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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 567471, 10 pages
Nonconstant Periodic Solutions of Discrete -Laplacian System via Clark Duality and Computations of the Critical Groups
1College of Mathematics and Information Sciences, Guangzhou University, Guangzhou, Guangdong 510006, China
2Key Laboratory of Mathematics and Interdisciplinary Sciences of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou, Guangdong 510006, China
Received 18 October 2013; Accepted 2 January 2014; Published 12 February 2014
Academic Editor: Svatoslav Staněk
Copyright © 2014 Bo Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Z. Guo and J. Yu, “The existence of periodic and subharmonic solutions of subquadratic second order difference equations,” Journal of the London Mathematical Society, vol. 68, no. 2, pp. 419–430, 2003.
- J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, vol. 74 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989.
- Z. Guo and J. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear difference equations,” Science in China A, vol. 46, no. 4, pp. 506–515, 2003.
- Z. Zhou, J. Yu, and Z. Guo, “Periodic solutions of higher-dimensional discrete systems,” Proceedings of the Royal Society of Edinburgh A, vol. 134, no. 5, pp. 1013–1022, 2004.
- H.-H. Bin, J.-S. Yu, and Z.-M. Guo, “Nontrivial periodic solutions for asymptotically linear resonant difference problem,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 477–488, 2006.
- X. Deng, H. Shi, and X. Xie, “Periodic solutions of second order discrete Hamiltonian systems with potential indefinite in sign,” Applied Mathematics and Computation, vol. 218, no. 1, pp. 148–156, 2011.
- H. Shi, “Boundary value problems of second order nonlinear functional difference equations,” Journal of Difference Equations and Applications, vol. 16, no. 9, pp. 1121–1130, 2010.
- Y.-F. Xue and C.-L. Tang, “Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 67, no. 7, pp. 2072–2080, 2007.
- J. Yu, H. Bin, and Z. Guo, “Multiple periodic solutions for discrete Hamiltonian systems,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 66, no. 7, pp. 1498–1512, 2007.
- J. Yu, Y. Long, and Z. Guo, “Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation,” Journal of Dynamics and Differential Equations, vol. 16, no. 2, pp. 575–586, 2004.
- B. Zheng, “Multiple periodic solutions to nonlinear discrete Hamiltonian systems,” Advances in Difference Equations, vol. 2007, Article ID 41830, 13 pages, 2007.
- Z. Zhou, J. Yu, and Y. Chen, “On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity,” Nonlinearity, vol. 23, no. 7, pp. 1727–1740, 2010.
- Z. Zhou, J. Yu, and Y. Chen, “Periodic solutions of a th-order nonlinear difference equation,” Science China, vol. 53, no. 1, pp. 41–50, 2010.
- S. Amghibech, “Bounds for the largest -Laplacian eigenvalue for graphs,” Discrete Mathematics, vol. 306, no. 21, pp. 2762–2771, 2006.
- R. P. Agarwal, D. O'Regan, and S. Stanêk, “An existence principle for nonlocal difference boundary value problems with -Laplacian and its application to singular problems,” Advances in Difference Equations, vol. 2008, Article ID 154302, 10 pages, 2008.
- R. P. Agarwal, K. Perera, and D. O'Regan, “Multiple positive solutions of singular discrete -Laplacian problems via variational methods,” Advances in Difference Equations, no. 2, pp. 93–99, 2005.
- P. Candito and N. Giovannelli, “Multiple solutions for a discrete boundary value problem involving the -Laplacian,” Computers & Mathematics with Applications, vol. 56, no. 4, pp. 959–964, 2008.
- Z. He, “On the existence of positive solutions of -Laplacian difference equations,” Journal of Computational and Applied Mathematics, vol. 161, no. 1, pp. 193–201, 2003.
- L. Jiang and Z. Zhou, “Three solutions to Dirichlet boundary value problems for -Laplacian difference equations,” Advances in Difference Equations, vol. 2008, Article ID 345916, 10 pages, 2008.
- J.-H. Park and S.-Y. Chung, “The Dirichlet boundary value problems for -Schrödinger operators on finite networks,” Journal of Difference Equations and Applications, vol. 17, no. 5, pp. 795–811, 2011.
- G. Bonanno and P. Candito, “Nonlinear difference equations investigated via critical point methods,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 70, no. 9, pp. 3180–3186, 2009.
- T. He and W. Chen, “Periodic solutions of second order discrete convex systems involving the -Laplacian,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 124–132, 2008.
- Z. Luo and X. Zhang, “Existence of nonconstant periodic solutions for a nonlinear discrete system involving the -Laplacian,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 35, no. 2, pp. 373–382, 2012.
- K. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, vol. 6 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, Mass, USA, 1993.
- T. Bartsch and S. Li, “Critical point theory for asymptotically quadratic functionals and applications to problems with resonance,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 28, no. 3, pp. 419–441, 1997.
- F. H. Clarke and I. Ekeland, “Hamiltonian trajectories having prescribed minimal period,” Communications on Pure and Applied Mathematics, vol. 33, no. 2, pp. 103–116, 1980.
- G. Cerami, “An existence criterion for the critical points on unbounded manifolds,” Istituto Lombardo. Accademia di Scienze e Lettere. Rendiconti. Scienze Matematiche, Fisiche, Chimiche e Geologiche A, vol. 112, no. 2, pp. 332–336, 1978.
- P. Bartolo, V. Benci, and D. Fortunato, “Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 7, no. 9, pp. 981–1012, 1983.
- K. C. Chang, “Solutions of asymptotically linear operator equations via Morse theory,” Communications on Pure and Applied Mathematics, vol. 34, no. 5, pp. 693–712, 1981.
- E. Spanier, Algebraic Topology, Springer, 1994.
- M. Reed and B. I. Simon, Functional Analysis, Elsevier, 1981.