# American Institute of Mathematical Sciences

• Previous Article
Singular limit for reactive transport through a thin heterogeneous layer including a nonlinear diffusion coefficient
• CPAA Home
• This Issue
• Next Article
Radial quasilinear elliptic problems with singular or vanishing potentials
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. Fei, S. 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. Liu, L. 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. Fei, S. 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. Liu, L. 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.
 [1] Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053 [2] Zhiping Fan, Duanzhi Zhang. Minimal period solutions in asymptotically linear Hamiltonian system with symmetries. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2095-2124. doi: 10.3934/dcds.2020354 [3] Yavdat Il'yasov, Nadir Sari. Solutions of minimal period for a Hamiltonian system with a changing sign potential. Communications on Pure and Applied Analysis, 2005, 4 (1) : 175-185. doi: 10.3934/cpaa.2005.4.175 [4] Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139 [5] Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064 [6] Chungen Liu. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 337-355. doi: 10.3934/dcds.2010.27.337 [7] Xingyong Zhang, Xianhua Tang. Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems. Communications on Pure and Applied Analysis, 2014, 13 (1) : 75-95. doi: 10.3934/cpaa.2014.13.75 [8] Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 [9] Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116 [10] B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217 [11] Norimichi Hirano, Zhi-Qiang Wang. Subharmonic solutions for second order Hamiltonian systems. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 467-474. doi: 10.3934/dcds.1998.4.467 [12] Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 [13] Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040 [14] Duanzhi Zhang. Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2227-2272. doi: 10.3934/dcds.2015.35.2227 [15] V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 [16] Shi Jin, Min Tang, Houde Han. A uniformly second order numerical method for the one-dimensional discrete-ordinate transport equation and its diffusion limit with interface. Networks and Heterogeneous Media, 2009, 4 (1) : 35-65. doi: 10.3934/nhm.2009.4.35 [17] S. Aubry, G. Kopidakis, V. Kadelburg. Variational proof for hard Discrete breathers in some classes of Hamiltonian dynamical systems. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 271-298. doi: 10.3934/dcdsb.2001.1.271 [18] Martin Redmann, Peter Benner. Approximation and model order reduction for second order systems with Levy-noise. Conference Publications, 2015, 2015 (special) : 945-953. doi: 10.3934/proc.2015.0945 [19] Paola Buttazzoni, Alessandro Fonda. Periodic perturbations of scalar second order differential equations. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 451-455. doi: 10.3934/dcds.1997.3.451 [20] Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778

2020 Impact Factor: 1.916

## Metrics

• HTML views (182)
• Cited by (0)

• on AIMS