LAUSR

The main goal of this paper is to investigate the boundedness, invariant intervals, semi-cycles and global attractivity of all nonnegative solutions of the equation $$\begin{aligned} x_{n+1}=\frac{\beta x_{n}+\gamma x_{n-k}}{A+Bx_{n}+C x_{n-k}},\quad n\in \mathbb {N}_0 , \end{aligned}$$xn+1=βxn+γxn-kA+Bxn+Cxn-k,n∈N0,where the parameters $$\beta , \gamma , A, B$$β,γ,A,B and C and the initial conditions $$x_{-k},x_{-k+1},\ldots ,x_0$$x-k,x-k+1,…,x0 are non-negative real numbers, $$k=\{1,2,\ldots \}$$k={1,2,…}. We give a detailed description of the semi-cycles of solutions, and determine conditions that satisfy the global asymptotic stability of the equilibrium points.