# American Institute of Mathematical Sciences

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

 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.
Citation: Zhiguo Wang, Yiqian Wang, Daxiong Piao. A new method for the boundedness of semilinear Duffing equations at resonance. Discrete & Continuous Dynamical Systems - A, 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.  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.  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.   Google Scholar [4] R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem,, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79.   Google Scholar [5] T. Ding, Nonlinear oscillations at a point of resonance,, Sci. Sin., 25 (1982), 918.   Google Scholar [6] R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations,, Lecture Notes in Math 568, (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.  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), 32 (2000), 1445.  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.  doi: 10.1007/BF02410787.  Google Scholar [10] M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials,, Commun. Math. Phys., 143 (1991), 43.  doi: 10.1007/BF02100285.  Google Scholar [11] B. Liu, Boundedness in nonlinear oscillations at resonance,, J. Differential Equations, 153 (1999), 142.  doi: 10.1006/jdeq.1998.3553.  Google Scholar [12] B. Liu, Boundedness in asymmetric oscillations,, J. Math. Anal. Appl., 231 (1999), 355.  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.  doi: 10.1360/04ys0019.  Google Scholar [14] B. Liu, Quasi-periodic solutions of forced isochronous oscillators at resonance,, J. Differential Equations, 246 (2009), 3471.  doi: 10.1016/j.jde.2009.02.015.  Google Scholar [15] J. Mawhin, Resonance and nonlinearity: A survey,, Ukrainian Math. J., 59 (2007), 197.  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, (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.   Google Scholar [18] R. Ortega, Asymmetric oscillators and twist mappings,, J. London Math. Soc., 53 (1996), 325.  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.  doi: 10.1112/S0024611599012034.  Google Scholar [20] C. Pan and X. Yu, Magnitude Estimates,, Shandong Science and Technology Press, (1983).   Google Scholar [21] H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, Lecture Notes in Math., 1007 (1983), 677.  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.  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.  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.   Google Scholar [25] X. Xing, The Lagrangian Stability of Solution for Nonlinear Equations,, Ph.D. thesis, (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.  doi: 10.1016/S0021-7824(01)01221-1.  Google Scholar

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##### References:
 [1] J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance,, Nonlinearity, 9 (1996), 1099.  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.  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.   Google Scholar [4] R. Dieckerhoff and E. Zehnder, Boundedness of solutions via the twist theorem,, Ann. Scuola Norm. Sup. Pisa, 14 (1987), 79.   Google Scholar [5] T. Ding, Nonlinear oscillations at a point of resonance,, Sci. Sin., 25 (1982), 918.   Google Scholar [6] R. E. Gaines and J. Mawhin, Coincidence Degree, and Nonlinear Differential Equations,, Lecture Notes in Math 568, (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.  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), 32 (2000), 1445.  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.  doi: 10.1007/BF02410787.  Google Scholar [10] M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials,, Commun. Math. Phys., 143 (1991), 43.  doi: 10.1007/BF02100285.  Google Scholar [11] B. Liu, Boundedness in nonlinear oscillations at resonance,, J. Differential Equations, 153 (1999), 142.  doi: 10.1006/jdeq.1998.3553.  Google Scholar [12] B. Liu, Boundedness in asymmetric oscillations,, J. Math. Anal. Appl., 231 (1999), 355.  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.  doi: 10.1360/04ys0019.  Google Scholar [14] B. Liu, Quasi-periodic solutions of forced isochronous oscillators at resonance,, J. Differential Equations, 246 (2009), 3471.  doi: 10.1016/j.jde.2009.02.015.  Google Scholar [15] J. Mawhin, Resonance and nonlinearity: A survey,, Ukrainian Math. J., 59 (2007), 197.  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, (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.   Google Scholar [18] R. Ortega, Asymmetric oscillators and twist mappings,, J. London Math. Soc., 53 (1996), 325.  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.  doi: 10.1112/S0024611599012034.  Google Scholar [20] C. Pan and X. Yu, Magnitude Estimates,, Shandong Science and Technology Press, (1983).   Google Scholar [21] H. Rüssman, On the existence of invariant curves of twist mappings of an annulus,, Lecture Notes in Math., 1007 (1983), 677.  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.  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.  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.   Google Scholar [25] X. Xing, The Lagrangian Stability of Solution for Nonlinear Equations,, Ph.D. thesis, (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.  doi: 10.1016/S0021-7824(01)01221-1.  Google Scholar
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