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.

## References

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. 1160.34313 MR1740241 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. 1160.34313 MR1740241 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. 0894.65036 MR1423802 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. 0894.65036 MR1423802 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. 0871.39002 MR1343651 10.1016/0895-7177(95)00096-K 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. 0871.39002 MR1343651 10.1016/0895-7177(95)00096-K 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. 1098.39001 MR2197124 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. 1098.39001 MR2197124 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. 1215.39001 MR2014482 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. 1215.39001 MR2014482 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. 1053.39011 MR2017238 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. 1053.39011 MR2017238 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. 1046.39005 MR1994691 10.1112/S0024610703004563 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. 1046.39005 MR1994691 10.1112/S0024610703004563 P. Jebelean and C. Şerban, “Ground state periodic solutions for difference equations with discrete

*p*-Laplacian,”*Applied Mathematics and Computation*, vol. 217, no. 23, pp. 9820–9827, 2011. 1228.39011 MR2811254 10.1016/j.amc.2011.04.076 P. Jebelean and C. Şerban, “Ground state periodic solutions for difference equations with discrete*p*-Laplacian,”*Applied Mathematics and Computation*, vol. 217, no. 23, pp. 9820–9827, 2011. 1228.39011 MR2811254 10.1016/j.amc.2011.04.076 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. 1112.39008 MR2277799 10.1016/j.jmaa.2006.03.017 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. 1112.39008 MR2277799 10.1016/j.jmaa.2006.03.017 J. Mawhin, “Periodic solutions of second order nonlinear difference systems with $\phi $-Laplacian: a variational approach,”

*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 12, pp. 4672–4687, 2012. MR2927127 J. Mawhin, “Periodic solutions of second order nonlinear difference systems with $\phi $-Laplacian: a variational approach,”*Nonlinear Analysis: Theory, Methods & Applications*, vol. 75, no. 12, pp. 4672–4687, 2012. MR2927127 J. Rodriguez and D. L. Etheridge, “Periodic solutions of nonlinear second-order difference equations,”

*Advances in Difference Equations*, no. 2, pp. 173–192, 2005. 1098.39004 MR2197131 J. Rodriguez and D. L. Etheridge, “Periodic solutions of nonlinear second-order difference equations,”*Advances in Difference Equations*, no. 2, pp. 173–192, 2005. 1098.39004 MR2197131 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. 1129.39008 MR2331858 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. 1129.39008 MR2331858 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. 1073.39009 MR2108253 10.1112/S0024610704005939 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. 1073.39009 MR2108253 10.1112/S0024610704005939 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. 1067.39022 MR2105789 10.1007/s10884-004-4292-2 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. 1067.39022 MR2105789 10.1007/s10884-004-4292-2 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. 1106.39022 MR2266548 10.1016/j.jmaa.2006.01.013 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. 1106.39022 MR2266548 10.1016/j.jmaa.2006.01.013 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. 1073.39010 MR2099576 10.1017/S0308210500003607 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. 1073.39010 MR2099576 10.1017/S0308210500003607 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. MR2893463 1248.39004 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. MR2893463 1248.39004 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. MR845785 0609.58002 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. MR845785 0609.58002*p*-Laplacian," Abstract and Applied Analysis 2012(SI09), 1-18, (2012). https://doi.org/10.1155/2012/135903