On the Rational Recursive Sequence $x_{n+1}=Ax_{n}+Bx_{n-k}+\frac{\beta x_{n}+\gamma x_{n-k}}{Cx_{n}+Dx_{n-k}}$

Abstract

In this article, we study the global and asymptotic properties of the solutions of the difference equation $$x_{n+1}=Ax_{n}+Bx_{n-k}+(\beta x_{n}+\gamma x_{n-k})/(Cx_{n}+Dx_{n-k}),\quad n=0,1,2,\ldots,$$ where the initial conditions x−k,…,x−1,x0 are arbitrary positive real numbers and the coefficients A,B,C,D,β and γ are positive constants, while k is a positive integer number. Some numerical examples will be given to illustrate our results.