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

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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

Abstract
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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 and 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.
[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.
[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.
[4]	R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79-95.
[5]	T. Ding, Nonlinear oscillations at a point of resonance, Sci. Sin., 25 (1982), 918-931.
[6]	R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Math 568, Springer-Verlag, Berlin, 1977.
[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.
[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.
[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.
[10]	M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Commun. Math. Phys., 143 (1991), 43-83. doi: 10.1007/BF02100285.
[11]	B. Liu, Boundedness in nonlinear oscillations at resonance, J. Differential Equations, 153 (1999), 142-174. doi: 10.1006/jdeq.1998.3553.
[12]	B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373. doi: 10.1006/jmaa.1998.6219.
[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.
[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.
[15]	J. Mawhin, Resonance and nonlinearity: A survey, Ukrainian Math. J., 59 (2007), 197-214. doi: 10.1007/s11253-007-0016-1.
[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.
[17]	J. Moser, On invariant curves of area preserving mappings of an annulus, Nachr. Acad. Wiss. Gottingen Math. Phys., 1962 (1962), 1-20.
[18]	R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342. doi: 10.1112/jlms/53.2.325.
[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.
[20]	C. Pan and X. Yu, Magnitude Estimates, Shandong Science and Technology Press, Jinan, 1983(Chinese version).
[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.
[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.
[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.
[24]	X. Xing and Y. Wang, Boundedness for semilinear Duffing equations at resonance, Taiwanese J. Math., 16 (2012), 1923-1949.
[25]	X. Xing, The Lagrangian Stability of Solution for Nonlinear Equations, Ph.D. thesis, Nanjing University, Nanjing, 2012.
[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.

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.
[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.
[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.
[4]	R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79-95.
[5]	T. Ding, Nonlinear oscillations at a point of resonance, Sci. Sin., 25 (1982), 918-931.
[6]	R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Math 568, Springer-Verlag, Berlin, 1977.
[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.
[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.
[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.
[10]	M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Commun. Math. Phys., 143 (1991), 43-83. doi: 10.1007/BF02100285.
[11]	B. Liu, Boundedness in nonlinear oscillations at resonance, J. Differential Equations, 153 (1999), 142-174. doi: 10.1006/jdeq.1998.3553.
[12]	B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373. doi: 10.1006/jmaa.1998.6219.
[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.
[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.
[15]	J. Mawhin, Resonance and nonlinearity: A survey, Ukrainian Math. J., 59 (2007), 197-214. doi: 10.1007/s11253-007-0016-1.
[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.
[17]	J. Moser, On invariant curves of area preserving mappings of an annulus, Nachr. Acad. Wiss. Gottingen Math. Phys., 1962 (1962), 1-20.
[18]	R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342. doi: 10.1112/jlms/53.2.325.
[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.
[20]	C. Pan and X. Yu, Magnitude Estimates, Shandong Science and Technology Press, Jinan, 1983(Chinese version).
[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.
[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.
[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.
[24]	X. Xing and Y. Wang, Boundedness for semilinear Duffing equations at resonance, Taiwanese J. Math., 16 (2012), 1923-1949.
[25]	X. Xing, The Lagrangian Stability of Solution for Nonlinear Equations, Ph.D. thesis, Nanjing University, Nanjing, 2012.
[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.

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