On a General Non-Linear Difference Equation of Third-Order

Year 2024, , 126 - 136, 30.06.2024

Merve Kara

https://doi.org/10.47000/tjmcs.1366596

Abstract

In this paper, we investigate the following general difference equations
\begin{equation*}
x_{n+1}=h^{-1}\left( h\left( x_{n}\right) \frac{Ah\left( x_{n-1}\right)+Bh\left( x_{n-2}\right) }{Ch\left( x_{n-1}\right)+Dh\left( x_{n-2}\right)}\right) ,\ n\in \mathbb{N}_{0},
\end{equation*}
where the parameters $A, B, C, D$ and the initial values $x_{-\Phi}$, for $\Phi=\overline{0,2}$ are real numbers, $h$ is a continuous and strictly monotone function, $h\left( \mathbb{R}\right) =\mathbb{R}$, $h\left( 0\right) =0$. In addition, we obtain closed-form solutions of aforementioned difference equations. Finally, numerical applications are given.

Keywords

Riccati difference equation, solution, closed form

Ethical Statement

List of Reviewers: 1. Prof. Dr. Yasin Yazlik Turkey, Nevsehir Haci Bektas Veli University, E-mail: [e-mail] 2. Prof. Dr. Raafat Abo-Zeid Egypt, The Higher Institute for Engineering &Technology Al-Obour, E-mail: [e-mail] 3. Prof. Dr. Necati Taskara Turkey, Selcuk University, E-mail: [e-mail] 4. Prof. Dr. Tarek Fawzi Ibrahim Egypt, Mansoura University, E mail: [e-mail] 5. Prof. Dr. Nouressadat Touafek Algeria, Mohamed Seddik Ben Yahia University, Email: [e-mail]

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