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

Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part

Qinqin Zhang 1,

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.
Keywords: Discrete Hamiltonian system, homoclinic orbit, strongly indefinite functional., periodicity.
Mathematics Subject Classification: Primary: 39A11; Secondary: 58E05, 70H0.
Citation: 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
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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

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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

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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

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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.  Google Scholar

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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.  Google Scholar

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

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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

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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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[21]

Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems,, \emph{Proc. Amer. Math. Soc.}, ().   Google Scholar

[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.  Google Scholar

[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.  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 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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems,, \emph{Proc. Amer. Math. Soc.}, ().   Google Scholar

[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.  Google Scholar

[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.  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 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.  Google Scholar

[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.  Google Scholar

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