• Previous Article
    Study on the stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous systems
  • DCDS-B Home
  • This Issue
  • Next Article
    Global convergence of a predator-prey model with stage structure and spatio-temporal delay
January  2011, 15(1): 255-271. doi: 10.3934/dcdsb.2011.15.255

Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China, China

Received  June 2009 Revised  September 2010 Published  October 2010

The existence and multiplicity of homoclinic orbits for a class of the second order Hamiltonian systems $\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \forall t \in \mathbb{R}$, are obtained via the concentration-compactness principle and the fountain theorem respectively, where $W(t, x)$ is superquadratic and need not satisfy the (AR) condition with respect to the second variable $ x\in\mathbb{R}^{N}$.
Citation: Li-Li Wan, Chun-Lei Tang. Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 255-271. doi: 10.3934/dcdsb.2011.15.255
References:
[1]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation,, Appl. Math. Lett., 16 (2003), 639. doi: doi:10.1016/S0893-9659(03)00059-4. Google Scholar

[2]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity,, Nonlinear Anal., 7 (1983), 981. doi: doi:10.1016/0362-546X(83)90115-3. Google Scholar

[3]

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: doi:10.1090/S0894-0347-1991-1119200-3. Google Scholar

[4]

Y. H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,, Nonlinear Anal., 25 (1995), 1095. doi: doi:10.1016/0362-546X(94)00229-B. Google Scholar

[5]

Y. H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms,, Commun. Contemp. Math., 8 (2006), 453. doi: doi:10.1142/S0219199706002192. Google Scholar

[6]

M. F. Furtado, L. A. Maia and E. A. B. Silva, Systems with coupling in $R^N$ for a class of noncoercive potentials,, Discrete Contin. Dyn. Syst., (2003), 295. Google Scholar

[7]

P. L. Felmer and E. A. B. Silva, Homoclinic and periodic orbits for Hamiltonian systems,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 285. Google Scholar

[8]

P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems,, Electron. J. Differential Equations, (1994). Google Scholar

[9]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1995). Google Scholar

[10]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223. Google Scholar

[11]

Y. Lv and C. L. Tang, Existence of even homoclinic orbits for second-order Hamiltonian systems,, Nonlinear Anal., 67 (2007), 2189. doi: doi:10.1016/j.na.2006.08.043. Google Scholar

[12]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems,, Differential Integral Equations, 5 (1992), 1115. Google Scholar

[13]

Z. Q. Ou and C. L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems,, J. Math. Anal. Appl., 291 (2004), 203. doi: doi:10.1016/j.jmaa.2003.10.026. Google Scholar

[14]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33. Google Scholar

[15]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, Math. Z., 206 (1991), 473. doi: doi:10.1007/BF02571356. Google Scholar

[16]

P. H. Rabinowitz, Homoclinic and heteroclinic orbits for a class of Hamiltonian systems,, Calc. Variations, 1 (1993), 1. doi: doi:10.1007/BF02163262. Google Scholar

[17]

M. Willem, "Minimax Theorems,", 24,, (1996). Google Scholar

[18]

J. Yang and F. B. Zhao, Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials,, Nonlinear Analysis: Real World Applications, 10 (2009), 1417. doi: doi:10.1016/j.nonrwa.2008.01.013. Google Scholar

[19]

W. M. Zou and S. J. Li, Infinitely many homoclinic orbits for the second-order hamiltonian systems,, Appl. Math. Lett., 16 (2003), 1283. doi: doi:10.1016/S0893-9659(03)90130-3. Google Scholar

[20]

W. M. Zou, Variant fountain theorems and their applications,, Manuscripta Math., 104 (2001), 343. doi: doi:10.1007/s002290170032. Google Scholar

show all references

References:
[1]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation,, Appl. Math. Lett., 16 (2003), 639. doi: doi:10.1016/S0893-9659(03)00059-4. Google Scholar

[2]

P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity,, Nonlinear Anal., 7 (1983), 981. doi: doi:10.1016/0362-546X(83)90115-3. Google Scholar

[3]

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: doi:10.1090/S0894-0347-1991-1119200-3. Google Scholar

[4]

Y. H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems,, Nonlinear Anal., 25 (1995), 1095. doi: doi:10.1016/0362-546X(94)00229-B. Google Scholar

[5]

Y. H. Ding, Multiple homoclinics in a Hamiltonian system with asymptotically or super linear terms,, Commun. Contemp. Math., 8 (2006), 453. doi: doi:10.1142/S0219199706002192. Google Scholar

[6]

M. F. Furtado, L. A. Maia and E. A. B. Silva, Systems with coupling in $R^N$ for a class of noncoercive potentials,, Discrete Contin. Dyn. Syst., (2003), 295. Google Scholar

[7]

P. L. Felmer and E. A. B. Silva, Homoclinic and periodic orbits for Hamiltonian systems,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 285. Google Scholar

[8]

P. Korman and A. C. Lazer, Homoclinic orbits for a class of symmetric Hamiltonian systems,, Electron. J. Differential Equations, (1994). Google Scholar

[9]

T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1995). Google Scholar

[10]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223. Google Scholar

[11]

Y. Lv and C. L. Tang, Existence of even homoclinic orbits for second-order Hamiltonian systems,, Nonlinear Anal., 67 (2007), 2189. doi: doi:10.1016/j.na.2006.08.043. Google Scholar

[12]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems,, Differential Integral Equations, 5 (1992), 1115. Google Scholar

[13]

Z. Q. Ou and C. L. Tang, Existence of homoclinic solution for the second order Hamiltonian systems,, J. Math. Anal. Appl., 291 (2004), 203. doi: doi:10.1016/j.jmaa.2003.10.026. Google Scholar

[14]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems,, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33. Google Scholar

[15]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems,, Math. Z., 206 (1991), 473. doi: doi:10.1007/BF02571356. Google Scholar

[16]

P. H. Rabinowitz, Homoclinic and heteroclinic orbits for a class of Hamiltonian systems,, Calc. Variations, 1 (1993), 1. doi: doi:10.1007/BF02163262. Google Scholar

[17]

M. Willem, "Minimax Theorems,", 24,, (1996). Google Scholar

[18]

J. Yang and F. B. Zhao, Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials,, Nonlinear Analysis: Real World Applications, 10 (2009), 1417. doi: doi:10.1016/j.nonrwa.2008.01.013. Google Scholar

[19]

W. M. Zou and S. J. Li, Infinitely many homoclinic orbits for the second-order hamiltonian systems,, Appl. Math. Lett., 16 (2003), 1283. doi: doi:10.1016/S0893-9659(03)90130-3. Google Scholar

[20]

W. M. Zou, Variant fountain theorems and their applications,, Manuscripta Math., 104 (2001), 343. doi: doi:10.1007/s002290170032. Google Scholar

[1]

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

[2]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807

[3]

Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353

[4]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128

[5]

Dong-Lun Wu, Chun-Lei Tang, Xing-Ping Wu. Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 57-72. doi: 10.3934/cpaa.2016.15.57

[6]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623

[7]

Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139

[8]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929

[9]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions. Communications on Pure & Applied Analysis, 2019, 18 (1) : 425-434. doi: 10.3934/cpaa.2019021

[10]

Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for a class of Hamiltonian systems with superquadratic or asymptotically quadratic potentials. Communications on Pure & Applied Analysis, 2011, 10 (1) : 269-286. doi: 10.3934/cpaa.2011.10.269

[11]

Norimichi Hirano, Zhi-Qiang Wang. Subharmonic solutions for second order Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 467-474. doi: 10.3934/dcds.1998.4.467

[12]

Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765

[13]

Amadeu Delshams, Pere Gutiérrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 757-783. doi: 10.3934/dcds.2004.11.757

[14]

Jun Wang, Junxiang Xu, Fubao Zhang. Homoclinic orbits for superlinear Hamiltonian systems without Ambrosetti-Rabinowitz growth condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1241-1257. doi: 10.3934/dcds.2010.27.1241

[15]

Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153

[16]

Xingyong Zhang, Xianhua Tang. Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 75-95. doi: 10.3934/cpaa.2014.13.75

[17]

S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493

[18]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097

[19]

Shanjian Tang. A second-order maximum principle for singular optimal stochastic controls. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1581-1599. doi: 10.3934/dcdsb.2010.14.1581

[20]

Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]