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Addendum to: Symbolic dynamics for the $N$-centre problem at negative energies
Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero
1. | School of Science, Shandong University of Technology, Zibo 255049, China |
2. | Department of Mathematics, College of Science, Hohai University, Nanjing 210098, China |
3. | Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539 |
References:
[1] |
N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part,, Math. Z., 248 (2004), 423.
doi: 10.1007/s00209-004-0663-y. |
[2] |
N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations,, J. Funct. Anal., 234 (2006), 277.
doi: 10.1016/j.jfa.2005.11.010. |
[3] |
G. Arioli and A. Szulkin, Homoclinic solutions of Hamiltonian systems with symmetry,, J. Differential Equations, 158 (1999), 291.
doi: 10.1006/jdeq.1999.3639. |
[4] |
T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory,, Math. Nachr., 279 (2006), 1267.
doi: 10.1002/mana.200410420. |
[5] |
T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods,, in, (2005), 77.
|
[6] |
G. Bonanno and R. Livrea, Multiple periodic solutions for Hamiltonian systems with not coercive potential,, J. Math. Anal. Appl., 363 (2010), 627.
doi: 10.1016/j.jmaa.2009.09.025. |
[7] |
C. Chen and X. Hu, Maslov index for homoclinic orbits of Hamiltonian systems,, Ann. I. H. Poincaré Anal. Linéaire, 24 (2007), 589.
doi: 10.1016/j.anihpc.2006.06.002. |
[8] |
J. F. Chu, J. Z. Lei and M. R. Zhang, Lyapunov stability for conservative systems with lower degrees of freedom,, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 423.
doi: 10.3934/dcdsb.2011.16.423. |
[9] |
V. Coti Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems,, Math. Ann., 228 (1990), 133.
doi: 10.1007/BF01444526. |
[10] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials,, J. Amer. Math. Soc., 4 (1991), 693.
doi: 10.2307/2939286. |
[11] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbbR^N$,, Comm. Pure Appl. Math., 45 (1992), 1217.
doi: 10.1002/cpa.3160451002. |
[12] |
Y. Ding, "Variational Methods for Strongly Indefinite Problems,", Interdiscip. Math. Sci., 7 (2007).
doi: 10.1142/9789812709639. |
[13] |
Y. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms,, Commun. Contemp. Math., 8 (2006), 453.
doi: 10.1142/S0219199706002192. |
[14] |
Y. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry,, Nonlinear Anal., 38 (1999), 391.
doi: 10.1016/S0362-546X(98)00204-1. |
[15] |
Y. Ding and L. Jeanjean, Homoclinic orbits for a nonperiodic Hamiltonian system,, J. Differential Equations, 237 (2007), 473.
doi: 10.1016/j.jde.2007.03.005. |
[16] |
Y. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system,, J. Differential Equations, 246 (2009), 2829.
doi: 10.1016/j.jde.2008.12.013. |
[17] |
Y. Ding and S. Li, Homoclinic orbits for first order Hamiltonian systems,, J. Math. Anal. Appl., 189 (1995), 585.
doi: 10.1006/jmaa.1995.1037. |
[18] |
Y. Ding and M. Willem, Homoclinic orbits of a Hamiltonian system,, Z. Angew. Math. Phys., 50 (1999), 759.
doi: 10.1007/s000330050177. |
[19] |
Z. Feng and D. Y. Gao, An asymptotic expression of the Schrödinger equation,, Z. Angew. Math. Phys., 60 (2009), 363.
doi: 10.1007/s00033-007-7102-y. |
[20] |
Z. Feng and Y. Huang, Approximate solution of the Burgers-Korteweg-de Vries equation,, Commun. Pure Appl. Anal., 6 (2007), 429.
doi: 10.3934/cpaa.2007.6.429. |
[21] |
M. Izydorek and J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems,, J. Differential Equations, 219 (2005), 375.
doi: 10.1016/j.jde.2005.06.029. |
[22] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences, 74 (1989).
|
[23] |
A. Mielke, Weak-convergence methods for hamiltonian multiscale problems,, Discrete Contin. Dyn. Syst., 20 (2008), 53.
doi: 10.3934/dcds.2008.20.53. |
[24] |
O. Koltsova and L. Lerman, Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium,, Discrete Contin. Dyn. Syst., 25 (2009), 883.
doi: 10.3934/dcds.2009.25.883. |
[25] |
I. SenGupta and M. C. Mariani, Spherical harmonics applied to differential and integro-differential equations arising in mathematical finance,, Differ. Equ. Dyn. Syst., 20 (2012), 93.
doi: 10.1007/s12591-012-0107-9. |
[26] |
J. Sun, H. Chen, J. J. Nieto and M. Otero-Novoa, Multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects,, Nonlinear Anal., 72 (2010), 4575.
doi: 10.1016/j.na.2010.02.034. |
[27] |
J. Sun, H. Chen and J. J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems,, J. Math. Anal. Appl., 373 (2011), 20.
doi: 10.1016/j.jmaa.2010.06.038. |
[28] |
J. Sun, H. Chen and J. J. Nieto, Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero,, J. Math. Anal. Appl., 378 (2011), 117.
doi: 10.1016/j.jmaa.2010.12.044. |
[29] |
A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems,, J. Funct. Anal., 187 (2001), 25.
doi: 10.1006/jfan.2001.3798. |
[30] |
J. Wang, J. Xu and F. Zhang, Homoclinic orbits of superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition,, Discrete Contin. Dyn. Syst., 27 (2010), 1241.
doi: 10.3934/dcds.2010.27.1241. |
[31] |
S. Zhang, Symmetrically homoclinic orbits for symmetric Hamiltonian systems,, J. Math. Anal. Appl., 247 (2000), 645.
doi: 10.1006/jmaa.2000.6839. |
show all references
References:
[1] |
N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part,, Math. Z., 248 (2004), 423.
doi: 10.1007/s00209-004-0663-y. |
[2] |
N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations,, J. Funct. Anal., 234 (2006), 277.
doi: 10.1016/j.jfa.2005.11.010. |
[3] |
G. Arioli and A. Szulkin, Homoclinic solutions of Hamiltonian systems with symmetry,, J. Differential Equations, 158 (1999), 291.
doi: 10.1006/jdeq.1999.3639. |
[4] |
T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory,, Math. Nachr., 279 (2006), 1267.
doi: 10.1002/mana.200410420. |
[5] |
T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods,, in, (2005), 77.
|
[6] |
G. Bonanno and R. Livrea, Multiple periodic solutions for Hamiltonian systems with not coercive potential,, J. Math. Anal. Appl., 363 (2010), 627.
doi: 10.1016/j.jmaa.2009.09.025. |
[7] |
C. Chen and X. Hu, Maslov index for homoclinic orbits of Hamiltonian systems,, Ann. I. H. Poincaré Anal. Linéaire, 24 (2007), 589.
doi: 10.1016/j.anihpc.2006.06.002. |
[8] |
J. F. Chu, J. Z. Lei and M. R. Zhang, Lyapunov stability for conservative systems with lower degrees of freedom,, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 423.
doi: 10.3934/dcdsb.2011.16.423. |
[9] |
V. Coti Zelati, I. Ekeland and E. Séré, A variational approach to homoclinic orbits in Hamiltonian systems,, Math. Ann., 228 (1990), 133.
doi: 10.1007/BF01444526. |
[10] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials,, J. Amer. Math. Soc., 4 (1991), 693.
doi: 10.2307/2939286. |
[11] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbbR^N$,, Comm. Pure Appl. Math., 45 (1992), 1217.
doi: 10.1002/cpa.3160451002. |
[12] |
Y. Ding, "Variational Methods for Strongly Indefinite Problems,", Interdiscip. Math. Sci., 7 (2007).
doi: 10.1142/9789812709639. |
[13] |
Y. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms,, Commun. Contemp. Math., 8 (2006), 453.
doi: 10.1142/S0219199706002192. |
[14] |
Y. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry,, Nonlinear Anal., 38 (1999), 391.
doi: 10.1016/S0362-546X(98)00204-1. |
[15] |
Y. Ding and L. Jeanjean, Homoclinic orbits for a nonperiodic Hamiltonian system,, J. Differential Equations, 237 (2007), 473.
doi: 10.1016/j.jde.2007.03.005. |
[16] |
Y. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system,, J. Differential Equations, 246 (2009), 2829.
doi: 10.1016/j.jde.2008.12.013. |
[17] |
Y. Ding and S. Li, Homoclinic orbits for first order Hamiltonian systems,, J. Math. Anal. Appl., 189 (1995), 585.
doi: 10.1006/jmaa.1995.1037. |
[18] |
Y. Ding and M. Willem, Homoclinic orbits of a Hamiltonian system,, Z. Angew. Math. Phys., 50 (1999), 759.
doi: 10.1007/s000330050177. |
[19] |
Z. Feng and D. Y. Gao, An asymptotic expression of the Schrödinger equation,, Z. Angew. Math. Phys., 60 (2009), 363.
doi: 10.1007/s00033-007-7102-y. |
[20] |
Z. Feng and Y. Huang, Approximate solution of the Burgers-Korteweg-de Vries equation,, Commun. Pure Appl. Anal., 6 (2007), 429.
doi: 10.3934/cpaa.2007.6.429. |
[21] |
M. Izydorek and J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems,, J. Differential Equations, 219 (2005), 375.
doi: 10.1016/j.jde.2005.06.029. |
[22] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences, 74 (1989).
|
[23] |
A. Mielke, Weak-convergence methods for hamiltonian multiscale problems,, Discrete Contin. Dyn. Syst., 20 (2008), 53.
doi: 10.3934/dcds.2008.20.53. |
[24] |
O. Koltsova and L. Lerman, Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium,, Discrete Contin. Dyn. Syst., 25 (2009), 883.
doi: 10.3934/dcds.2009.25.883. |
[25] |
I. SenGupta and M. C. Mariani, Spherical harmonics applied to differential and integro-differential equations arising in mathematical finance,, Differ. Equ. Dyn. Syst., 20 (2012), 93.
doi: 10.1007/s12591-012-0107-9. |
[26] |
J. Sun, H. Chen, J. J. Nieto and M. Otero-Novoa, Multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects,, Nonlinear Anal., 72 (2010), 4575.
doi: 10.1016/j.na.2010.02.034. |
[27] |
J. Sun, H. Chen and J. J. Nieto, Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems,, J. Math. Anal. Appl., 373 (2011), 20.
doi: 10.1016/j.jmaa.2010.06.038. |
[28] |
J. Sun, H. Chen and J. J. Nieto, Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero,, J. Math. Anal. Appl., 378 (2011), 117.
doi: 10.1016/j.jmaa.2010.12.044. |
[29] |
A. Szulkin and W. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems,, J. Funct. Anal., 187 (2001), 25.
doi: 10.1006/jfan.2001.3798. |
[30] |
J. Wang, J. Xu and F. Zhang, Homoclinic orbits of superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition,, Discrete Contin. Dyn. Syst., 27 (2010), 1241.
doi: 10.3934/dcds.2010.27.1241. |
[31] |
S. Zhang, Symmetrically homoclinic orbits for symmetric Hamiltonian systems,, J. Math. Anal. Appl., 247 (2000), 645.
doi: 10.1006/jmaa.2000.6839. |
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