Abstract
By using critical point theory, some new existence results of at least one periodic solution with minimal period pM for fourth-order nonlinear difference equations are obtained. Our approach used in this paper is a variational method.
Funding source: Scientific Research Fund of Jiangxi Provincial Education Department
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: The project is funded by the Scientific Research Fund of Jiangxi Provincial Education Department (No. GJJ170889).
Employment or leadership: None declared.
Honorarium: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, Springer, New York, 2010.Search in Google Scholar
[2] G. M. Bisci and D. Repovš, “Existence of solutions for p-Laplacian discrete equations,” Appl. Math. Comput., vol. 242, pp. 454–461, 2014, https://doi.org/10.1016/j.amc.2014.05.118.Search in Google Scholar
[3] G. M. Bisci and D. Repovš, “On sequences of solutions for discrete anisotropic equations,” Expo. Math., vol. 32, no. 3, pp. 284–295, 2014, https://doi.org/10.1016/j.exmath.2013.12.001.Search in Google Scholar
[4] A. Cabada and V. D. Dimitrov, “Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities,” J. Math. Anal. Appl., vol. 371, no. 2, pp. 518–533, 2010, https://doi.org/10.1016/j.jmaa.2010.05.052.Search in Google Scholar
[5] X. C. Cai, J. S. Yu, and Z. M. Guo, “Existence of periodic solutions for fourth-order difference equations,” Comput. Math. Appl., vol. 50, no. 1–2, pp. 49–55, 2005, https://doi.org/10.1016/j.camwa.2005.03.004.Search in Google Scholar
[6] P. Candito and G. M. Bisci, “Existence of two solutions for a second-order discrete boundary value problem,” Adv. Nonlinear Stud., vol. 11, no. 2, pp. 443–453, 2011, https://doi.org/10.1515/ans-2011-0212.Search in Google Scholar
[7] P. Chen and H. Fang, “Existence of periodic and subharmonic solutions for second-order p-Laplacian difference equations,” Adv. Differ. Equ., vol. 2007, pp. 1–9, 2007, https://doi.org/10.1155/2007/42530.Search in Google Scholar
[8] P. Chen and X. H. Tang, “Existence and multiplicity of homoclinic orbits for 2nth-order nonlinear difference equations containing both many advances and retardations,” J. Math. Anal. Appl., vol. 381, no. 2, pp. 485–505, 2011, https://doi.org/10.1016/j.jmaa.2011.02.016.Search in Google Scholar
[9] H. Fang and D. P. Zhao, “Existence of nontrivial homoclinic orbits for fourth-order difference equations,” Appl. Math. Comput., vol. 214, no. 1, pp. 163–170, 2009, https://doi.org/10.1016/j.amc.2009.03.061.Search in Google Scholar
[10] Z. M. He and J. S. Yu, “On the existence of positive solutions of fourth-order difference equations,” Appl. Math. Comput., vol. 161, no. 1, pp. 139–148, 2005, https://doi.org/10.1016/j.amc.2003.12.016.Search in Google Scholar
[11] A. Peterson and J. Ridenhour, “The (2,2)-disconjugacy of a fourth order difference equation,” J. Differ. Equ. Appl., vol. 1, no. 1, pp. 87–93, 1995, https://doi.org/10.1080/10236199508808009.Search in Google Scholar
[12] X. H. Tang and X. Y. Zhang, “Periodic solutions for second-order discrete Hamiltonian systems,” J. Differ. Equ. Appl., vol. 17, no. 10, pp. 1413–1430, 2011, https://doi.org/10.1080/10236190903555237.Search in Google Scholar
[13] J. S. Yu, Y. H. Long, and Z. M. Guo, “Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation,” J. Dynam. Differ. Equ., vol. 16, no. 2, pp. 575–586, 2004, https://doi.org/10.1007/s10884-004-4292-2.Search in Google Scholar
[14] Z. Zhou, and D. F. Ma, “Multiplicity results of breathers for the discrete nonlinear Schrödinger equations with unbounded potentials,” Sci. China Math., vol. 58, no. 4, pp. 781–790, 2015.10.1007/s11425-014-4883-2Search in Google Scholar
[15] Z. Zhou, J. S. Yu, and Y. M. Chen, “Homoclinic solutions in periodic difference equations with saturable nonlinearity,” Sci. China Math., vol. 54, no. 1, pp. 83–93, 2011.10.1007/s11425-010-4101-9Search in Google Scholar
[16] P. Agarwal, “Some inequalities involving Hadamard-type k-fractional integral operators,” Math. Methods Appl. Sci., vol. 40, no. 11, pp. 3882–3891, 2017, https://doi.org/10.1002/mma.4270.Search in Google Scholar
[17] P. Agarwal, S. K. Q. Al-Omari, and J. Choi, “Real covering of the generalized Hankel-Clifford transform of Fox kernel type of a class of Boehmians,” Bull. Korean Math. Soc., vol. 52, no. 5, pp. 1607–1619, 2015, https://doi.org/10.4134/bkms.2015.52.5.1607.Search in Google Scholar
[18] P. Agarwal, S. K. Q. Al-Omari, and P. Park, “An extension of some variant of Meijer type integrals in the class of Boehmians,” J. Inequal. Appl., vol. 70, no. 1, pp. 1–10, 2016, https://doi.org/10.1186/s13660-016-0998-z.Search in Google Scholar
[19] P. Agarwal and A. A. El-Sayed, “Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation,” Phys. A., vol. 500, no. 15, pp. 40–49, 2018, https://doi.org/10.1016/j.physa.2018.02.014.Search in Google Scholar
[20] P. Chen and X. H. Tang, “Existence of solutions for a class of second-order p-Laplacian systems with impulsive effects,” Appl. Math., vol. 59, no. 5, pp. 543–570, 2014, https://doi.org/10.1007/s10492-014-0071-5.Search in Google Scholar
[21] A. A. El-Sayed and P. Agarwal, “Numerical solution of multiterm variable-order fractional differential equations via shifted Legendre polynomials,” Math. Methods Appl. Sci., vol. 42, no. 11, 2019, https://doi.org/10.1002/mma.5627.Search in Google Scholar
[22] C. J. Guo, D. O’Regan, and R. P. Agarwal, “Existence of multiple periodic solutions for a class of first-order neutral differential equations,” Appl. Anal. Discrete Math., vol. 5, no. 1, pp. 147–158, 2011, https://doi.org/10.2298/aadm100914028g.Search in Google Scholar
[23] C. J. Guo, D. O’Regan, Y. T. Xu, and R. P. Agarwal, “Existence of homoclinic orbits for a class of first-order differential difference equations,” Acta Math. Sci. Ser. B Engl. Ed., vol. 35, no. 5, pp. 1077–1094, 2015, https://doi.org/10.1016/s0252-9602(15)30041-2.Search in Google Scholar
[24] C. J. Guo, D. O’Regan, Y. T. Xu, and R. P. Agarwal, “Existence of periodic solutions for a class of second-order superquadratic delay differential equations,” Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., vol. 21, no. 5, pp. 405–419, 2014.Search in Google Scholar
[25] C. J. Guo, D. O’Regan, Y. T. Xu, and R. P. Agarwal, “Homoclinic orbits for a singular second-order neutral differential equation,” J. Math. Anal. Appl., vol. 366, no. 2, pp. 550–560, 2010, https://doi.org/10.1016/j.jmaa.2009.12.038.Search in Google Scholar
[26] C. J. Guo and Y. T. Xu, “Existence of periodic solutions for a class of second order differential equation with deviating argument,” J. Appl. Math. Comput., vol. 28, no. 1-2, pp. 425–433, 2008, https://doi.org/10.1007/s12190-008-0116-6.Search in Google Scholar
[27] S. Jain and P. Agarwal, “On new applications of fractional calculus,” Bol. Soc. Parana. Mat., vol. 37, no. 3, pp. 113–118, 2019, https://doi.org/10.5269/bspm.v37i3.18626.Search in Google Scholar
[28] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989.10.1007/978-1-4757-2061-7Search in Google Scholar
[29] M. Ruzhansky, C. Y. Je, P. Agarwal, and I. Area, Advances in Real and Complex Analysis with Applications, Springer, Singapore, 2017.10.1007/978-981-10-4337-6Search in Google Scholar
[30] S. Sitho, S. K. Ntouyas, P. Agarwal, and J. Tariboon, “Noninstantaneous impulsive inequalities via conformable fractional calculus,” J. Inequal. Appl, vol. 2018, no. 261, pp. 1–14, 2018, https://doi.org/10.1186/s13660-018-1855-z.Search in Google Scholar PubMed PubMed Central
[31] X. H. Tang, “Non-Nehari manifold method for asymptotically periodic Schrödinger equations,” Sci. China Math., vol. 58, no. 4, pp. 715–728, 2015, https://doi.org/10.1007/s11425-014-4957-1.Search in Google Scholar
[32] X. H. Tang and S. T. Chen, “Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials,” Discrete Contin. Dyn. Syst., vol. 37, no. 9, pp. 4973–5002, 2017, https://doi.org/10.3934/dcds.2017214.Search in Google Scholar
[33] X. H. Tang and S. T. Chen, “Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials,” Calc. Var. Partial Differ. Equ., vol. 56, no. 4, pp. 110–134, 2017, https://doi.org/10.1007/s00526-017-1214-9.Search in Google Scholar
[34] J. Tariboon, S. Ntouyas, and P. Agarwal, “New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equation,” Adv. Differ. Equ., vol. 2015, no. 18, pp. 1–19, 2015, https://doi.org/10.1186/s13662-014-0348-8.Search in Google Scholar
[35] X. M. Zhang, P. Agarwal, Z. H. Liu, and H. Peng, “The general solution for impulsive differential equations with Riemann-Liouville fractional-order q∈(1, 2).,” Open Math., vol. 13, no. 908 C23, 2015, https://doi.org/10.1515/math-2015-0073.Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
