September  2015, 14(5): 1929-1940. doi: 10.3934/cpaa.2015.14.1929

Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part

1. 

College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China

Received  November 2014 Revised  January 2015 Published  June 2015

Based on a generalized linking theorem for the strongly indefinite functionals, we study the existence of homoclinic orbits of the second order self-adjoint discrete Hamiltonian system \begin{eqnarray} \triangle [p(n)\triangle u(n-1)]-L(n)u(n)+\nabla W(n, u(n))=0, \end{eqnarray} where $p(n), L(n)$ and $W(n, x)$ are $N$-periodic on $n$, and $0$ lies in a gap of the spectrum $\sigma(\mathcal{A})$ of the operator $\mathcal{A}$, which is bounded self-adjoint in $l^2(\mathbb{Z}, \mathbb{R}^{\mathcal{N}})$ defined by $(\mathcal{A}u)(n)=\triangle [p(n)\triangle u(n-1)]-L(n)u(n)$. Under weak superquadratic conditions, we establish the existence of homoclinic orbits.
Citation: Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929
References:
[1]

R. P. Agarwal, Difference Equations and Inequalities: Theory, Method and Applications, second edition, Monographs and Textbooks in Pure and Applied Mathematics, 228, Marcel Dekker, Inc., 2000.

[2]

C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations, Kluwer Texts in the Mathematical Sciences, 16, Kluwer Academic, Dordrecht, 1996. doi: 10.1007/978-1-4757-2467-7.

[3]

C. J. Batkam, Homoclinic orbits of first-order superquadratic Hamiltonian systems, Discrete Contin. Dyn. Syst., 34 (2014), 3353-3369. doi: 10.3934/dcds.2014.34.3353.

[4]

J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger Hamiltonian with potentials of order zero, Discrete Contin. Dyn. Syst., 33 (2013), 1061-1076. doi: 10.3934/dcds.2013.33.1061.

[5]

X. Q. Deng and G. Cheng, Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign, Acta Appl. Math., 103 (2008), 301-314. doi: 10.1007/s10440-008-9237-z.

[6]

D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford, 1987.

[7]

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.

[8]

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.

[9]

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.

[10]

X. Y. Lin and X. H. Tang, Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential, Advances in Difference Equations, 154 (2013), http://www. advancesindifferenceequations.com/content/2013/1/154.

[11]

M. Ma and Z. M. Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlinear Anal., 67 (2007), 1737-1745. doi: 10.1016/j.na.2006.08.014.

[12]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.

[13]

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.

[14]

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.

[15]

J. Sun, J. Chu and Z. Feng, Homoclinic orbits for first order periodic Hamiltonian systems with spectrum zero, Discrete Contin. Dyn. Syst., 33 (2013), 3807-3824. doi: 10.3934/dcds.2013.33.3807.

[16]

X. H. Tang and J. Chen, Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems, Advances in Difference Equations, 242 (2013), http://www. advancesindifferenceequations.com/content/2013/1/242. doi: 10.1186/1687-1847-2013-242.

[17]

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.

[18]

X. H. Tang and X. Y. Lin, Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential, J. Differ. Equ. Appl., 19 (2013), 796-813. doi: 10.1080/10236198.2012.691168.

[19]

X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116. doi: 10.1017/S144678871400041X.

[20]

J. S. Yu, H. Shi 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.

[21]

Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., acceped for publication.

[22]

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.

[23]

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.

[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 and J. S. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Mathematica Sinica, 29 (2013), 1809-1822. doi: 10.1007/s10114-013-0736-0.

[26]

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, Method and Applications, second edition, Monographs and Textbooks in Pure and Applied Mathematics, 228, Marcel Dekker, Inc., 2000.

[2]

C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fraction and Riccati Equations, Kluwer Texts in the Mathematical Sciences, 16, Kluwer Academic, Dordrecht, 1996. doi: 10.1007/978-1-4757-2467-7.

[3]

C. J. Batkam, Homoclinic orbits of first-order superquadratic Hamiltonian systems, Discrete Contin. Dyn. Syst., 34 (2014), 3353-3369. doi: 10.3934/dcds.2014.34.3353.

[4]

J. Cruz-Sampedro, Boundary values of the resolvent of Schrödinger Hamiltonian with potentials of order zero, Discrete Contin. Dyn. Syst., 33 (2013), 1061-1076. doi: 10.3934/dcds.2013.33.1061.

[5]

X. Q. Deng and G. Cheng, Homoclinic orbits for second order discrete Hamiltonian systems with potential changing sign, Acta Appl. Math., 103 (2008), 301-314. doi: 10.1007/s10440-008-9237-z.

[6]

D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford, 1987.

[7]

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.

[8]

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.

[9]

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.

[10]

X. Y. Lin and X. H. Tang, Homoclinic orbits for discrete Hamiltonian systems with subquadratic potential, Advances in Difference Equations, 154 (2013), http://www. advancesindifferenceequations.com/content/2013/1/154.

[11]

M. Ma and Z. M. Guo, Homoclinic orbits and subharmonics for nonlinear second order difference equations, Nonlinear Anal., 67 (2007), 1737-1745. doi: 10.1016/j.na.2006.08.014.

[12]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.

[13]

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.

[14]

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.

[15]

J. Sun, J. Chu and Z. Feng, Homoclinic orbits for first order periodic Hamiltonian systems with spectrum zero, Discrete Contin. Dyn. Syst., 33 (2013), 3807-3824. doi: 10.3934/dcds.2013.33.3807.

[16]

X. H. Tang and J. Chen, Infinitely many homoclinic orbits for a class of discrete Hamiltonian systems, Advances in Difference Equations, 242 (2013), http://www. advancesindifferenceequations.com/content/2013/1/242. doi: 10.1186/1687-1847-2013-242.

[17]

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.

[18]

X. H. Tang and X. Y. Lin, Infinitely many homoclinic orbits for discrete Hamiltonian systems with subquadratic potential, J. Differ. Equ. Appl., 19 (2013), 796-813. doi: 10.1080/10236198.2012.691168.

[19]

X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116. doi: 10.1017/S144678871400041X.

[20]

J. S. Yu, H. Shi 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.

[21]

Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., acceped for publication.

[22]

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.

[23]

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.

[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 and J. S. Yu, Homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity, Acta Mathematica Sinica, 29 (2013), 1809-1822. doi: 10.1007/s10114-013-0736-0.

[26]

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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