Invariant intervals and global behavior of a higher order difference equation

In this paper, we study the boundedness, the invariant intervals,the periodic character and the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation yn+1=p+qyn−k1+yn−k+ryn−k−1,n=0,1,…, where the parameters p,q,r∈(0,∞)p,q,r∈(0,∞),k∈{0,1,2,…}k∈{0,1,2,…}, and the initial conditions y−k−1y−k−1,y−k,…y0∈[0,∞)y−k,…y0∈[0,∞). We show that the unique positive equilibrium of the above equation is globally asymptotically stable under certain conditions.