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January  2022, 21(1): 47-59. doi: 10.3934/cpaa.2021166

Periodic solutions with prescribed minimal period for second order even Hamiltonian systems

1. 

School of Mathematics and Computational Sciences, Wuyi University, Jiangmen, 529020, China

2. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

* Corresponding author

Received  April 2021 Revised  June 2021 Published  January 2022 Early access  September 2021

Fund Project: The first author is supported by National Natural Science Foundation of China grant 11901438 and Natural Science Foundation of Guangdong Province, China grant 2018A0303130058, 2021A1515010062. The third author is supported by National Natural Science Foundation of China grant 11771104, 12171110 and Science and Technology Planning Project of Guangdong Province of China grant 2020A1414010106

In this paper, we develop a new method to study Rabinowitz's conjecture on the existence of periodic solutions with prescribed minimal period for second order even Hamiltonian system without any convexity assumptions. Specifically, we first study the associated homogenous Dirichlet boundary value problems for the discretization of the Hamiltonian system with given step length and obtain a sequence of nonnegative solutions corresponding to different step lengths by using discrete variational methods. Then, using the sequence of nonnegative solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, by utilizing the limit function of convergent subsequence and the symmetry of the potential, we will obtain the desired periodic solution. In particular, we prove Rabinowitz's conjecture in the case when the potential satisfies a certain symmetric assumption. Moreover, our main result greatly improves the related results in the literature in the case where $ N = 1 $.

Citation: Juhong Kuang, Weiyi Chen, Zhiming Guo. Periodic solutions with prescribed minimal period for second order even Hamiltonian systems. Communications on Pure and Applied Analysis, 2022, 21 (1) : 47-59. doi: 10.3934/cpaa.2021166
References:
[1]

A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann., 255 (1981), 405-421.  doi: 10.1007/BF01450713.

[2]

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

[3]

I. Ekeland, Convexity Method in Hamiltonian Mechanics, Spinger, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.

[4]

G. Fei and Q. Qiu, Minimal periodic solutions of nonlinear Hamiltonian systems, Nonlinear Anal., 27 (1996), 821-839.  doi: 10.1016/0362-546X(95)00077-9.

[5]

G. Fei and T. Wang, The minimal period problem for nonconvex even second order Hamiltonian systems, J. Math. Anal. Appl., 215 (1997), 543-559.  doi: 10.1006/jmaa.1997.5666.

[6]

G. FeiS. Kim and T. Wang, Solutions of minimal period for even classical Hamiltonian systems, Nonlinear Anal., 43 (2001), 363-375.  doi: 10.1016/S0362-546X(99)00199-6.

[7]

M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian systems, Nonlinear Anal., 7 (1983), 475-482.  doi: 10.1016/0362-546X(83)90039-1.

[8]

M. Girardi and M. Matzeu, Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem, Nonlinear Anal., 10 (1986), 371-383.  doi: 10.1016/0362-546X(86)90134-3.

[9]

Z. Guo and J. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68 (2003), 419-430.  doi: 10.1112/S0024610703004563.

[10]

Z. Guo and J. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A, 46 (2003), 506-515.  doi: 10.1360/03ys9051.

[11]

Z. Guo and J. Yu, Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems, Nonlinear Anal., 55 (2003), 969-983.  doi: 10.1016/j.na.2003.07.019.

[12]

J. Kuang and L. Kong, Positive solutions for a class of singular discrete Dirichlet problems with a parameter, Appl. Math. Lett., 109 (2020), 106548.  doi: 10.1016/j.aml.2020.106548.

[13]

C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian system, Discrete Contin. Dyn. Syst., 27 (2010), 337-355.  doi: 10.3934/dcds.2010.27.337.

[14]

C. Liu and X. Zhang, Subharmonic solutions and minimal perodic solutions of first-order Hamiltonian systems with anisotropic growth, Discrete Contin. Dyn. Syst., 37 (2017), 1559-1574.  doi: 10.3934/dcds.2017064.

[15]

C. LiuL. Zuo and X. Zhang, Minimal periodic solutions of first-order convex Hamiltonian systems with anisotropic growth, Front. Math. China, 13 (2018), 1063-1073.  doi: 10.1007/s11464-018-0721-0.

[16]

C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications, J. Differ. Equ., 165 (2000), 355-376.  doi: 10.1006/jdeq.2000.3775.

[17]

Y. Long, The minimal period problem of periodic solutions for autonomous superquadratic second order Hamiltonian systems, J. Differ. Equ., 111 (1994), 147-174.  doi: 10.1006/jdeq.1994.1079.

[18]

Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials, Ann. Inst. H. Poincare Anal. Non Lineaire, 10 (1993), 605-626.  doi: 10.1016/S0294-1449(16)30199-8.

[19]

Y. Long, On the minimal period for periodic solutions of nonlinear Hamiltonian systems, Chinese Ann. Math. Ser. B, 18 (1997), 481-485. 

[20]

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

[21]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65, Am. Math. Soc., Providence, 1986. doi: 10.1090/cbms/065.

[22]

Y. Xiao, Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials, Acta Math. Sin. English Ser., 26 (2010), 825-830.  doi: 10.1007/s10114-009-8305-2.

[23]

D. Zhang, Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems, Sci. China Math., 57 (2014), 81-96.  doi: 10.1007/s11425-013-4598-9.

[24]

D. Zhang, Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems, Discrete Contin. Dyn. Syst., 35 (2015), 2227-2272.  doi: 10.3934/dcds.2015.35.2227.

show all references

References:
[1]

A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann., 255 (1981), 405-421.  doi: 10.1007/BF01450713.

[2]

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

[3]

I. Ekeland, Convexity Method in Hamiltonian Mechanics, Spinger, Berlin, 1990. doi: 10.1007/978-3-642-74331-3.

[4]

G. Fei and Q. Qiu, Minimal periodic solutions of nonlinear Hamiltonian systems, Nonlinear Anal., 27 (1996), 821-839.  doi: 10.1016/0362-546X(95)00077-9.

[5]

G. Fei and T. Wang, The minimal period problem for nonconvex even second order Hamiltonian systems, J. Math. Anal. Appl., 215 (1997), 543-559.  doi: 10.1006/jmaa.1997.5666.

[6]

G. FeiS. Kim and T. Wang, Solutions of minimal period for even classical Hamiltonian systems, Nonlinear Anal., 43 (2001), 363-375.  doi: 10.1016/S0362-546X(99)00199-6.

[7]

M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian systems, Nonlinear Anal., 7 (1983), 475-482.  doi: 10.1016/0362-546X(83)90039-1.

[8]

M. Girardi and M. Matzeu, Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem, Nonlinear Anal., 10 (1986), 371-383.  doi: 10.1016/0362-546X(86)90134-3.

[9]

Z. Guo and J. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68 (2003), 419-430.  doi: 10.1112/S0024610703004563.

[10]

Z. Guo and J. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A, 46 (2003), 506-515.  doi: 10.1360/03ys9051.

[11]

Z. Guo and J. Yu, Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems, Nonlinear Anal., 55 (2003), 969-983.  doi: 10.1016/j.na.2003.07.019.

[12]

J. Kuang and L. Kong, Positive solutions for a class of singular discrete Dirichlet problems with a parameter, Appl. Math. Lett., 109 (2020), 106548.  doi: 10.1016/j.aml.2020.106548.

[13]

C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian system, Discrete Contin. Dyn. Syst., 27 (2010), 337-355.  doi: 10.3934/dcds.2010.27.337.

[14]

C. Liu and X. Zhang, Subharmonic solutions and minimal perodic solutions of first-order Hamiltonian systems with anisotropic growth, Discrete Contin. Dyn. Syst., 37 (2017), 1559-1574.  doi: 10.3934/dcds.2017064.

[15]

C. LiuL. Zuo and X. Zhang, Minimal periodic solutions of first-order convex Hamiltonian systems with anisotropic growth, Front. Math. China, 13 (2018), 1063-1073.  doi: 10.1007/s11464-018-0721-0.

[16]

C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications, J. Differ. Equ., 165 (2000), 355-376.  doi: 10.1006/jdeq.2000.3775.

[17]

Y. Long, The minimal period problem of periodic solutions for autonomous superquadratic second order Hamiltonian systems, J. Differ. Equ., 111 (1994), 147-174.  doi: 10.1006/jdeq.1994.1079.

[18]

Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials, Ann. Inst. H. Poincare Anal. Non Lineaire, 10 (1993), 605-626.  doi: 10.1016/S0294-1449(16)30199-8.

[19]

Y. Long, On the minimal period for periodic solutions of nonlinear Hamiltonian systems, Chinese Ann. Math. Ser. B, 18 (1997), 481-485. 

[20]

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

[21]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 65, Am. Math. Soc., Providence, 1986. doi: 10.1090/cbms/065.

[22]

Y. Xiao, Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials, Acta Math. Sin. English Ser., 26 (2010), 825-830.  doi: 10.1007/s10114-009-8305-2.

[23]

D. Zhang, Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems, Sci. China Math., 57 (2014), 81-96.  doi: 10.1007/s11425-013-4598-9.

[24]

D. Zhang, Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems, Discrete Contin. Dyn. Syst., 35 (2015), 2227-2272.  doi: 10.3934/dcds.2015.35.2227.

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