Periodicity and Solution of Rational Recurrence Relation of Order Six

Tarek F. Ibrahim Ibrahim

Department of Mathematics, Faculty of Sciences and arts (S. A.) King Khalid University, Abha , Saudi Arabia.

DOI: 10.4236/am.2012.37107                

Abstract

Difference equations or discrete dynamical systems is diverse field whose impact almost every branch of pure and ap- plied mathematics. Every dynamical system a_n+1=f(a_n) determines a difference equation and vise versa. We ob-tain in this paper the solution and periodicity of the following difference equation. xn+1=(x_nx_n-2x_n-4)/(x_n-1x_n-3x_n-5, (1) n=0,1,... where the initial conditions x_-5,x_-4,x_-3,x_-2,x_-1 and x₀ are arbitrary real numbers with x_-1,x_-3 and x_-5 not equal to be zero. On the other hand, we will study the local stability of the solutions of Equation (1). Moreover, we give graphically the behavior of some numerical examples for this difference equation with some initial conditions.

Keywords

Difference Equation; Solutions; Periodicity; Local Stability

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	C. Cinar, “On the Positive Solutions of the Difference Equation ,” Applied Mathematics and Computation, Vol. 158, No. 3, 2004, pp. 793-797. doi:10.1016/j.amc.2003.08.139
[2]	C. Cinar, “On the Positive Solutions of the Difference Equation ,” Applied Mathematics and Computation, Vol. 156, No. 2, 2004, pp. 587-590. doi:10.1016/j.amc.2003.08.010
[3]	S. N. Elaydi, “An Introduction to Difference Equations,” Springer-Verlag Inc., New York, 1996.
[4]	R. Karatas, C. Cinar and D. Simsek, “On Positive Solutions of the Difference Equation ,” International Journal of Contemporary Mathematical Sciences, Vol. 1, No. 10, 2006, pp. 495-500.
[5]	T. F. Ibrahim, “On the Third Order Rational Difference Equation ,” International Journal of Contemporary Mathematical Sciences, Vol. 4, No. 27, 2009, pp. 1321-1334.
[6]	T. F. Ibrahim, “Global Asymptotic Stability of a Nonlinear Difference Equation with Constant Coefficients,” Mathematical Modelling and Applied Computing, Vol. 1, No. 1, 2009.
[7]	T. F. Ibrahim, “Dynamics of a Rational Recursive Sequence of Order Two,” International Journal of Mathematics and Computation, Vol. 5, No. D09, 2009, pp. 98105.
[8]	T. F. Ibrahim, “Solvability and Attractivity of the Solutions of a Rational Difference Equation,” Journal of Pure and Applied Mathematics: Advances and Applications, Vol. 2, No. 2, 2009, pp. 227-237.
[9]	T. F. Ibrahim, “Periodicity and Analytic Solution of a Recursive Sequence with Numerical Examples,” Journal of Interdisciplinary Mathematics, Vol. 12, No. 5, 2009, pp. 701-708.
[10]	V. L. Kocic and G. Ladas, “Global Behavior of Nonlinear Difference Equations of Higher Order with Applications,” Kluwer Academic Publishers, Dordrecht, 1993.
[11]	M. R. S. Kulenovic and G. Ladas, “Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures,” Chapman & Hall/CRC Press, Boca Raton, 2001. doi:10.1201/9781420035384
[12]	G. ladas and M. Kulenovic, “On Period Two Solutions of ,” Journal of Difference Equations and Applications, Vol. 6, 2000, pp. 641-646.
[13]	D. Simsek, C. Cinar and I. Yalcinkaya, “On the Recursive Sequence ,” International Journal of Contemporary Mathematical Sciences, Vol. 1, No. 10, 2006, pp. 475-480.