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November  2019, 12(7): 2085-2095. doi: 10.3934/dcdss.2019134

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

Peng Mei , Zhan Zhou , and  Genghong Lin

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.

Keywords: Periodic and subharmonic solution, 2$n$th-order, nonlinear difference equation, $\phi_c$-Laplacian, critical point theory.
Mathematics Subject Classification: 39A23.
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, 2019, 12 (7) : 2085-2095. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[8]

Z. M. Guo and J. S. Yu, Applications of critical point theory to difference equations, Differences and Differential Equations, 42 (2004), 187-200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Z. M. Guo and J. S. Yu, Applications of critical point theory to difference equations, Differences and Differential Equations, 42 (2004), 187-200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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