Research Article
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Year 2019, , 212 - 217, 26.12.2019
https://doi.org/10.32323/ujma.626465

Abstract

References

  • [1] R. Abo-Zeid, Behavior of solutions of a second order rational difference equation, Math. Morav., 23 (1) (2019) , 11-25 .
  • [2] R. Abo-Zeid, Global behavior of two third order rational difference equations with quadratic terms, Math. Slovaca, 69 (1) (2019) , 147-158 .
  • [3] R. Abo-Zeid, Global Behavior of a fourth order difference equation with quadratic term, Bol. Soc. Mat. Mexicana, 25 (1) (2019) , 187-194 .
  • [4] R. Abo-Zeid, Behavior of solutions of a higher order difference equation, Alabama J. Math., 42 (2018) , 1-10 .
  • [5] R. Abo-Zeid, On the solutions of a higher order difference equation, Georgian Math. J., doi:10.1515/gmj-2018-0008.
  • [6] R. Abo-Zeid, On a third order difference equation, Acta Univ. Apulensis, 55 (2018) , 89-103 .
  • [7] R. Abo-Zeid Forbidden sets and stability in some rational difference equations, J. Difference Equ. Appl., 24 (2) (2018) , 220-239 .
  • [8] R. Abo-Zeid, On the solutions of a second order difference equation, Math. Morav., 21 (2) (2017), 61-75 .
  • [9] R. Abo-Zeid, Global behavior of a higher order rational difference equation, Filomat 30 (12) (2016), 3265-3276 .
  • [10] R. Abo-Zeid, Global behavior of a third order rational difference equation, Math. Bohem., 139 (1) (2014) , 25-37 .
  • [11] R. Abo-Zeid, On the solutions of two third order recursive sequences, Armenian J. Math., 6 (2) (2014), 64-66 .
  • [12] R. Abo-Zeid, Global behavior of a fourth order difference equation, Acta Commentaiones Univ. Tartuensis Math., 18 (2) (2014) , 211-220 .
  • [13] A.M. Amleh, E. Camouzis and G. Ladas, On the dynamics of a rational difference equation, Part 2, Int. J. Difference Equ., 3 (2) (2008) , 195-225 .
  • [14] A.M. Amleh, E. Camouzis and G. Ladas, On the dynamics of a rational difference equation, Part 1, Int. J. Difference Equ., 3 (1) (2008) , 1-35 .
  • [15] I. Bajo, Forbidden sets of planar rational systems of difference equations with common denominator, Appl. Anal. Discrete Math., 8 (2014) , 16-32 .
  • [16] F. Balibrea and A. Cascales, On forbidden sets, J. Difference Equ. Appl. 21 (10) (2015) , 974-996 .
  • [17] E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2008 .
  • [18] H. El-Metwally and E. M. Elsayed, Qualitative study of solutions of some difference equations, Abstr. Appl. Anal., 2012 (2012) , Article ID 248291 , 16 pages, doi: 10.1155/2012/248291 .
  • [19] E.M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., 2011 (2011) , Article ID 982309 , 18 pages, doi: 10.1155/2011/982309 .
  • [20] M. Gümüş, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl., 24 (6) (2018) , 976-991 .
  • [21] M. Gümüş and Ö . Öcalan, The qualitative analysis of a rational system of difference equations, J. Fract. Calc. Appl., 9 (2) (2018) , 113-126 .
  • [22] Inci Okumuş and Yüksel Soykan, A review on dynamical nature of systems of nonlinear difference equations, J. Inform. Math. Sci., 11 (2) (2019) , 235–251 .
  • [23] R. Khalaf-Allah, Asymptotic behavior and periodic nature of two difference equations, Ukrainian Math. J., 61 (6) (2009) , 988-993 .
  • [24] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht, 1993 .
  • [25] M. R. S. Kulenovic, and M. Mehuljic, Global behavior of some rational second order difference equations, Int. J. Difference Equ., 7 (2) (2012) , 153–162 .
  • [26] M.R.S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman and Hall/HRC, Boca Raton, 2002 .
  • [27] H. Sedaghat, On third order rational equations with quadratic terms, J. Difference Equ. Appl., 14 (8) (2008) , 889-897 .
  • [28] I. Szalkai, Avoiding forbidden Sequences by finding suitable initial values, Int. J. Difference Equ., 3 (2) (2008) , 305-315 .

Global Behavior of Two Rational Third Order Difference Equations

Year 2019, , 212 - 217, 26.12.2019
https://doi.org/10.32323/ujma.626465

Abstract

In this paper, we solve and study the global behavior of all admissible solutions of  the two difference equations $$x_{n+1}=\frac{x_{n}x_{n-2}}{x_{n-1}-x_{n-2}}, \quad n=0,1,...,$$ and $$x_{n+1}=\frac{x_{n}x_{n-2}}{-x_{n-1}+x_{n-2}}, \quad n=0,1,...,$$ where  the initial values $x_{-2}$, $x_{-1}$, $x_{0}$ are real numbers.\\ We show that every admissible solution for the first equation converges to zero. For the other equation, we show that every admissible solution is periodic with prime period  six.  Finally we give some illustrative examples.

References

  • [1] R. Abo-Zeid, Behavior of solutions of a second order rational difference equation, Math. Morav., 23 (1) (2019) , 11-25 .
  • [2] R. Abo-Zeid, Global behavior of two third order rational difference equations with quadratic terms, Math. Slovaca, 69 (1) (2019) , 147-158 .
  • [3] R. Abo-Zeid, Global Behavior of a fourth order difference equation with quadratic term, Bol. Soc. Mat. Mexicana, 25 (1) (2019) , 187-194 .
  • [4] R. Abo-Zeid, Behavior of solutions of a higher order difference equation, Alabama J. Math., 42 (2018) , 1-10 .
  • [5] R. Abo-Zeid, On the solutions of a higher order difference equation, Georgian Math. J., doi:10.1515/gmj-2018-0008.
  • [6] R. Abo-Zeid, On a third order difference equation, Acta Univ. Apulensis, 55 (2018) , 89-103 .
  • [7] R. Abo-Zeid Forbidden sets and stability in some rational difference equations, J. Difference Equ. Appl., 24 (2) (2018) , 220-239 .
  • [8] R. Abo-Zeid, On the solutions of a second order difference equation, Math. Morav., 21 (2) (2017), 61-75 .
  • [9] R. Abo-Zeid, Global behavior of a higher order rational difference equation, Filomat 30 (12) (2016), 3265-3276 .
  • [10] R. Abo-Zeid, Global behavior of a third order rational difference equation, Math. Bohem., 139 (1) (2014) , 25-37 .
  • [11] R. Abo-Zeid, On the solutions of two third order recursive sequences, Armenian J. Math., 6 (2) (2014), 64-66 .
  • [12] R. Abo-Zeid, Global behavior of a fourth order difference equation, Acta Commentaiones Univ. Tartuensis Math., 18 (2) (2014) , 211-220 .
  • [13] A.M. Amleh, E. Camouzis and G. Ladas, On the dynamics of a rational difference equation, Part 2, Int. J. Difference Equ., 3 (2) (2008) , 195-225 .
  • [14] A.M. Amleh, E. Camouzis and G. Ladas, On the dynamics of a rational difference equation, Part 1, Int. J. Difference Equ., 3 (1) (2008) , 1-35 .
  • [15] I. Bajo, Forbidden sets of planar rational systems of difference equations with common denominator, Appl. Anal. Discrete Math., 8 (2014) , 16-32 .
  • [16] F. Balibrea and A. Cascales, On forbidden sets, J. Difference Equ. Appl. 21 (10) (2015) , 974-996 .
  • [17] E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2008 .
  • [18] H. El-Metwally and E. M. Elsayed, Qualitative study of solutions of some difference equations, Abstr. Appl. Anal., 2012 (2012) , Article ID 248291 , 16 pages, doi: 10.1155/2012/248291 .
  • [19] E.M. Elsayed, Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., 2011 (2011) , Article ID 982309 , 18 pages, doi: 10.1155/2011/982309 .
  • [20] M. Gümüş, The global asymptotic stability of a system of difference equations, J. Difference Equ. Appl., 24 (6) (2018) , 976-991 .
  • [21] M. Gümüş and Ö . Öcalan, The qualitative analysis of a rational system of difference equations, J. Fract. Calc. Appl., 9 (2) (2018) , 113-126 .
  • [22] Inci Okumuş and Yüksel Soykan, A review on dynamical nature of systems of nonlinear difference equations, J. Inform. Math. Sci., 11 (2) (2019) , 235–251 .
  • [23] R. Khalaf-Allah, Asymptotic behavior and periodic nature of two difference equations, Ukrainian Math. J., 61 (6) (2009) , 988-993 .
  • [24] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht, 1993 .
  • [25] M. R. S. Kulenovic, and M. Mehuljic, Global behavior of some rational second order difference equations, Int. J. Difference Equ., 7 (2) (2012) , 153–162 .
  • [26] M.R.S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman and Hall/HRC, Boca Raton, 2002 .
  • [27] H. Sedaghat, On third order rational equations with quadratic terms, J. Difference Equ. Appl., 14 (8) (2008) , 889-897 .
  • [28] I. Szalkai, Avoiding forbidden Sequences by finding suitable initial values, Int. J. Difference Equ., 3 (2) (2008) , 305-315 .
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

R. Abo-zeid 0000-0002-1858-5583

H. Kamal This is me

Publication Date December 26, 2019
Submission Date September 29, 2019
Acceptance Date November 5, 2019
Published in Issue Year 2019

Cite

APA Abo-zeid, R., & Kamal, H. (2019). Global Behavior of Two Rational Third Order Difference Equations. Universal Journal of Mathematics and Applications, 2(4), 212-217. https://doi.org/10.32323/ujma.626465
AMA Abo-zeid R, Kamal H. Global Behavior of Two Rational Third Order Difference Equations. Univ. J. Math. Appl. December 2019;2(4):212-217. doi:10.32323/ujma.626465
Chicago Abo-zeid, R., and H. Kamal. “Global Behavior of Two Rational Third Order Difference Equations”. Universal Journal of Mathematics and Applications 2, no. 4 (December 2019): 212-17. https://doi.org/10.32323/ujma.626465.
EndNote Abo-zeid R, Kamal H (December 1, 2019) Global Behavior of Two Rational Third Order Difference Equations. Universal Journal of Mathematics and Applications 2 4 212–217.
IEEE R. Abo-zeid and H. Kamal, “Global Behavior of Two Rational Third Order Difference Equations”, Univ. J. Math. Appl., vol. 2, no. 4, pp. 212–217, 2019, doi: 10.32323/ujma.626465.
ISNAD Abo-zeid, R. - Kamal, H. “Global Behavior of Two Rational Third Order Difference Equations”. Universal Journal of Mathematics and Applications 2/4 (December 2019), 212-217. https://doi.org/10.32323/ujma.626465.
JAMA Abo-zeid R, Kamal H. Global Behavior of Two Rational Third Order Difference Equations. Univ. J. Math. Appl. 2019;2:212–217.
MLA Abo-zeid, R. and H. Kamal. “Global Behavior of Two Rational Third Order Difference Equations”. Universal Journal of Mathematics and Applications, vol. 2, no. 4, 2019, pp. 212-7, doi:10.32323/ujma.626465.
Vancouver Abo-zeid R, Kamal H. Global Behavior of Two Rational Third Order Difference Equations. Univ. J. Math. Appl. 2019;2(4):212-7.

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