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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

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 & 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.  Google Scholar

[2]

T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations," Higher Education Press, Beijing, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

Z. Opial, Sur les périodes des solutions de l'équation différentielle $ x''+g(x)= 0$, Ann. Polon. Math., 10 (1961), 49-72.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. L. Ren, Z. B. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing system,, Discrete Continuous Dynam. Systems B, ().   Google Scholar

[14]

S. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal., 3 (1979), 897-904. doi: 10.1016/0362-546X(79)90057-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

T. R. Ding, "Applications of Qualitative Methods of Ordinary Differential Equations," Higher Education Press, Beijing, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

Z. Opial, Sur les périodes des solutions de l'équation différentielle $ x''+g(x)= 0$, Ann. Polon. Math., 10 (1961), 49-72.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

J. L. Ren, Z. B. Cheng and S. Siegmund, Positive periodic solution for Brillouin electron beam focusing system,, Discrete Continuous Dynam. Systems B, ().   Google Scholar

[14]

S. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal., 3 (1979), 897-904. doi: 10.1016/0362-546X(79)90057-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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