# The Dynamics and Attractivity for a Rational Recursive Sequence of Order Three

## Keywords:

Stability, Periodicity, Boundedness, Recursive Sequence, Difference Equation## Abstract

This paper is concerned with the behavior of solution of the nonlinear difference equation

where the initial conditions are arbitrary positive real numbers and a,b,c,d,e are positive constants.

### Downloads

## References

E. M. Elsayed and Abdul Khaliq, Global attractivity and periodicity behavior of a recursive sequence, J.Comp.Anal.Appl,22 (2017), 369–379.

E.M.Elabbasy,H.El-Metwally and E.M.Elsayed, Utilitas Mathematica, 87 (2012), 93-110.

E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, Global attractivity and periodic character of a fractional differenceequation of order three, Yokohama Math. J., 53 (2007), 89-100.

E. M. Elabbasy, H. El-Metwally and E. M. Elsayed, On the difference equationxn+1=axn−bxncxn−dxn−1,Adv.Differ. Equ., Volume 2006 (2006), Article ID 82579,1–10.

E.M.E.Zayed, Dynamics of the nonlinear rational difference equationxn+1=Axn+Bxn−k+pxn+xn−kq+xn−k,European journal of Pure and Applied Mathematics, 3 (2010) 254-268.

E. M. Elsayed, M.M.El-Dessoky and Asim Asiri, Dynamics and behavior a second order rational difference equation,J.Comp.Anal. Appl, 16(2014), 794-807.

E. M. Elabbasy and E. M. Elsayed, Dynamics of a Rational Difference Equation, Chinese Annals of Mathematics, Series B,30 B (2), (2009), 187–198.

E. M. Elabbasy and E. M. Elsayed, Global attractivity and periodic nature of a difference equation, World Applied SciencesJournal, 12 (1) (2011), 39–47.

E. M. Elsayed, On the solution of recursive sequence of order two, Fasciculi Mathematici, 40 (2008), 5–13.

E. M. Elsayed, Dynamics of a recursive sequence of higher order, Communications on Applied Nonlinear Analysis, 16 (2)(2009), 37–50.

E. M. Elsayed, Dynamics of recursive sequence of order two, Kyungpook Mathematical Journal, 50(2010), 483-497.

E. M. Elsayed, On the Difference Equationxn+1=xn−5−1 +xn−2xn−5, International Journal of Contemporary Math-ematical Sciences, 3 (33) (2008), 1657-1664.

E. M. Elsayed, Qualitative behavior of difference equation of order three, Acta Scientiarum Mathematicarum (Szeged), 75(1-2), 113–129.

E. M. Elsayed, Qualitative behavior of s rational recursive sequence, Indagationes Mathematicae, New Series, 19(2) (2008),189–201.

E. M. Elsayed, On the Global attractivity and the solution of recursive sequence, Studia Scientiarum MathematicarumHungarica, 47 (3) (2010), 401-418.

E. M. Elsayed, Qualitative properties for a fourth order rational difference equation, Acta Applicandae Mathematicae, 110(2) (2010), 589–604.114

E. M. Elsayed, On the global attractivity and the periodic character of a recursive sequence, Opuscula Mathematica, 30(4) (2010), 431–446.

E. M. Elsayad, Bratislav Iricanin and Stevo Stevic, On the max-type equation, Ars Combinatoria, 95 (2010) 187–192.

H. El-Metwally, Global behavior of an economic model, Chaos, Solitons and Fractals, 33 (2007), 994–1005.

H. El-Metwally, E. A. Grove, G. Ladas and H. D. Voulov, On the global attractivity and the periodic character of somedifference equations, J. Differ. Equations Appl., 7 (2001), 1-14.

M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comp., 176(2) (2006), 768-774.

N. Battaloglu, C. Cinar, and I. Yal ̧cınkaya, The dynamics of the difference equation,ARS Combinatoria, XCVII (2010).

A. E. Hamza and A. Morsy, On the recursive sequencexn+1=A∏ki=lxn−2i−1B+C∏k−1i=lxn−2i,Computers andMathematics with Applications, 56 (7) (2008), 1726-1731.

T. F. Ibrahim, On the third order rational difference equationxn+1=xnxn−2xn−1(a+bxnxn−2),Int. J. Contemp.Math. Sciences, 4 (27) (2009), 1321-1334.

R. Karatas, C. Cinar and D. Simsek, On positive solutions of the difference equationxn+1=xn−51 +xn−2xn−5,Int. J.Contemp. Math. Sci., Vol. 1, 2006, no. 10, 495-500.

V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, KluwerAcademic Publishers, Dordrecht, 1993.

M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems andConjectures, Chapman & Hall / CRC Press, 2001.

M. R. S. Kulenovic and Z. Nurkanovic, Global behavior of a three-dimensional linear fractional system of difference equations,J. Math. Anal. Appl., 310 (2005), 673–689.

W. Li and H. R. Sun, Dynamics of a rational difference equation Appl. Math. Comp., 163 (2005), 577–591.

A. Rafiq, Convergence of an iterative scheme due to Agarwal et al., Rostock. Math. Kolloq., 61 (2006), 95–105.

M. Saleh and M. Aloqeili, On the difference equationyn+1=A+ynyn−kwithA <0, Appl. Math. Comp., 176(1) (2006), 359–363.

D. Simsek, C. Cinar and I. Yalcinkaya, On the Recursive Sequencexn+1=xn−31 +xn−1,Int. J. Contemp. Math. Sci.,Vol. 1, 2006, no. 10, 475-480.

C. Wang and S. Wang. ,Global Behavior of Equilibrium Point for A Class of Fractional Difference Equation. Proceedingof the 7th Asian Control Conference. Hong Kong, China, August 27-29, (2009), 288-291.

C. Wang, S. Wang and X. Yan, Global asymptotic stability of 3-species mutualism models with diffusion and delay effects,Discrete Dynamics in Natural and Science, Volume 2009, Article ID 317298, 20 pages.

C. Wang, F. Gong, S. Wang, L. LI and Q. Shi, Asymptotic behavior of equilibrium point for a class of nonlinear differenceequation, Advances in Difference Equations, Volume 2009, Article ID 214309, 8 pages.

E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence xn+1=α+βxn+γxn−1A+Bxn+Cxn−1,Communications on Applied Nonlinear Analysis, 12 (4) (2005), 15–28.[38] E. M. E. Zayed and M. A. El-Moneam, On the rational recursive sequence xn+1=αxn+βxn−1+γxn−2+δxn−3Axn+Bxn−1+Cxn−2+Dxn−3, Comm. Appl. Nonlinear Analysis, 12 (2005),15-28.[39] I. Yal ̧cınkaya, Global asymptotic stability in a rational equation, Sel ̧cuk Journal of Applied Mathematics, Summer-Autumn,6 (2) (2005), 59-68.

I. Yal ̧cınkaya, On the difference equationxn+1=α+xn−2xkn,Fasciculi Mathematici, 42 (2009), 133-140.[41] I. Yal ̧cınkaya, On the difference equationxn+1=α+xn−mxkn, Discrete Dynamics in Nature and Society, Vol. 2008,Article ID 805460,8 pages, doi: 10.1155/2008/ 805460.

I. Yal ̧cınkaya, On the global asymptotic stability of a second-order system of difference equations, Discrete Dynamics inNature and Society, Vol. 2008, Article ID 860152, 12 pages, doi: 10.1155/2008/ 860152.[43]I. Yal ̧cınkaya, and C. Cinar, On the dynamics of the difference equationxn+1=axn−kb+cxpn, Fasciculi Mathematici, 42115, 141-148.

I. Yal ̧cınkaya, C. Cinar and M. Atalay, On the solutions of systems of difference equations, Advances in Difference Equations,Vol. 2008, Article ID 143943, 9 pages, DOI: 10.1155/2008/ 143943.

Lin-Xia Hua,b, Wan-Tong Lia and Hong-Wu Xu,Global asymptotical stability of a second order rational difference equation,Computers and Mathematics with Applications, 54 (2007), 1260–1266.

Qamar Din,On a system of a rational difference equation, Demonstratio Mathematica, Demonostratio Mathematica, XLVII ,2 (20140.324-335.

Qi Wang a, Fanping Zeng b, Xinhe Liuc, Weiling Youa, Stability of a rational difference equation, Applied MathematicsLetters 25 (2012) 2232–2239

Stevo Stevic ́, On positive solutions of a (k + 1)th order difference equation, Applied Mathematics Letters 19 (2006) 427–431

## Published

## How to Cite

*Computer Reviews Journal*,

*5*, 106-116. Retrieved from https://purkh.com/index.php/tocomp/article/view/528