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Year 2018, , 194 - 204, 25.12.2018
https://doi.org/10.33401/fujma.454999

Abstract

References

  • 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.

Qualitative Behavior of Two Rational Difference Equations

Year 2018, , 194 - 204, 25.12.2018
https://doi.org/10.33401/fujma.454999

Abstract

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.

References

  • 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.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mohammed Almatrafi

E. M. Elsayed

Faris Alzahrani This is me

Publication Date December 25, 2018
Submission Date August 24, 2018
Acceptance Date November 12, 2018
Published in Issue Year 2018

Cite

APA Almatrafi, M., Elsayed, E. M., & Alzahrani, F. (2018). Qualitative Behavior of Two Rational Difference Equations. Fundamental Journal of Mathematics and Applications, 1(2), 194-204. https://doi.org/10.33401/fujma.454999
AMA Almatrafi M, Elsayed EM, Alzahrani F. Qualitative Behavior of Two Rational Difference Equations. FUJMA. December 2018;1(2):194-204. doi:10.33401/fujma.454999
Chicago Almatrafi, Mohammed, E. M. Elsayed, and Faris Alzahrani. “Qualitative Behavior of Two Rational Difference Equations”. Fundamental Journal of Mathematics and Applications 1, no. 2 (December 2018): 194-204. https://doi.org/10.33401/fujma.454999.
EndNote Almatrafi M, Elsayed EM, Alzahrani F (December 1, 2018) Qualitative Behavior of Two Rational Difference Equations. Fundamental Journal of Mathematics and Applications 1 2 194–204.
IEEE M. Almatrafi, E. M. Elsayed, and F. Alzahrani, “Qualitative Behavior of Two Rational Difference Equations”, FUJMA, vol. 1, no. 2, pp. 194–204, 2018, doi: 10.33401/fujma.454999.
ISNAD Almatrafi, Mohammed et al. “Qualitative Behavior of Two Rational Difference Equations”. Fundamental Journal of Mathematics and Applications 1/2 (December 2018), 194-204. https://doi.org/10.33401/fujma.454999.
JAMA Almatrafi M, Elsayed EM, Alzahrani F. Qualitative Behavior of Two Rational Difference Equations. FUJMA. 2018;1:194–204.
MLA Almatrafi, Mohammed et al. “Qualitative Behavior of Two Rational Difference Equations”. Fundamental Journal of Mathematics and Applications, vol. 1, no. 2, 2018, pp. 194-0, doi:10.33401/fujma.454999.
Vancouver Almatrafi M, Elsayed EM, Alzahrani F. Qualitative Behavior of Two Rational Difference Equations. FUJMA. 2018;1(2):194-20.

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