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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

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1006/jdeq.1997.3367.  Google Scholar

[2]

D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance,, Discrete Contin. Dynam. Systems, 8 (2002), 907.   Google Scholar

[3]

D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance,, Differential Integral equations, 15 (2002), 1139.   Google Scholar

[4]

A. Capietto and Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance,, J. London Math. Soc., 68 (2003), 119.  doi: 10.1112/S0024610703004459.  Google Scholar

[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.   Google Scholar

[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, ().   Google Scholar

[7]

N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations,, Bull. Austral. Math. Soc., 15 (1976), 321.  doi: 10.1017/S0004972700022747.  Google Scholar

[8]

P. O. Frederickson and A. C. Lazer, Necessary and sufficient damping in a second order oscillator,, J. Differential Equations, 5 (1969), 262.   Google Scholar

[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.  doi: 10.1007/s00033-011-0152-1.  Google Scholar

[10]

T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions,", Springer, (1975).   Google Scholar

[11]

C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators,, J. Differential Equations, 147 (1998), 58.  doi: 10.1006/jdeq.1998.3441.  Google Scholar

[12]

C. Fabry and A. Fonda, Periodic solutions of perturbed isochronous Hamiltonian systems at resonance,, J. Differential Equations, 214 (2005), 299.  doi: 10.1016/j.jde.2005.02.003.  Google Scholar

[13]

C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillators at resonance,, Nonlinearity, 13 (2000), 493.  doi: 10.1088/0951-7715/13/3/302.  Google Scholar

[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.  doi: 10.1016/j.jde.2010.08.006.  Google Scholar

[15]

A. Fonda and J. Mawhin, Planar differential systems at resonance,, Adv. Diff. Eqns, 11 (2006), 1111.   Google Scholar

[16]

S. Fucik, "Solvability of Nonlinear Equations and Boundary Value Problems,", Reidel, (1980).   Google Scholar

[17]

A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillations at resonance,, Ann. Mat. Pura. Appl., 82 (1969), 49.  doi: 10.1007/BF02410787.  Google Scholar

[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.  doi: 10.1090/S0002-9947-1989-0979963-1.  Google Scholar

[19]

J. J. Landau and E. M. Lifshitz, "Mechanics, Course of Theoretical Physics,", Vol. 1, (1960).   Google Scholar

[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.  doi: 10.3934/dcds.2003.9.751.  Google Scholar

[21]

D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach-Dancer condition,, J. Differential Equations, 171 (2001), 233.  doi: 10.1006/jdeq.2000.3847.  Google Scholar

[22]

M. Krasnosel'skii and P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,", Springer-Verlag, (1984).   Google Scholar

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.  doi: 10.1006/jdeq.1997.3367.  Google Scholar

[2]

D. Bonheure, C. Fabry and D. Smets, Periodic solutions of forced isochronous oscillator at resonance,, Discrete Contin. Dynam. Systems, 8 (2002), 907.   Google Scholar

[3]

D. Bonheure and C. Fabry, Unbounded solutions of forced isochronous oscillators at resonance,, Differential Integral equations, 15 (2002), 1139.   Google Scholar

[4]

A. Capietto and Z. Wang, Periodic solutions of Liénard equations with asymmetric nonlinearities at resonance,, J. London Math. Soc., 68 (2003), 119.  doi: 10.1112/S0024610703004459.  Google Scholar

[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.   Google Scholar

[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, ().   Google Scholar

[7]

N. Dancer, Boundary value problems for weakly nonlinear ordinary differential equations,, Bull. Austral. Math. Soc., 15 (1976), 321.  doi: 10.1017/S0004972700022747.  Google Scholar

[8]

P. O. Frederickson and A. C. Lazer, Necessary and sufficient damping in a second order oscillator,, J. Differential Equations, 5 (1969), 262.   Google Scholar

[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.  doi: 10.1007/s00033-011-0152-1.  Google Scholar

[10]

T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions,", Springer, (1975).   Google Scholar

[11]

C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators,, J. Differential Equations, 147 (1998), 58.  doi: 10.1006/jdeq.1998.3441.  Google Scholar

[12]

C. Fabry and A. Fonda, Periodic solutions of perturbed isochronous Hamiltonian systems at resonance,, J. Differential Equations, 214 (2005), 299.  doi: 10.1016/j.jde.2005.02.003.  Google Scholar

[13]

C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillators at resonance,, Nonlinearity, 13 (2000), 493.  doi: 10.1088/0951-7715/13/3/302.  Google Scholar

[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.  doi: 10.1016/j.jde.2010.08.006.  Google Scholar

[15]

A. Fonda and J. Mawhin, Planar differential systems at resonance,, Adv. Diff. Eqns, 11 (2006), 1111.   Google Scholar

[16]

S. Fucik, "Solvability of Nonlinear Equations and Boundary Value Problems,", Reidel, (1980).   Google Scholar

[17]

A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillations at resonance,, Ann. Mat. Pura. Appl., 82 (1969), 49.  doi: 10.1007/BF02410787.  Google Scholar

[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.  doi: 10.1090/S0002-9947-1989-0979963-1.  Google Scholar

[19]

J. J. Landau and E. M. Lifshitz, "Mechanics, Course of Theoretical Physics,", Vol. 1, (1960).   Google Scholar

[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.  doi: 10.3934/dcds.2003.9.751.  Google Scholar

[21]

D. Qian, Infinity of subharmonics for asymmetric Duffing equations with the Lazer-Leach-Dancer condition,, J. Differential Equations, 171 (2001), 233.  doi: 10.1006/jdeq.2000.3847.  Google Scholar

[22]

M. Krasnosel'skii and P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,", Springer-Verlag, (1984).   Google Scholar

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