Dynamics of the Rational Difference Equation

Authors

M. A. El-Moneam, Mathematics Department, Faculty of Science and Arts in Farasan, Jazan University, Kingdom of Saudi ArabiaFollow
E. M. E. Zayed, Mathematics Department, Faculty of Science, Zagazig University, Zagazig, EgyptFollow

Abstract

In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation xn+1 = Axn +Bxn−k +Cxn−l +Dxn−s + bxn−k +hxn−l dxn−k +exn−l , n = 0,1,2, ....., where the coefficients A,B,C,D,b,d, e,h ∈ (0,¥), while k, l and s are positive integers. The initial conditions x−s ,..., x−l ,..., x−k, ..., x−1, x0 are arbitrary positive real numbers such that k < l < s . We will prove that the equilibrium points of this equation are locally asymptotically stable, global attractor and hence they are global stability. This equation will have ( or have not ) prime period two solution under suitable conditions on these coefficients. The solutions of this equation will be proved to be bounded. Some numerical examples will be given to illustrate our results.

Recommended Citation

A. El-Moneam, M. and M. E. Zayed, E. (2014) "Dynamics of the Rational Difference Equation," Information Sciences Letters: Vol. 3 : Iss. 2 , PP -.
Available at: https://digitalcommons.aaru.edu.jo/isl/vol3/iss2/2