Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part

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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

Abstract
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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 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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