American Institute of Mathematical Sciences

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July  2016, 36(7): 3961-3991. doi: 10.3934/dcds.2016.36.3961

A new method for the boundedness of semilinear Duffing equations at resonance

Zhiguo Wang 1, , Yiqian Wang 2, and  Daxiong Piao 3,

1. 

School of Mathematical Sciences, Soochow University, Suzhou 215006
2. 

Department of Mathematics, Nanjing University, Nanjing 210093, China
3. 

School of Mathematical Sciences, Ocean University of China, Qingdao 266100

Received  February 2015 Revised  November 2015 Published  March 2016

We introduce a new method for the boundedness problem of semilinear Duffing equations at resonance. In particular, it can be used to study a class of semilinear equations at resonance without the polynomial-like growth condition. As an application, we prove the boundedness of all the solutions for the equation $\ddot{x}+n^2x+g(x)+\psi(x)=p(t)$ under the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $\psi(x)$ are periodic and $g(x)$ is bounded.
Keywords: periodic nonlinearity, Moser's theorem., boundedness, Hamiltonian system, at resonance.
Mathematics Subject Classification: Primary: 34C15; Secondary: 70H0.
Citation: Zhiguo Wang, Yiqian Wang, Daxiong Piao. A new method for the boundedness of semilinear Duffing equations at resonance. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3961-3991. doi: 10.3934/dcds.2016.36.3961
References:
[1]

J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity, 9 (1996), 1099-1111. doi: 10.1088/0951-7715/9/5/003.  Google Scholar

[2]

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.  Google Scholar

[3]

V. I. Arnold, On the behavior of an adiabatic invariant under slow periodic variation of the Hamiltonian, Sov. Math. Dokl., 3 (1962), 136-140.  Google Scholar

[4]

R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79-95.  Google Scholar

[5]

T. Ding, Nonlinear oscillations at a point of resonance, Sci. Sin., 25 (1982), 918-931.  Google Scholar

[6]

R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Math 568, Springer-Verlag, Berlin, 1977.  Google Scholar

[7]

L. Jiao, D. Piao and Y. Wang, Boundedness for general semilinear Duffing equations via the twist theorem, J. Differential Equations, 252 (2012), 91-113. doi: 10.1016/j.jde.2011.09.019.  Google Scholar

[8]

A. M. Krssnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Math. Comput. Model. 32 (2000), 1445-1455. doi: 10.1016/S0895-7177(00)00216-8.  Google Scholar

[9]

A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl., 82 (1969), 49-68. doi: 10.1007/BF02410787.  Google Scholar

[10]

M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Commun. Math. Phys., 143 (1991), 43-83. doi: 10.1007/BF02100285.  Google Scholar

[11]

B. Liu, Boundedness in nonlinear oscillations at resonance, J. Differential Equations, 153 (1999), 142-174. doi: 10.1006/jdeq.1998.3553.  Google Scholar

[12]

B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373. doi: 10.1006/jmaa.1998.6219.  Google Scholar

[13]

B. Liu, Quasi-periodic solutions of a semilinear Liénard equation at resonance, Sci. China Ser. A: Mathematics, 48 (2005), 1234-1244. doi: 10.1360/04ys0019.  Google Scholar

[14]

B. Liu, Quasi-periodic solutions of forced isochronous oscillators at resonance, J. Differential Equations, 246 (2009), 3471-3495. doi: 10.1016/j.jde.2009.02.015.  Google Scholar

[15]

J. Mawhin, Resonance and nonlinearity: A survey, Ukrainian Math. J., 59 (2007), 197-214. doi: 10.1007/s11253-007-0016-1.  Google Scholar

[16]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[17]

J. Moser, On invariant curves of area preserving mappings of an annulus, Nachr. Acad. Wiss. Gottingen Math. Phys., 1962 (1962), 1-20.  Google Scholar

[18]

R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342. doi: 10.1112/jlms/53.2.325.  Google Scholar

[19]

R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc., 79 (1999), 381-413. doi: 10.1112/S0024611599012034.  Google Scholar

[20]

C. Pan and X. Yu, Magnitude Estimates, Shandong Science and Technology Press, Jinan, 1983(Chinese version). Google Scholar

[21]

H. Rüssman, On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math., Springer-Verlag, Berlin, 1007 (1983), 677-718. doi: 10.1007/BFb0061441.  Google Scholar

[22]

Y. Wang, Boundedness of solutions in a class of Duffing equations with oscillating potentials, Nonlinear Anal.TAM, 71 (2009), 2906-2917. doi: 10.1016/j.na.2009.01.172.  Google Scholar

[23]

X. Wang, Invariant tori and boundedness in asymmetric oscillations, Acta Math. Sinica(Engl. Ser.),19 (2003), 765-782. doi: 10.1007/s10114-003-0249-3.  Google Scholar

[24]

X. Xing and Y. Wang, Boundedness for semilinear Duffing equations at resonance, Taiwanese J. Math., 16 (2012), 1923-1949.  Google Scholar

[25]

X. Xing, The Lagrangian Stability of Solution for Nonlinear Equations, Ph.D. thesis, Nanjing University, Nanjing, 2012. Google Scholar

[26]

J. Xu and J. You, Persistence of lower-dimensional tori under the first Melnikov's nonresonance condition, J. Math. Pures. Appl., 80 (2001), 1045-1067. doi: 10.1016/S0021-7824(01)01221-1.  Google Scholar

show all references

References:
[1]

J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity, 9 (1996), 1099-1111. doi: 10.1088/0951-7715/9/5/003.  Google Scholar

[2]

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.  Google Scholar

[3]

V. I. Arnold, On the behavior of an adiabatic invariant under slow periodic variation of the Hamiltonian, Sov. Math. Dokl., 3 (1962), 136-140.  Google Scholar

[4]

R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79-95.  Google Scholar

[5]

T. Ding, Nonlinear oscillations at a point of resonance, Sci. Sin., 25 (1982), 918-931.  Google Scholar

[6]

R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Math 568, Springer-Verlag, Berlin, 1977.  Google Scholar

[7]

L. Jiao, D. Piao and Y. Wang, Boundedness for general semilinear Duffing equations via the twist theorem, J. Differential Equations, 252 (2012), 91-113. doi: 10.1016/j.jde.2011.09.019.  Google Scholar

[8]

A. M. Krssnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Math. Comput. Model. 32 (2000), 1445-1455. doi: 10.1016/S0895-7177(00)00216-8.  Google Scholar

[9]

A. C. Lazer and D. E. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl., 82 (1969), 49-68. doi: 10.1007/BF02410787.  Google Scholar

[10]

M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Commun. Math. Phys., 143 (1991), 43-83. doi: 10.1007/BF02100285.  Google Scholar

[11]

B. Liu, Boundedness in nonlinear oscillations at resonance, J. Differential Equations, 153 (1999), 142-174. doi: 10.1006/jdeq.1998.3553.  Google Scholar

[12]

B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373. doi: 10.1006/jmaa.1998.6219.  Google Scholar

[13]

B. Liu, Quasi-periodic solutions of a semilinear Liénard equation at resonance, Sci. China Ser. A: Mathematics, 48 (2005), 1234-1244. doi: 10.1360/04ys0019.  Google Scholar

[14]

B. Liu, Quasi-periodic solutions of forced isochronous oscillators at resonance, J. Differential Equations, 246 (2009), 3471-3495. doi: 10.1016/j.jde.2009.02.015.  Google Scholar

[15]

J. Mawhin, Resonance and nonlinearity: A survey, Ukrainian Math. J., 59 (2007), 197-214. doi: 10.1007/s11253-007-0016-1.  Google Scholar

[16]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7.  Google Scholar

[17]

J. Moser, On invariant curves of area preserving mappings of an annulus, Nachr. Acad. Wiss. Gottingen Math. Phys., 1962 (1962), 1-20.  Google Scholar

[18]

R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342. doi: 10.1112/jlms/53.2.325.  Google Scholar

[19]

R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc., 79 (1999), 381-413. doi: 10.1112/S0024611599012034.  Google Scholar

[20]

C. Pan and X. Yu, Magnitude Estimates, Shandong Science and Technology Press, Jinan, 1983(Chinese version). Google Scholar

[21]

H. Rüssman, On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math., Springer-Verlag, Berlin, 1007 (1983), 677-718. doi: 10.1007/BFb0061441.  Google Scholar

[22]

Y. Wang, Boundedness of solutions in a class of Duffing equations with oscillating potentials, Nonlinear Anal.TAM, 71 (2009), 2906-2917. doi: 10.1016/j.na.2009.01.172.  Google Scholar

[23]

X. Wang, Invariant tori and boundedness in asymmetric oscillations, Acta Math. Sinica(Engl. Ser.),19 (2003), 765-782. doi: 10.1007/s10114-003-0249-3.  Google Scholar

[24]

X. Xing and Y. Wang, Boundedness for semilinear Duffing equations at resonance, Taiwanese J. Math., 16 (2012), 1923-1949.  Google Scholar

[25]

X. Xing, The Lagrangian Stability of Solution for Nonlinear Equations, Ph.D. thesis, Nanjing University, Nanjing, 2012. Google Scholar

[26]

J. Xu and J. You, Persistence of lower-dimensional tori under the first Melnikov's nonresonance condition, J. Math. Pures. Appl., 80 (2001), 1045-1067. doi: 10.1016/S0021-7824(01)01221-1.  Google Scholar

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