American Institute of Mathematical Sciences

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

Periodic and subharmonic solutions for duffing equation with a singularity

 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.
Citation: Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete & Continuous Dynamical Systems - A, 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.  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, (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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.   Google Scholar

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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.  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, (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.  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.  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.  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.  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.  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.  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.  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.   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.  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.  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.  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.  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.  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.   Google Scholar
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