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

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

1. Abo-Zeid, R., Kamal, H., Global behavior of two rational third order difference equations, Univers. J. Math. Appl., 2(4)(2019), 212–217.
2. Abo-Zeid, R., Global behavior and oscillation of a third order difference equation, Quaest. Math., 44(9)(2021), 1261–1280.
3. Almatrafi, M.B., Elsayed, E.M., Alzahrani, F., Qualitative behavior of two rational difference equations, Fundam. J. Math. Appl., 1(2)(2018), 198–204.
4. De Moivre, A., The Doctrine of Chances, 3nd edition, In Landmark Writings in Western Mathematics, London, 1756.
5. Elabbasy, E.M., Elsayed, E.M., Dynamics of a rational difference equation, Chin. Ann. Math., 30(2)(2009), 187–198.
6. 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.
7. Elsayed, E.M., Qualitative behavior of a rational recursive sequence, Indag. Math., 19(2)(2008), 189–201.
8. Elsayed, E.M., Qualitative properties for a fourth order rational difference equation, Acta Appl. Math., 110(2)(2010), 589–604.

9. Elsayed, E.M., Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., (2011), 1–17.
10. 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.
11. Ghezal, A., Zemmouri, I., On a solvable p-dimensional system of nonlinear difference equations, J. Math. Comput. Sci., 12(2022).
12. Ghezal, A., Note on a rational system of (4k + 4)−order difference equations: periodic solution and convergence, J. Appl. Math. Comput., (2022), 1–9.
13. Halim, Y., Touafek, N., Yazlik, Y., Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39(6)(2015), 1004–1018.
14. 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.
15. 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.
16. 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.
17. Kara, M., Yazlik, Y., Solvability of a nonlinear three-dimensional system of difference equations with constant coefficients, Math. Slovaca., 71(5)(2021), 1133–1148.
18. Kara, M., Yazlik, Y., Solvable three-dimensional system of higher-order nonlinear difference equations, Filomat, 36(10)(2022), 3453–3473.
19. 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.
20. Khatibzadeh, H., Ibrahim, T.F., Asymptotic stability and oscillatory behavior of a difference equation, Electron. J. Math. Anal. Appl., 4(2)(2016), 227–233.
21. 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.
22. Stevic, S., Alghamdi, M.A., Shahzad, N., Maturi, D.A., On a class of solvable difference equations, Abstr. Appl. Anal., (2013), 1–7.
23. 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.
24. 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.
25. Taskara, N., Tollu, D.T., Touafek, N., Yazlik, Y., A solvable system of difference equations, Commun. Korean Math. Soc., 35(1)(2020), 301–319.
26. 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.
27. 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.
28. Tollu, D.T., Yazlik, Y., Taskara, N., On a solvable nonlinear difference equation of higher order, Turkish J. Math., 42(2018), 1765–1778.
29. Touafek, N., On a general system of difference equations defined by homogeneous functions, Math. Slovaca., 71(3)(2021), 697–720.
30. Yalcinkaya, I., Cinar, C., Simsek, D., Global asymptotic stability of a system of difference equations, Appl. Anal., 87(2008), 677–687.
31. Yalcinkaya, I., On the global asymptotic behavior of a system of two nonlinear difference equations, Ars Combin., 95(2010), 151–159.
32. Yalcinkaya, I., Tollu, D.T., Global behavior of a second order system of difference equations, Adv. Stud. Contemp. Math., 26(4)(2016), 653–667.
33. 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.
34. Yazlik, Y., Tollu, D.T., Taskara, N., On the solutions of difference equation systems with Padovan numbers, Appl. Math., 4(2013), 15–20.
35. 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.