Periodic and subharmonic solutions for duffing equation with a singularity

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May 2012, 32(5): 1557-1574. doi: 10.3934/dcds.2012.32.1557

Periodic and subharmonic solutions for duffing equation with a singularity

Zhibo Cheng ^1, and Jingli Ren ^1,

1.	Dept. of Math., Zhengzhou University, Zhengzhou 450001

Received December 2010 Revised April 2011 Published January 2012

Abstract
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This paper is devoted to the existence and multiplicity of periodic and subharmonic solutions for a superlinear Duffing equation with a singularity. In this manner, various preceding theorems are improved and sharpened. Our proof is based on a generalized version of the Poincaré-Birkhoff twist theorem.

Keywords: superlinear; singularity, harmonic and subharmonic solution, Poincaré-Birkhoff twist theorem., Duffing equation.

Mathematics Subject Classification: 34C15, 34C2.

Citation: Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1557-1574. doi: 10.3934/dcds.2012.32.1557

References:

[1]	P. Le Calvez and J. Wang, Some remarks on the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 138 (2010), 703-715. doi: 10.1090/S0002-9939-09-10105-3.
[2]	T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations," Higher Education Press, Beijing, 2004.
[3]	T. R. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378. doi: 10.1016/0022-0396(92)90076-Y.
[4]	T. R. Ding, R. Iannacci and F. Zanolin, Existence and multiplicity results for periodic solution of semilinear Duffing equation, J. Differential Equations, 105 (1993), 364-409. doi: 10.1006/jdeq.1993.1093.
[5]	W. Y. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88 (1983), 341-346. doi: 10.1090/S0002-9939-1983-0695272-2.
[6]	A. Fonda, R. Manásevich and F. Zanolin, Subharmonic solutions for some second-order differential equatins with singularities, SIAM J. Math. Anal., 24 (1993), 1294-1311. doi: 10.1137/0524074.
[7]	A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth, Nonlinear Analy., 74 (2011), 2485-2496. doi: 10.1016/j.na.2010.12.004.
[8]	P. Habets and L. Sanchez, Periodic solution of some Liénard equations with singularities, Proc. Amer. Math. Soc., 109 (1990), 1035-1044. doi: 10.2307/2048134.
[9]	D. Jiang, J. Chu and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. Differential Equations, 211 (2005), 282-302. doi: 10.1016/j.jde.2004.10.031.
[10]	Z. Opial, Sur les périodes des solutions de l'équation différentielle $ x''+g(x)= 0$, Ann. Polon. Math., 10 (1961), 49-72.
[11]	M. del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equations with singularities, Proc. R. Soc. Edinb. Sect. A, 120 (1992), 231-243. doi: 10.1017/S030821050003211X.
[12]	M. del Pino and R. Manásevich, Infinitely many $T$-periodic solutions for a problem ariding in nonlinear elasticity, J. Differential Equations, 103 (1993), 260-277. doi: 10.1006/jdeq.1993.1050.
[13]	J. L. Ren, Z. B. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing system,, Discrete Continuous Dynam. Systems B, ().
[14]	S. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal., 3 (1979), 897-904. doi: 10.1016/0362-546X(79)90057-9.
[15]	P. J. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations, 232 (2007), 277-284. doi: 10.1016/j.jde.2006.08.006.
[16]	Z.-H. Wang, Periodic solutions of the second-order differential equations with singularity, Nonlinear Anal., 58 (2004), 319-331. doi: 10.1016/j.na.2004.05.006.
[17]	J. Xia and Z.-H. Wang, Existence and multiplicity of periodic solutions for the Duffing equation with singularity, Proc. R. Soc. Edinb. Sect. A, 137 (2007), 625-645.

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References:

[1]	P. Le Calvez and J. Wang, Some remarks on the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 138 (2010), 703-715. doi: 10.1090/S0002-9939-09-10105-3.
[2]	T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations," Higher Education Press, Beijing, 2004.
[3]	T. R. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378. doi: 10.1016/0022-0396(92)90076-Y.
[4]	T. R. Ding, R. Iannacci and F. Zanolin, Existence and multiplicity results for periodic solution of semilinear Duffing equation, J. Differential Equations, 105 (1993), 364-409. doi: 10.1006/jdeq.1993.1093.
[5]	W. Y. Ding, A generalization of the Poincaré-Birkhoff theorem, Proc. Amer. Math. Soc., 88 (1983), 341-346. doi: 10.1090/S0002-9939-1983-0695272-2.
[6]	A. Fonda, R. Manásevich and F. Zanolin, Subharmonic solutions for some second-order differential equatins with singularities, SIAM J. Math. Anal., 24 (1993), 1294-1311. doi: 10.1137/0524074.
[7]	A. Fonda and R. Toader, Radially symmetric systems with a singularity and asymptotically linear growth, Nonlinear Analy., 74 (2011), 2485-2496. doi: 10.1016/j.na.2010.12.004.
[8]	P. Habets and L. Sanchez, Periodic solution of some Liénard equations with singularities, Proc. Amer. Math. Soc., 109 (1990), 1035-1044. doi: 10.2307/2048134.
[9]	D. Jiang, J. Chu and M. Zhang, Multiplicity of positive periodic solutions to superlinear repulsive singular equations, J. Differential Equations, 211 (2005), 282-302. doi: 10.1016/j.jde.2004.10.031.
[10]	Z. Opial, Sur les périodes des solutions de l'équation différentielle $ x''+g(x)= 0$, Ann. Polon. Math., 10 (1961), 49-72.
[11]	M. del Pino, R. Manásevich and A. Montero, $T$-periodic solutions for some second order differential equations with singularities, Proc. R. Soc. Edinb. Sect. A, 120 (1992), 231-243. doi: 10.1017/S030821050003211X.
[12]	M. del Pino and R. Manásevich, Infinitely many $T$-periodic solutions for a problem ariding in nonlinear elasticity, J. Differential Equations, 103 (1993), 260-277. doi: 10.1006/jdeq.1993.1050.
[13]	J. L. Ren, Z. B. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing system,, Discrete Continuous Dynam. Systems B, ().
[14]	S. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal., 3 (1979), 897-904. doi: 10.1016/0362-546X(79)90057-9.
[15]	P. J. Torres, Weak singularities may help periodic solutions to exist, J. Differential Equations, 232 (2007), 277-284. doi: 10.1016/j.jde.2006.08.006.
[16]	Z.-H. Wang, Periodic solutions of the second-order differential equations with singularity, Nonlinear Anal., 58 (2004), 319-331. doi: 10.1016/j.na.2004.05.006.
[17]	J. Xia and Z.-H. Wang, Existence and multiplicity of periodic solutions for the Duffing equation with singularity, Proc. R. Soc. Edinb. Sect. A, 137 (2007), 625-645.

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