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

Mohamed Abd El-moneam

doi:10.33401/fujma.1239100

Research Article

Year 2023, Volume: 6 Issue: 2, 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, Volume: 6 Issue: 2, 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 Volume: 6 Issue: 2

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 ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$. FUJMA. 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 ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$”, FUJMA, 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 ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$. FUJMA. 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 ${x_{n+1}}=\frac{{ \alpha {x_{n-m}+\eta {x_{n-k}{+\sigma {x_{n-l}}}}+}}\delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}}{x_{n-l}}\left( {{x_{n-k}}+{x_{n-l}}}\right) }}$. FUJMA. 2023;6(2):128-36.

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