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Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems
1. | School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 |
References:
[1] |
A. Ambrosetti, V. Benci and Y. Long, A note on the existence of multiple brake orbits,, Nonlinear Anal. T. M. A., 21 (1993), 643.
doi: 10.1016/0362-546X(93)90061-V. |
[2] |
A. Ambrosetti and V. Coti Zelati, Solutions with minimal period for Hamiltonian systems in a potential well,, Ann. I. H. P. Anal. non linéaire, 4 (1987), 275.
|
[3] |
A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems,, Math. Ann., 255 (1981), 405.
doi: 10.1007/BF01450713. |
[4] |
T. An and Y. Long, Index theories of second order Hamiltonian systems,, Nonlinear Anal., 34 (1998), 585.
doi: 10.1016/S0362-546X(97)00572-5. |
[5] |
V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems,, Ann. I. H. P. Analyse Nonl., 1 (1984), 401.
|
[6] |
V. Benci and F. Giannoni, A new proof of the existence of a brake orbit. In "Advanced Topics in the Theory of Dynamical Systems",, Notes Rep. Math. Sci. Eng., 6 (1989), 37.
|
[7] |
S. Bolotin, Libration motions of natural dynamical systems,, Vestnik Moskov Univ. Ser. I. Mat. Mekh., 6 (1978), 72.
|
[8] |
S. Bolotin and V. V. Kozlov, Librations with many degrees of freedom,, J. Appl. Math. Mech., 42 (1978), 245.
|
[9] |
B. Booss and K. Furutani, The Maslov-type index - a functional analytical definition and the spectral flow formula,, Tokyo J. Math., 21 (1998), 1.
doi: 10.3836/tjm/1270041982. |
[10] |
B. Booss and C. Zhu, General spectral flow formula for fixed maximal domain,, Central Eur. J. Math., 3 (2005), 558.
doi: 10.2478/BF02475923. |
[11] |
S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov-type index,, Comm. Pure Appl. Math., 47 (1994), 121.
doi: 10.1002/cpa.3160470202. |
[12] |
C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations,, Commu. Pure. Appl. Math., 37 (1984), 207.
doi: 10.1002/cpa.3160370204. |
[13] |
D. Dong and Y. Long, The Iteration Theory of the Maslov-type Index Theory with Applications to Nonlinear Hamiltonian Systems,, Trans. Amer. Math. Soc., 349 (1997), 2619.
doi: 10.1090/S0002-9947-97-01718-2. |
[14] |
J. J. Duistermaat, Fourier Integral Operators,, Birkhäuser, (1996). Google Scholar |
[15] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics,, Spring-Verlag. Berlin, (1990).
doi: 10.1007/978-3-642-74331-3. |
[16] |
I. Ekeland and E. Hofer, Periodic solutions with percribed period for convex autonomous Hamiltonian systems,, Invent. Math., 81 (1985), 155.
doi: 10.1007/BF01388776. |
[17] |
G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems,, Nonlinear Anal., 27 (1996), 811.
doi: 10.1016/0362-546X(95)00077-9. |
[18] |
G. Fei, S.-K. Kim and T. Wang, Minimal Period Estimates of Period Solutions for Superquadratic Hamiltonian Syetems,, J. Math. Anal. Appl., 238 (1999), 216.
doi: 10.1006/jmaa.1999.6527. |
[19] |
G. Fei, S.-K. Kim and T. Wang, Solutions of minimal period for even classical Hamiltonian systems,, Nonlinear Anal., 43 (2001), 363.
doi: 10.1016/S0362-546X(99)00199-6. |
[20] |
M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian equations,, Nonlinear Anal., 7 (1983), 475.
doi: 10.1016/0362-546X(83)90039-1. |
[21] |
M. 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. TMA., 10 (1986), 371.
doi: 10.1016/0362-546X(86)90134-3. |
[22] |
M. Girardi and M. Matzeu, Periodic solutions of convex Hamiltonian systems with a quadratic growth at the origin and superquadratic at infinity,, Ann. Math. Pura ed App., 147 (1987), 21.
doi: 10.1007/BF01762410. |
[23] |
M. Girardi and M. Matzeu, Dual Morse index estimates for periodic solutions of Hamiltonian systems in some nonconvex superquadratic case,, Nonlinear Anal. TMA., 17 (1991), 481.
doi: 10.1016/0362-546X(91)90143-O. |
[24] |
H. Gluck and W. Ziller, Existence of periodic solutions of conservtive systems,, Seminar on Minimal Submanifolds, (1983), 65.
|
[25] |
E. W. C. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy,, J. Math. Anal. Appl., 132 (1988), 1.
doi: 10.1016/0022-247X(88)90039-X. |
[26] |
K. Hayashi, Periodic solution of classical Hamiltonian systems,, Tokyo J. Math., 6 (1983), 473.
doi: 10.3836/tjm/1270213886. |
[27] |
C. Liu, A note on the monotonicity of Maslov-type index of Linear Hamiltonian systems with applications,, Proceedings of the royal Society of Edinburg, 135 (2005), 1263.
doi: 10.1017/S0308210500004364. |
[28] |
C. Liu, Maslov-type index theory for symplectic paths with Lagrangian boundary conditions,, Adv. Nonlinear Stud., 7 (2007), 131.
|
[29] |
C. Liu, Asymptotically linear hamiltonian systems with largrangian boundary conditions,, Pacific J. Math., 232 (2007), 233.
doi: 10.2140/pjm.2007.232.233. |
[30] |
C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems,, Discrete Contin. Dyn. Syst., 27 (2010), 337.
doi: 10.3934/dcds.2010.27.337. |
[31] |
C. Liu and Y. Long, An optimal increasing estimate for iterated Maslov-type indices,, Chinese Sci. Bull., 42 (1997), 2275.
|
[32] |
C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications,, J. Diff. Equa., 165 (2000), 355.
doi: 10.1006/jdeq.2000.3775. |
[33] |
C. Liu and D. Zhang, An iteration theory of Maslov-type index for symplectic paths associated with a Lagranfian subspace and Multiplicity of brake orbits in bounded convex symmetric domains,, , (). Google Scholar |
[34] |
Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems,, Science in China, 7 (1990), 673.
|
[35] |
Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials,, Ann. I. H. P. Anal. non linéaire, 10 (1993), 605.
|
[36] |
Y. Long, The minimal period problem of period solutions for autonomous superquadratic second Hamiltonian systems,, J. Diff. Equa., 111 (1994), 147.
doi: 10.1006/jdeq.1994.1079. |
[37] |
Y. Long, On the minimal period for periodic solution problem of nonlinear Hamiltonian systems,, Chinese Ann. of math., 18 (1997), 481.
|
[38] |
Y. Long, Bott formula of the Maslov-type index theory,, Pacific J. Math., 187 (1999), 113.
doi: 10.2140/pjm.1999.187.113. |
[39] |
Y. long, Index Theory for Symplectic Paths with Applications,, Birkhäuser, (2002).
doi: 10.1007/978-3-0348-8175-3. |
[40] |
Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains,, Advances in Math., 203 (2006), 568.
doi: 10.1016/j.aim.2005.05.005. |
[41] |
P. H. Rabinowitz, Periodic solution of Hamiltonian systems,, Commu. Pure Appl. Math., 31 (1978), 157.
doi: 10.1002/cpa.3160310203. |
[42] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Regional Conf. Ser. in Math., 45 (1986), 287.
|
[43] |
P. H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems,, Nonlinear Anal. T. M. A., 11 (1987), 599.
doi: 10.1016/0362-546X(87)90075-7. |
[44] |
J. Robbin and D. Salamon, The Maslov indices for paths,, Topology, 32 (1993), 827.
doi: 10.1016/0040-9383(93)90052-W. |
[45] |
H. Seifert, Periodische Bewegungen mechanischer Systeme,, Math. Z., 51 (1948), 197.
doi: 10.1007/BF01291002. |
[46] |
A. Szulkin, Cohomology and Morse theory for strongly indefinite functions,, Math. Z., 209 (1992), 375.
doi: 10.1007/BF02570842. |
[47] |
Y. Xiao, Periodic Solutions with Prescribed Minimal Period for Second Order Hamiltonian Systems with Even Potentials,, Acta Math. Sinica, 26 (2010), 825.
doi: 10.1007/s10114-009-8305-2. |
[48] |
D. Zhang, Symmetric period solutions with prescribed period for even autonomous semipositive hamiltonian systems,, Sci. China Math., 57 (2014), 81.
doi: 10.1007/s11425-013-4598-9. |
[49] |
D. Zhang, Maslov-type index and brake orbits in nonlinear Hamiltonian systems,, Science in China, 50 (2007), 761.
doi: 10.1007/s11425-007-0034-3. |
[50] |
C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I),, Chinese Ann. of Math., 20 (1999), 413.
doi: 10.1142/S0252959999000485. |
show all references
References:
[1] |
A. Ambrosetti, V. Benci and Y. Long, A note on the existence of multiple brake orbits,, Nonlinear Anal. T. M. A., 21 (1993), 643.
doi: 10.1016/0362-546X(93)90061-V. |
[2] |
A. Ambrosetti and V. Coti Zelati, Solutions with minimal period for Hamiltonian systems in a potential well,, Ann. I. H. P. Anal. non linéaire, 4 (1987), 275.
|
[3] |
A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems,, Math. Ann., 255 (1981), 405.
doi: 10.1007/BF01450713. |
[4] |
T. An and Y. Long, Index theories of second order Hamiltonian systems,, Nonlinear Anal., 34 (1998), 585.
doi: 10.1016/S0362-546X(97)00572-5. |
[5] |
V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems,, Ann. I. H. P. Analyse Nonl., 1 (1984), 401.
|
[6] |
V. Benci and F. Giannoni, A new proof of the existence of a brake orbit. In "Advanced Topics in the Theory of Dynamical Systems",, Notes Rep. Math. Sci. Eng., 6 (1989), 37.
|
[7] |
S. Bolotin, Libration motions of natural dynamical systems,, Vestnik Moskov Univ. Ser. I. Mat. Mekh., 6 (1978), 72.
|
[8] |
S. Bolotin and V. V. Kozlov, Librations with many degrees of freedom,, J. Appl. Math. Mech., 42 (1978), 245.
|
[9] |
B. Booss and K. Furutani, The Maslov-type index - a functional analytical definition and the spectral flow formula,, Tokyo J. Math., 21 (1998), 1.
doi: 10.3836/tjm/1270041982. |
[10] |
B. Booss and C. Zhu, General spectral flow formula for fixed maximal domain,, Central Eur. J. Math., 3 (2005), 558.
doi: 10.2478/BF02475923. |
[11] |
S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov-type index,, Comm. Pure Appl. Math., 47 (1994), 121.
doi: 10.1002/cpa.3160470202. |
[12] |
C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations,, Commu. Pure. Appl. Math., 37 (1984), 207.
doi: 10.1002/cpa.3160370204. |
[13] |
D. Dong and Y. Long, The Iteration Theory of the Maslov-type Index Theory with Applications to Nonlinear Hamiltonian Systems,, Trans. Amer. Math. Soc., 349 (1997), 2619.
doi: 10.1090/S0002-9947-97-01718-2. |
[14] |
J. J. Duistermaat, Fourier Integral Operators,, Birkhäuser, (1996). Google Scholar |
[15] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics,, Spring-Verlag. Berlin, (1990).
doi: 10.1007/978-3-642-74331-3. |
[16] |
I. Ekeland and E. Hofer, Periodic solutions with percribed period for convex autonomous Hamiltonian systems,, Invent. Math., 81 (1985), 155.
doi: 10.1007/BF01388776. |
[17] |
G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems,, Nonlinear Anal., 27 (1996), 811.
doi: 10.1016/0362-546X(95)00077-9. |
[18] |
G. Fei, S.-K. Kim and T. Wang, Minimal Period Estimates of Period Solutions for Superquadratic Hamiltonian Syetems,, J. Math. Anal. Appl., 238 (1999), 216.
doi: 10.1006/jmaa.1999.6527. |
[19] |
G. Fei, S.-K. Kim and T. Wang, Solutions of minimal period for even classical Hamiltonian systems,, Nonlinear Anal., 43 (2001), 363.
doi: 10.1016/S0362-546X(99)00199-6. |
[20] |
M. Girardi and M. Matzeu, Some results on solutions of minimal period to superquadratic Hamiltonian equations,, Nonlinear Anal., 7 (1983), 475.
doi: 10.1016/0362-546X(83)90039-1. |
[21] |
M. 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. TMA., 10 (1986), 371.
doi: 10.1016/0362-546X(86)90134-3. |
[22] |
M. Girardi and M. Matzeu, Periodic solutions of convex Hamiltonian systems with a quadratic growth at the origin and superquadratic at infinity,, Ann. Math. Pura ed App., 147 (1987), 21.
doi: 10.1007/BF01762410. |
[23] |
M. Girardi and M. Matzeu, Dual Morse index estimates for periodic solutions of Hamiltonian systems in some nonconvex superquadratic case,, Nonlinear Anal. TMA., 17 (1991), 481.
doi: 10.1016/0362-546X(91)90143-O. |
[24] |
H. Gluck and W. Ziller, Existence of periodic solutions of conservtive systems,, Seminar on Minimal Submanifolds, (1983), 65.
|
[25] |
E. W. C. van Groesen, Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy,, J. Math. Anal. Appl., 132 (1988), 1.
doi: 10.1016/0022-247X(88)90039-X. |
[26] |
K. Hayashi, Periodic solution of classical Hamiltonian systems,, Tokyo J. Math., 6 (1983), 473.
doi: 10.3836/tjm/1270213886. |
[27] |
C. Liu, A note on the monotonicity of Maslov-type index of Linear Hamiltonian systems with applications,, Proceedings of the royal Society of Edinburg, 135 (2005), 1263.
doi: 10.1017/S0308210500004364. |
[28] |
C. Liu, Maslov-type index theory for symplectic paths with Lagrangian boundary conditions,, Adv. Nonlinear Stud., 7 (2007), 131.
|
[29] |
C. Liu, Asymptotically linear hamiltonian systems with largrangian boundary conditions,, Pacific J. Math., 232 (2007), 233.
doi: 10.2140/pjm.2007.232.233. |
[30] |
C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems,, Discrete Contin. Dyn. Syst., 27 (2010), 337.
doi: 10.3934/dcds.2010.27.337. |
[31] |
C. Liu and Y. Long, An optimal increasing estimate for iterated Maslov-type indices,, Chinese Sci. Bull., 42 (1997), 2275.
|
[32] |
C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications,, J. Diff. Equa., 165 (2000), 355.
doi: 10.1006/jdeq.2000.3775. |
[33] |
C. Liu and D. Zhang, An iteration theory of Maslov-type index for symplectic paths associated with a Lagranfian subspace and Multiplicity of brake orbits in bounded convex symmetric domains,, , (). Google Scholar |
[34] |
Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems,, Science in China, 7 (1990), 673.
|
[35] |
Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials,, Ann. I. H. P. Anal. non linéaire, 10 (1993), 605.
|
[36] |
Y. Long, The minimal period problem of period solutions for autonomous superquadratic second Hamiltonian systems,, J. Diff. Equa., 111 (1994), 147.
doi: 10.1006/jdeq.1994.1079. |
[37] |
Y. Long, On the minimal period for periodic solution problem of nonlinear Hamiltonian systems,, Chinese Ann. of math., 18 (1997), 481.
|
[38] |
Y. Long, Bott formula of the Maslov-type index theory,, Pacific J. Math., 187 (1999), 113.
doi: 10.2140/pjm.1999.187.113. |
[39] |
Y. long, Index Theory for Symplectic Paths with Applications,, Birkhäuser, (2002).
doi: 10.1007/978-3-0348-8175-3. |
[40] |
Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains,, Advances in Math., 203 (2006), 568.
doi: 10.1016/j.aim.2005.05.005. |
[41] |
P. H. Rabinowitz, Periodic solution of Hamiltonian systems,, Commu. Pure Appl. Math., 31 (1978), 157.
doi: 10.1002/cpa.3160310203. |
[42] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,, CBMS Regional Conf. Ser. in Math., 45 (1986), 287.
|
[43] |
P. H. Rabinowitz, On the existence of periodic solutions for a class of symmetric Hamiltonian systems,, Nonlinear Anal. T. M. A., 11 (1987), 599.
doi: 10.1016/0362-546X(87)90075-7. |
[44] |
J. Robbin and D. Salamon, The Maslov indices for paths,, Topology, 32 (1993), 827.
doi: 10.1016/0040-9383(93)90052-W. |
[45] |
H. Seifert, Periodische Bewegungen mechanischer Systeme,, Math. Z., 51 (1948), 197.
doi: 10.1007/BF01291002. |
[46] |
A. Szulkin, Cohomology and Morse theory for strongly indefinite functions,, Math. Z., 209 (1992), 375.
doi: 10.1007/BF02570842. |
[47] |
Y. Xiao, Periodic Solutions with Prescribed Minimal Period for Second Order Hamiltonian Systems with Even Potentials,, Acta Math. Sinica, 26 (2010), 825.
doi: 10.1007/s10114-009-8305-2. |
[48] |
D. Zhang, Symmetric period solutions with prescribed period for even autonomous semipositive hamiltonian systems,, Sci. China Math., 57 (2014), 81.
doi: 10.1007/s11425-013-4598-9. |
[49] |
D. Zhang, Maslov-type index and brake orbits in nonlinear Hamiltonian systems,, Science in China, 50 (2007), 761.
doi: 10.1007/s11425-007-0034-3. |
[50] |
C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I),, Chinese Ann. of Math., 20 (1999), 413.
doi: 10.1142/S0252959999000485. |
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