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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), 423-443.
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), 277-320.
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), 291-313.
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), 1267-1288.
doi: 10.1002/mana.200410420. |
[5] |
T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods, in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. II, Elsevier B. V., Amsterdam, (2005), 77-146. |
[6] |
G. Bonanno and R. Livrea, Multiple periodic solutions for Hamiltonian systems with not coercive potential, J. Math. Anal. Appl., 363 (2010), 627-638.
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), 589-603.
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), 423-443.
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), 133-160.
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), 693-727.
doi: 10.2307/2939286. |
[11] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^N2$, Comm. Pure Appl. Math., 45 (1992), 1217-1269.
doi: 10.1002/cpa.3160451002. |
[12] |
Y. Ding, "Variational Methods for Strongly Indefinite Problems," Interdiscip. Math. Sci., 7, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 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), 453-480.
doi: 10.1142/S0219199706002192. |
[14] |
Y. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal., 38 (1999), 391-415.
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), 473-490.
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), 2829-2848.
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), 585-601.
doi: 10.1006/jmaa.1995.1037. |
[18] |
Y. Ding and M. Willem, Homoclinic orbits of a Hamiltonian system, Z. Angew. Math. Phys., 50 (1999), 759-778.
doi: 10.1007/s000330050177. |
[19] |
Z. Feng and D. Y. Gao, An asymptotic expression of the Schrödinger equation, Z. Angew. Math. Phys., 60 (2009), 363-375.
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), 429-440.
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), 375-389.
doi: 10.1016/j.jde.2005.06.029. |
[22] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. |
[23] |
A. Mielke, Weak-convergence methods for hamiltonian multiscale problems, Discrete Contin. Dyn. Syst., 20 (2008), 53-79.
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), 883-913.
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), 93-109.
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), 4575-4586.
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), 20-29.
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), 117-127.
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), 25-41.
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), 1241-1257.
doi: 10.3934/dcds.2010.27.1241. |
[31] |
S. Zhang, Symmetrically homoclinic orbits for symmetric Hamiltonian systems, J. Math. Anal. Appl., 247 (2000), 645-652.
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), 423-443.
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), 277-320.
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), 291-313.
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), 1267-1288.
doi: 10.1002/mana.200410420. |
[5] |
T. Bartsch and A. Szulkin, Hamiltonian Systems: Periodic and Homoclinic Solutions by Variational Methods, in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. II, Elsevier B. V., Amsterdam, (2005), 77-146. |
[6] |
G. Bonanno and R. Livrea, Multiple periodic solutions for Hamiltonian systems with not coercive potential, J. Math. Anal. Appl., 363 (2010), 627-638.
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), 589-603.
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), 423-443.
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), 133-160.
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), 693-727.
doi: 10.2307/2939286. |
[11] |
V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^N2$, Comm. Pure Appl. Math., 45 (1992), 1217-1269.
doi: 10.1002/cpa.3160451002. |
[12] |
Y. Ding, "Variational Methods for Strongly Indefinite Problems," Interdiscip. Math. Sci., 7, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 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), 453-480.
doi: 10.1142/S0219199706002192. |
[14] |
Y. Ding and M. Girardi, Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonlinear Anal., 38 (1999), 391-415.
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), 473-490.
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), 2829-2848.
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), 585-601.
doi: 10.1006/jmaa.1995.1037. |
[18] |
Y. Ding and M. Willem, Homoclinic orbits of a Hamiltonian system, Z. Angew. Math. Phys., 50 (1999), 759-778.
doi: 10.1007/s000330050177. |
[19] |
Z. Feng and D. Y. Gao, An asymptotic expression of the Schrödinger equation, Z. Angew. Math. Phys., 60 (2009), 363-375.
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), 429-440.
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), 375-389.
doi: 10.1016/j.jde.2005.06.029. |
[22] |
J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989. |
[23] |
A. Mielke, Weak-convergence methods for hamiltonian multiscale problems, Discrete Contin. Dyn. Syst., 20 (2008), 53-79.
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), 883-913.
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), 93-109.
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), 4575-4586.
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), 20-29.
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), 117-127.
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), 25-41.
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), 1241-1257.
doi: 10.3934/dcds.2010.27.1241. |
[31] |
S. Zhang, Symmetrically homoclinic orbits for symmetric Hamiltonian systems, J. Math. Anal. Appl., 247 (2000), 645-652.
doi: 10.1006/jmaa.2000.6839. |
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