Periodic solutions of Liénard equations with resonant isochronous potentials

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April 2013, 33(4): 1563-1581. doi: 10.3934/dcds.2013.33.1563

Periodic solutions of Liénard equations with resonant isochronous potentials

Tiantian Ma ^1, and Zaihong Wang ^1,

1.	School of Mathematical Sciences, Capital Normal University, Beijing 100048, China, China

Received September 2011 Revised September 2012 Published October 2012

Abstract
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In this paper, we study the existence and multiplicity of periodic solutions of Liénard equations $$ x''+f(x)x'+V'(x)+g(x)=p(t), $$ where $V$ is a $2\pi/n$-isochronous potential. When $F(F(x)=\int_0^xf(s)ds)$ and $g$ are bounded, we provide new sufficient conditions to ensure the existence of periodic solutions of this equations. Moreover, we prove the multiplicity of periodic solutions of the given equations under certain bounded conditions by using topological degree method. When $F, g$ satisfy a certain class of unbounded conditions, we also give sufficient conditions to ensure the existence of $2\pi$-periodic solutions of the given equations.

Keywords: Liénard equation, periodic solution., isochronous potential.

Mathematics Subject Classification: Primary: 34C25; Secondary: 34B1.

Citation: Tiantian Ma, Zaihong Wang. Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1563-1581. doi: 10.3934/dcds.2013.33.1563

References:

[1]	J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations, 143 (1998), 201-220. doi: 10.1006/jdeq.1997.3367.
[2]	D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance, Discrete Contin. Dynam. Systems, 8 (2002), 907-930.
[3]	D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance, Differential Integral equations, 15 (2002), 1139-1152.
[4]	A. Capietto and Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, J. London Math. Soc., 68 (2003), 119-132. doi: 10.1112/S0024610703004459.
[5]	A. Capietto, W. Dambrosio and Z. Wang, Coexistence of unbounded and periodic solutions to perturbed damped isochronous oscillators at resonance, Proc. Edinburgh Math. Soc., 138A (2008), 15-32.
[6]	A. Capietto, W. Dambrosio, T. Ma and Z. Wang, Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance, Discrete Contin. Dynam. Systems, (to appear).
[7]	N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321-328. doi: 10.1017/S0004972700022747.
[8]	P. O. Frederickson and A. C. Lazer, Necessary and sufficient damping in a second order oscillator, J. Differential Equations, 5 (1969), 262-270.
[9]	T. Ma and Z. Wang, Existence and infinity of periodic solutions of some second order differential equations with isochronous potentials, Z. Angew. Math. Phys., 63 (2012), 25-49. doi: 10.1007/s00033-011-0152-1.
[10]	T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Springer, New York, 1975.
[11]	C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78. doi: 10.1006/jdeq.1998.3441.
[12]	C. Fabry and A. Fonda, Periodic solutions of perturbed isochronous Hamiltonian systems at resonance, J. Differential Equations, 214 (2005), 299-325. doi: 10.1016/j.jde.2005.02.003.
[13]	C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillators at resonance, Nonlinearity, 13 (2000), 493-505. doi: 10.1088/0951-7715/13/3/302.
[14]	A. Fonda and M. Garrione, Double resonance with Landesman-Lazer conditions for planar systems of ordinary differential equations, J. Differential Equations, 250 (2011), 1052-1082. doi: 10.1016/j.jde.2010.08.006.
[15]	A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Diff. Eqns, 11 (2006), 1111-1133.
[16]	S. Fucik, "Solvability of Nonlinear Equations and Boundary Value Problems," Reidel, Boston, 1980.
[17]	A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillations at resonance, Ann. Mat. Pura. Appl., 82 (1969), 49-68. doi: 10.1007/BF02410787.
[18]	A. C. Lazer and P. J. McKenna, Existence, uniqueness and stability of oscillators in differential equations with asymmetric nonlinearities, Trans. Amer. Math. Soc., 315 (1989), 721-739. doi: 10.1090/S0002-9947-1989-0979963-1.
[19]	J. J. Landau and E. M. Lifshitz, "Mechanics, Course of Theoretical Physics," Vol. 1, Oxford: Pergamon, 1960.
[20]	Z. Wang, Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives, Discrete Contin. Dynam. Systems, 9 (2003), 751-770. doi: 10.3934/dcds.2003.9.751.
[21]	D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach-Dancer condition, J. Differential Equations, 171 (2001), 233-250. doi: 10.1006/jdeq.2000.3847.
[22]	M. Krasnosel'skii and P. Zabreiko, "Geometrical Methods of Nonlinear Analysis," Springer-Verlag, 1984.

show all references

References:

[1]	J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator, J. Differential Equations, 143 (1998), 201-220. doi: 10.1006/jdeq.1997.3367.
[2]	D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance, Discrete Contin. Dynam. Systems, 8 (2002), 907-930.
[3]	D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance, Differential Integral equations, 15 (2002), 1139-1152.
[4]	A. Capietto and Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance, J. London Math. Soc., 68 (2003), 119-132. doi: 10.1112/S0024610703004459.
[5]	A. Capietto, W. Dambrosio and Z. Wang, Coexistence of unbounded and periodic solutions to perturbed damped isochronous oscillators at resonance, Proc. Edinburgh Math. Soc., 138A (2008), 15-32.
[6]	A. Capietto, W. Dambrosio, T. Ma and Z. Wang, Unbounded solutions and periodic solutions of perturbed isochronous Hamiltonian systems at resonance, Discrete Contin. Dynam. Systems, (to appear).
[7]	N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations, Bull. Austral. Math. Soc., 15 (1976), 321-328. doi: 10.1017/S0004972700022747.
[8]	P. O. Frederickson and A. C. Lazer, Necessary and sufficient damping in a second order oscillator, J. Differential Equations, 5 (1969), 262-270.
[9]	T. Ma and Z. Wang, Existence and infinity of periodic solutions of some second order differential equations with isochronous potentials, Z. Angew. Math. Phys., 63 (2012), 25-49. doi: 10.1007/s00033-011-0152-1.
[10]	T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Springer, New York, 1975.
[11]	C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78. doi: 10.1006/jdeq.1998.3441.
[12]	C. Fabry and A. Fonda, Periodic solutions of perturbed isochronous Hamiltonian systems at resonance, J. Differential Equations, 214 (2005), 299-325. doi: 10.1016/j.jde.2005.02.003.
[13]	C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillators at resonance, Nonlinearity, 13 (2000), 493-505. doi: 10.1088/0951-7715/13/3/302.
[14]	A. Fonda and M. Garrione, Double resonance with Landesman-Lazer conditions for planar systems of ordinary differential equations, J. Differential Equations, 250 (2011), 1052-1082. doi: 10.1016/j.jde.2010.08.006.
[15]	A. Fonda and J. Mawhin, Planar differential systems at resonance, Adv. Diff. Eqns, 11 (2006), 1111-1133.
[16]	S. Fucik, "Solvability of Nonlinear Equations and Boundary Value Problems," Reidel, Boston, 1980.
[17]	A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillations at resonance, Ann. Mat. Pura. Appl., 82 (1969), 49-68. doi: 10.1007/BF02410787.
[18]	A. C. Lazer and P. J. McKenna, Existence, uniqueness and stability of oscillators in differential equations with asymmetric nonlinearities, Trans. Amer. Math. Soc., 315 (1989), 721-739. doi: 10.1090/S0002-9947-1989-0979963-1.
[19]	J. J. Landau and E. M. Lifshitz, "Mechanics, Course of Theoretical Physics," Vol. 1, Oxford: Pergamon, 1960.
[20]	Z. Wang, Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives, Discrete Contin. Dynam. Systems, 9 (2003), 751-770. doi: 10.3934/dcds.2003.9.751.
[21]	D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach-Dancer condition, J. Differential Equations, 171 (2001), 233-250. doi: 10.1006/jdeq.2000.3847.
[22]	M. Krasnosel'skii and P. Zabreiko, "Geometrical Methods of Nonlinear Analysis," Springer-Verlag, 1984.

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