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

Year 2024, Volume: 16 Issue: 1, 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]

References

• Abo-Zeid, R., Kamal, H., Global behavior of two rational third order difference equations, Univers. J. Math. Appl., 2(4)(2019), 212–217.
• Abo-Zeid, R., Global behavior and oscillation of a third order difference equation, Quaest. Math., 44(9)(2021), 1261–1280.
• Almatrafi, M.B., Elsayed, E.M., Alzahrani, F., Qualitative behavior of two rational difference equations, Fundam. J. Math. Appl., 1(2)(2018), 198–204.
• De Moivre, A., The Doctrine of Chances, 3nd edition, In Landmark Writings in Western Mathematics, London, 1756.
• Elabbasy, E.M., Elsayed, E.M., Dynamics of a rational difference equation, Chin. Ann. Math., 30(2)(2009), 187–198.
• Elsayed, E.M., El-Metwally, H.A., Elsayed, E.M., Global behavior of the solutions of some difference equations, Adv. Difference Equ., 2011(1)(2011), 1–16.
• Elsayed, E.M., Qualitative behavior of a rational recursive sequence, Indag. Math., 19(2)(2008), 189–201.
• Elsayed, E.M., Qualitative properties for a fourth order rational difference equation, Acta Appl. Math., 110(2)(2010), 589–604.
• Elsayed, E.M., Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., (2011), 1–17.
• Elsayed, E.M., Alzahrani, F., Abbas, I., Alotaibi, N.H., Dynamical behavior and solution of nonlinear difference equation via Fibonacci sequence, J. Appl. Anal. Comput., 10(1)(2020), 282–296.
• Ghezal, A., Zemmouri, I., On a solvable p-dimensional system of nonlinear difference equations, J. Math. Comput. Sci., 12(2022).
• Ghezal, A., Note on a rational system of (4k + 4)−order difference equations: periodic solution and convergence, J. Appl. Math. Comput., (2022), 1–9.
• Halim, Y., Touafek, N., Yazlik, Y., Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39(6)(2015), 1004–1018.
• Ibrahim, T.F., Touafek, N., On a third order rational difference equation with variable coefficients, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms., 20(2)(2013), 251–264.
• Kara, M., Yazlik, Y., Tollu, D.T., Solvability of a system of higher order nonlinear difference equations, Hacet. J. Math. Stat., 49(5)(2020), 1566–1593.
• Kara, M., Yazlik, Y., On eight solvable systems of difference equations in terms of generalized Padovan sequences, Miskolc Math. Notes., 22(2)(2021), 695–708.
• Kara, M., Yazlik, Y., Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients, Math. Slovaca., 71(5)(2021), 1133–1148.
• Kara, M., Yazlik, Y., Solvable three-dimensional system of higher-order nonlinear difference equations, Filomat, 36(10)(2022), 3453–3473.
• Kara, M., Yazlik, Y., Solutions formulas for three-dimensional difference equations system with constant coefficients, Turk. J. Math. Comput. Sci., 14(1)(2022), 107–116.
• Khatibzadeh, H., Ibrahim, T.F., Asymptotic stability and oscillatory behavior of a difference equation, Electron. J. Math. Anal. Appl., 4(2)(2016), 227–233.
• Sanbo, A., Elsayed, E.M., Some properties of the solutions of the difference equation xn+1 = axn + bxn xn−4cxn−3+dxn−4, Open J. Discrete Appl. Math., 2(2)(2019), 31–47.
• Stevic, S., Alghamdi, M.A., Shahzad, N., Maturi, D.A., On a class of solvable difference equations, Abstr. Appl. Anal., (2013), 1–7.
• Stevic, S., Iricanin, B., Kosmola, W., ˇSmarda, Z., On a solvable class of nonlinear difference equations of fourth order, Electron. J. Qual. Theory Differ. Equ., 37(2022), 1–47.
• Taskara, N., Tollu, D.T., Yazlik, Y., Solutions of rational difference system of order three in terms of Padovan numbers, J. Adv. Res. Appl. Math., 7(3)(2015), 18–29.
• Taskara, N., Tollu, D.T., Touafek, N., Yazlik, Y., A solvable system of difference equations, Commun. Korean Math. Soc., 35(1)(2020), 301–319.
• Tollu, D.T., Yazlik, Y., Taskara, N., The solutions of four Riccati difference equations associated with Fibonacci numbers, Balkan J. Math., 2(1)(2014), 163–172.
• Tollu, D.T., Yazlik, Y., Taskara, N., Behavior of positive solutions of a difference equation, J. Appl. Math. Inform., 35(3-4)(2017), 217–230.
• Tollu, D.T., Yazlik, Y., Taskara, N., On a solvable nonlinear difference equation of higher order, Turkish J. Math., 42(2018), 1765–1778.
• Touafek, N., On a general system of difference equations defined by homogeneous functions, Math. Slovaca., 71(3)(2021), 697–720.
• Yalcinkaya, I., Cinar, C., Simsek, D., Global asymptotic stability of a system of difference equations, Appl. Anal., 87(2008), 677–687.
• Yalcinkaya, I., On the global asymptotic behavior of a system of two nonlinear difference equations, Ars Combin., 95(2010), 151–159.
• Yalcinkaya, I., Tollu, D.T., Global behavior of a second order system of difference equations, Adv. Stud. Contemp. Math., 26(4)(2016), 653–667.
• Yalcinkaya, I., Ahmad, H., Tollu, D.T., Li, Y., On a system of k−difference equations of order three, Math. Probl. Eng., (2020), 1–11.
• Yazlik, Y., Tollu, D.T., Taskara, N., On the solutions of difference equation systems with Padovan numbers, Appl. Math., 4(2013), 15–20.
• Yazlik, Y., Tollu, D.T., Taskara N. On the solutions of a three-dimensional system of difference equations, Kuwait J. Sci., 43(1)(2016), 95–111.