Frontiers of Mathematics in China >
Boundedness of semilinear Duffing equations with singularity
Received date: 20 Dec 2011
Accepted date: 04 Aug 2014
Published date: 29 Oct 2014
Copyright
We prove the boundedness of all solutions for the equation x″ + V′(x) = DxG(x, t), where V(x) is of singular potential, i.e., limx→-1V(x) = +∞, and G(x, t) is bounded and periodic in t. We give sufficient conditions on V(x) and G(x, t) to ensure that all solutions are bounded.
Xiumei XING , Lei JIAO . Boundedness of semilinear Duffing equations with singularity[J]. Frontiers of Mathematics in China, 2014 , 9(6) : 1427 -1452 . DOI: 10.1007/s11464-014-0424-0
1 |
Alonso J, Ortega R. Unbounded solutions of semilinear equations at resonance. Nonlinearity, 1996, 9: 1099-1111
|
2 |
Arnold V. On the behavior of an adiabatic invariant under a slow periodic change of the Hamiltonian. Dokl Akad Nauk, 1962, 142(4): 758-761 (Transl Sov Math Dokl, 3: 136-139)
|
3 |
Capietto A, Dambrosio W, Liu B. On the boundedness of solutions to a nonlinear singular oscillator. Z Angew Math Phys, 2009, 60: 1007-1034
|
4 |
Dieckerhoff R, Zehnder E. Boundedness of solutions via the twist theorem. Ann Sc Norm Super Pisa Cl Sci, 1987, 14: 79-95
|
5 |
Jiao L, Piao D, Wang Y. Boundedness for general semilinear Duffing equations via the twist theorem. J Differential Equations, 2012, 252: 91-113
|
6 |
Lazer A, Leach D. Bounded perturbations of forced harmonic oscillators at resonance. Ann Mat Pura Appl, 1969, 82: 49-68
|
7 |
Levi M. Quasiperiodic motions in superquadratic time-periodic potentials. Comm Math Phys, 1991, 143: 43-83
|
8 |
Littlewood J. Unbounded solutions of y″ + g(y) = p(t). J Lond Math Soc, 1966, 41: 491-496
|
9 |
Liu B. Boundedness in nonlinear oscillations at resonance. J Differential Equations, 1999, 153: 142-174
|
10 |
Liu B. Boundedness in asymmetric oscillations. J Math Anal Appl, 1999, 231: 355-373
|
11 |
Liu B. Quasi-periodic solutions of forced isochronous oscillators at resonance. J Differential Equa<?Pub Caret?>tions, 2009, 246: 3471-3495
|
12 |
Ma S, Wu J. A small twist theorem and boundedness of solutions for semilinear Duffing equations at resonance. Nonlinear Anal, 2007, 67(1): 200-237
|
13 |
Mawhin J. Resonance and nonlinearity: A survey. Ukrainian Math J, 2007, 59(2): 197-214
|
14 |
Morris G. A case of boundedness in Littlewood’s problem on oscillatory differential equations. Bull Aust Math Soc, 1976, 14: 71-93
|
15 |
Moser J. On invariant curves of area-preserving mappings of an annulus. Nachr Akad Wiss Göttingen Math-Phys Kl II, 1962, 1: 1-20
|
16 |
Ortega R. Boundedness in a piecewise linear oscillator and a variant of the small twist theorem. Proc Lond Math Soc, 1999, 79: 381-413
|
17 |
Xu J, You J. Persistence of lower-dimensional tori under the first Melnikov’s nonresonance condition. J Math Pures Appl, 2001, 80(10): 1045-1067
|
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