Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form xn+1 = k i=0 i =j,j−1 xn−i + xn−j+1xn−j + 1 k i=0 xn−i, j= 1, 2, . . . , k, where n ∈ {0, 1, . . .}, k ∈ {1, 2, . . .} and the initial values x−k,x−k+1, . . . , x0 are positive real numbers. For these difference equations, we investigate the oscillatory behavior of the positive solutions and prove that the unique equilibrium ¯x = 1 is globally asymptotically stable.  2004 Elsevier Inc. All rights reserved