonautonomous second order differential equations with singular nonlinearities. Abstr. Appl. Anal. 2012, Article ID 903281 (2012) View ArticleMathSciNetGoogle Scholar
- Fonda, A, Manásevich, R, Zanolin, F: Subharmonic solutions for some second-order differential equations with singularities. SIAM J. Math. Anal. 24, 1294-1311 (1993) View ArticleMATHMathSciNetGoogle Scholar
- Chu, JF, Fan, N, Torres, PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 388, 665-675 (2012) View ArticleMATHMathSciNetGoogle Scholar
- Chen, ZB, Ren, JL: Studies on a damped differential equation with repulsive singularity. Math. Methods Appl. Sci. 36, 983-992 (2013) View ArticleMathSciNetGoogle Scholar
- Li, X, Zhang, ZH: Periodic solutions for damped differential equations with a weak repulsive singularity. Nonlinear Anal. 70, 2395-2399 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989) View ArticleMATHGoogle Scholar
- Rabinowitz, PH: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. in. Math., vol. 65. Am. Math. Soc., Providence (1986) Google Scholar
- Schechter, M: Linking Methods in Critical Point Theory. Birkhäuser, Boston (1999) View ArticleMATHGoogle Scholar
- Lazer, AC, Solimini, S: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 99, 109-114 (1987) View ArticleMATHMathSciNetGoogle Scholar
- Wu, X, Chen, SX, Teng, KM: On variational methods for a class of damped vibration problems. Nonlinear Anal. 68, 1432-1441 (2008) View ArticleMATHMathSciNetGoogle Scholar