American Institute of Mathematical Sciences

Advanced
July  2010, 9(4): 1069-1082. doi: 10.3934/cpaa.2010.9.1069

Periodic solutions of Hamiltonian systems with anisotropic growth

Tianqing An 1, and  Zhi-Qiang Wang 2,

1. 

Department of Mathematics, Hohai University, Nanjing, 210098, China
2. 

Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, United States

Received  August 2009 Revised  February 2010 Published  April 2010

In this paper we obtain some existence and multiplicity results for periodic solutions of nonautonomous Hamiltonian systems $\dot z(t)=J\nabla H(z(t),t)$ whose Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Our results generalize some earlier work [3] of P. Felmer and [5] of P. Felmer and Z.-Q. Wang.
Keywords: growth conditions, Hamiltonian system, minimax theorems., Periodic solution.
Mathematics Subject Classification: Primary: 34C25; 58E05; Secondary: 58F0.
Citation: Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069
[1]

Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053
[2]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[3]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[4]

Stanislav Antontsev, Michel Chipot, Sergey Shmarev. Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1527-1546. doi: 10.3934/cpaa.2013.12.1527
[5]

Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1
[6]

Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064
[7]

Ran Zhuo, Fengquan Li, Boqiang Lv. Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space. Communications on Pure & Applied Analysis, 2014, 13 (3) : 977-990. doi: 10.3934/cpaa.2014.13.977
[8]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361
[9]

Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1133-1151. doi: 10.3934/jimo.2018089
[10]

Jingli Ren, Zhibo Cheng, Stefan Siegmund. Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 385-392. doi: 10.3934/dcdsb.2011.16.385
[11]

Ely Kerman. On primes and period growth for Hamiltonian diffeomorphisms. Journal of Modern Dynamics, 2012, 6 (1) : 41-58. doi: 10.3934/jmd.2012.6.41
[12]

Tomasz Adamowicz, Przemysław Górka. The Liouville theorems for elliptic equations with nonstandard growth. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2377-2392. doi: 10.3934/cpaa.2015.14.2377
[13]

Fatima Ezzahra Lembarki, Jaume Llibre. Periodic orbits for a generalized Friedmann-Robertson-Walker Hamiltonian system in dimension $6$. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1165-1211. doi: 10.3934/dcdss.2015.8.1165
[14]

Harald Garcke, Kei Fong Lam. Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4277-4308. doi: 10.3934/dcds.2017183
[15]

Saeid Abbasi-Parizi, Majid Aminnayeri, Mahdi Bashiri. Robust solution for a minimax regret hub location problem in a fuzzy-stochastic environment. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1271-1295. doi: 10.3934/jimo.2018083
[16]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[17]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789
[18]

Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399
[19]

Zuohuan Zheng, Jing Xia, Zhiming Zheng. Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 409-417. doi: 10.3934/dcds.2006.14.409
[20]

Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy