The characteristics of a higher-order rational difference equation

In this paper, we study the global behavior of the higher order nonlinear difference equationxn+1=α+βxn-k+1+γxn-2k+1A+Bxn-k+1,n=0,1,…,where A, B and α, β, γ are positive, k ∈ {1, 2, 3, …}, and the initial conditions x−2k+1, … , x−1, x0 are positive real numbers. We investigate the character of semicycles, periodicity, global stability and boundedness of above mentioned equation.