American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities
  • CPAA Home
  • This Issue
  • Next Article
    High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems
January  2014, 13(1): 75-95. doi: 10.3934/cpaa.2014.13.75

Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems

Xingyong Zhang 1, and  Xianhua Tang 2,

1. 

Department of Mathematics, Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, China
2. 

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Received  May 2011 Revised  April 2012 Published  July 2013

In this paper, some existence theorems are obtained for periodic solutions of second order Hamiltonian systems under non-quadratic conditions by using the minimax principle. Our results unite, extend and improve those relative works in some known literature.
Keywords: critical point, linking theorem., Second order Hamiltonian system, nonconstant periodic solution, nontrivial periodic solution.
Mathematics Subject Classification: Primary: 34C25, 37J45; Secondary: 35J20, 35J2.
Citation: Xingyong Zhang, Xianhua Tang. Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 75-95. doi: 10.3934/cpaa.2014.13.75
References:
[1]

E. A. B. Ailva, Subharmonic solutions for subquadratic Hamiltonian systems,, J. Diff. Eqs., 115 (1995), 120. doi: 10.1006/jdeq.1995.1007. Google Scholar

[2]

K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems,", Birkh\, (1993). doi: 10.1007/978-1-4612-0385-8. Google Scholar

[3]

Y. Ding, "Variational Methods for Strongly Indefinite Problems,", World Scientific Publishing Co. Pte. Ltd., (2007). Google Scholar

[4]

I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems,, Invent. Math., 81 (1985), 155. doi: 10.1007/BF01388776. Google Scholar

[5]

I. Ekeland, "Convexity Method in Hamiltonian Mechanics,", Springer-Verlag, (1990). doi: 10.1007/978-3-642-74331-3. Google Scholar

[6]

G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems,, Nonlinear Anal., 27 (1996), 821. doi: 10.1016/0362-546X(95)00077-9. Google Scholar

[7]

G. Fei, S. K. Kim and T. Wang, Minimal period estimates of periodic solutions for superquadratic Hamiltonian systems,, J. Math. Anal. Appl., 238 (1999), 216. doi: 10.1006/jmaa.1999.6527. Google Scholar

[8]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems,, Electronic Journal of Differential Equations, 2002 (2002), 1. Google Scholar

[9]

Q. Jiang and S. Ma, Periodic solution for a class of subquadratic second order Hamiltonian system,, Jounal of Southwest China normal University (Natural Science), 32 (2007), 6. Google Scholar

[10]

Sophia Th. Kyritsi and Nikolaos S. Papageorgiou, On superquadratic periodic systems with indefinite linear part,, Nonlinear Anal. TMA., 72 (2010), 946. doi: 10.1016/j.na.2009.07.035. Google Scholar

[11]

S. Luan and A. Mao, Periodic solutions for a class of non-autonomous Hamiltonian systems,, Nonlinear Anal., 61 (2005), 1413. doi: 10.1016/j.na.2005.01.108. Google Scholar

[12]

S. Luan and A. Mao, Periodic solutions of nonautonomous second order Hamiltonian systems,, Acta Mathematica Sinica, 21 (2005), 685. doi: 10.1007/s10114-005-0532-6. Google Scholar

[13]

S. Li and M. Willem, Applications of local linking to critical point theory,, J. Math. Anal. Appl., 189 (1995), 6. doi: 10.1006/jmaa.1995.1002. Google Scholar

[14]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer-Verlag, (1989). doi: 10.1007/978-1-4757-2061-7. Google Scholar

[15]

K. Perera, Critical groups of critical points produced by local linking with applications,, Abstr. Appl. Anal., 3 (1998), 437. doi: 10.1155/S1085337598000657. Google Scholar

[16]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", in: CBMS Regional Conf. Ser. in Math., (1986). Google Scholar

[17]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 31 (1978), 157. doi: 10.1002/cpa.3160310203. Google Scholar

[18]

M. Schechter, Periodic non-autonomous second-order dynamical systems,, J. Differential Equations, 223 (2006), 290. doi: 10.1016/j.jde.2005.02.022. Google Scholar

[19]

M. Schechter, "Minimax Systems and Critical Point Theory,", Birkh\, (2009). doi: 10.1007/978-0-8176-4902-9. Google Scholar

[20]

C. L. Tang, Periodic solutions of nonautonomous second order systems with sublinear nonlinearity,, Proc. Amer. Math. Soc. \textbf{126} (1998), 126 (1998), 3263. doi: 10.1090/S0002-9939-98-04706-6. Google Scholar

[21]

Z. L. Tao and C. L. Tang, Periodic and subharmonic solutions of second order Hamiltonian systems,, J. Math. Anal. Appl., 293 (2004), 435. doi: 10.1016/j.jmaa.2003.11.007. Google Scholar

[22]

Z. L. Tao and C. L. Tang, Periodic solutions of nonquadratic second order Hamiltonian systems,, (Chinese), 27 (2002), 841. Google Scholar

[23]

Z. L. Tao, S. Yan and S. L. Wu, Periodic solutions for a class of superquadratic Hamiltonian systems,, J. Math. Anal. Appl., 331 (2007), 152. doi: 10.1016/j.jmaa.2006.08.041. Google Scholar

[24]

Y. W. Ye and C. L. Tang, Periodic and subharmonic soltions for a class of superquadratic second order Hamiltonian systems,, Nonlinear Anal., 71 (2009), 2298. doi: 10.1016/j.na.2009.01.064. Google Scholar

show all references

References:
[1]

E. A. B. Ailva, Subharmonic solutions for subquadratic Hamiltonian systems,, J. Diff. Eqs., 115 (1995), 120. doi: 10.1006/jdeq.1995.1007. Google Scholar

[2]

K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems,", Birkh\, (1993). doi: 10.1007/978-1-4612-0385-8. Google Scholar

[3]

Y. Ding, "Variational Methods for Strongly Indefinite Problems,", World Scientific Publishing Co. Pte. Ltd., (2007). Google Scholar

[4]

I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems,, Invent. Math., 81 (1985), 155. doi: 10.1007/BF01388776. Google Scholar

[5]

I. Ekeland, "Convexity Method in Hamiltonian Mechanics,", Springer-Verlag, (1990). doi: 10.1007/978-3-642-74331-3. Google Scholar

[6]

G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems,, Nonlinear Anal., 27 (1996), 821. doi: 10.1016/0362-546X(95)00077-9. Google Scholar

[7]

G. Fei, S. K. Kim and T. Wang, Minimal period estimates of periodic solutions for superquadratic Hamiltonian systems,, J. Math. Anal. Appl., 238 (1999), 216. doi: 10.1006/jmaa.1999.6527. Google Scholar

[8]

G. Fei, On periodic solutions of superquadratic Hamiltonian systems,, Electronic Journal of Differential Equations, 2002 (2002), 1. Google Scholar

[9]

Q. Jiang and S. Ma, Periodic solution for a class of subquadratic second order Hamiltonian system,, Jounal of Southwest China normal University (Natural Science), 32 (2007), 6. Google Scholar

[10]

Sophia Th. Kyritsi and Nikolaos S. Papageorgiou, On superquadratic periodic systems with indefinite linear part,, Nonlinear Anal. TMA., 72 (2010), 946. doi: 10.1016/j.na.2009.07.035. Google Scholar

[11]

S. Luan and A. Mao, Periodic solutions for a class of non-autonomous Hamiltonian systems,, Nonlinear Anal., 61 (2005), 1413. doi: 10.1016/j.na.2005.01.108. Google Scholar

[12]

S. Luan and A. Mao, Periodic solutions of nonautonomous second order Hamiltonian systems,, Acta Mathematica Sinica, 21 (2005), 685. doi: 10.1007/s10114-005-0532-6. Google Scholar

[13]

S. Li and M. Willem, Applications of local linking to critical point theory,, J. Math. Anal. Appl., 189 (1995), 6. doi: 10.1006/jmaa.1995.1002. Google Scholar

[14]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Springer-Verlag, (1989). doi: 10.1007/978-1-4757-2061-7. Google Scholar

[15]

K. Perera, Critical groups of critical points produced by local linking with applications,, Abstr. Appl. Anal., 3 (1998), 437. doi: 10.1155/S1085337598000657. Google Scholar

[16]

P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", in: CBMS Regional Conf. Ser. in Math., (1986). Google Scholar

[17]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems,, Comm. Pure Appl. Math., 31 (1978), 157. doi: 10.1002/cpa.3160310203. Google Scholar

[18]

M. Schechter, Periodic non-autonomous second-order dynamical systems,, J. Differential Equations, 223 (2006), 290. doi: 10.1016/j.jde.2005.02.022. Google Scholar

[19]

M. Schechter, "Minimax Systems and Critical Point Theory,", Birkh\, (2009). doi: 10.1007/978-0-8176-4902-9. Google Scholar

[20]

C. L. Tang, Periodic solutions of nonautonomous second order systems with sublinear nonlinearity,, Proc. Amer. Math. Soc. \textbf{126} (1998), 126 (1998), 3263. doi: 10.1090/S0002-9939-98-04706-6. Google Scholar

[21]

Z. L. Tao and C. L. Tang, Periodic and subharmonic solutions of second order Hamiltonian systems,, J. Math. Anal. Appl., 293 (2004), 435. doi: 10.1016/j.jmaa.2003.11.007. Google Scholar

[22]

Z. L. Tao and C. L. Tang, Periodic solutions of nonquadratic second order Hamiltonian systems,, (Chinese), 27 (2002), 841. Google Scholar

[23]

Z. L. Tao, S. Yan and S. L. Wu, Periodic solutions for a class of superquadratic Hamiltonian systems,, J. Math. Anal. Appl., 331 (2007), 152. doi: 10.1016/j.jmaa.2006.08.041. Google Scholar

[24]

Y. W. Ye and C. L. Tang, Periodic and subharmonic soltions for a class of superquadratic second order Hamiltonian systems,, Nonlinear Anal., 71 (2009), 2298. doi: 10.1016/j.na.2009.01.064. Google Scholar

[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]

Zhirong He, Weinian Zhang. Critical periods of a periodic annulus linking to equilibria at infinity in a cubic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 841-854. doi: 10.3934/dcds.2009.24.841
[3]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[4]

Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222
[5]

Shiwang Ma. Nontrivial periodic solutions for asymptotically linear hamiltonian systems at resonance. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2361-2380. doi: 10.3934/cpaa.2013.12.2361
[6]

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
[7]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277
[8]

Yingte Sun, Xiaoping Yuan. Quasi-periodic solution of quasi-linear fifth-order KdV equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6241-6285. doi: 10.3934/dcds.2018268
[9]

Paola Buttazzoni, Alessandro Fonda. Periodic perturbations of scalar second order differential equations. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 451-455. doi: 10.3934/dcds.1997.3.451
[10]

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
[11]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335
[12]

Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139
[13]

Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775
[14]

Claudianor O. Alves. Existence of periodic solution for a class of systems involving nonlinear wave equations. Communications on Pure & Applied Analysis, 2005, 4 (3) : 487-498. doi: 10.3934/cpaa.2005.4.487
[15]

Kaifa Wang, Aijun Fan. Uniform persistence and periodic solution of chemostat-type model with antibiotic. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 789-795. doi: 10.3934/dcdsb.2004.4.789
[16]

Mi-Young Kim. Uniqueness and stability of positive periodic numerical solution of an epidemic model. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 365-375. doi: 10.3934/dcdsb.2007.7.365
[17]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121
[18]

Xuelei Wang, Dingbian Qian, Xiying Sun. Periodic solutions of second order equations with asymptotical non-resonance. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4715-4726. doi: 10.3934/dcds.2018207
[19]

Xiaojun Chang, Yong Li. Rotating periodic solutions of second order dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 643-652. doi: 10.3934/dcds.2016.36.643
[20]

Chiara Zanini, Fabio Zanolin. Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4045-4067. doi: 10.3934/dcds.2012.32.4045

2018 Impact Factor: 0.925

Tools

Metrics

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

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences