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Existence of periodic solutions for a class of difference systems with \(p\)-Laplacian. (English) Zbl 1254.39009

Summary: By applying the least action principle and minimax methods in critical point theory, we prove the existence of periodic solutions for a class of difference systems with \(p\)-Laplacian and obtain some existence theorems.

MSC:

39A23 Periodic solutions of difference equations
39A12 Discrete version of topics in analysis
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
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[1] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, Chapman & Hall/CRC Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. · Zbl 1160.34313 · doi:10.4171/ZAA/964
[2] C. D. Ahlbrandt and A. C. Peterson, Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, vol. 16 of Kluwer Texts in the Mathematical Sciences, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1996. · Zbl 0894.65036
[3] S. N. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1999. · Zbl 0930.39001
[4] R. P. Agarwal and J. Popenda, “Periodic solutions of first order linear difference equations,” Mathematical and Computer Modelling, vol. 22, no. 1, pp. 11-19, 1995. · Zbl 0871.39002 · doi:10.1016/0895-7177(95)00096-K
[5] R. P. Agarwal, K. Perera, and D. O’Regan, “Multiple positive solutions of singular discrete p-Laplacian problems via variational methods,” Advances in Difference Equations, vol. 2005, no. 2, pp. 93-99, 2005. · Zbl 1098.39001 · doi:10.1155/ADE.2005.93
[6] Z. Guo and J. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear difference equations,” Science in China Series A, vol. 46, no. 4, pp. 506-515, 2003. · Zbl 1215.39001 · doi:10.1007/BF02884022
[7] Z. Guo and J. Yu, “Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 55, no. 7-8, pp. 969-983, 2003. · Zbl 1053.39011 · doi:10.1016/j.na.2003.07.019
[8] 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. · Zbl 1046.39005 · doi:10.1112/S0024610703004563
[9] P. Jebelean and C. \cSerban, “Ground state periodic solutions for difference equations with discrete p-Laplacian,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9820-9827, 2011. · Zbl 1228.39011 · doi:10.1016/j.amc.2011.04.076
[10] H. Liang and P. Weng, “Existence and multiple solutions for a second-order difference boundary value problem via critical point theory,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 511-520, 2007. · Zbl 1112.39008 · doi:10.1016/j.jmaa.2006.03.017
[11] J. Mawhin, “Periodic solutions of second order nonlinear difference systems with \varphi -Laplacian: a variational approach,” Nonlinear Analysis: Theory, Methods & Applications, vol. 75, no. 12, pp. 4672-4687, 2012. · Zbl 1252.39016 · doi:10.1016/j.jmaa.2006.03.017
[12] J. Rodriguez and D. L. Etheridge, “Periodic solutions of nonlinear second-order difference equations,” Advances in Difference Equations, no. 2, pp. 173-192, 2005. · Zbl 1098.39004 · doi:10.1155/ADE.2005.173
[13] Y.-F. Xue and C.-L. Tang, “Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 7, pp. 2072-2080, 2007. · Zbl 1129.39008 · doi:10.1016/j.na.2006.08.038
[14] J. Yu, Z. Guo, and X. Zou, “Periodic solutions of second order self-adjoint difference equations,” Journal of the London Mathematical Society, vol. 71, no. 1, pp. 146-160, 2005. · Zbl 1073.39009 · doi:10.1112/S0024610704005939
[15] 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. · Zbl 1067.39022 · doi:10.1007/s10884-004-4292-2
[16] J. Yu, X. Deng, and Z. Guo, “Periodic solutions of a discrete Hamiltonian system with a change of sign in the potential,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1140-1151, 2006. · Zbl 1106.39022 · doi:10.1016/j.jmaa.2006.01.013
[17] 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. · Zbl 1073.39010 · doi:10.1017/S0308210500003607
[18] Z. M. Luo and X. Y. Zhang, “Existence of nonconstant periodic solutions for a nonlinear discrete system involving the p-Laplacian,” Bulletin of the Malaysian Mathematical Science Society, vol. 35, no. 2, pp. 373-382, 2012. · Zbl 1248.39004 · doi:10.1017/S0308210500003607
[19] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, USA, 1986. · Zbl 0609.58002
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