On the rational recursive sequence $ x_{n+1}=\dfrac {\alpha_0x_n+\alpha_1x_{n-l}+\alpha _2x_{n-k}} {\beta_0x_n+\beta_1x_{n-l}+\beta_2x_{n-k}}$

Abstract

The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation $$ x_{n+1}=\frac {\alpha _0x_n+\alpha _1x_{n-l}+\alpha _2x_{n-k}} {\beta _0x_n+\beta _1x_{n-l}+\beta _2x_{n-k}}, \quad n=0,1,2,\dots $$ where the coefficients $\alpha _i,\beta _i\in (0,\infty )$ for $ i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $ x_{-k}, \dots , x_{-l}, \dots , x_{-1}, x_0 $ are arbitrary positive real numbers such that $l<k$. Some numerical experiments are presented.