ON A SECOND ORDER RATIONAL DIFFERENCE EQUATION

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

1. Bozkurt, F., Ozturk, I. and Ozen, S. The global behavior of the difference equation, Stud. Univ. Babes-Bolyai Math. 54 (2), 3–12, 2009.
2. Cinar, C. On the positive solutions of the difference equation xn+1= +xnxn−1 , Appl. Math. Comp. 150 (1), 21–24, 2004.
3. 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.
4. Clark, C. W., A delayed recruitment of a population dynamics with an application to baleen whale populations, J. Math. Biol. 3, 381–391, 1976.
5. 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.
6. 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.
7. 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.
8. Feuer, J. Periodic solutions of the Lyness max equation, J. Math. Anal. Appl. 288 (1), –160, 2003.

9. Feuer, J. Two classes of piecewise-linear difference equations with eventual periodicity three, J. Math. Anal. Appl. 332 (1), 564–569, 2007.
10. Kocic, V. L. and Ladas, G. Global behavior of nonlinear difference equations of higher order with applications(Kluwer Academic Publishers, Dordrecht, 1993).
11. 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).
12. 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.
13. Li, X. and Zhu, D. Global asymptotic stability in a rational equation, J. Difference Equ. Appl. 9, 833–839, 2003.
14. 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.
15. Stevi´c, S. A note on periodic character of a higher order difference equation, Rostocker Math. Kolloq. 61, 21–30, 2006.
16. Stevi´c, S. On a class of higher-order difference equations, Chaos Solitons and Fractals 42, –145, 2009.
17. 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.
18. Yal¸cinkaya, I. On the difference equation xn+1= α +xn−2, Fasc. Math. 42, 133–139, 2009. xk n