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Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators
Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions
Center for Applied Mathematics, Guangzhou University, Guangzhou, Guangdong, 510006, China |
$\triangle [p(n)\triangle u(n-1)]-L(n)u(n)+\nabla W(n, u(n)) = 0, $ |
$p(n), L(n)$ |
$W(n, x)$ |
$N$ |
$n$ |
zhongwenzy$ |
$σ(\mathcal{A})$ |
$\mathcal{A}$ |
$l^2(\mathbb{Z}, \mathbb{R}^{\mathcal{N}})$ |
$(\mathcal{A}u)(n) = \triangle [p(n)\triangle u(n-1)]-L(n)u(n)$ |
$\lim_{|x|\to ∞}\frac{W(n, x)}{|x|^2} = ∞$ |
$ n∈ \mathbb{Z}$ |
References:
[1] |
R. P. Agarwal, Difference Equations and Inequalities: Theory, Metho and Applications, second edition, Marcel Dekker, Inc. 2000. |
[2] |
C. D. Ahlbran and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations, Kluwer Academic, Dordrecht, 1996.
doi: 10.1007/978-1-4757-2467-7. |
[3] |
Z. Balanov, C. Carcía - Azpeitia and W. Krawcewicz,
On Variational and Topological Methods in Nonlinear Difference Equations, Commun. Pure Appl. Anal., 17 (2018), 2813-2844.
doi: 10.3934/cpaa.2018133. |
[4] |
V. Coti Zelati, I. Ekeland and E. Sere,
A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.
doi: 10.1007/BF01444526. |
[5] |
V. Coti Zelati and P. H. Rabinowitz,
Homoclinic orbits for second second order Hamiltonian
systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727.
doi: 10.2307/2939286. |
[6] |
D. E. Edmunds, W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987. |
[7] |
Z. M. Guo and J. S. Yu,
The existence of periodic and subharmonic solutions of subquadratic second order difference equation, J. London Math. Soc., 68 (2003), 419-430.
doi: 10.1112/S0024610703004563. |
[8] |
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. |
[9] |
G. B. Li and A. Szulkin,
An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
doi: 10.1142/S0219199702000853. |
[10] |
G. H. Lin and Z. Zhou,
Homoclinic solutions of discrete φ- Laplacian equations with mixed nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1723-1747.
doi: 10.3934/cpaa.2018082. |
[11] |
X. Y. Lin and X. H. Tang,
Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 59-72.
doi: 10.1016/j.jmaa.2010.06.008. |
[12] |
Z. L. Liu and Z.-Q. Wang,
On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud., 4 (2004), 561-572.
doi: 10.1515/ans-2004-0411. |
[13] |
W. Omana and M. Willem,
Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.
|
[14] |
P. H. Rabinowitz,
Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.
doi: 10.1017/S0308210500024240. |
[15] |
P. H. Rabinowitz and K. Tanaka,
Some results on connecting orbits for a class of Hamiltonian
systems, Math. Z., 206 (1991), 473-499.
doi: 10.1007/BF02571356. |
[16] |
X. H. Tang,
Non-Nehari manifold method for asymptotically linear Schr¨odinger equation, J. Aust. Math. Soc., 98 (2015), 104-116.
doi: 10.1017/S144678871400041X. |
[17] |
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for SchrödingerPoisson problems with general potentials, Disc. Contin. Dyn. Syst.-Series A., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[18] |
X. H. Tang and X. Y. Lin,
Existence and multiplicity of homoclinic solutions for second-order
discrete Hamiltonian systems with subquadratic potential, J. Differ. Equ. Appl., 17 (2011), 1617-1634.
doi: 10.1080/10236191003730514. |
[19] |
X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Diff. Equat..
doi: 10.1007/s10884-018-9662-2. |
[20] |
H. F. Xiao and J. S. Yu,
Heteroclinic orbits for a discrete pendulum equation, J. Difference Equ. Appl., 17 (2011), 1267-1280.
doi: 10.1080/10236190903167991. |
[21] |
J. S. Yu and Z. M. Guo,
Homoclinic orbits for nonlinear difference equations containing
both advance and retardation, J. Math. Anal. Appl., 352 (2009), 799-806.
doi: 10.1016/j.jmaa.2008.11.043. |
[22] |
Q. Zhang,
Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., 143 (2015), 3155-3163.
doi: 10.1090/S0002-9939-2015-12107-7. |
[23] |
Q. Zhang,
Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part and
super linear terms, Commun. Pure Appl. Anal., 14 (2015), 1929-1940.
doi: 10.3934/cpaa.2015.14.1929. |
[24] |
Z. Zhou and J. S. Yu,
On the existence of homoclinic solutions of a class of discrete nonlinear
periodic systems, J. Differential Equations, 249 (2010), 1199-1212.
doi: 10.1016/j.jde.2010.03.010. |
[25] |
Z. Zhou, J. S. Yu and Y. Chen,
Homoclinic solutions in periodic difference equations with
saturable nonlinearity, Science China, Mathematics, 54 (2011), 83-93.
doi: 10.1007/s11425-010-4101-9. |
show all references
References:
[1] |
R. P. Agarwal, Difference Equations and Inequalities: Theory, Metho and Applications, second edition, Marcel Dekker, Inc. 2000. |
[2] |
C. D. Ahlbran and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations, Kluwer Academic, Dordrecht, 1996.
doi: 10.1007/978-1-4757-2467-7. |
[3] |
Z. Balanov, C. Carcía - Azpeitia and W. Krawcewicz,
On Variational and Topological Methods in Nonlinear Difference Equations, Commun. Pure Appl. Anal., 17 (2018), 2813-2844.
doi: 10.3934/cpaa.2018133. |
[4] |
V. Coti Zelati, I. Ekeland and E. Sere,
A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.
doi: 10.1007/BF01444526. |
[5] |
V. Coti Zelati and P. H. Rabinowitz,
Homoclinic orbits for second second order Hamiltonian
systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727.
doi: 10.2307/2939286. |
[6] |
D. E. Edmunds, W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987. |
[7] |
Z. M. Guo and J. S. Yu,
The existence of periodic and subharmonic solutions of subquadratic second order difference equation, J. London Math. Soc., 68 (2003), 419-430.
doi: 10.1112/S0024610703004563. |
[8] |
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. |
[9] |
G. B. Li and A. Szulkin,
An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.
doi: 10.1142/S0219199702000853. |
[10] |
G. H. Lin and Z. Zhou,
Homoclinic solutions of discrete φ- Laplacian equations with mixed nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1723-1747.
doi: 10.3934/cpaa.2018082. |
[11] |
X. Y. Lin and X. H. Tang,
Existence of infinitely many homoclinic orbits in discrete Hamiltonian systems, J. Math. Anal. Appl., 373 (2011), 59-72.
doi: 10.1016/j.jmaa.2010.06.008. |
[12] |
Z. L. Liu and Z.-Q. Wang,
On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud., 4 (2004), 561-572.
doi: 10.1515/ans-2004-0411. |
[13] |
W. Omana and M. Willem,
Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.
|
[14] |
P. H. Rabinowitz,
Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.
doi: 10.1017/S0308210500024240. |
[15] |
P. H. Rabinowitz and K. Tanaka,
Some results on connecting orbits for a class of Hamiltonian
systems, Math. Z., 206 (1991), 473-499.
doi: 10.1007/BF02571356. |
[16] |
X. H. Tang,
Non-Nehari manifold method for asymptotically linear Schr¨odinger equation, J. Aust. Math. Soc., 98 (2015), 104-116.
doi: 10.1017/S144678871400041X. |
[17] |
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for SchrödingerPoisson problems with general potentials, Disc. Contin. Dyn. Syst.-Series A., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[18] |
X. H. Tang and X. Y. Lin,
Existence and multiplicity of homoclinic solutions for second-order
discrete Hamiltonian systems with subquadratic potential, J. Differ. Equ. Appl., 17 (2011), 1617-1634.
doi: 10.1080/10236191003730514. |
[19] |
X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Diff. Equat..
doi: 10.1007/s10884-018-9662-2. |
[20] |
H. F. Xiao and J. S. Yu,
Heteroclinic orbits for a discrete pendulum equation, J. Difference Equ. Appl., 17 (2011), 1267-1280.
doi: 10.1080/10236190903167991. |
[21] |
J. S. Yu and Z. M. Guo,
Homoclinic orbits for nonlinear difference equations containing
both advance and retardation, J. Math. Anal. Appl., 352 (2009), 799-806.
doi: 10.1016/j.jmaa.2008.11.043. |
[22] |
Q. Zhang,
Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., 143 (2015), 3155-3163.
doi: 10.1090/S0002-9939-2015-12107-7. |
[23] |
Q. Zhang,
Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part and
super linear terms, Commun. Pure Appl. Anal., 14 (2015), 1929-1940.
doi: 10.3934/cpaa.2015.14.1929. |
[24] |
Z. Zhou and J. S. Yu,
On the existence of homoclinic solutions of a class of discrete nonlinear
periodic systems, J. Differential Equations, 249 (2010), 1199-1212.
doi: 10.1016/j.jde.2010.03.010. |
[25] |
Z. Zhou, J. S. Yu and Y. Chen,
Homoclinic solutions in periodic difference equations with
saturable nonlinearity, Science China, Mathematics, 54 (2011), 83-93.
doi: 10.1007/s11425-010-4101-9. |
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