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

Urszula Ostaszewska; Ewa Schmeidel; Małgorzata Zdanowicz

doi:10.3934/dcdsb.2025070

Article Contents

2025, Volume 30, Issue 11: 4417-4425. Doi: 10.3934/dcdsb.2025070

This issue Previous Article Existence of nonoscillatory solution on $ k $-dimensional system of delayed nonlinear discrete equations with $ p $-Laplacian Next Article Positive solutions for resonant singular non-autonomous $ (p, q) $-equations

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

1.
Faculty of Mathematics, University of Bialystok, Poland
2.
Faculty of Computer Science, University of Bialystok, Poland

^*Corresponding author: Urszula Ostaszewska
^*Corresponding author: Urszula Ostaszewska

Received: September 2024

Revised: April 2025

Early access: May 2025

Published: November 2025

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[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. Amorosoa, P. 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, 2^nd 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. You, Y. Tian, Y. 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.