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

Our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equationxn+1=α+βxn-k+1+γxn-2k+1Bxn-k+1+Cxn-2k+1,n=0,1,2,…where B, C 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 show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients. Our concentration is on invariant intervals, the character of semicycles, and the boundedness of the above mentioned equation. Our final comments are about informative examples.