Some complementary results on asymptotic behavior of solutions of neutral difference equations

Authors

NOUR H. M. ALSHARIFFollow
BAŞAK KARPUZFollow

Author ORCID Identifier

NOUR ALSHARIF: 0009-0008-5793-5946

BAŞAK KARPUZ: 0000-0002-0242-972X

DOI

10.55730/1300-0098.3564

Abstract

This paper focuses on examining the boundedness and asymptotic behavior of all solutions of the neutral difference equations\begin{equation}\Delta[x_{n}-p_{n}x_{n-\kappa}]+q_{n}x_{n-\ell}=0\quad\text{for}\ n=0,1,\cdots,\label{abseq1}\tag{$\star$}\end{equation}and\begin{equation}\Delta[x_{n}-px_{n-\kappa}]+q_{n}x_{n-\ell}=0\quad\text{for}\ n=0,1,\cdots,\label{abseq2}\tag{$\star\star$}\end{equation}where $\kappa,\ell\in\N$, $\{p_{n}\}\subset[0,1)$, $p\in[0,1)$ and $\{q_{n}\}\subset[0,\infty)$.Diverging from much of the existing literature, our results accommodate the scenario where$\{p_{n}\}\subset[\frac{1}{2},1)$ and $p\in[\frac{1}{2},1)$ for \eqref{abseq1} and \eqref{abseq2}, respectively.Furthermore, we underscore the practical implications of our results through the presentation of numerical examples. This paper focuses on examining the boundedness and asymptotic behavior of all solutions of the neutral difference equations \begin{equation} \Delta[x_{n}-p_{n}x_{n-\kappa}]+q_{n}x_{n-\ell}=0\quad\text{for}\ n=0,1,\cdots,\label{abseq1}\tag{$\star$} \end{equation} and \begin{equation} \Delta[x_{n}-px_{n-\kappa}]+q_{n}x_{n-\ell}=0\quad\text{for}\ n=0,1,\cdots,\label{abseq2}\tag{$\star\star$} \end{equation} where $\kappa,\ell\in\N$, $\{p_{n}\}\subset[0,1)$, $p\in[0,1)$ and $\{q_{n}\}\subset[0,\infty)$. Diverging from much of the existing literature, our results accommodate the scenario where $\{p_{n}\}\subset[\frac{1}{2},1)$ and $p\in[\frac{1}{2},1)$ for \eqref{abseq1} and \eqref{abseq2}, respectively. Furthermore, we underscore the practical implications of our results through the presentation of numerical examples.

Keywords

Neutral difference equations, variable coefficients, boundedness, asymptotic behavior

First Page

1110

Last Page

1126

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Recommended Citation

ALSHARIF, NOUR H. M. and KARPUZ, BAŞAK (2024) "Some complementary results on asymptotic behavior of solutions of neutral difference equations," Turkish Journal of Mathematics: Vol. 48: No. 6, Article 9. https://doi.org/10.55730/1300-0098.3564
Available at: https://journals.tubitak.gov.tr/math/vol48/iss6/9