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Year 2023, , 128 - 136, 30.06.2023

Mohamed Abd El-moneam

https://doi.org/10.33401/fujma.1239100

Abstract

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 Mathematics, 7(5) (2022), 7374--7384.
  • [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., 95 (2013).
  • [5] Q. Din, On a system of rational difference equation, Demonstratio Mathematica, (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, and 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, Br. J. Math. Comput. Sci., 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, Article ID 982309, (2011).
  • [14] E. M. Elsayed, T. F. İbrahim, Solutions and periodicity of a rational recursive sequences of order five, (Accepted and to appear 2012-2013, Bull. Malaysian Math. Sci. Soc.).
  • [15] E. A. Grove, G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall CRC, Vol. 4, 2005.
  • [16] T. F. İbrahim, Boundedness and stability of a rational difference equation with delay, Rev. Roum. Math. Pures Appl. 57 (2012), 215--224.
  • [17] T. F. İbrahim, Periodicity and global attractivity of difference equation of higher order, J. Comput. Anal. Appl., 16, (2014).
  • [18] T. F. İbrahim, Three-dimensional max-type cyclic system of difference equations, Int. J. Phys. Sci., 8(15) 2013, 629--634.
  • [19] T. F. İbrahim, N. Touafek, On a third-order rational difference equation with variable coefficients, DCDIS Series B: Applications & Algorithms (Dyn. Contin. Discret. I.) 20(2) (2013), 251--264.
  • [20] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
  • [21] D. Şimsek, C. Çınar, İ Yalçınkaya, 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.
  • [22] S. Stević, Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl., 316 (2006) 60--68.
  • [23] N. Touafek, E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Modelling, 55 (2012), 1987--1997.
  • [24] N. Touafek, E. M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sr 2012., 55(103), 217--224.
  • [25] İç Yalçınkaya, On the max-type equation $x_(n+1)=max(1/x(n),A(n) x(n-1))$, Discrete Dyn. Nat. Soc., 2012, Article ID 327437, (2012), 9 pages.
  • [26] M. E. Erdoğan, C. Çınar, İ. Yalçınkaya, \textit{On the dynamics of the recursive 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.
  • [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., A4 (2011), 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.

Qualitative Behavior of the difference equation ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$

Year 2023, , 128 - 136, 30.06.2023

Mohamed Abd El-moneam

https://doi.org/10.33401/fujma.1239100

Abstract

In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation ${x_{n+1}}=% \frac{{\alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{% {\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$, $% n=0,1,2,...$ where the parameters $\alpha ,\beta ,\gamma ,\delta ,{\eta },{% \sigma }\in (0,\infty )$, while $m,k,l$ are positive integers, such that $% m

Keywords

Difference equations; Rational difference equations; qualitative properties of solutions of difference equations Equilibrium Globally asymptotically stable. Oscillates Prime period two solution

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 Mathematics, 7(5) (2022), 7374--7384.
  • [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., 95 (2013).
  • [5] Q. Din, On a system of rational difference equation, Demonstratio Mathematica, (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, and 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, Br. J. Math. Comput. Sci., 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, Article ID 982309, (2011).
  • [14] E. M. Elsayed, T. F. İbrahim, Solutions and periodicity of a rational recursive sequences of order five, (Accepted and to appear 2012-2013, Bull. Malaysian Math. Sci. Soc.).
  • [15] E. A. Grove, G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman & Hall CRC, Vol. 4, 2005.
  • [16] T. F. İbrahim, Boundedness and stability of a rational difference equation with delay, Rev. Roum. Math. Pures Appl. 57 (2012), 215--224.
  • [17] T. F. İbrahim, Periodicity and global attractivity of difference equation of higher order, J. Comput. Anal. Appl., 16, (2014).
  • [18] T. F. İbrahim, Three-dimensional max-type cyclic system of difference equations, Int. J. Phys. Sci., 8(15) 2013, 629--634.
  • [19] T. F. İbrahim, N. Touafek, On a third-order rational difference equation with variable coefficients, DCDIS Series B: Applications & Algorithms (Dyn. Contin. Discret. I.) 20(2) (2013), 251--264.
  • [20] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
  • [21] D. Şimsek, C. Çınar, İ Yalçınkaya, 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.
  • [22] S. Stević, Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl., 316 (2006) 60--68.
  • [23] N. Touafek, E. M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Modelling, 55 (2012), 1987--1997.
  • [24] N. Touafek, E. M. Elsayed, On the periodicity of some systems of nonlinear difference equations, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sr 2012., 55(103), 217--224.
  • [25] İç Yalçınkaya, On the max-type equation $x_(n+1)=max(1/x(n),A(n) x(n-1))$, Discrete Dyn. Nat. Soc., 2012, Article ID 327437, (2012), 9 pages.
  • [26] M. E. Erdoğan, C. Çınar, İ. Yalçınkaya, \textit{On the dynamics of the recursive 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.
  • [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., A4 (2011), 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.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences, Applied Mathematics (Other)
Journal Section Articles
Authors

Mohamed Abd El-moneam 0000-0002-1676-2662

Early Pub Date June 25, 2023
Publication Date June 30, 2023
Submission Date January 19, 2023
Acceptance Date May 5, 2023
Published in Issue Year 2023

Cite

APA Abd El-moneam, M. (2023). Qualitative Behavior of the difference equation ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$. Fundamental Journal of Mathematics and Applications, 6(2), 128-136. https://doi.org/10.33401/fujma.1239100
AMA Abd El-moneam M. Qualitative Behavior of the difference equation ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$. Fundam. J. Math. Appl. June 2023;6(2):128-136. doi:10.33401/fujma.1239100
Chicago Abd El-moneam, Mohamed. “Qualitative Behavior of the Difference Equation ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$”. Fundamental Journal of Mathematics and Applications 6, no. 2 (June 2023): 128-36. https://doi.org/10.33401/fujma.1239100.
EndNote Abd El-moneam M (June 1, 2023) Qualitative Behavior of the difference equation ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$. Fundamental Journal of Mathematics and Applications 6 2 128–136.
IEEE M. Abd El-moneam, “Qualitative Behavior of the difference equation ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$”, Fundam. J. Math. Appl., vol. 6, no. 2, pp. 128–136, 2023, doi: 10.33401/fujma.1239100.
ISNAD Abd El-moneam, Mohamed. “Qualitative Behavior of the Difference Equation ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$”. Fundamental Journal of Mathematics and Applications 6/2 (June 2023), 128-136. https://doi.org/10.33401/fujma.1239100.
JAMA Abd El-moneam M. Qualitative Behavior of the difference equation ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$. Fundam. J. Math. Appl. 2023;6:128–136.
MLA Abd El-moneam, Mohamed. “Qualitative Behavior of the Difference Equation ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$”. Fundamental Journal of Mathematics and Applications, vol. 6, no. 2, 2023, pp. 128-36, doi:10.33401/fujma.1239100.
Vancouver Abd El-moneam M. Qualitative Behavior of the difference equation ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$. Fundam. J. Math. Appl. 2023;6(2):128-36.

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