# American Institute of Mathematical Sciences

January  2019, 18(1): 425-434. doi: 10.3934/cpaa.2019021

## Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions

 Center for Applied Mathematics, Guangzhou University, Guangzhou, Guangdong, 510006, China

* Corresponding author

Received  February 2018 Revised  April 2018 Published  August 2018

Fund Project: This work was partially supported by National Natural Science Foundation of China grant 11471085 and the Program for Changjiang Scholars and Innovative Research Team in University grant IRT1226

Consider the second order self-adjoint discrete Hamiltonian system
 $\triangle [p(n)\triangle u(n-1)]-L(n)u(n)+\nabla W(n, u(n)) = 0,$
where
 $p(n), L(n)$
and
 $W(n, x)$
are
 $N$
-periodic on
 $n$
, and
 zhongwenzy$lies in a gap of the spectrum $σ(\mathcal{A})$of the operator $\mathcal{A}$, which is bounded self-adjoint in $l^2(\mathbb{Z}, \mathbb{R}^{\mathcal{N}})$defined by $(\mathcal{A}u)(n) = \triangle [p(n)\triangle u(n-1)]-L(n)u(n)$. We obtain a sufficient condition on the existence of nontrivial homoclinic orbits for the above system under a much weaker condition than $\lim_{|x|\to ∞}\frac{W(n, x)}{|x|^2} = ∞$uniformly in $ n∈ \mathbb{Z}$, which has been a common condition used in the existing literature. We also give three examples to illustrate our result. Citation: 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 ##### References:  [1] R. P. Agarwal, Difference Equations and Inequalities: Theory, Metho and Applications, second edition, Marcel Dekker, Inc. 2000. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] D. E. Edmunds, W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120. Google Scholar [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. Google Scholar [15] P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499. doi: 10.1007/BF02571356. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar show all references ##### References:  [1] R. P. Agarwal, Difference Equations and Inequalities: Theory, Metho and Applications, second edition, Marcel Dekker, Inc. 2000. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] D. E. Edmunds, W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [13] W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120. Google Scholar [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. Google Scholar [15] P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499. doi: 10.1007/BF02571356. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar  [1] Xiao-Fei Zhang, Fei Guo. Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1625-1642. doi: 10.3934/cpaa.2016005 [2] 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 [3] Xiangjin Xu. Multiple solutions of super-quadratic second order dynamical systems. Conference Publications, 2003, 2003 (Special) : 926-934. doi: 10.3934/proc.2003.2003.926 [4] Alain Bensoussan, Jens Frehse. On diagonal elliptic and parabolic systems with super-quadratic Hamiltonians. Communications on Pure & Applied Analysis, 2009, 8 (1) : 83-94. doi: 10.3934/cpaa.2009.8.83 [5] Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293 [6] 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 [7] Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883 [8] 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 [9] 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 [10] Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the$L^\infty\$ bound of solutions for a strongly indefinite differential superlinear system. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107 [11] 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 [12] 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 [13] Christian Bonatti, Lorenzo J. Díaz, Todd Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 589-604. doi: 10.3934/dcds.2008.20.589 [14] Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485 [15] W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351 [16] 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 [17] Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829 [18] B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217 [19] Fabrizio Colombo, Davide Guidetti. Identification of the memory kernel in the strongly damped wave equation by a flux condition. Communications on Pure & Applied Analysis, 2009, 8 (2) : 601-620. doi: 10.3934/cpaa.2009.8.601 [20] Sergey Gonchenko, Ivan Ovsyannikov. Homoclinic tangencies to resonant saddles and discrete Lorenz attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 273-288. doi: 10.3934/dcdss.2017013

2018 Impact Factor: 0.925