Mohammed ALMATRAFİ, E. M. ELSAYED, Faris ALZAHRANİ

Qualitative Behavior of Two Rational Difference Equations

Obtaining the exact solutions of most rational recursive equations is sophisticated sometimes. Therefore, a considerable number of nonlinear difference equations is often investigated by studying the qualitative behavior of the governing forms of these equations. The prime purpose of this work is to analyse the equilibria, local stability, global stability character, boundedness character and the solution behavior of the following fourth order fractional difference equations: $$ x_{n+1}=\frac{\alpha x_{n}x_{n-3}}{\beta x_{n-3}-\gamma x_{n-2}},\ \ \ \ x_{n+1}=\frac{\alpha x_{n}x_{n-3}}{-\beta x_{n-3}+\gamma x_{n-2}},\ \ \ n=0,1,..., $$ where the constants $\alpha ,\ \beta ,\ \gamma \in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{+}$ and the initial values $x_{-3},\ x_{-2},\ x_{-1}$\ and $x_{0}$ are required to be arbitrary non zero real numbers. Furthermore, some numerical figures will be obviously shown in this paper.

Keywords:

Equilibria, Local stability, Periodicity Boundedness character, Global behavior, Recursive equations,

PDF

___

1] M. Avotina, On three second-order rational difference equations with period-two solutions, Int. J. Difference Equ., 9 (1) (2014), 23-35.
[2] I. Bajo, E. Liz, Global behaviour of a second-order nonlinear difference equation, J. Difference Equ. Appl., 17 (10) (2011), 1471-1486.
[3] C. Cinar, On the positive solutions of the difference equation $x_{n+1}=ax_{n-1}/(1+bx_{n}x_{n-1}),$; Appl. Math. Comput., 156 (2004) 587-590.
[4] Q. Din, On a system of rational difference equation, Demonstratio Math., XLVII (2) 2014, 324-335.
[5] M. A. El-Moneam, E. M. E. Zayed, Dynamics of the rational difference equation, Inf. Sci. Lett., 3 (2) (2014), 45-53.
[6] E. M. Elsayed, On the solution of recursive sequence of order two, Fasc. Math., 40 (2008), 5-13.
[7] T. F. Ibrahim, Global asymptotic stability and solutions of a nonlinear rational difference equation, Int. J. Math. Comput., 5(D09) (2009), 98-105.
[8] C. Cinar, On the positive solutions of the difference equation $x_{n+1}=x_{n-1}/(1+x_{n}x_{n-1}),$; Appl. Math. Comput., 150 (2004), 21-24.
[9] R. Karatas, Global behavior of a higher order difference equation, Int. J. Contemp. Math. Sci., 12(3) (2017),133-138.
[10] A. Khaliq, F. Alzahrani, E. M. Elsayed, Global attractivity of a rational difference equation of order ten, J. Nonlinear Sci. Appl., 9 (2016), 4465-4477.
[11] A. Khaliq, E. Elsayed, The dynamics and solution of some difference equations, J. Nonlinear Sci. Appl., 9(3) (2016), 1052-1063.
[12] Y. Kostrov, On a second-order rational difference equation with a quadratic term, Int. J. Difference Equ., 11(2) (2016), 179-202.
[13] M. R. S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall CRC Press, 2001.
[14] M. A. Obaid, E. M. Elsayed, M. M. El-Dessoky, Global attractivity and periodic character of difference equation of order four, Discrete Dyn. Nat. Soc., 2012 (2012), Article ID 746738, 20 pages.
[15] D. T. Tollu, Y. Yazlik, N. Taskara, Behavior of positive solutions of a difference equation, J. Appl. Math. Inform., 35(3-4) (2017), 217-230.
[16] E. M. Elabbasy, H. El-Metawally, E. M. Elsayed, On the difference equation $x_{n+1}=(ax_{n}^{2}+bx_{n-1}x_{n-k})/(cx_{n}^{2}+dx_{n-1}x_{n-k})$, Sarajevo J. Math., 4(17)(2008), 1-10.
[17] E. M. Elsayed, A. Alghamdi, Dynamics and global stability of higher order nonlinear difference equation, J. Comput. Anal. Appl., 21(3) (2016), 493-503.