View More View Less
  • 1 Guangdong Construction Polytechnic, Guangzhou 510440, China
  • 2 Hunan Agricultural University, Changsha 410128, China
  • 3 Hunan Agricultural University, Changsha 410128, China
  • 4 Jinan University, Zhuhai 519070, China
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

By making use of the critical point theory, the existence of periodic solutions for fourth-order nonlinear p-Laplacian difference equations is obtained. The main approach used in our paper is a variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions of fourth-order nonlinear p-Laplacian difference equations. The results obtained successfully generalize and complement the existing one.

  • [1]

    Agarwal, R. P., Difference Equations and Inequalities: Theory, Methods and Applications, Marcel Dekker, New York (1992).

  • [2]

    Agarwal, R. P., Perera, K. and O’Regan, D., Multiple positive solutions of singular and nonsingular discrete problems via variational methods, Nonlinear Anal., 58(1–2) (2004), 6973.

    • Search Google Scholar
    • Export Citation
  • [3]

    Agarwal, R. P., Perera, K. and O’Regan, D., Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Adv. Difference Equ., 2005 (2005), 9399.

    • Search Google Scholar
    • Export Citation
  • [4]

    Ahlbrandt, C. D., Dominant and recessive solutions of symmetric three term recurrences. J. Differential Equations, 107(2) (1994) 238-258.

    • Search Google Scholar
    • Export Citation
  • [5]

    Anderson, D. R., Avery, R. I. and Henderson, J., Existence of solutions for a one dimensional p-Laplacian on time-scales, J. Difference Equ. Appl., 10(10) (2006), 889896.

    • Search Google Scholar
    • Export Citation
  • [6]

    Avery, R. I. and Henderson, J., Existence of three positive pseudo-symmetric solutions for a one dimensional discrete p-Laplacian, J. Difference Equ. Appl., 10(6) (2004), 529539.

    • Search Google Scholar
    • Export Citation
  • [7]

    Cai, X. C., Yu, J. S. and Guo, Z. M., Existence of periodic solutions for fourthorder difference equations, Comput. Math. Appl., 50(1–2) (2005), 4955.

    • Search Google Scholar
    • Export Citation
  • [8]

    Cecchi, M., Marini, M. and Villari, G., On the monotonicity property for a certain class of second order differential equations, J. Differential Equations, 82(2) (1989), 1527.

    • Search Google Scholar
    • Export Citation
  • [9]

    Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston (1993).

  • [10]

    Chen, P. and Fang, H., Existence of periodic and subharmonic solutions for secondorder p-Laplacian difference equations, Adv. Difference Equ., 2007 (2007), 19.

    • Search Google Scholar
    • Export Citation
  • [11]

    Chen, P. and Tang, X. H., Existence of infinitely many homoclinic orbits for fourth-order difference systems containing both advance and retardation, Appl. Math. Comput., 217(9) (2011), 44084415.

    • Search Google Scholar
    • Export Citation
  • [12]

    Chen, P. and Tang, X. H., New existence and multiplicity of solutions for some Dirichlet problems with impulsive effects, Math. Comput. Modelling, 55(3–4) (2012), 723739.

    • Search Google Scholar
    • Export Citation
  • [13]

    Chen, P. and Tang, X. H., Infinitely many homoclinic solutions for the secondorder discrete p-Laplacian systems, Bull. Belg. Math. Soc., 20(2) (2013), 193212.

    • Search Google Scholar
    • Export Citation
  • [14]

    Chen, P. and Tang, X. H., Existence of homoclinic solutions for some secondorder discrete Hamiltonian systems, J. Difference Equ. Appl., 19(4) (2013), 633648.

    • Search Google Scholar
    • Export Citation
  • [15]

    Erbe, L. H., Xia, H. and Yu, J. S., Global stability of a linear nonautonomous delay difference equations, J. Difference Equ. Appl., 1(2) (1995), 151161.

    • Search Google Scholar
    • Export Citation
  • [16]

    Fang, H. and Zhao, D. P., Existence of nontrivial homoclinic orbits for fourthorder difference equations, Appl. Math. Comput., 214(1) (2009), 163170.

    • Search Google Scholar
    • Export Citation
  • [17]

    Guo, C. J., O’Regan, D. and Agarwal, R. P., Existence of multiple periodic solutions for a class of first-order neutral differential equations, Appl. Anal. Discrete Math., 5(1) (2011), 147158.

    • Search Google Scholar
    • Export Citation
  • [18]

    Guo, C. J., O’Regan, D., Xu, Y. T. and Agarwal, R. P., Existence of homoclinic orbits of a class of second order differential difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 20(6) (2013), 675690.

    • Search Google Scholar
    • Export Citation
  • [19]

    Guo, C. J., O’Regan, D., Xu, Y. T. and Agarwal, R. P., Existence of infinite periodic solutions for a class of first-order delay differential equations, Funct. Differ. Equ., 19 (2012), 115123.

    • Search Google Scholar
    • Export Citation
  • [20]

    Guo, C. J., O’Regan, D., Xu, Y. T. and Agarwal, R. P., Existence and multiplicity of homoclinic orbits of a second-order differential difference equation via variational methods, Appl. Math. Inform. Mech., 4(1) (2012), 115.

    • Search Google Scholar
    • Export Citation
  • [21]

    Guo, C. J. and Xu, Y. T., Existence of periodic solutions for a class of second order differential equation with deviating argument, J. Appl. Math. Comput., 28(1–2) (2008), 425433.

    • Search Google Scholar
    • Export Citation
  • [22]

    Guo, Z. M. and Yu, J. S., Applications of critical point theory to difference equations, Fields Inst. Commun., 42 (2004), 187200.

  • [23]

    Guo, Z. M. and Yu, J. S., Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Math, 46(4) (2003), 506515.

    • Search Google Scholar
    • Export Citation
  • [24]

    Guo, Z. M. and Yu, J. S., The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc., 68(2) (2003), 419430.

    • Search Google Scholar
    • Export Citation
  • [25]

    He, Z. M., On the existence of positive solutions of p-Laplacian difference equations, J. Comput. Appl. Math., 161(1) (2003), 193201.

    • Search Google Scholar
    • Export Citation
  • [26]

    Jiang, D., Chu, J., O’Regan, D. and Agarwal, R. P., Positive solutions for continuous and discrete boundary value problems to the one-dimension pLaplacian, Math. Inequal. Appl., 7(4) (2004), 523534.

    • Search Google Scholar
    • Export Citation
  • [27]

    Kocic, V. L. and Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht (1993).

    • Search Google Scholar
    • Export Citation
  • [28]

    Li, Y. and Lu, L., Existence of positive solutions of p-Laplacian difference equations, Appl. Math. Lett., 19(10) (2006), 10191023.

  • [29]

    Liu, Y. J. and Ge, W. G., Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator, J. Math. Anal. Appl., 278(2) (2003), 551561.

    • Search Google Scholar
    • Export Citation
  • [30]

    Matsunaga, H., Hara, T. and Sakata, S., Global attractivity for a nonlinear difference equation with variable delay, Computers Math. Appl., 41(5–6) (2001), 543551.

    • Search Google Scholar
    • Export Citation
  • [31]

    Mawhin, J. and Willem, M., Critical Point Theory and Hamiltonian Systems, Springer, New York (1989).

  • [32]

    Mickens, R. E., Difference Equations: Theory and Application, Van Nostrand Reinhold, New York (1990).

  • [33]

    Pankov, A. and Zakhrchenko, N., On some discrete variational problems, Acta Appl. Math., 65(1–3) (2001), 295303.

  • [34]

    Peterson, A. and Ridenhour, J., The (2, 2)-disconjugacy of a fourth order difference equation, J. Difference Equ. Appl., 1(1) (1995), 8793.

    • Search Google Scholar
    • Export Citation
  • [35]

    Popenda, J. and Schmeidel, E., On the solutions of fourth order difference equations, Rocky Mountazn J. Math., 25(4) (1995), 14851499.

    • Search Google Scholar
    • Export Citation
  • [36]

    Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, Amer. Math. Soc., Providence, RI, New York (1986).

    • Search Google Scholar
    • Export Citation
  • [37]

    Shi, H. P., Ling, W. P., Long, Y. H. and Zhang, H. Q., Periodic and subharmonic solutions for second order nonlinear functional difference equations, Commun. Math. Anal., 5(2) (2008), 5059.

    • Search Google Scholar
    • Export Citation
  • [38]

    Smets, D. and Willem, M., Solitary waves with prescribed speed on infinite lattices, J. Funct. Anal., 149(1) (1997), 266275.

  • [39]

    Thandapani, E. and Arockiasamy, I. M., Fourth-order nonlinear oscillations of difference equations, Comput. Math. Appl., 42(3–5) (2001), 357368.

    • Search Google Scholar
    • Export Citation
  • [40]

    Tian, Y., Du, Z. J. and Ge, W. G., Existence results for discrete Sturm-Liuville problem via variational methods, J. Difference Equ. Appl., 13(6) (2007), 467478.

    • Search Google Scholar
    • Export Citation
  • [41]

    Tian, Y. and Ge, W. G., The existence of solutions for a second-order discrete Neumann problem with a p-Laplacian, J. Appl. Math. Comput., 26(1–2) (2008), 333340.

    • Search Google Scholar
    • Export Citation
  • [42]

    Yan, J. and Liu, B., Oscillatory and asymptotic behavior of fourth order nonlinear difference equations, Acta. Math. Sinica, 13(1) (1997), 105115.

    • Search Google Scholar
    • Export Citation
  • [43]

    Yu, J. S. and Guo, Z. M., On boundary value problems for a discrete generalized Emden-Fowler equation, J. Differential Equations, 231(1) (2006), 1831.

    • Search Google Scholar
    • Export Citation
  • [44]

    Yu, J. S., Long, Y. H. and Guo, Z. M., Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation, J. Dynam. Differential Equations, 16(2) (2004), 575586.

    • Search Google Scholar
    • Export Citation
  • [45]

    Zhou, Z., Yu, J. S. and Chen, Y. M., Homoclinic solutions in periodic difference equations with saturable nonlinearity, Sci. China Math, 54(1) (2011), 8393.

    • Search Google Scholar
    • Export Citation
  • [46]

    Zhou, Z., Yu, J. S. and Guo, Z. M., Periodic solutions of higher-dimensional discrete systems, Proc. Roy. Soc. Edinburgh (Section A), 134(5) (2004), 10131022.

    • Search Google Scholar
    • Export Citation
  • [47]

    Zhou, Z. and Zhang, Q., Uniform stability of nonlinear difference systems, J. Math. Anal. Appl., 225(2) (1998), 486500.

The author instruction is available in PDF.

Please, download the file from HERE

Manuscript submission: HERE

 

  • Impact Factor (2019): 0.486
  • Scimago Journal Rank (2019): 0.234
  • SJR Hirsch-Index (2019): 23
  • SJR Quartile Score (2019): Q3 Mathematics (miscellaneous)
  • Impact Factor (2018): 0.309
  • Scimago Journal Rank (2018): 0.253
  • SJR Hirsch-Index (2018): 21
  • SJR Quartile Score (2018): Q3 Mathematics (miscellaneous)

Language: English, French, German

Founded in 1966
Publication: One volume of four issues annually
Publication Programme: 2020. Vol. 57.
Indexing and Abstracting Services:

  • CompuMath Citation Index
  • Mathematical Reviews
  • Referativnyi Zhurnal/li>
  • Research Alert
  • Science Citation Index Expanded (SciSearch)/li>
  • SCOPUS
  • The ISI Alerting Services

 

Subscribers can access the electronic version of every printed article.

Senior editors

Editor(s)-in-Chief: Pálfy Péter Pál

Managing Editor(s): Sági, Gábor

Editorial Board

  • Biró, András (Number theory)
  • Csáki, Endre (Probability theory and stochastic processes, Statistics)
  • Domokos, Mátyás (Algebra (Ring theory, Invariant theory))
  • Győri, Ervin (Graph and hypergraph theory, Extremal combinatorics, Designs and configurations)
  • O. H. Katona, Gyula (Combinatorics)
  • Márki, László (Algebra (Semigroup theory, Category theory, Ring theory))
  • Némethi, András (Algebraic geometry, Analytic spaces, Analysis on manifolds)
  • Pach, János (Combinatorics, Discrete and computational geometry)
  • Rásonyi, Miklós (Probability theory and stochastic processes, Financial mathematics)
  • Révész, Szilárd Gy. (Analysis (Approximation theory, Potential theory, Harmonic analysis, Functional analysis))
  • Ruzsa, Imre Z. (Number theory)
  • Soukup, Lajos (General topology, Set theory, Model theory, Algebraic logic, Measure and integration)
  • Stipsicz, András (Low dimensional topology and knot theory, Manifolds and cell complexes, Differential topology)
  • Szász, Domokos (Dynamical systems and ergodic theory, Mechanics of particles and systems)
  • Tóth, Géza (Combinatorial geometry)

STUDIA SCIENTIARUM MATHEMATICARUM HUNGARICA
Gábor Sági
Address: P.O. Box 127, H–1364 Budapest, Hungary
Phone: (36 1) 483 8344 ---- Fax: (36 1) 483 8333
E-mail: smh.studia@renyi.mta.hu