Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author

    * Corresponding author

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

Abstract Full Text(HTML) Related Papers Cited by
  • 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 0 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.

    Mathematics Subject Classification: Primary: 39A11, 58E05; Secondary: 70H05.


    \begin{equation} \\ \end{equation}
  • 加载中
  •   R. P. Agarwal, Difference Equations and Inequalities: Theory, Metho and Applications, second edition, Marcel Dekker, Inc. 2000.
      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.
      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.
      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.
      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.
      D. E. Edmunds, W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
      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.
      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.
      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.
      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.
      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.
      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.
      W. Omana  and  M. Willem , Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992) , 1115-1120. 
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
      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.
  • 加载中

Article Metrics

HTML views(756) PDF downloads(367) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint