Infinitely Many Solutions for Discrete Fourth-Order Boundary Value Problem with Four Parameters

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran

Abstract

The aim of this paper is to study the existence of infinitely many solutions for discrete fourth-order boundary value problem with four parameters involving oscillatory behaviors of nonlinearity at infinity. The approach is based on variational methods. In addition, one example is presented to illustrate the feasibility and effectiveness of the main result.

Keywords

References

  1. D. R. Anderson, F. Minhós, A discrete fourth-order Lidstone problem with parameters, Appl. Math. Comput., 214, 523–533 (2009).
  2. G. Bonanno, G. Molica Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., 2009, 1–20 (2009).
  3. A. Cabada, A. Iannizzotto, S. Tersian, Multiple solutions for discrete boundary value problem, J. Math. Anal. Appl., 356, 418–428 (2009).
  4. G. D’Aguì, S. Heidarkhani, A. Sciammetta, Infinitely many solutions for a class of quasilinear two-point boundary value problems, Electron. J. Qual. Theory Differ. Equ., 8, 1–15 (2015).
  5. M. Galewski, S. Głaffb, On the discrete boundary value problem for anisotropic equation, J. Math. Anal. Appl., 386, 956–965 (2012).
  6. J. R. Graef, S. Heidarkhani, L. Kong, M. Wang, Existence of solutions to a discrete fourth order boundary value problem, J. Differ. Equ. Appl., 24, 849–858 (2018).
  7. J. R. Graef, L. Kong, M. Wang, Multiple solutions to a periodic boundary value problem for a nonlinear discrete fourth order equation, Adv. Dyn. Syst. Appl., 8, 203–215 (2013).
  8. J. R. Graef, L. Kong, M. Wang, Two nontrivial solutions for a discrete fourth order periodic boundary value problem, Commun. Appl. Anal., 19, 487–496 (2015).
  9. J. R. Graef, L. Kong, M. Wang, B. Yang, Uniqueness and parameter dependence of positive solutions of a discrete fourth-order problem, J. Differ. Equ. Appl., 19, 1133– 1146 (2013).
  10. S. Heidarkhani, G. A. Afrouzi, G. Caristi, J. Henderson, S. Moradi, A variational ap[1]proach to difference equations, J. Differ. Equ. Appl., 22, 1761–1776 (2016).
  11. S. Heidarkhani, G.A. Afrouzi, A. Salari, G. Caristi, Discrete fourth-order boundary value problems with four parameters, Appl. Math. Comput., 346, 167–182 (2019).
  12. S. Heidarkhani, M. Bohner, S. Moradi, G. Caristi, Three solutions for discrete fourth[1]order boundary value problem with four parameters, preprint.
  13. S. Heidarkhani, S. Moradi, A variational approache to the discrete fourth-order bound[1]ary value problem with four parameters, preprint.
  14. S. Heidarkhani, Y. Zhao, G. Caristi, G.A. Afrouzi, S. Moradi, Infinitely many solutions for perturbed impulsive fractional differential problems, Appl. Anal., 96, 1401–1424 (2017).
  15. L. Kong, Solutions of a class of discrete fourth order boundary value problems, Minimax Theory Appl., 3, 35–46 (2018).
  16. A. Kristály, M. Mihăilescu, V. Rădulescu, Discrete boundary value problems involving oscillatory nonlinearities: small and large solutions, J. Differ. Equ. Appl., 17, 1431–1440 (2011).
  17. X. Liu, T. Zhou, H. Shi, Existence of solutions to boundary value problems for a fourth[1]order difference equation, Discrete Dynamics in Nature and Society, 2018, 9 pages (2018).
  18. M. Mihăilescu, V. Rădulescu, S. Tersian, Eigenvalue problems for anisotropic discrete boundary value problems, J. Differ. Equ. Appl., 15, 557–567 (2009).
  19. M. Ousbika, Z. EL Allali, Existence of three solutions to the discrete fourth-order boundary value problem with four parameters, Bol. Soc. Paran. Mat., 38, 177–189 (2020).
  20. B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113, 401–410 (2000).
  21. J. Yang, Sign-changing solutions to discrete fourth-order Neumann boundary value problems, Adv. Differ. Equ., 2013, 1–11 (2013).
  22. S. B. Zhang, L. Kong, Y. Sun, X. Deng, Existence of positive solutions for BVPs of fourth-order difference equations, Appl. Math. Comput., 131, 583–591 (2002).
Global Analysis and Discrete Mathematics
Volume 7, Issue 1
November 2022
Pages 117-129

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History

  • Receive Date: 08 June 2022
  • Revise Date: 25 October 2022
  • Accept Date: 29 October 2022

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