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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 and 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.

[2]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801–810.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic, Matematiche, 65 (2010), 97–107.

[18]

R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equation, Lecture notes in Mathematics, vol. 568 Berlin: Springer-Verlag, 1977.

[19]

R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type, Differential Integral Equations, 3 (1990), 275–284.

[20]

R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. B, 3 (1989), 533–546.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

M. Zhang, Periodic solutions of damped differential systems with repulsive singular forces, Proc. Amer. Math. Soc., 127 (1999), 401–407.

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.

[2]

H. Brezis and J. Mawhin, Periodic solutions of the forced relativistic pendulum, Differential Integral Equations, 23 (2010), 801–810.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

J. Mawhin, Periodic solutions of the forced pendulum: classical vs relativistic, Matematiche, 65 (2010), 97–107.

[18]

R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equation, Lecture notes in Mathematics, vol. 568 Berlin: Springer-Verlag, 1977.

[19]

R. Ortega, Stability of a periodic problem of Ambrosetti-Prodi type, Differential Integral Equations, 3 (1990), 275–284.

[20]

R. Ortega, Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. B, 3 (1989), 533–546.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

M. Zhang, Periodic solutions of damped differential systems with repulsive singular forces, Proc. Amer. Math. Soc., 127 (1999), 401–407.

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