Global behavior of the nonlinear difference equation xn+1 = f (xn−s, xn−t ) ✩

In this note we consider a nonlinear difference equation of the form xn+1 = f (xn−s, xn−t ), n = 0, 1, . . . , under some certain assumptions, where s, t ∈ {0, 1, 2, . . .} with s < t and the initial values x−t , x−t+1, . . . , x0 ∈ (0,+∞). We prove that the length of its finite semicycle is less than or equal to t and give sufficient conditions under which every positive solution of this equation converges to the positive equilibrium. Some known results are included and improved.  2005 Elsevier Inc. All rights reserved.