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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), 120-145. doi: 10.1006/jdeq.1995.1007.  Google Scholar

[2]

K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhäuser Boston, Inc., Boston, MA, 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., Hackensack, NJ, 2007.  Google Scholar

[4]

I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems, Invent. Math., 81 (1985), 155-188. doi: 10.1007/BF01388776.  Google Scholar

[5]

I. Ekeland, "Convexity Method in Hamiltonian Mechanics," Springer-Verlag, Berlin, 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), 821-839. 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), 216-233. 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), 1-12.  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), 6-10. Google Scholar

[10]

Sophia Th. Kyritsi and Nikolaos S. Papageorgiou, On superquadratic periodic systems with indefinite linear part, Nonlinear Anal. TMA., 72 (2010), 946-954. 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), 1413-1426. 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, English version, 21 (2005), 685-690. 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), 6-32. doi: 10.1006/jmaa.1995.1002.  Google Scholar

[14]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 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), 437-446. 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., 65, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[17]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.  Google Scholar

[18]

M. Schechter, Periodic non-autonomous second-order dynamical systems, J. Differential Equations, 223 (2006), 290-302. doi: 10.1016/j.jde.2005.02.022.  Google Scholar

[19]

M. Schechter, "Minimax Systems and Critical Point Theory," Birkhäuser Boston, Inc., Boston, MA, 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. 126 (1998), 3263-3270. 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), 435-445. 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), Jounal of Southwest China normal University (Natural Science), 27 (2002), 841-846. 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), 152-158. 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), 2298-2307. 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), 120-145. doi: 10.1006/jdeq.1995.1007.  Google Scholar

[2]

K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problems," Birkhäuser Boston, Inc., Boston, MA, 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., Hackensack, NJ, 2007.  Google Scholar

[4]

I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems, Invent. Math., 81 (1985), 155-188. doi: 10.1007/BF01388776.  Google Scholar

[5]

I. Ekeland, "Convexity Method in Hamiltonian Mechanics," Springer-Verlag, Berlin, 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), 821-839. 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), 216-233. 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), 1-12.  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), 6-10. Google Scholar

[10]

Sophia Th. Kyritsi and Nikolaos S. Papageorgiou, On superquadratic periodic systems with indefinite linear part, Nonlinear Anal. TMA., 72 (2010), 946-954. 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), 1413-1426. 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, English version, 21 (2005), 685-690. 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), 6-32. doi: 10.1006/jmaa.1995.1002.  Google Scholar

[14]

J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, New York, 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), 437-446. 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., 65, American Mathematical Society, Providence, RI, 1986.  Google Scholar

[17]

P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.  Google Scholar

[18]

M. Schechter, Periodic non-autonomous second-order dynamical systems, J. Differential Equations, 223 (2006), 290-302. doi: 10.1016/j.jde.2005.02.022.  Google Scholar

[19]

M. Schechter, "Minimax Systems and Critical Point Theory," Birkhäuser Boston, Inc., Boston, MA, 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. 126 (1998), 3263-3270. 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), 435-445. 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), Jounal of Southwest China normal University (Natural Science), 27 (2002), 841-846. 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), 152-158. 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), 2298-2307. doi: 10.1016/j.na.2009.01.064.  Google Scholar

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