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Periodic and subharmonic solutions for duffing equation with a singularity
1. | Dept. of Math., Zhengzhou University, Zhengzhou 450001 |
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. |
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. |
[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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