Boundedness of solutions for semilinear reversible systems

Author:
Xiong Li

Translated by:

Journal:
Proc. Amer. Math. Soc. **132** (2004), 2057-2066

MSC (2000):
Primary 34C11

DOI:
https://doi.org/10.1090/S0002-9939-04-07284-3

Published electronically:
January 20, 2004

MathSciNet review:
2053978

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we will study the boundedness of all solutions for second-order differential equations

where and satisfies the sublinear growth condition. Since the system in general is non-Hamiltonian, we have to introduce reversibility assumptions to apply the twist theorem for reversible mappings. Under some suitable conditions we then obtain the existence of invariant tori and thus the boundedness of all solutions.

**1.**R. Dieckerhoff and E. Zehnder,*Boundedness of solutions via the twist theorem*, Ann. Scuola Norm. Sup. Pisa Cl. Sci.,**14**(1987), 79-95. MR**89e:34066****2.**T. Ding,*Nonlinear oscillations at a point of resonance*, Scientia Sinica Ser. A,**25**(1982), 918-931. MR**84c:34058****3.**M. Kunze, T. Kupper and B. Liu,*Boundedness and unboundedness of solutions for reversible oscillators at resonance*, Nonlinearity,**14**(5) (2001), 1105-1122. MR**2002g:34079****4.**T. Kupper and J. You,*Existence of quasiperiodic solutions and Littlewood's boundedness problem of Duffing equations with subquadratic potentials*, Nonlinear Anal.,**35**(1999), 549-559. MR**99i:34064****5.**M. Levi,*Quasi-periodic motions in superquadratic time-periodic potentials*, Comm. Math. Phys.,**143**(1) (1991), 43-83. MR**93i:34080****6.**X. Li,*Boundedness of solutions for sublinear reversible systems*, Science in China (Series A),**44**(2) (2001), 137-144. MR**2002a:34054****7.**X. Li,*Boundedness of solutions for superlinear reversible systems*, Chinese Ann. Math. (Series B),**22B**(1) (2001), 31-46. MR**2002a:34053****8.**X. Li,*Invariant tori for semilinear reversible systems*, preprint.**9.**B. Liu,*On Littlewood's boundedness problem for sublinear Duffing equations*, Trans. Amer. Math. Soc.,**353**(4) (2001), 1567-1585. MR**2001m:34084****10.**B. Liu,*Boundedness in asymmetric oscillations*, J. Math. Anal. Appl.,**231**(1999), 355-373. MR**2000c:34093****11.**B. Liu,*Boundedness in nonlinear oscillations at resonance*, J. Differential Equations,**153**(1999), 142-174. MR**2000d:34075****12.**B. Liu,*Boundedness of solutions for semilinear Duffing equations*, J. Differential Equations,**145**(1998), 119-144. MR**99e:34041****13.**B. Liu and F. Zanolin,*Boundedness of solutions for second order quasilinear ODEs*, preprint.**14.**B. Liu,*Invariant curves of reversible mappings with small twist*, preprint.**15.**B. Liu,*An application of KAM theorem of reversible systems*, Sci. China Ser. A,**34**(1991), 1068-1078. MR**93d:58143****16.**G. Morris,*A case of boundedness in Littlewood's problem on oscillatory differential equations*, Bull. Austral. Math. Soc.,**14**(1976), 71-93. MR**53:6019****17.**R. Ortega,*Invariant curves of mappings with averaged small twist*, Advanced Nonlinear Studies,**1**(2001), 14-39. MR**2002h:37067****18.**R. Ortega,*Boundedness in a piecewise linear oscillator and a variant of the small twist theorem*, Proceedings London Math. Soc.,**79**(1999), 381-413. MR**2000g:34055****19.**R. Ortega,*Asymmetric oscillators and twist mappings*, J. London Math. Soc.,**53**(1996), 325-342. MR**96k:34093****20.**M. Sevryuk,*Reversible systems*, Lecture Notes in Mathematics, Vol. 1211, Springer-Verlag, Berlin, 1986. MR**88b:58058****21.**J. You,*Boundedness for solutions of superlinear Duffing equations via the twist theorem*, Sci. China Ser. A,**35**(1992), 399-412. MR**95c:34067****22.**R. Yuan,*Quasiperiodic solutions and boundedness of solutions for a class of nonlinear differential equations of second order*, Nonlinear Anal.,**31**(1998), 649-664. MR**99a:34130****23.**X. Yuan,*Invariant tori of Duffing-type equations*, J. Differential Equations,**142**(1998), 231-262. MR**99a:34135**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
34C11

Retrieve articles in all journals with MSC (2000): 34C11

Additional Information

**Xiong Li**

Affiliation:
Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China

Email:
xli@bnu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-04-07284-3

Keywords:
Boundedness of solutions,
KAM theory,
reversible systems

Received by editor(s):
November 5, 2002

Received by editor(s) in revised form:
March 2, 2003, and March 21, 2003

Published electronically:
January 20, 2004

Additional Notes:
This project (10301006) was supported by the NSFC

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2004
American Mathematical Society