On the recursive sequence xn+1=α+βxn-k+1A+Bxn-k+1+Cxn-2k+1

In this paper, we study the global asymptotic stability, invariant intervals, the character of semicycles, and the boundedness of all positive solutions of the higher order nonlinear difference equationxn+1=α+βxn-k+1A+Bxn-k+1+Cxn-2k+1,n=0,1,…,where A, B, C and α, β are positive, k ∈ {1, 2, 3, … }, and the initial conditions x−2k+1, … , x−1, x0 are positive real numbers. We show that the unique positive equilibrium of the equation is globally asymptotically stable. Finally, we present some informative examples.