On the Global of the Difference Equation ${x_{n+1}}=\frac{{\alpha {x_{n-m}+\eta {x_{n-k}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$

Year 2022, Volume: 5 Issue: 4, 189 - 198, 30.12.2022

Mohamed Abd El-moneam

https://doi.org/10.33434/cams.1182861

Abstract

In this article, we consider and discuss some properties of the positive solutions to the following rational nonlinear DE ${x_{n+1}}=\frac{{\alpha { x_{n-m}+\eta {x_{n-k}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}} \left( {{x_{n-k}}+{x_{n-l}}}\right) }}$, $n=0,1,...,$ where the parameters $ \alpha ,\beta ,\gamma ,\delta ,{\eta }\in (0,\infty )$, while $m,k,l$ are positive integers, such that $m

Keywords

Difference equations, Qualitative properties of solutions of difference equations, Rational difference equations, Globally asymptotically stable, \ Prime period two solution., Equilibrium

References

• [1] R. P. Agarwal, E. M. Elsayed, On the solution of fourth-order rational recursive sequence, Adv. Stud. Contemp. Math., 20(4) (2010), 525–545.
• [2] A. M. Alotaibi, M. A. El-Moneam, On the dynamics of the nonlinear rational difference equation ${\ x_{n+1}}=\frac{{\alpha {x_{n-m}+}}% \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{\ x_{n-l}}\left( {{x_{n-k}}+{ x_{n-l}}}\right) }}$ , AIMS Math., 7(5) (2022), 7374–7384, DOI: 10.3934/math.2022411.
• [3] R. Devault, W. Kosmala, G. Ladas, S. W. Schaultz, Global behavior of $y_{n+1}=\dfrac{p+y_{n-k}}{qy_{n}+y_{n-k}}$ , Nonlinear Anal. Theory Methods Appl., 47 (2004) 83–89.
• [4] Q. Din,Dynamics of a discrete Lotka-Volterra model, Adv. Differ. Equ., (2013), 95.
• [5] Q. Din, On a system of rational difference equation, Demonstr. Math., in press.
• [6] E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, On the difference equation $\ x_{n+1}=ax_{n}-\dfrac{bx_{n}}{cx_{n}-dx_{n-1}}$ , Adv. Differ. Equ., 2006, Article ID 82579, 1-10.
• [7] H. El-Metwally, E. A. Grove, G. Ladas, H. D. Voulov, On the global attractivity and the periodic character of some difference equations, J. Differ. Equations Appl., 7 (2001), 837-850.
• [8] M. A. El-Moneam, On the dynamics of the higher order nonlinear rational difference equation, Math. Sci. Lett., 3(2)(2014), 121-129.
• [9] M. A. El-Moneam, On the dynamics of the solutions of the rational recursive sequences, British Journal of Mathematics & Computer Science, 5(5) (2015), 654-665.
• [10] M. A. El-Moneam, S.O. Alamoudy, On study of the asymptotic behavior of some rational difference equations, DCDIS Series A: Mathematical Analysis, 21(2014), 89-109.
• [11] M. A. El-Moneam, E. M. E. Zayed, Dynamics of the rational difference equation, Inf. Sci. Lett., 3(2) (2014), 1-9.
• [12] M. A. El-Moneam, E. M. E. Zayed, On the dynamics of the nonlinear rational difference equation $x_{n+1}=Ax_{n}+Bx_{n-k}+Cx_{n-l}+% \frac{bx_{n-k}}{dx_{n-k}-ex_{n-l}}$ , J. Egypt. Math. Soc., 23 (2015), 494-499.
• [13] E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., 2011 (2011), Article ID 982309.
• [14] E. M. Elsayed, T. F. Ibrahim, Solutions and periodicity of a rational recursive sequences of order five, (Accepted and to appear 2012-2013, Bull. Malays. Math. Sci. Soc.)
• [15] M. E. Erdogan, C. Cinar, I. Yalcinkaya, On the dynamics of the rescursive sequence ${x_{n+1}}=\frac{{{x_{n-1}}}}{{\beta +\gamma x_{n-2}^{2}{x_{n-4}}+\gamma {x_{n-2}}x_{n-4}^{2}}}$ , Comput. Math. Appl., 61 2011), 533-537.
• [16] E. A. Grove, G. Ladas, Periodicities in Nonlinear Difference Equations, 4, Chapman & Hall / CRC, 2005.
• [17] T. F. Ibrahim, Boundedness and stability of a rational difference equation with delay Rev. Roum. Math. Pures Appl., 57 (2012), 215-224.
• [18] T. F. Ibrahim, Periodicity and global attractivity of difference equation of higher order, Accepted 2013 and to Appear in: J. Comput. Anal. Appl., 16 (2014).
• [19] T. F. Ibrahim, Three-dimensional max-type cyclic system of difference equations, Int. J. Phys. Sci., 8(15) (2013), 629-634.
• [20] T. F. Ibrahim, N. Touafek, On a third-order rational difference equation with variable coefficients, Dyn. Contin. Discret. I. Series B: Applications & Algorithms, 20(2)(2013), 251-264
• [21] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
• [22] D. Simsek, C. Cinar, I. Yalcinkaya, On the recursive sequence $x_{n+1}=\dfrac{x_{n-3}}{1+x_{n-1}}$ , Int. J. Contemp. Math. Sci., 1(10) (2006), 475-480.
• [23] S. Stevic,Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl., 316 (2006) 60–68.
• [24] N. Touafek, E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Modelling, 55 (2012), 1987-1997.
• [25] N. Touafek, E. M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sr, 55(103) (2012), 217-224.
• [26] I. Yalcinkaya, Global asymptotic stability of a system of difference equations, Appl. Anal., 87(6) (2008), 677-687 .
• [27] E. M. E. Zayed, On the dynamics of the nonlinear rational difference equation, DCDIS Series A: Mathematical Analysis, (to appear).
• [28] E. M. E. Zayed, M. A. El-Moneam, On the rational recursive two sequences $x_{n+1}=ax_{n-k}+bx_{n-k}/\left( cx_{n}+\delta dx_{n-k}\right)$, Acta Math. Vietnamica, 35(2010), 355-369.
• [29] E. M. E. Zayed, M. A. El-Moneam, On the global attractivity of two nonlinear difference equations, J. Math. Sci., 177(2011), 487-499.
• [30] E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence $x_{n+1}=\left( A+\alpha _{0}x_{n}+\alpha _{1}x_{n-\sigma }\right) /\left( B+\beta _{0}x_{n}+\beta _{1}x_{n-\tau }\right)$, Acta Math. Vietnamica, 36(2011), 73-87.
• [31] E. M. E. Zayed, M. A. El-Moneam, On the global asymptotic stability for a rational recursive sequence, Iran J. Sci. Technol. Trans. A Sci., (2011), A4, 333-339.
• [32] E. M. E. Zayed, M. A. El-Moneam, On the rational recursive sequence $x_{n+1}=\frac{\alpha _{0}x_{n}+\alpha _{1}x_{n-l}+\alpha _{2}x_{n-m}+\alpha _{3}x_{n-k}}{\beta _{0}x_{n}+\beta _{1}x_{n-l}+\beta _{2}x_{n-m}+\beta _{3}x_{n-k}}$ , WSEAS Trans. Math., 11(5)(2012), 373-382.
• [33] E. M. E. Zayed, M. A. El-Moneam, On the qualitative study of the nonlinear difference equation $x_{n+1}=\frac{\alpha x_{n-\sigma }}{\beta +\gamma x_{n-\tau }^{p}}$ , Fasc. Math., 50(2013), 137-147.
• [34] E. M. E. Zayed, M. A. El-Moneam, Dynamics of the rational difference equation $x_{n+1}=\gamma x_{n}+\frac{\alpha x_{n-l}+\beta x_{n-k}% }{Ax_{n-l}+Bx_{n-k}}$ , Comm. Appl. Nonl. Anal., 21(2014), 43-53.