\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Existence of solutions to nonlinear $ 2n $-th order discrete boundary value problem via variational method

  • *Corresponding author: Urszula Ostaszewska

    *Corresponding author: Urszula Ostaszewska 
Abstract / Introduction Full Text(HTML) Related Papers Cited by
  • The paper deals with the boundary value problem for the $ 2n $-th order difference equation of the form

    $ \Delta^n(p(t-n)\Delta^n x(t-n)) +f(t,x(t)) = 0, $

    where $ t \in [1,T]_{{\mathbb Z}} $, and $ \Delta ^{k}x(1-n) = \Delta^{k} x(T-n+1) $, $ k = 0,1,\dots,2n-\; 1 $. Existence theorems for this equation are obtained based on the variational methods. A few examples illustrate the main results.

    Mathematics Subject Classification: Primary: 39A27, 47J30.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] R. P. Agarwal, Difference Equations and Inequalities, Theory, Methods, and Applications, 2$^{nd}$ edition, Monogr. Textbooks Pure Appl. Math., 228, Marcel Dekker, New York, 2000.
    [2] Z. AlSharawi, J. M. Cushing and S. Elaydi, Theory and Applications of Difference Equationsand Discrete Dynamical Systems, Springer Proc. Math. Stat., 102, Springer, Heidelberg, 2014.
    [3] E. AmorosoaP. Candito and J. Mawhin, Existence of apriori bounded solutions for discrete two-point boundary value problems, J. Math. Anal. Appl., 519 (2023), 1-18. 
    [4] A. Cabada and N. Dimitrov, Nontrivial solutions of inverse discrete problems with sign-changing nonlinearities, Adv. Differ. Equ., 2019 (2019), Article ID 450, 16 pp. doi: 10.1186/s13662-019-2383-y.
    [5] A. Cabada and L. López-Somoza, Lower and upper solutions for even order boundaryvalue problems, Mathematics, 7 (2019), 878.  doi: 10.3390/math7100878.
    [6] A. Cabada and L. Saavedra, Constant sign solution for a simply supported beam equation, EJQTDE, 2017 (2017), Paper No. 59, 17 pp. doi: 10.14232/ejqtde.2017.1.59.
    [7] X. Cai and J. Yu, Existence of periodic solutions for a $2n$th-order nonlinear difference equation, J. Math. Anal. Appl., 329 (2007), 870-878.  doi: 10.1016/j.jmaa.2006.07.022.
    [8] P. Drábek and J. Milota, Methods of Nonlinear Analysis. Applications to Differential Equations, Birkhäuser Verlag, Basel, 2007.
    [9] M. Galewski, Uniqueness of solutions for discrete boundary value problems, Dynam. Systems Appl., 22 (2013), 105-113. 
    [10] M. Galewski and J. Smejda, Some remarks on nonlinear discrete boundary value problems, Demonstratio Math., 45 (2012), 575-583.  doi: 10.1515/dema-2013-0402.
    [11] M. Galewski and J. Smejda, A note on a fourth order discrete boundary value problem, Opuscula Math., 32 (2012), 115-123.  doi: 10.7494/OpMath.2012.32.1.115.
    [12] I. Győri and L. Horváth, Sharp algebraic periodicity conditions for linear higher order difference equations, Comput. Math. Appl., 64 (2012), 2262-2274.  doi: 10.1016/j.camwa.2012.02.018.
    [13] O. Hammouti, Existence and multiplicity of solutions for nonlinear $2n$-th order difference boundary value problems, J. Elliptic Parabol., 8 (2022), 1081-1097.  doi: 10.1007/s41808-022-00166-9.
    [14] W. G. Kelley and A. C. Peterson, Difference Equations. An Introduction with Applications, 2nd edition, Harcourt/Academic Press, San Diego, CA, 2001.
    [15] Y. Liu and X. Liu, The existence of periodic solutions of higherorder nonlinear periodic difference equations, Math. Methods Appl. Sci., 36 (2013), 1459-1470.  doi: 10.1002/mma.2700.
    [16] P. Stehlík, On periodic disrete boundary value problems, J. Difference Equ. Appl., 14 (2008), 259-273. 
    [17] P. Stehlík, On variational methods for second order discrete periodic problems, Advances in Discrete Dynamical Systems, Adv. Stud. Pure Math., Mathematical Society of Japan, Tokyo, 53 (2009), 339-346.
    [18] M. YouY. TianY. Yue and J. Liu, Existence Results of Multiple Solutions for a $2n$th-OrderFinite Difference Equation, Bull. Malays. Math. Sci. Soc., 43 (2020), 2887-2907.  doi: 10.1007/s40840-019-00836-3.
  • 加载中
SHARE

Article Metrics

HTML views(421) PDF downloads(88) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return