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Analytical and Solutions of Fourth Order Difference Equations

Year 2019, Volume: 2 Issue: 1, 9 - 21, 22.03.2019

Marwa M. Alzubaidi , Elsayed M. Elsayed

https://doi.org/10.33434/cams.447757
Cited By: 2

Abstract

In this article, we presented the solutions of the following recursive sequences\[ x_{n+1}=\frac{x_{n-2}x_{n-3}}{x_{n}(\pm1\pm x_{n-2}x_{n-3})}, \] where the initial conditions $x_{-3}\ ,x_{-2}\ ,x_{-1}$\ and $x_{0}\ $are arbitrary real numbers. Also, we studied some dynamic behavior of these equations.

Keywords

Stability Global attractor Difference equation Linearized stability Periodicity

References

  • [1] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
  • [2] A. Asiri, M. M. El-Dessoky, E. M. Elsayed, Solution of a third order fractional system of difference equations, J. Comput. Anal. Appl., 24 (3) (2018), 444-453.
  • [3] E. Camouzis, G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004.
  • [4] Q. Din, Global Stability and Neimark-Sacker bifurcation of a host-parasitoid model, Int. J. Syst. Sct., 48 (6) (2017), 1194-1202.
  • [5] Q. Din, E. M. Elsayed, Stability analysis of a discrete ecological model, Computational Ecology and Software, 2 (2014), 89–103.
  • [6] Q. Din, M. A. Khan, Global stability of a system of exponential difference equations, CMMPG, 1 (2) (2016), 71-85.
  • [7] Q. Din, M. A. Khan, U. Saeed, Qualitative behaviour of generalised Beddington model, Zeitschrift f¨ur Naturforschung A, 71 (2) (2016), 145-155.
  • [8] E. M. Elabbasy, A. A. Elsadany, S. Ibrahim, Qualitative behavior of rational difference equations of Higher Order, Malaya J. Mat., 3 (4) (2015), 530–539.
  • [9] H. El-Metwally, E. M. Elsayed, Qualitative Behavior of some rational difference equations, J. Comput. Anal. Appl., 20 (2) (2016), 226-236.
  • [10] M. A. El-Moneam, On The dynamics of the solutions of the rational recursive sequences, Brit. J. Math. Comput. Sci., 5 (2015), 654–665.
  • [11] M. A. El-Moneam, S. O. Alamoudy, On study of the asymptotic behavior of some rational difference equations, DCDIS, A: Mathematical Analysis, 22 (2015), 157-176.
  • [12] E. M. Elsayed, On the solutions and periodicity of some rational systems of difference equations, B. Math. Soc. Sci. Math., 60 (108) (2) (2017), 159–171.
  • [13] E. M. Elsayed, A. Khaliq, Global attractivity and periodicity behavior of a recursive sequence, J. Comput. Anal. Appl., 22 (2) (2017), 369-379.
  • [14] E. M. Elsayed, A. M. Ahmed, Dynamics of a three-dimensional systems of rational difference equations, Math. Method Appl. Sci., 39 (5) (2016), 1026–1038.
  • [15] E. M. Elsayed, A. Alghamdi, Dynamics and global stability of higher order nonlinear difference equation, J. Comput. Anal. Appl., 21 (3) (2016), 493-503.
  • [16] E. M. Elsayed, M. M. El-Dessoky, Dynamics and global behavior for a fourth-order rational difference equation, Hacet. J. Math. Stat., 42 (5) (2013), 479-494.
  • [17] E. M. Elsayed, M. Ghazel, A. E. Matouk, Dynamical analysis of the rational difference equation $x_{n+1}=Cx_{n-3}/(A+Bx_{n-1} x_{n-3})$, J. Comput. Anal. Appl, 23 (3) (2017), 496-507.
  • [18] E. M. Elsayed, T. F. Ibrahim, Solutions and periodicity of a rational recursive sequences of order five, B. Malaysian Math. Sci. Soc., 38 (1) (2015), 95-112.
  • [19] E. M. Elsayed, T. F. Ibrahim, Periodicity and Solutions for Some Systems of Nonlinear Rational Difference Equations, Hacet. J. Math. Stat., 44 (6) (2015), 1361–1390.
  • [20] M. Erdogan, K. Uslu, Behavior of a Nonlinear Difference Equation $x_{n+1}=\dfrac{1-x_{n}}{A+\underset{i=1}{\overset{k}{\sum}}x_{n-i}},$; Journal of Life Sciences, 10 (2016), 215-219.
  • [21] T. F. Ibrahim, Periodicity and global attractivity of difference equation of higher order, J. Comput. Anal. Appl., 16 (2014), 552-564.
  • [22] T. F. Ibrahim, Solving a class of third-order max-type difference equations, DCDIS, Series A:Mathematical Analysis , 21(2014), 333-342.
  • [23] T. F. Ibrahim, Closed form expression of tractable semi-max-type two-dimensional system of difference equations with variable coefficients, JOEMS, 24 (4) (2016), 538-544.
  • [24] T. F. Ibrahim , M. A. El-Moneam, Global stability of a higher-order difference equation, Iran. J. Sci. Technol. A , 1 (41) (2017), 51-58
  • [25] R. Karatas, C. Cinar, D. Simsek, On positive solutions of the difference equation $x_{n+1}=\frac{x_{n-5}}{1+x_{n-2}x_{n-5}},\ $ ; Int. J. Contemp. Math. Sci., 1 (10) (2006) 495 500.
  • [26] A. Q. Khan, Q. Din, M. N. Qureshi, T. F. Ibrahim, Global behavior of an anti-competitive system of fourth-order rational difference equations, Computational Ecology and Software, 2014, 4(1).
  • [27] K. Liu, P. Li, F. Han, W. Zhong, Global dynamics of nonlinear difference equation $x_{n+1}=A+x_{n}/x_{n-1}x_{n-2}$, J. Comput. Anal. Appl., 24 (6) (2018), 1125-1132.
  • [28] R. Mostafaei, N. Rastegar, On a recurrence relation, QScience Connect, 10 (2014), 1-11.
  • [29] U. Saeed, M. Ozair, T. Hussain, Q. Din, Fractional-order vector-host disease model, DCDIS, Series B: Applications & Algorithms, 24 (2017), 97-111.
  • [30] H. Sedaghat, Nonlinear Difference Equations, Theory and Applications to Social Science Models, Kluwer Academic Publishers, Dordrecht, 2003.
  • [31] D. T. Tollou, Y. Yazlik, N. Taskara, Behavior of positive solutions of a difference equation, J. Appl. Math. & Informatics, 35 (3-4) (2017), 217 - 230.
  • [32] N. Touafek, On a second order rational difference equation, Hacet. J. Math. Stat., 41 (6) (2012), 867–874.
  • [33] C. Wang, M. Hu, On the solutions of a rational recursive sequence, JMI, 1 (2013), 25-33.
  • [34] C. Y. Wang, X. J. Fang, R. Li, On the dynamics of a certain four-order fractional difference equations, J. Comput. Anal. Appl., 22 (5) (2017), 968-976.
  • [35] Y. Yazlik, E. M. Elsayed, N. Taskara, On the behaviour of the solutions of difference equation systems, J. Comput. Anal. Appl., 16 (5) (2014), 932–941.
  • [36] E. M. E. Zayed, Qualitative behavior of the rational recursive sequence $x_{n+1}=Ax_{n}+Bx_{n-k}+\dfrac{p+x_{n-k}}{qx_{n}+x_{n-k} }$ , Inter. J. Adv. Math., 1 (1) (2014), 44-55.
  • [37] R. Karatas, Global behavior of a higher order difference equation , Inter. J. Contemp. Math. Sci., 12 (3) (2017), 133-138.
  • [38] M. Gumus, R. Abo-Zeid, O . Ocalan, Dynamical behavior of a third-order difference equation with arbitrary powers, Kyungpook Math. J., 57 (2017), 251-263.
  • [39] E. M. Elsayed, Dynamics and behavior of a higher order rational difference equation, J. Nonlinear Sci. App., 9 (4) (2016), 1463-1474.
  • [40] M. R. S. Kulenovic, G. Ladas,N. R. Prokup, A rational difference Equation, Comput. Math. Appl., 41 (2001), 671-678.
  • [41] E. M. Elsayed, New method to obtain periodic solutions of period two and three of a rational difference equation, Nonlinear Dynam., 79 (1) (2015), 241-250.
  • [42] M. A. Al-Shabi, R. Abo-Zeid, Global asymptotic stability of a higher order difference equation, Appl. Math. Sci., 4(17) (2010), 839-847.
  • [43] A. M. Amleh, E. Drymonis, Eventual Monotonicity in Nonlinear Difference Equation, International Journal of Difference Equation, 11 (2) (2016), 123-137.
  • [44] S. Nirmaladevi, N. Karthikeyan, Dynamics and behavior of higher order nonlinear rational difference equation, IJARIIE, 3 (4) (2017), 2395-4396.
  • [45] E. M. Elsayed, H. El-Metwally, Global Behavior and periodicity of some difference equations, J. Comput. Anal. Appl, 19 (2) (2015), 298-309.
  • [46] M. Akbar, N. Ali, The improved F-expansion method with Riccati equation and its applications in mathematical physics, Cogent Math., 4 (1) (2017), 1-19.

Year 2019, Volume: 2 Issue: 1, 9 - 21, 22.03.2019

Marwa M. Alzubaidi , Elsayed M. Elsayed

https://doi.org/10.33434/cams.447757
Cited By: 2

Abstract

References

  • [1] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
  • [2] A. Asiri, M. M. El-Dessoky, E. M. Elsayed, Solution of a third order fractional system of difference equations, J. Comput. Anal. Appl., 24 (3) (2018), 444-453.
  • [3] E. Camouzis, G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004.
  • [4] Q. Din, Global Stability and Neimark-Sacker bifurcation of a host-parasitoid model, Int. J. Syst. Sct., 48 (6) (2017), 1194-1202.
  • [5] Q. Din, E. M. Elsayed, Stability analysis of a discrete ecological model, Computational Ecology and Software, 2 (2014), 89–103.
  • [6] Q. Din, M. A. Khan, Global stability of a system of exponential difference equations, CMMPG, 1 (2) (2016), 71-85.
  • [7] Q. Din, M. A. Khan, U. Saeed, Qualitative behaviour of generalised Beddington model, Zeitschrift f¨ur Naturforschung A, 71 (2) (2016), 145-155.
  • [8] E. M. Elabbasy, A. A. Elsadany, S. Ibrahim, Qualitative behavior of rational difference equations of Higher Order, Malaya J. Mat., 3 (4) (2015), 530–539.
  • [9] H. El-Metwally, E. M. Elsayed, Qualitative Behavior of some rational difference equations, J. Comput. Anal. Appl., 20 (2) (2016), 226-236.
  • [10] M. A. El-Moneam, On The dynamics of the solutions of the rational recursive sequences, Brit. J. Math. Comput. Sci., 5 (2015), 654–665.
  • [11] M. A. El-Moneam, S. O. Alamoudy, On study of the asymptotic behavior of some rational difference equations, DCDIS, A: Mathematical Analysis, 22 (2015), 157-176.
  • [12] E. M. Elsayed, On the solutions and periodicity of some rational systems of difference equations, B. Math. Soc. Sci. Math., 60 (108) (2) (2017), 159–171.
  • [13] E. M. Elsayed, A. Khaliq, Global attractivity and periodicity behavior of a recursive sequence, J. Comput. Anal. Appl., 22 (2) (2017), 369-379.
  • [14] E. M. Elsayed, A. M. Ahmed, Dynamics of a three-dimensional systems of rational difference equations, Math. Method Appl. Sci., 39 (5) (2016), 1026–1038.
  • [15] E. M. Elsayed, A. Alghamdi, Dynamics and global stability of higher order nonlinear difference equation, J. Comput. Anal. Appl., 21 (3) (2016), 493-503.
  • [16] E. M. Elsayed, M. M. El-Dessoky, Dynamics and global behavior for a fourth-order rational difference equation, Hacet. J. Math. Stat., 42 (5) (2013), 479-494.
  • [17] E. M. Elsayed, M. Ghazel, A. E. Matouk, Dynamical analysis of the rational difference equation $x_{n+1}=Cx_{n-3}/(A+Bx_{n-1} x_{n-3})$, J. Comput. Anal. Appl, 23 (3) (2017), 496-507.
  • [18] E. M. Elsayed, T. F. Ibrahim, Solutions and periodicity of a rational recursive sequences of order five, B. Malaysian Math. Sci. Soc., 38 (1) (2015), 95-112.
  • [19] E. M. Elsayed, T. F. Ibrahim, Periodicity and Solutions for Some Systems of Nonlinear Rational Difference Equations, Hacet. J. Math. Stat., 44 (6) (2015), 1361–1390.
  • [20] M. Erdogan, K. Uslu, Behavior of a Nonlinear Difference Equation $x_{n+1}=\dfrac{1-x_{n}}{A+\underset{i=1}{\overset{k}{\sum}}x_{n-i}},$; Journal of Life Sciences, 10 (2016), 215-219.
  • [21] T. F. Ibrahim, Periodicity and global attractivity of difference equation of higher order, J. Comput. Anal. Appl., 16 (2014), 552-564.
  • [22] T. F. Ibrahim, Solving a class of third-order max-type difference equations, DCDIS, Series A:Mathematical Analysis , 21(2014), 333-342.
  • [23] T. F. Ibrahim, Closed form expression of tractable semi-max-type two-dimensional system of difference equations with variable coefficients, JOEMS, 24 (4) (2016), 538-544.
  • [24] T. F. Ibrahim , M. A. El-Moneam, Global stability of a higher-order difference equation, Iran. J. Sci. Technol. A , 1 (41) (2017), 51-58
  • [25] R. Karatas, C. Cinar, D. Simsek, On positive solutions of the difference equation $x_{n+1}=\frac{x_{n-5}}{1+x_{n-2}x_{n-5}},\ $ ; Int. J. Contemp. Math. Sci., 1 (10) (2006) 495 500.
  • [26] A. Q. Khan, Q. Din, M. N. Qureshi, T. F. Ibrahim, Global behavior of an anti-competitive system of fourth-order rational difference equations, Computational Ecology and Software, 2014, 4(1).
  • [27] K. Liu, P. Li, F. Han, W. Zhong, Global dynamics of nonlinear difference equation $x_{n+1}=A+x_{n}/x_{n-1}x_{n-2}$, J. Comput. Anal. Appl., 24 (6) (2018), 1125-1132.
  • [28] R. Mostafaei, N. Rastegar, On a recurrence relation, QScience Connect, 10 (2014), 1-11.
  • [29] U. Saeed, M. Ozair, T. Hussain, Q. Din, Fractional-order vector-host disease model, DCDIS, Series B: Applications & Algorithms, 24 (2017), 97-111.
  • [30] H. Sedaghat, Nonlinear Difference Equations, Theory and Applications to Social Science Models, Kluwer Academic Publishers, Dordrecht, 2003.
  • [31] D. T. Tollou, Y. Yazlik, N. Taskara, Behavior of positive solutions of a difference equation, J. Appl. Math. & Informatics, 35 (3-4) (2017), 217 - 230.
  • [32] N. Touafek, On a second order rational difference equation, Hacet. J. Math. Stat., 41 (6) (2012), 867–874.
  • [33] C. Wang, M. Hu, On the solutions of a rational recursive sequence, JMI, 1 (2013), 25-33.
  • [34] C. Y. Wang, X. J. Fang, R. Li, On the dynamics of a certain four-order fractional difference equations, J. Comput. Anal. Appl., 22 (5) (2017), 968-976.
  • [35] Y. Yazlik, E. M. Elsayed, N. Taskara, On the behaviour of the solutions of difference equation systems, J. Comput. Anal. Appl., 16 (5) (2014), 932–941.
  • [36] E. M. E. Zayed, Qualitative behavior of the rational recursive sequence $x_{n+1}=Ax_{n}+Bx_{n-k}+\dfrac{p+x_{n-k}}{qx_{n}+x_{n-k} }$ , Inter. J. Adv. Math., 1 (1) (2014), 44-55.
  • [37] R. Karatas, Global behavior of a higher order difference equation , Inter. J. Contemp. Math. Sci., 12 (3) (2017), 133-138.
  • [38] M. Gumus, R. Abo-Zeid, O . Ocalan, Dynamical behavior of a third-order difference equation with arbitrary powers, Kyungpook Math. J., 57 (2017), 251-263.
  • [39] E. M. Elsayed, Dynamics and behavior of a higher order rational difference equation, J. Nonlinear Sci. App., 9 (4) (2016), 1463-1474.
  • [40] M. R. S. Kulenovic, G. Ladas,N. R. Prokup, A rational difference Equation, Comput. Math. Appl., 41 (2001), 671-678.
  • [41] E. M. Elsayed, New method to obtain periodic solutions of period two and three of a rational difference equation, Nonlinear Dynam., 79 (1) (2015), 241-250.
  • [42] M. A. Al-Shabi, R. Abo-Zeid, Global asymptotic stability of a higher order difference equation, Appl. Math. Sci., 4(17) (2010), 839-847.
  • [43] A. M. Amleh, E. Drymonis, Eventual Monotonicity in Nonlinear Difference Equation, International Journal of Difference Equation, 11 (2) (2016), 123-137.
  • [44] S. Nirmaladevi, N. Karthikeyan, Dynamics and behavior of higher order nonlinear rational difference equation, IJARIIE, 3 (4) (2017), 2395-4396.
  • [45] E. M. Elsayed, H. El-Metwally, Global Behavior and periodicity of some difference equations, J. Comput. Anal. Appl, 19 (2) (2015), 298-309.
  • [46] M. Akbar, N. Ali, The improved F-expansion method with Riccati equation and its applications in mathematical physics, Cogent Math., 4 (1) (2017), 1-19.
There are 46 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Marwa M. Alzubaidi

Elsayed M. Elsayed

Publication Date March 22, 2019
Submission Date July 25, 2018
Acceptance Date November 12, 2018
Published in Issue Year 2019 Volume: 2 Issue: 1

Cite

APA Alzubaidi, M. M., & Elsayed, E. M. (2019). Analytical and Solutions of Fourth Order Difference Equations. Communications in Advanced Mathematical Sciences, 2(1), 9-21. https://doi.org/10.33434/cams.447757
AMA Alzubaidi MM, Elsayed EM. Analytical and Solutions of Fourth Order Difference Equations. Communications in Advanced Mathematical Sciences. March 2019;2(1):9-21. doi:10.33434/cams.447757
Chicago Alzubaidi, Marwa M., and Elsayed M. Elsayed. “Analytical and Solutions of Fourth Order Difference Equations”. Communications in Advanced Mathematical Sciences 2, no. 1 (March 2019): 9-21. https://doi.org/10.33434/cams.447757.
EndNote Alzubaidi MM, Elsayed EM (March 1, 2019) Analytical and Solutions of Fourth Order Difference Equations. Communications in Advanced Mathematical Sciences 2 1 9–21.
IEEE M. M. Alzubaidi and E. M. Elsayed, “Analytical and Solutions of Fourth Order Difference Equations”, Communications in Advanced Mathematical Sciences, vol. 2, no. 1, pp. 9–21, 2019, doi: 10.33434/cams.447757.
ISNAD Alzubaidi, Marwa M. - Elsayed, Elsayed M. “Analytical and Solutions of Fourth Order Difference Equations”. Communications in Advanced Mathematical Sciences 2/1 (March 2019), 9-21. https://doi.org/10.33434/cams.447757.
JAMA Alzubaidi MM, Elsayed EM. Analytical and Solutions of Fourth Order Difference Equations. Communications in Advanced Mathematical Sciences. 2019;2:9–21.
MLA Alzubaidi, Marwa M. and Elsayed M. Elsayed. “Analytical and Solutions of Fourth Order Difference Equations”. Communications in Advanced Mathematical Sciences, vol. 2, no. 1, 2019, pp. 9-21, doi:10.33434/cams.447757.
Vancouver Alzubaidi MM, Elsayed EM. Analytical and Solutions of Fourth Order Difference Equations. Communications in Advanced Mathematical Sciences. 2019;2(1):9-21.

Cited By

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