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EIGENVALUE COMPARISON FOR THE DISCRETE (3, 3) CONJUGATE BOUNDARY VALUE PROBLEM

  • Jun Ji (Department of Mathematics Kennesaw State University) ;
  • Bo Yang (Department of Mathematics Kennesaw State University)
  • Received : 2021.12.24
  • Accepted : 2023.02.15
  • Published : 2023.07.31

Abstract

In this paper, we consider a boundary value problem for a sixth order difference equation. We prove the monotone behavior of the eigenvalue of the problem as the coefficients in the difference equation change values and the existence of a positive solution for a class of problems.

Keywords

References

  1. R. P. Agarwal, B. Kovacs, and D. O'Regan, Positive solutions for a sixth-order boundary value problem with four parameters, Bound. Value Probl. 2013 (2013), 184, 22 pp. https://doi.org/10.1186/1687-2770-2013-184
  2. R. E. Bellman, Introduction to Matrix Analysis, reprint of the second (1970) edition, Classics in Applied Mathematics, 19, SIAM, Philadelphia, PA, 1997. https://doi.org/10.1137/1.9781611971170
  3. J. V. Chaparova, L. A. Peletier, and S. A. Tersian, Existence and nonexistence of nontrivial solutions of semilinear sixth-order ordinary differential equations, Appl. Math. Lett. 17 (2004), no. 10, 1207-1212. https://doi.org/10.1016/j.aml.2003.05.014
  4. F. R. Gantmakher, The Theory of Matrices, Vols. 1, Chelsea, New York, 1960.
  5. J. R. Graef, S. Heidarkhani, L. Kong, and M. Wang, Existence of solutions to a discrete fourth order boundary value problem, J. Difference Equ. Appl. 24 (2018), no. 6, 849-858. https://doi.org/10.1080/10236198.2018.1428963
  6. J. R. Graef and B. Yang, Boundary value problems for sixth order nonlinear ordinary differential equations, Dynam. Systems Appl. 10 (2001), no. 4, 465-475.
  7. T. B. Gyulov, Trivial and nontrivial solutions of a boundary value problem for a sixth-order ordinary differential equation, C. R. Acad. Bulgare Sci. 58 (2005), no. 9, 1013-1018.
  8. T. B. Gyulov, G. Moro,sanu, and S. A. Tersian, Existence for a semilinear sixth-order ODE, J. Math. Anal. Appl. 321 (2006), no. 1, 86-98. https://doi.org/10.1016/j.jmaa.2005.08.007
  9. J. Ji and B. Yang, Eigenvalue comparisons for boundary value problems of the discrete beam equation, Adv. Difference Equ. 2006 (2006), Art. ID 81025, 9 pp.
  10. J. Ji and B. Yang, Eigenvalue comparisons for second order difference equations with Neumann boundary conditions, Linear Algebra Appl. 425 (2007), no. 1, 171-183. https://doi.org/10.1016/j.laa.2007.03.021
  11. J. Ji and B. Yang, Positive solutions for boundary value problems of second order difference equations and their computation, J. Math. Anal. Appl. 367 (2010), no. 2, 409-415. https://doi.org/10.1016/j.jmaa.2010.01.026
  12. J. Ji and B. Yang, Eigenvalue comparisons for a class of boundary value problems of discrete beam equation, Appl. Math. Comput. 218 (2012), no. 9, 5402-5408. https://doi.org/10.1016/j.amc.2011.11.024
  13. J. Ji and B. Yang, Spectral properties of a boundary value problem for the discrete beam equation, J. Appl. Math. Comput. 54 (2017), no. 1-2, 95-108. https://doi.org/10.1007/s12190-016-0999-6
  14. X.-G. Lv and T.-Z. Huang, A note on inversion of Toeplitz matrices, Appl. Math. Lett. 20 (2007), no. 12, 1189-1193. https://doi.org/10.1016/j.aml.2006.10.008
  15. M. K.-P. Ng, K. Rost, and Y.-W. Wen, On inversion of Toeplitz matrices, Linear Algebra Appl. 348 (2002), 145-151. https://doi.org/10.1016/S0024-3795(01)00592-4
  16. M. Sohaib, S. Haq, S. Mukhtar, and I. Khan, Numerical solution of sixth-order boundary-value problems using Legendre wavelet collocation method, Results in Phys. 8 (2018), 1204-1208. https://doi.org/10.1016/j.rinp.2018.01.065
  17. R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962.
  18. B. Yang, Positive solutions to a nonlinear sixth order boundary value problem, Differ. Equ. Appl. 11 (2019), no. 2, 307-317. https://doi.org/10.7153/dea-2019-11-13
  19. B. G. Zhang, L. Kong, Y. Sun, and X. Deng, Existence of positive solutions for BVPs of fourth-order difference equations, Appl. Math. Comput. 131 (2002), no. 2-3, 583-591. https://doi.org/10.1016/S0096-3003(01)00171-0