Periodic solutions for a singular Liénard equation with indefinite weight

Shiping Lu, Runyu Xue

DOI: http://dx.doi.org/10.12775/TMNA.2019.037

Abstract


In this paper, the existence of positive periodic solutions is studied for a singular Liénard equation where the weight function has an indefinite sign. Due to the lack of a priori estimates over the set of all possible positive periodic solutions in this equation, a new method is proposed for estimating a priori bounds of positive periodic solutions. By the use of a continuation theorem of the Mawhin coincidence degree, new conditions for existence of positive periodic solutions to the equation are obtained. The main results show that the singularity of coefficient function associated to the friction term at $x=0$ may help the existence of periodic solutions.

Keywords


Liénard equation; Mawhin's continuation theorem; periodic solution; singularity

Full Text:

PREVIEW FULL TEXT

References


A. Boscaggin, G. Feltrin and F. Zanolin, Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the superlinear case, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 449–474.

A. Boscaggin and F. Zanolin, Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight, J. Differential Equations 252 (2012), 2900–2921.

A. Boscaggin and F. Zanolin, Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem, Ann. Mat. Pura Appl. 194 (2015), 451–478.

J.L. Bravo and P.J. Torres, Periodic solutions of a singular equation with indefinite weight, Adv. Nonlinear Stud. 10 (2010), 927–938.

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.

G. Feltrin and F. Zanolin, Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems, Adv. Differential Equations 20 (2015), 937–982.

A. Fonda and A. Sfecci, On a singular periodic Ambrosetti–Prodi problem, Nonlinear Anal. 149 (2017), 146–155.

R.E. Gaines and J.L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Math., Springer, Berlin, vol. 568, 1997.

P. Habets and L. Sanchez, Periodic solutions of some Liénard equation swith singularities, Proc. Amer. Math. Soc. 109 (1990), 1035–1044.

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.

R. Hakl, P.J. Torres and M. Zamora, Periodic solutions of singular second order differential equations: the repulsive case, Topol. Methods Nonlinear Anal. 39 (2012), 199–220.

R. Hakl, P.J. Torres and M. Zamora, Periodic solutions of singular second order differential equations: upper and lower functions, Nonlinear Anal. 74 (2011), 7078–7093.

R. Hakl and M. Zamora, On the open problems connected to the results of Lazer and Solimini, Proc. Roy. Soc. Edinburgh Sect. A. Math. 144 (2014), 109–118.

R. Hakl and M. Zamora, Existence and uniqueness of a periodic solution to an indefinite attractive singular equation, Ann. Mat. Pura Appl. 195 (2016), 995–1009.

R. Hakl and M. Zamora, Periodic solutions to second-order indefinite singular equations, J. Differential Equations 263 (2017), 451–469.

R. Hakl and M. Zamora, Periodic solutions of an indefinite singular equation arising from the Kepler problem on the sphere, Canad. J. Math. 70 (2018), 173–190.

R. Hakl and M. Zamora, Existence and multiplicity of periodic solutions to indefinite singular equations having a non-monotone term with two singularities, Adv. Nonlinear Stud. (to appear).

P. Jebelean and J. Mawhin, Periodic solutions of singular nonlinear perturbations of the ordinary p-Laplacian, J. Adv. Nonlinear Stud. 2 (2002), no. 3, 299–312.

A.C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc. 99 (1987), 109–114.

X. Li and Z. Zhang, Periodic solutions for second order differential equations with a singular nonlinearity, Nonlinear Anal. 69 (2008), 3866–3876.

S. Lu, Y. Guo and L. Chen, Periodic solutions for Liénard equation with an indefinite singularity, Nonlinear Anal. Real World Appl. 45 (2019), 542–556.

S. Lu, Y. Wang and Y. Guo, Existence of periodic solutions of a Liénard equation with a singularity of repulsive type, Boundary Value Problems, 95 (2017), DOI: 10.1186/s13661017-0826-5.

J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991) (M. Furi and P. Zecca, eds.), Lecture Notes in Mathematics, vol. 1537, Springer, Berlin, 1993, 74–142.

M. Nagumo, On the periodic solution of an ordinary differential equation of second order, Zenkoku Shijou Suugaku Danwakai (1944), 54–61 (Japanese); English transl. in Mitio Nagumo Collected Papers, Springer–Verlag, 1993.

P.J. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations 232 (2007), 277–284.

P.J. Torres, Mathematical Models with Singularities – A Zoo of Singular Creatures, Atlantis Press, 2015.

A.J. Ureña, Periodic solutions of singular equations, Topol. Methods Nonlinear Anal. 47 (2016), 55–72.

Z. Wang, Periodic solutions of Liénard equations with a singularity and a deviating argument, Nonlinear Anal. 16 (2014), 227–234.

M. Zhang, Periodic solutions of Liénard equations with singular forces of repulsive type, J. Math. Anal. Appl. 203 (1996), no. 1, 254–269.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism