# American Institute of Mathematical Sciences

## Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations

 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

* Corresponding author: Zhan Zhou

Received  December 2017 Revised  May 2018 Published  December 2018

We consider a 2$n$th-order nonlinear difference equation containing both many advances and retardations with $\phi_c$-Laplacian. Using the critical point theory, we obtain some new and concrete criteria for the existence and multiplicity of periodic and subharmonic solutions in the more general case of the nonlinearity.

Citation: Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019134
##### References:
 [1] Z. AlSharawi, J. M. Cushing and S. Elaydi, Theory and Applications of Difference Equations and Discrete Dynamical Systems, Springer Proceedings in Mathematics & Statistics, 102. Springer, Heidelberg, 2014. [2] Z. Balanov, C. Garcia-Azpeitia and W. Krawcewicz, On variational and topological methods in nonlinear difference equations, Communications on Pure and Applied Analysis, 17 (2018), 2813-2844. doi: 10.3934/cpaa.2018133. [3] X. C. Cai and J. S. Yu, Existence of periodic solutions for a 2$n$th-order nonlinear difference equation, Journal of Mathematical Analysis and Applications, 329 (2007), 870-878. doi: 10.1016/j.jmaa.2006.07.022. [4] P. Chen and X. H. Tang, Existence of homoclinic orbits for 2$n$th-order nonlinear difference equations containing both many advances and retardations, Journal of Mathematical Analysis and Applications, 381 (2011), 485-505. doi: 10.1016/j.jmaa.2011.02.016. [5] L. H. Erbe, H. Xia and J. S. Yu, Global stability of a linear nonautonomous delay difference equations, Journal of Difference Equations and Applications, 1 (1995), 151-161. doi: 10.1080/10236199508808016. [6] Z. M. Guo and J. S. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Science China Mathematics, 46 (2003), 506-515. doi: 10.1007/BF02884022. [7] Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, Journal of the London Mathematical Society, 68 (2003), 419-430. doi: 10.1112/S0024610703004563. [8] Z. M. Guo and J. S. Yu, Applications of critical point theory to difference equations, Differences and Differential Equations, 42 (2004), 187-200. [9] J. H. Leng, Periodic and subharmonic solutions for 2$n$th-order $\phi_{c}$-Laplacian difference equations containing both advance and retardation, Indagationes Mathematicae, 27 (2016), 902-913. doi: 10.1016/j.indag.2016.05.002. [10] G. H. Lin and Z. Zhou, Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities, Communications on Pure and Applied Analysis, 17 (2018), 1723-1747. doi: 10.3934/cpaa.2018082. [11] X. Liu, Y. B. Zhang, H. P. Shi and X. Q. Deng, Periodic and subharmonic solutions for fourth-order nonlinear difference equations, Applied Mathematics and Computation, 236 (2014), 613-620. doi: 10.1016/j.amc.2014.03.086. [12] X. H. Liu, L. H. Zhang, P. Agarwal and G. T. Wang, On some new integral inequalities of Gronwall-Bellman-Bihari type with delay for discontinuous functions and their applications, Indagationes Mathematicae, 27 (2016), 1-10. doi: 10.1016/j.indag.2015.07.001. [13] A. Mai and Z. Zhou, Discrete solitons for periodic discrete nonlinear Schrödinger equations, Applied Mathematics and Computation, 222 (2013), 34-41. doi: 10.1016/j.amc.2013.07.042. [14] H. Matsunaga, T. Hara and S. Sakata, Global attractivity for a nonlinear difference equation with variable delay, Computers and Mathematics with Applications, 41 (2001), 543-551. doi: 10.1016/S0898-1221(00)00297-2. [15] J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: a variational approach, Nonlinear Analysis, 75 (2012), 4672-4687. doi: 10.1016/j.na.2011.11.018. [16] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conference Series in Mathematics, American Mathematical Society, 1986. doi: 10.1090/cbms/065. [17] H. P. Shi, Periodic and subharmonic solutions for second-order nonlinear difference equations, Journal of Applied Mathematics and Computing, 48 (2015), 157-171. doi: 10.1007/s12190-014-0796-z. [18] H. P. Shi and Y. B. Zhang, Existence of periodic solutions for a 2$n$th-order nonlinear difference equation, Taiwanese Journal of Mathematics, 20 (2016), 143-160. doi: 10.11650/tjm.20.2016.5844. [19] J. S. Yu and Z. M. Guo, On boundary value problems for a discrete generalized Emden-Fowler equation, Journal of Differential Equations, 231 (2006), 18-31. doi: 10.1016/j.jde.2006.08.011. [20] Q. Q. Zhang, Boundary value problems for forth order nonlinear $p$-Laplacian difference equations, Journal of Applied Mathematics, 2014 (2014), Article ID 343129, 6 pages. doi: 10.1155/2014/343129. [21] Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proceedings of the American Mathematical Society, 143 (2015), 3155-3163. doi: 10.1090/S0002-9939-2015-12107-7. [22] Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part, Communications on Pure and Applied Analysis, 14 (2015), 1929-1940. doi: 10.3934/cpaa.2015.14.1929. [23] Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions, Communications on Pure and Applied Analysis, 18 (2019), 425-434. doi: 10.3934/cpaa.2019021. [24] Z. Zhou and D. F. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Science China Mathematics, 58 (2015), 781-790. doi: 10.1007/s11425-014-4883-2. [25] Z. Zhou and M. T. Su, Boundary value problems for 2$n$th-order $\phi_{c}$-Laplacian difference equations containing both advance and retardation, Applied Mathematics Letters, 41 (2015), 7-11. doi: 10.1016/j.aml.2014.10.006. [26] Z. Zhou and J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, Journal of Differential Equations, 249 (2010), 1199-1212. doi: 10.1016/j.jde.2010.03.010. [27] Z. Zhou, J. S. Yu and Y. M. Chen, Periodic solutions for a 2$n$th-order nonlinear difference equation, Science China Mathematics, 53 (2010), 41-50. doi: 10.1007/s11425-009-0167-7. [28] Z. Zhou, J. S. Yu and Y. M. 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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##### References:
 [1] Z. AlSharawi, J. M. Cushing and S. Elaydi, Theory and Applications of Difference Equations and Discrete Dynamical Systems, Springer Proceedings in Mathematics & Statistics, 102. Springer, Heidelberg, 2014. [2] Z. Balanov, C. Garcia-Azpeitia and W. Krawcewicz, On variational and topological methods in nonlinear difference equations, Communications on Pure and Applied Analysis, 17 (2018), 2813-2844. doi: 10.3934/cpaa.2018133. [3] X. C. Cai and J. S. Yu, Existence of periodic solutions for a 2$n$th-order nonlinear difference equation, Journal of Mathematical Analysis and Applications, 329 (2007), 870-878. doi: 10.1016/j.jmaa.2006.07.022. [4] P. Chen and X. H. Tang, Existence of homoclinic orbits for 2$n$th-order nonlinear difference equations containing both many advances and retardations, Journal of Mathematical Analysis and Applications, 381 (2011), 485-505. doi: 10.1016/j.jmaa.2011.02.016. [5] L. H. Erbe, H. Xia and J. S. Yu, Global stability of a linear nonautonomous delay difference equations, Journal of Difference Equations and Applications, 1 (1995), 151-161. doi: 10.1080/10236199508808016. [6] Z. M. Guo and J. S. Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Science China Mathematics, 46 (2003), 506-515. doi: 10.1007/BF02884022. [7] Z. M. Guo and J. S. Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, Journal of the London Mathematical Society, 68 (2003), 419-430. doi: 10.1112/S0024610703004563. [8] Z. M. Guo and J. S. Yu, Applications of critical point theory to difference equations, Differences and Differential Equations, 42 (2004), 187-200. [9] J. H. Leng, Periodic and subharmonic solutions for 2$n$th-order $\phi_{c}$-Laplacian difference equations containing both advance and retardation, Indagationes Mathematicae, 27 (2016), 902-913. doi: 10.1016/j.indag.2016.05.002. [10] G. H. Lin and Z. Zhou, Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities, Communications on Pure and Applied Analysis, 17 (2018), 1723-1747. doi: 10.3934/cpaa.2018082. [11] X. Liu, Y. B. Zhang, H. P. Shi and X. Q. Deng, Periodic and subharmonic solutions for fourth-order nonlinear difference equations, Applied Mathematics and Computation, 236 (2014), 613-620. doi: 10.1016/j.amc.2014.03.086. [12] X. H. Liu, L. H. Zhang, P. Agarwal and G. T. Wang, On some new integral inequalities of Gronwall-Bellman-Bihari type with delay for discontinuous functions and their applications, Indagationes Mathematicae, 27 (2016), 1-10. doi: 10.1016/j.indag.2015.07.001. [13] A. Mai and Z. Zhou, Discrete solitons for periodic discrete nonlinear Schrödinger equations, Applied Mathematics and Computation, 222 (2013), 34-41. doi: 10.1016/j.amc.2013.07.042. [14] H. Matsunaga, T. Hara and S. Sakata, Global attractivity for a nonlinear difference equation with variable delay, Computers and Mathematics with Applications, 41 (2001), 543-551. doi: 10.1016/S0898-1221(00)00297-2. [15] J. Mawhin, Periodic solutions of second order nonlinear difference systems with $\phi$-Laplacian: a variational approach, Nonlinear Analysis, 75 (2012), 4672-4687. doi: 10.1016/j.na.2011.11.018. [16] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conference Series in Mathematics, American Mathematical Society, 1986. doi: 10.1090/cbms/065. [17] H. P. Shi, Periodic and subharmonic solutions for second-order nonlinear difference equations, Journal of Applied Mathematics and Computing, 48 (2015), 157-171. doi: 10.1007/s12190-014-0796-z. [18] H. P. Shi and Y. B. Zhang, Existence of periodic solutions for a 2$n$th-order nonlinear difference equation, Taiwanese Journal of Mathematics, 20 (2016), 143-160. doi: 10.11650/tjm.20.2016.5844. [19] J. S. Yu and Z. M. Guo, On boundary value problems for a discrete generalized Emden-Fowler equation, Journal of Differential Equations, 231 (2006), 18-31. doi: 10.1016/j.jde.2006.08.011. [20] Q. Q. Zhang, Boundary value problems for forth order nonlinear $p$-Laplacian difference equations, Journal of Applied Mathematics, 2014 (2014), Article ID 343129, 6 pages. doi: 10.1155/2014/343129. [21] Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proceedings of the American Mathematical Society, 143 (2015), 3155-3163. doi: 10.1090/S0002-9939-2015-12107-7. [22] Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part, Communications on Pure and Applied Analysis, 14 (2015), 1929-1940. doi: 10.3934/cpaa.2015.14.1929. [23] Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions, Communications on Pure and Applied Analysis, 18 (2019), 425-434. doi: 10.3934/cpaa.2019021. [24] Z. Zhou and D. F. Ma, Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials, Science China Mathematics, 58 (2015), 781-790. doi: 10.1007/s11425-014-4883-2. [25] Z. Zhou and M. T. Su, Boundary value problems for 2$n$th-order $\phi_{c}$-Laplacian difference equations containing both advance and retardation, Applied Mathematics Letters, 41 (2015), 7-11. doi: 10.1016/j.aml.2014.10.006. [26] Z. Zhou and J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, Journal of Differential Equations, 249 (2010), 1199-1212. doi: 10.1016/j.jde.2010.03.010. [27] Z. Zhou, J. S. Yu and Y. M. Chen, Periodic solutions for a 2$n$th-order nonlinear difference equation, Science China Mathematics, 53 (2010), 41-50. doi: 10.1007/s11425-009-0167-7. [28] Z. Zhou, J. S. Yu and Y. M. 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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