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March  2017, 16(2): 591-609. doi: 10.3934/cpaa.2017029

## Existence and stability of periodic solutions for relativistic singular equations

 1 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 2 College of Science, Hohai University, Nanjing 210098, China 3 Department of Mathematics, Southeast University, Nanjing 211189, China 4 College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author

Received  July 2016 Revised  November 2016 Published  January 2017

In this paper, we study the existence, multiplicity and stability of positive periodic solutions of relativistic singular differential equations. The proof of the existence and multiplicity is based on the continuation theorem of coincidence degree theory, and the proof of stability is based on a known connection between the index of a periodic solution and its stability.

Citation: Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure & Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029
##### References:
 [1] C. Bereanu and J. Mawhin, Periodic solutions of nonlinear perturbations of Φ-Laplacians with possibly bounded Φ, Nonlinear Anal., 68 (2008), 1668–1681. doi: 10.1016/j.na.2006.12.049.  Google Scholar [2] H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801–810.  Google Scholar [3] C. Bereanu and P.J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., 140 (2012), 2713–2719. doi: 10.1090/S0002-9939-2011-11101-8.  Google Scholar [4] C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded Φ-Laplacians, J. Dynam. Differential Equations, 22 (2010), 463–471. doi: 10.1007/s10884-010-9172-3.  Google Scholar [5] C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular Φ-Laplacian, J. Differential Equations, 243 (2007), 536–557. doi: 10.1016/j.jde.2007.05.014.  Google Scholar [6] J. Chu and M. Li, Twist periodic solutions of second order singular differential equations, J. Math. Anal. Appl., 355 (2009), 830–838. doi: 10.1016/j.jmaa.2009.02.033.  Google Scholar [7] J. Chu, P. J. Torres and F. Wang, Radial stability of periodic solutions of the GyldenMeshcherskii-type problem, Discrete Contin. Dyn. Syst., 35 (2015), 1921–1932. doi: 10.3934/dcds.2015.35.1921.  Google Scholar [8] J. Chu, P.J. Torres and F. Wang, Twist periodic solutions for differential equations with a combined attractive-repulsive singularity, J. Math. Anal. Appl., 437 (2016), 1070–1083. doi: 10.1016/j.jmaa.2016.01.057.  Google Scholar [9] J. Chu, N. Fan and P. J. Torres, Periodic solutions for second order singular damped differential equations, J. Math. Anal. Appl., 388 (2012), 665–675. doi: 10.1016/j.jmaa.2011.09.061.  Google Scholar [10] J. Chu, P.J. Torres and M. Zhang, Periodic solutions of second order non-autonomous singular dynamical systems, J. Differential Equations, 239 (2007), 196–212. doi: 10.1016/j.jde.2007.05.007.  Google Scholar [11] J.Á. Cid and P.J. Torres, On the existence and stability of periodic solutions for pendulum-like equations with friction and Φ-Laplacian, Discrete Contin. Dyn. Syst., 33 (2013), 141–152. doi: 10.3934/dcds.2013.33.141.  Google Scholar [12] F. Forbat and A. Huaux, Détermination approchée et stabilité locale de la solution périodique d'une équation différentielle non linéaire, Mém. Publ. Soc. Sci. Arts Lett. Hainaut., 76 (1962), 3–13. Google Scholar [13] A. Fonda and R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach, J. Differential Equations, 244 (2008), 3235–3264. doi: 10.1016/j.jde.2007.11.005.  Google Scholar [14] D. Franco and P. J. Torres, Periodic solutions of singular systems without the strong force condition, Proc. Amer. Math. Soc., 136 (2008), 1229–1236. doi: 10.1090/S0002-9939-07-09226-X.  Google Scholar [15] R. Hakl and P. J. Torres, On periodic solutions of second-order differential equations with attractive-repulsive singularities, J. Differential Equations, 248 (2010), 111–126. doi: 10.1016/j.jde.2009.07.008.  Google Scholar [16] A. Huaux, Sur L'existence d'une solution périodique de l'e quation différentielle non linéaire $\ddot x + (0, 2)\dot x + \frac{x}{{1 -x}} = (0, 5)$ cos ωt, Bull. Cl. Sci. Acad. R. Belguique, 48 (1962), 494–504.  Google Scholar [17] J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic, Matematiche, 65 (2010), 97–107.  Google Scholar [18] R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equation, Lecture notes in Mathematics, vol. 568 Berlin: Springer-Verlag, 1977.  Google Scholar [19] R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type, Differential Integral Equations, 3 (1990), 275–284.  Google Scholar [20] R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. B, 3 (1989), 533–546.  Google Scholar [21] R. Ortega, Some applications of the topological degree to stability theory, in Topological methods in differential equations and inclusions (Montreal, PQ, 1994), 377–409, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , 472, Kluwer Acad. Publ. , Dordrecht, 1995.  Google Scholar [22] R. Ortega, Stability and index of periodic solutions of a nonlinear telegraph equation, Commun. Pure Appl. Anal., 4 (2005), 823–837. doi: 10.3934/cpaa.2005.4.823.  Google Scholar [23] H. N. Pishkenari, M. Behzad and A. Meghdari, Nonlinear dynamic analysis of atomic force microscopy under deterministic and random excitation, Chaos Solitons Fractals, 37 (2008), 748–762. Google Scholar [24] S. Rützel, S. I. Lee and A. Raman, Nonlinear dynamics of atomic-force-microscope probes driven in Lennard-Jones potentials, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 459 (2003), 1925–1948. Google Scholar [25] P. J. Torres, Twist solutions of a Hill's equations with singular term, Adv. Nonlinear Stud., 2 (2002), 279–287. doi: 10.1515/ans-2002-0305.  Google Scholar [26] P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations, Nonlinear Anal., 56 (2004), 591–599. doi: 10.1016/j.na.2003.10.005.  Google Scholar [27] P. J. Torres, Existence and stability of periodic solutions for second-order semilinear differential equations with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 195–201. doi: 10.1017/S0308210505000739.  Google Scholar [28] P. J. Torres, Periodic oscillations of the relativistic pendulum with friction, Phys. Lett. A, 372 (2008), 6386–6387. doi: 10.1016/j.physleta.2008.08.060.  Google Scholar [29] P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with Φ-Laplacian, Commun. Contemp. Math., 13 (2011), 283–292. doi: 10.1142/S0219199711004208.  Google Scholar [30] M. Zhang, Periodic solutions of damped differential systems with repulsive singular forces, Proc. Amer. Math. Soc., 127 (1999), 401–407. Google Scholar

show all references

##### References:
 [1] C. Bereanu and J. Mawhin, Periodic solutions of nonlinear perturbations of Φ-Laplacians with possibly bounded Φ, Nonlinear Anal., 68 (2008), 1668–1681. doi: 10.1016/j.na.2006.12.049.  Google Scholar [2] H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801–810.  Google Scholar [3] C. Bereanu and P.J. Torres, Existence of at least two periodic solutions of the forced relativistic pendulum, Proc. Amer. Math. Soc., 140 (2012), 2713–2719. doi: 10.1090/S0002-9939-2011-11101-8.  Google Scholar [4] C. Bereanu, P. Jebelean and J. Mawhin, Periodic solutions of pendulum-like perturbations of singular and bounded Φ-Laplacians, J. Dynam. Differential Equations, 22 (2010), 463–471. doi: 10.1007/s10884-010-9172-3.  Google Scholar [5] C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular Φ-Laplacian, J. Differential Equations, 243 (2007), 536–557. doi: 10.1016/j.jde.2007.05.014.  Google Scholar [6] J. Chu and M. Li, Twist periodic solutions of second order singular differential equations, J. Math. Anal. Appl., 355 (2009), 830–838. doi: 10.1016/j.jmaa.2009.02.033.  Google Scholar [7] J. Chu, P. J. Torres and F. Wang, Radial stability of periodic solutions of the GyldenMeshcherskii-type problem, Discrete Contin. Dyn. Syst., 35 (2015), 1921–1932. doi: 10.3934/dcds.2015.35.1921.  Google Scholar [8] J. Chu, P.J. Torres and F. Wang, Twist periodic solutions for differential equations with a combined attractive-repulsive singularity, J. Math. Anal. Appl., 437 (2016), 1070–1083. doi: 10.1016/j.jmaa.2016.01.057.  Google Scholar [9] J. Chu, N. Fan and P. J. Torres, Periodic solutions for second order singular damped differential equations, J. Math. Anal. Appl., 388 (2012), 665–675. doi: 10.1016/j.jmaa.2011.09.061.  Google Scholar [10] J. Chu, P.J. Torres and M. Zhang, Periodic solutions of second order non-autonomous singular dynamical systems, J. Differential Equations, 239 (2007), 196–212. doi: 10.1016/j.jde.2007.05.007.  Google Scholar [11] J.Á. Cid and P.J. Torres, On the existence and stability of periodic solutions for pendulum-like equations with friction and Φ-Laplacian, Discrete Contin. Dyn. Syst., 33 (2013), 141–152. doi: 10.3934/dcds.2013.33.141.  Google Scholar [12] F. Forbat and A. Huaux, Détermination approchée et stabilité locale de la solution périodique d'une équation différentielle non linéaire, Mém. Publ. Soc. Sci. Arts Lett. Hainaut., 76 (1962), 3–13. Google Scholar [13] A. Fonda and R. Toader, Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach, J. Differential Equations, 244 (2008), 3235–3264. doi: 10.1016/j.jde.2007.11.005.  Google Scholar [14] D. Franco and P. J. Torres, Periodic solutions of singular systems without the strong force condition, Proc. Amer. Math. Soc., 136 (2008), 1229–1236. doi: 10.1090/S0002-9939-07-09226-X.  Google Scholar [15] R. Hakl and P. J. Torres, On periodic solutions of second-order differential equations with attractive-repulsive singularities, J. Differential Equations, 248 (2010), 111–126. doi: 10.1016/j.jde.2009.07.008.  Google Scholar [16] A. Huaux, Sur L'existence d'une solution périodique de l'e quation différentielle non linéaire $\ddot x + (0, 2)\dot x + \frac{x}{{1 -x}} = (0, 5)$ cos ωt, Bull. Cl. Sci. Acad. R. Belguique, 48 (1962), 494–504.  Google Scholar [17] J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic, Matematiche, 65 (2010), 97–107.  Google Scholar [18] R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equation, Lecture notes in Mathematics, vol. 568 Berlin: Springer-Verlag, 1977.  Google Scholar [19] R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type, Differential Integral Equations, 3 (1990), 275–284.  Google Scholar [20] R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. B, 3 (1989), 533–546.  Google Scholar [21] R. Ortega, Some applications of the topological degree to stability theory, in Topological methods in differential equations and inclusions (Montreal, PQ, 1994), 377–409, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , 472, Kluwer Acad. Publ. , Dordrecht, 1995.  Google Scholar [22] R. Ortega, Stability and index of periodic solutions of a nonlinear telegraph equation, Commun. Pure Appl. Anal., 4 (2005), 823–837. doi: 10.3934/cpaa.2005.4.823.  Google Scholar [23] H. N. Pishkenari, M. Behzad and A. Meghdari, Nonlinear dynamic analysis of atomic force microscopy under deterministic and random excitation, Chaos Solitons Fractals, 37 (2008), 748–762. Google Scholar [24] S. Rützel, S. I. Lee and A. Raman, Nonlinear dynamics of atomic-force-microscope probes driven in Lennard-Jones potentials, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 459 (2003), 1925–1948. Google Scholar [25] P. J. Torres, Twist solutions of a Hill's equations with singular term, Adv. Nonlinear Stud., 2 (2002), 279–287. doi: 10.1515/ans-2002-0305.  Google Scholar [26] P. J. Torres and M. Zhang, Twist periodic solutions of repulsive singular equations, Nonlinear Anal., 56 (2004), 591–599. doi: 10.1016/j.na.2003.10.005.  Google Scholar [27] P. J. Torres, Existence and stability of periodic solutions for second-order semilinear differential equations with a singular nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 195–201. doi: 10.1017/S0308210505000739.  Google Scholar [28] P. J. Torres, Periodic oscillations of the relativistic pendulum with friction, Phys. Lett. A, 372 (2008), 6386–6387. doi: 10.1016/j.physleta.2008.08.060.  Google Scholar [29] P. J. Torres, Nondegeneracy of the periodically forced Liénard differential equation with Φ-Laplacian, Commun. Contemp. Math., 13 (2011), 283–292. doi: 10.1142/S0219199711004208.  Google Scholar [30] M. Zhang, Periodic solutions of damped differential systems with repulsive singular forces, Proc. Amer. Math. Soc., 127 (1999), 401–407. Google Scholar
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