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

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, 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

[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

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

[1]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[2]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

[3]

Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265

[4]

Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317

[5]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293

[6]

Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ I: Analytical results and applications. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 1-60. doi: 10.3934/dcdsb.2020231

[7]

Ivan Bailera, Joaquim Borges, Josep Rifà. On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $. Advances in Mathematics of Communications, 2021, 15 (1) : 35-54. doi: 10.3934/amc.2020041

[8]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[9]

Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033

[10]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[11]

Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278

[12]

Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228

[13]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[14]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[15]

Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control & Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030

[16]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021002

[17]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052

[18]

Kengo Matsumoto. $ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms. Electronic Research Archive, , () : -. doi: 10.3934/era.2021006

[19]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[20]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (144)
  • HTML views (518)
  • Cited by (1)

Other articles
by authors

[Back to Top]