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Well-posedness of abstract distributed-order fractional diffusion equations
Infinitely many homoclinic solutions for damped vibration problems with subquadratic potentials
1. | Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China |
2. | School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875 |
References:
[1] |
C. O. Alves, P. C. Carrião and O. H. Miyagaki, paper title is capitalized., Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation,, Appl. Math. Lett., 16 (2003), 639.
doi: 10.1016/S0893-9659(03)00059-4. |
[2] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349.
doi: 10.1016/0022-1236(73)90051-7. |
[3] |
P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign,, Comm. Appl. Nonlinear Anal., 1 (1994), 97.
|
[4] |
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: 10.1090/S0894-0347-1991-1119200-3. |
[5] |
A. Daouas, Homoclinic solutions for superquadratic Hamiltonian systems without periodicity assumption,, Nonlinear Anal., 74 (2011), 3407.
doi: 10.1016/j.na.2011.03.001. |
[6] |
Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,, Nonlinear Anal., 25 (1995), 1095.
doi: 10.1016/0362-546X(94)00229-B. |
[7] |
Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and supequadratic Hamiltonian systems,, Nonlinear Anal., 71 (2009), 1395.
doi: 10.1016/j.na.2008.10.116. |
[8] |
M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems,, J. Differential Equations, 219 (2005), 375.
doi: 10.1016/j.jde.2005.06.029. |
[9] |
P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems,, Electron. J. Differential Equations, (1994), 1.
doi: 10.1.1.27.6093. |
[10] |
X. Lv and J. Jiang, Existence of homoclinic solutions for a class of second-order Hamiltonian systems with general potentials,, Nonlinear Analysis: Real World Applications, 13 (2012), 1152.
doi: 10.1016/j.nonrwa.2011.09.008. |
[11] |
X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems,, Nonlinear Anal., 72 (2010), 390.
doi: 10.1016/j.na.2009.06.073. |
[12] |
Y. Lv and C. Tang, Existence of even homoclinic orbits for a class of Hamiltonian systems,, Nonlinear Anal., 67 (2007), 2189.
doi: 10.1016/j.na.2006.08.043. |
[13] |
W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems,, Differential Integral Equations, 5 (1992), 1115.
|
[14] |
H. Poincaré, Les méthodes nouvelles de la mécanique céleste,, Gauthier-Villars, (): 1897.
|
[15] |
P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", CBMS Reg. Conf. Ser. in. Math., (1986).
|
[16] |
P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A., 114 (1990), 33.
doi: 10.1017/S0308210500024240. |
[17] |
P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, Math. Z., 206 (1991), 473.
doi: 10.1007/BF02571356. |
[18] |
A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems,, Proceedings of the Second World Congress of Nonlinear Analysis, 30 (1997), 4849.
doi: SO362-546X(97)00142-9. |
[19] |
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. |
[20] |
J. Sun, J. J. Nieto and M. Otero-Novoa, On homoclinic orbits for a class of damped vibration systems,, Advances in Difference Equations, 2012 ().
doi: 10.1186/1687-1847-2012-102. |
[21] |
X. Tang and X. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite subquadratic potentials,, Nonlinear Anal., 74 (2011), 6314.
doi: 10.1016/j.na.2011.06.010. |
[22] |
X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 1103.
doi: 10.1017/S0308210509001346. |
[23] |
L. Wan and C. Tang, Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition,, Discrete. Cont. Dyn. Syst. Ser. B, 15 (2011), 255.
doi: 10.3934/dcdsb.2011.15.255. |
[24] |
J. Wang, J. Xu and F. Zhang, Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials,, Comm. Pure. Appl. Anal., 10 (2011), 269.
doi: 10.3934/cpaa.2011.10.269. |
[25] |
J. Wang, F. Zhang and J. Xu, Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems,, J. Math. Anal. Appl., 366 (2010), 569.
doi: 10.1016/j.jmaa.2010.01.060. |
[26] |
J. Wei and J, Wang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials,, J. Math. Anal. Anal., 366 (2010), 694.
doi: 10.1016/j.jmaa.2009.12.024. |
[27] |
X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems,, Nonlinear Anal., 74 (2011), 4392.
doi: 10.1016/j.na.2011.03.059. |
[28] |
M. Yang and Z. Han, The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials,, Nonlinear Analysis: Real World Applications, 12 (2011), 2742.
doi: 10.1016/j.nonrwa.2011.03.019. |
[29] |
M. Yang and Z. Han, Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities,, Nonlinear Anal., 74 (2011), 2635.
doi: 10.1016/j.na.2010.12.019. |
[30] |
R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order non-autonomous systems,, Electron. J. of Differential Equations, 128 (2009), 1.
|
[31] |
Q. Zhang and C. Liu, Infinitely many homoclinic solutions for second order Hamiltonian systems,, Nonlinear Anal., 72 (2010), 894.
doi: 10.1016/j.na.2009.07.021. |
[32] |
Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems,, Nonlinear Anal., 71 (2009), 4125.
doi: 10.1016/j.na.2009.02.071. |
[33] |
Z. Zhang and R. Yuan, Homoclinic solutions of some second order non-autonomous systems,, Nonlinear Anal., 71 (2009), 5790.
doi: 10.1016/j.na.2009.05.003. |
[34] |
Z. Zhang and R. Yuan, Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems,, Nonlinear Analysis: Real World Applications, 11 (2010), 4185.
doi: 10.1016/j.nonrwa.2010.05.005. |
[35] |
W. Zhu, Existence of homoclinic solutions for a class of second order systems,, Nonlinear Anal., 75 (2012), 2455.
doi: 10.1016/j.na.2011.10.043. |
[36] |
W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems,, Appl. Math. Lett., 16 (2003), 1283.
doi: 10.1016/S0893-9659(03)90130-3. |
show all references
References:
[1] |
C. O. Alves, P. C. Carrião and O. H. Miyagaki, paper title is capitalized., Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation,, Appl. Math. Lett., 16 (2003), 639.
doi: 10.1016/S0893-9659(03)00059-4. |
[2] |
A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349.
doi: 10.1016/0022-1236(73)90051-7. |
[3] |
P. Caldiroli and P. Montecchiari, Homoclinic orbits for second order Hamiltonian systems with potential changing sign,, Comm. Appl. Nonlinear Anal., 1 (1994), 97.
|
[4] |
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: 10.1090/S0894-0347-1991-1119200-3. |
[5] |
A. Daouas, Homoclinic solutions for superquadratic Hamiltonian systems without periodicity assumption,, Nonlinear Anal., 74 (2011), 3407.
doi: 10.1016/j.na.2011.03.001. |
[6] |
Y. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,, Nonlinear Anal., 25 (1995), 1095.
doi: 10.1016/0362-546X(94)00229-B. |
[7] |
Y. Ding and C. Lee, Homoclinics for asymptotically quadratic and supequadratic Hamiltonian systems,, Nonlinear Anal., 71 (2009), 1395.
doi: 10.1016/j.na.2008.10.116. |
[8] |
M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second order Hamiltonian systems,, J. Differential Equations, 219 (2005), 375.
doi: 10.1016/j.jde.2005.06.029. |
[9] |
P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems,, Electron. J. Differential Equations, (1994), 1.
doi: 10.1.1.27.6093. |
[10] |
X. Lv and J. Jiang, Existence of homoclinic solutions for a class of second-order Hamiltonian systems with general potentials,, Nonlinear Analysis: Real World Applications, 13 (2012), 1152.
doi: 10.1016/j.nonrwa.2011.09.008. |
[11] |
X. Lv, S. Lu and P. Yan, Existence of homoclinic solutions for a class of second-order Hamiltonian systems,, Nonlinear Anal., 72 (2010), 390.
doi: 10.1016/j.na.2009.06.073. |
[12] |
Y. Lv and C. Tang, Existence of even homoclinic orbits for a class of Hamiltonian systems,, Nonlinear Anal., 67 (2007), 2189.
doi: 10.1016/j.na.2006.08.043. |
[13] |
W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems,, Differential Integral Equations, 5 (1992), 1115.
|
[14] |
H. Poincaré, Les méthodes nouvelles de la mécanique céleste,, Gauthier-Villars, (): 1897.
|
[15] |
P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", CBMS Reg. Conf. Ser. in. Math., (1986).
|
[16] |
P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A., 114 (1990), 33.
doi: 10.1017/S0308210500024240. |
[17] |
P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, Math. Z., 206 (1991), 473.
doi: 10.1007/BF02571356. |
[18] |
A. Salvatore, Homoclinic orbits for a special class of nonautonomous Hamiltonian systems,, Proceedings of the Second World Congress of Nonlinear Analysis, 30 (1997), 4849.
doi: SO362-546X(97)00142-9. |
[19] |
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. |
[20] |
J. Sun, J. J. Nieto and M. Otero-Novoa, On homoclinic orbits for a class of damped vibration systems,, Advances in Difference Equations, 2012 ().
doi: 10.1186/1687-1847-2012-102. |
[21] |
X. Tang and X. Lin, Infinitely many homoclinic orbits for Hamiltonian systems with indefinite subquadratic potentials,, Nonlinear Anal., 74 (2011), 6314.
doi: 10.1016/j.na.2011.06.010. |
[22] |
X. Tang and X. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A., 141 (2011), 1103.
doi: 10.1017/S0308210509001346. |
[23] |
L. Wan and C. Tang, Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition,, Discrete. Cont. Dyn. Syst. Ser. B, 15 (2011), 255.
doi: 10.3934/dcdsb.2011.15.255. |
[24] |
J. Wang, J. Xu and F. Zhang, Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials,, Comm. Pure. Appl. Anal., 10 (2011), 269.
doi: 10.3934/cpaa.2011.10.269. |
[25] |
J. Wang, F. Zhang and J. Xu, Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems,, J. Math. Anal. Appl., 366 (2010), 569.
doi: 10.1016/j.jmaa.2010.01.060. |
[26] |
J. Wei and J, Wang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with general potentials,, J. Math. Anal. Anal., 366 (2010), 694.
doi: 10.1016/j.jmaa.2009.12.024. |
[27] |
X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems,, Nonlinear Anal., 74 (2011), 4392.
doi: 10.1016/j.na.2011.03.059. |
[28] |
M. Yang and Z. Han, The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials,, Nonlinear Analysis: Real World Applications, 12 (2011), 2742.
doi: 10.1016/j.nonrwa.2011.03.019. |
[29] |
M. Yang and Z. Han, Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities,, Nonlinear Anal., 74 (2011), 2635.
doi: 10.1016/j.na.2010.12.019. |
[30] |
R. Yuan and Z. Zhang, Homoclinic solutions for a class of second order non-autonomous systems,, Electron. J. of Differential Equations, 128 (2009), 1.
|
[31] |
Q. Zhang and C. Liu, Infinitely many homoclinic solutions for second order Hamiltonian systems,, Nonlinear Anal., 72 (2010), 894.
doi: 10.1016/j.na.2009.07.021. |
[32] |
Z. Zhang and R. Yuan, Homoclinic solutions for a class of non-autonomous subquadratic second order Hamiltonian systems,, Nonlinear Anal., 71 (2009), 4125.
doi: 10.1016/j.na.2009.02.071. |
[33] |
Z. Zhang and R. Yuan, Homoclinic solutions of some second order non-autonomous systems,, Nonlinear Anal., 71 (2009), 5790.
doi: 10.1016/j.na.2009.05.003. |
[34] |
Z. Zhang and R. Yuan, Homoclinic solutions for a class of asymptotically quadratic Hamiltonian systems,, Nonlinear Analysis: Real World Applications, 11 (2010), 4185.
doi: 10.1016/j.nonrwa.2010.05.005. |
[35] |
W. Zhu, Existence of homoclinic solutions for a class of second order systems,, Nonlinear Anal., 75 (2012), 2455.
doi: 10.1016/j.na.2011.10.043. |
[36] |
W. Zou and S. Li, Infinitely many homoclinic orbits for the second-order Hamiltonian systems,, Appl. Math. Lett., 16 (2003), 1283.
doi: 10.1016/S0893-9659(03)90130-3. |
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