ON A SECOND ORDER RATIONAL DIFFERENCE EQUATION

Year 2012, Volume: 41 Issue: 6, 867 - 874, 01.06.2012

Nouressadat Touafek

Abstract

ON A SECOND ORDER RATIONAL DIFFERENCE EQUATION

Abstract

In this paper, we investigate the stability of the following difference
equation
xn+1 =
ax4
n + bxnx
3
n−1 + cx2
nx
2
n−1 + dx3
nxn−1 + ex4
n−1
Ax4
n + Bxnx
3
n−1 + Cx2
nx
2
n−1 + Dx3
nxn−1 + Ex4
n−1
,
n = 0, 1, . . . ,
where the parameters a, b, c, d, e, A, B, C, D, E are positive real
numbers and the initial values x0, x−1 are arbitrary positive numbers. 

Keywords

: Difference equations, Global stability

References

• Bozkurt, F., Ozturk, I. and Ozen, S. The global behavior of the difference equation, Stud. Univ. Babes-Bolyai Math. 54 (2), 3–12, 2009.
• Cinar, C. On the positive solutions of the difference equation xn+1= +xnxn−1 , Appl. Math. Comp. 150 (1), 21–24, 2004.
• Cinar, C., Karatas, R. and Yal¸cinkaya, I. On solutions of the difference equation xn+1= xn−3 −1+xnxn−1xn−2xn−3 , Math. Bohem. 132 (3), 257–261, 2007.
• Clark, C. W., A delayed recruitment of a population dynamics with an application to baleen whale populations, J. Math. Biol. 3, 381–391, 1976.
• Chen, D., Li, X. and Wang, Y. Dynamics for nonlinear difference equation xn+1= αxn−k β+γx n−l Elabbasy, E. M., El-Metwally, H. and Elsayed, E. M. On the difference equation xn+1= , Adv. Differ. Equ. Article ID 82579, 10 pages, 2006.
• Elabbasy, E. M. and Elsayed, E. M. On the global attractivity of difference equation of higher n−dxn−1 order, Carpathian J. Math. 24 (2), 45–53, 2008.
• Elsayed, E. M. Expressions of solutions for a class of differential equations, An. S¸tiint. Univ. ”Ovidius” Constanta. Ser. Mat. 18 (1), 99–114, 2010.
• Feuer, J. Periodic solutions of the Lyness max equation, J. Math. Anal. Appl. 288 (1), –160, 2003.
• Feuer, J. Two classes of piecewise-linear difference equations with eventual periodicity three, J. Math. Anal. Appl. 332 (1), 564–569, 2007.
• Kocic, V. L. and Ladas, G. Global behavior of nonlinear difference equations of higher order with applications(Kluwer Academic Publishers, Dordrecht, 1993).
• Kulenovic, M. R. S. and Ladas, G. Dynamics of second order rational difference equations with open problems and conjectures(Chapman and Hall, CRC Press, 2001).
• Ozturk, I., Bozkurt, F. and Ozen, S., On the difference equation yn+1=α+βe−yn, Appl. γ+yn−1 Math. Comput. 181 (2), 1387–1393, 2006.
• Li, X. and Zhu, D. Global asymptotic stability in a rational equation, J. Difference Equ. Appl. 9, 833–839, 2003.
• Li, X. Existence of solutions with a single semicycle for a general second-order rational difference equation, J. Math. Anal. Appl. 334 (1), 528–533, 2007.
• Stevi´c, S. A note on periodic character of a higher order difference equation, Rostocker Math. Kolloq. 61, 21–30, 2006.
• Stevi´c, S. On a class of higher-order difference equations, Chaos Solitons and Fractals 42, –145, 2009.
• Yal¸cinkaya, I., Iricanin, B. D. and Cinar, C. On a max-type difference equation, Discrete Dyn. Nat. Soc. Article ID 47264, 10 pages, 2007.
• Yal¸cinkaya, I. On the difference equation xn+1= α +xn−2, Fasc. Math. 42, 133–139, 2009. xk n