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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 5

Solvability of a system of higher order nonlinear difference equations

Merve Kara(AKSARAY UNIVERSITY, AKSARAY VOCATIONAL SCHOOL)0000-0001-8081-0254
Yasin Yazlik(NEVSEHIR HACI BEKTAS VELI UNIVERSITY)0000-0001-6369-540X
Durhasan Turgut Tollu(Necmettin Erbakan University)0000-0002-3313-8829
Pages
—
Publication Date
2020-10-06
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Our obtained results significantly extend and develop some recent results in the literature.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper we show that the system of difference equations \[ x_n= a y_{n-k}+\frac{dy_{n-k}x_{n-( k+l ) }}{b x_{n-(k+l)}+cy_{n-l}}=\alpha x_{n-k}+\frac{\delta x_{n-k}y_{n-(k+l)}}{\beta y_{n-(k+l)}}+\gamma x_{n-l}, \] where $n\in \mathbb{N}_{0},$ $k$ and $l$ are positive integers, the parameters $a$, $b$, $c$, $d$, $\alpha $, $\beta $, $\gamma $, $\delta $ are real numbers and the initial values $x_{-j}$, $y_{-j}$, $j=\overline{1,k+l}$, are real numbers, can be solved in the closed form.
📊
Key Findings & Data:We also determine the asymptotic behavior of solutions for the case $l=1$ and describe the forbidden set of the initial values using the obtained formulas.
💡
Conclusion & Impact:Our obtained results significantly extend and develop some recent results in the literature.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:System of difference equations · Asymptotic behavior · Fibonacci sequence

Abstract

In this paper we show that the system of difference equations \[ x_n= a y_{n-k}+\frac{dy_{n-k}x_{n-( k+l ) }}{b x_{n-(k+l)}+cy_{n-l}}=\alpha x_{n-k}+\frac{\delta x_{n-k}y_{n-(k+l)}}{\beta y_{n-(k+l)}}+\gamma x_{n-l}, \] where $n\in \mathbb{N}_{0},$ $k$ and $l$ are positive integers, the parameters $a$, $b$, $c$, $d$, $\alpha $, $\beta $, $\gamma $, $\delta $ are real numbers and the initial values $x_{-j}$, $y_{-j}$, $j=\overline{1,k+l}$, are real numbers, can be solved in the closed form. We also determine the asymptotic behavior of solutions for the case $l=1$ and describe the forbidden set of the initial values using the obtained formulas. Our obtained results significantly extend and develop some recent results in the literature.

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Cite This Article

Kara, M., Yazlik, Y., Tollu, D. T. (2020). Solvability of a system of higher order nonlinear difference equations. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 5. https://doi.org/10.15672/hujms.474649

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