| Title:
             | 
Existence of solutions for a nonlinear discrete system involving the $p$-Laplacian (English) | 
| Author:
             | 
Zhang, Xingyong | 
| Author:
             | 
Tang, Xianhua | 
| Language:
             | 
English | 
| Journal:
             | 
Applications of Mathematics | 
| ISSN:
             | 
0862-7940 (print) | 
| ISSN:
             | 
1572-9109 (online) | 
| Volume:
             | 
57 | 
| Issue:
             | 
1 | 
| Year:
             | 
2012 | 
| Pages:
             | 
11-30 | 
| Summary lang:
             | 
English | 
| . | 
| Category:
             | 
math | 
| . | 
| Summary:
             | 
The existence of solutions for boundary value problems for a nonlinear discrete system involving the $p$-Laplacian is investigated. The approach is based on critical point theory. (English) | 
| Keyword:
             | 
critical point theory | 
| Keyword:
             | 
boundary value problems | 
| Keyword:
             | 
discrete systems | 
| Keyword:
             | 
$p$-Laplacian | 
| Keyword:
             | 
variational method | 
| MSC:
             | 
35B38 | 
| MSC:
             | 
35K92 | 
| MSC:
             | 
37J45 | 
| MSC:
             | 
39A10 | 
| MSC:
             | 
39A12 | 
| MSC:
             | 
58E50 | 
| MSC:
             | 
70H05 | 
| idZBL:
             | 
Zbl 1249.39009 | 
| idMR:
             | 
MR2891303 | 
| DOI:
             | 
10.1007/s10492-012-0002-2 | 
| . | 
| Date available:
             | 
2012-01-09T19:23:07Z | 
| Last updated:
             | 
2020-07-02 | 
| Stable URL:
             | 
http://hdl.handle.net/10338.dmlcz/141816 | 
| . | 
| Reference:
             | 
[1] Bai, D., Xu, Y.: Nontrivial solutions of boundary value problems of second-order difference equations.J. Math. Anal. Appl. 326 (2007), 297-302. Zbl 1113.39018, MR 2277783, 10.1016/j.jmaa.2006.02.091 | 
| Reference:
             | 
[2] Bonanno, G., Candito, P.: Nonlinear difference equations investigated via critical points methods.Nonlinear Anal., Theory Methods Appl. 70 (2009), 3180-3186. MR 2503063, 10.1016/j.na.2008.04.021 | 
| Reference:
             | 
[3] Bonanno, G., Candito, P.: Infinitely many solutions for a class of discrete non-linear boundary value problems.Appl. Anal. 88 (2009), 605-616. Zbl 1176.39004, MR 2541143, 10.1080/00036810902942242 | 
| Reference:
             | 
[4] Candito, P., Giovannelli, N.: Multiple solutions for a discrete boundary value problem involving the $p$-Laplacian.Comput. Math. Appl. 56 (2008), 959-964. Zbl 1155.39301, MR 2437868, 10.1016/j.camwa.2008.01.025 | 
| Reference:
             | 
[5] Guo, Z. M., Yu, J. S.: Existence of periodic and subharmonic solutions for second-order superlinear difference equations.Sci. China Ser. A 46 (2003), 506-515. Zbl 1215.39001, MR 2014482, 10.1007/BF02884022 | 
| Reference:
             | 
[6] Guo, Z. M., Yu, J. S.: The existence of periodic and subharmonic solutions to subquadratic second order difference equations.J. Lond. Math. Soc., II. Ser. 68 (2003), 419-430. MR 1994691, 10.1112/S0024610703004563 | 
| Reference:
             | 
[7] Kuang, J.: Applied Inequalities.Shandong Science and Technology Press Jinan City (2004), Chinese. MR 1305610 | 
| Reference:
             | 
[8] Lu, W. D.: Variational Methods in Differential Equations.Scientific Publishing House in China (2002). | 
| Reference:
             | 
[9] Ma, J., Tang, C. L.: Periodic solutions for some nonautonomous second order systems.J. Math. Anal. Appl. 275 (2002), 482-494. Zbl 1024.34036, MR 1943760, 10.1016/S0022-247X(02)00636-4 | 
| Reference:
             | 
[10] Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems.Springer-Verlag New York (1989). Zbl 0676.58017, MR 0982267 | 
| Reference:
             | 
[11] Rabinowitz, P. H.: Minimax Methods in Critical Point Theory with Application to Differential Equations. Reg. Conf. Ser. Math, 65.Am. Math. Soc. Provindence (1986). MR 0845785 | 
| Reference:
             | 
[12] Tang, C.-L., Wu, X.-P.: Notes on periodic solutions of subquadratic second order systems.J. Math. Anal. Appl. 285 (2003), 8-16. Zbl 1054.34075, MR 2000135, 10.1016/S0022-247X(02)00417-1 | 
| Reference:
             | 
[13] Wu, J. F., Wu, X. P.: Existence of nontrivial periodic solutions for a class of superquadratic second-order Hamiltonian systems.J. Southwest Univ. (Natural Science Edition) 30 (2008), 26-31. MR 2131492 | 
| Reference:
             | 
[14] Wu, X.-P., Tang, C.-L.: Periodic solution of a class of non-autonomous second order systems.J. Math. Anal. Appl. 236 (1999), 227-235. MR 1704579, 10.1006/jmaa.1999.6408 | 
| Reference:
             | 
[15] Xue, Y.-F., Tang, C.-L.: Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems.Appl. Math. Comput. 196 (2008), 494-500. Zbl 1153.39024, MR 2388705, 10.1016/j.amc.2007.06.015 | 
| Reference:
             | 
[16] Xue, Y.-F., Tang, C.-L.: Existence of a periodic solution for subquadratic second-order discrete Hamiltonian system.Nonlinear Anal., Theory Methods Appl. 67 (2007), 2072-2080. Zbl 1129.39008, MR 2331858, 10.1016/j.na.2006.08.038 | 
| Reference:
             | 
[17] Zhang, X., Tang, X.: Existence of nontrivial solutions for boundary value problems of second-order discrete systems.Math. Slovaca 61 (2011), 769-778. Zbl 1274.39018, MR 2827213, 10.2478/s12175-011-0044-z | 
| Reference:
             | 
[18] Zhao, F., Wu, X.: Periodic solutions for a class of nonautonomous second order systems.J. Math. Anal. Appl. 296 (2004), 422-434. Zbl 1050.34062, MR 2075174, 10.1016/j.jmaa.2004.01.041 | 
| Reference:
             | 
[19] Zhou, Z., Yu, J.-S., Guo, Z.-M.: Periodic solutions of higher-dimensional discrete systems.Proc. R. Soc. Edinb., Sect. A, Math. 134 (2004), 1013-1022. Zbl 1073.39010, MR 2099576, 10.1017/S0308210500003607 | 
| . |