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

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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.

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

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References

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