Periodic and subharmonic solutions for \(2n\)th-order \(p\)-Laplacian difference equations. (English) Zbl 1320.39016
J. Contemp. Math. Anal., Armen. Acad. Sci. 49, No. 5, 223-231 (2014) and Izv. Nats. Akad. Nauk Armen., Mat. 49, No. 5, 40-52 (2014).
Authors’ abstract: By using the critical point theory, some new criteria for the existence and multiplicity of periodic and subharmonic solutions for \(2n\)th-order \(p\)-Laplacian difference equations are obtained. The proof is based on the linking theorem in combination with variational technique. Our results generalize and improve the known in the literature results.
Reviewer: Hui-Sheng Ding (Jiangxi)
Keywords:
periodic and subharmonic solutions; \(2n\)th-order nonlinear difference equations; discrete variational theory; \(p\)-Laplacian; critical point theory; linking theoremReferences:
| [1] | R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods and Applications (Marcel Dekker, New York, 1992). · Zbl 0925.39001 |
| [2] | C.D. Ahlbrandt and A.C. Peterson, “The (<Emphasis Type=”Italic“>n, n)-disconjugacy of a 2<Emphasis Type=”Italic“>nth-order linear difference equation”, Comput. Math. Appl., 28(1-3), 1-9, 1994. · Zbl 0815.39003 · doi:10.1016/0898-1221(94)00088-3 |
| [3] | D. Anderson, “A 2<Emphasis Type=”Italic“>nth-order linear difference equation”, Comput. Math. Appl., 2(4), 521-529, 1998. · Zbl 0903.39001 |
| [4] | X.C. Cai and J.S. Yu, “Existence of periodic solutions for a 2<Emphasis Type=”Italic“>nth-order nonlinear difference equation”, J. Math. Anal. Appl., 329(2), 870-878, 2007. · Zbl 1153.39302 · doi:10.1016/j.jmaa.2006.07.022 |
| [5] | X.C. Cai, J.S. Yu and Z. M. Guo, “Existence of periodic solutions for fourth-order difference equations”, Comput. Math. Appl., 50(1-2), 49-55, 2005. · Zbl 1086.39002 · doi:10.1016/j.camwa.2005.03.004 |
| [6] | P. Chen and H. Fang, “Existence of periodic and subharmonic solutions for second-order <Emphasis Type=”Italic“>p-Laplacian difference equations”, Adv. Difference Equ., 2007, 1-9, 2007. · Zbl 1148.39002 · doi:10.1155/2007/42530 |
| [7] | P. Chen and X.H. Tang, “Existence and multiplicity of homoclinic orbits for 2<Emphasis Type=”Italic“>nth-order nonlinear difference equations containing both many advances and retardations”, J. Math. Anal. Appl., 381(2), 485-505, 2011. · Zbl 1228.39005 · doi:10.1016/j.jmaa.2011.02.016 |
| [8] | G. Cordaro, “Existence and location of periodic solution to convex and non coercive Hamiltonian systems”, Discrete Contin. Dyn. Syst., 12(5), 983-996, 2005. · Zbl 1082.34012 · doi:10.3934/dcds.2005.12.983 |
| [9] | Z. M. Guo and J.S. Yu, “Applications of critical point theory to difference equations”, Fields Inst. Commun., 42, 187-200, 2004. · Zbl 1067.39007 |
| [10] | Z.M. Guo and J.S. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear difference equations”, Sci. China Math, 46(4), 506-515, 2003. · Zbl 1215.39001 · doi:10.1007/BF02884022 |
| [11] | Z.M. Guo and J.S. Yu, “The existence of periodic and subharmonic solutions of subquadratic second order difference equations”, J. London Math. Soc., 68(2), 419-430, 2003. · Zbl 1046.39005 · doi:10.1112/S0024610703004563 |
| [12] | H. Matsunaga, T. Hara and S. Sakata, “Global attractivity for a nonlinear difference equation with variable delay”, Computers Math. Appl., 41(5-6), 543-551, 2001. · Zbl 0985.39009 · doi:10.1016/S0898-1221(00)00297-2 |
| [13] | J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems (Springer, New York, 1989). · Zbl 0676.58017 · doi:10.1007/978-1-4757-2061-7 |
| [14] | M. Migda, “Existence of nonoscillatory solutions of some higher order difference equations”, Appl. Math. E-notes, 4(2), 33-39, 2004. · Zbl 1069.39010 |
| [15] | T. Peil and A. Peterson, “Asymptotic behavior of solutions of a two-term difference equation, Rocky Mountain J. Math., <Emphasis Type=”Bold”>24(1), 233-251, 1994. · Zbl 0809.39006 · doi:10.1216/rmjm/1181072463 |
| [16] | P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations (Amer. Math. Soc., Providence, 1986). · Zbl 0609.58002 |
| [17] | H.P. Shi, W.P. Ling, Y.H. Long and H.Q. Zhang, “Periodic and subharmonic solutions for second order nonlinear functional difference equations”, Commun. Math. Anal., 5(2), 50-59, 2008. · Zbl 1163.39005 |
| [18] | J.S. Yu and Z.M. Guo, “On boundary value problems for a discrete generalized Emden-Fowler equation”, J. Differential Equations, 231(1), 18-31, 2006. · Zbl 1112.39011 · doi:10.1016/j.jde.2006.08.011 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
