Global asymptotic stability and oscillation of a family of difference equations

We consider the family of difference equations of the form x(n+1) = (Sigma(inot equalj, j-1i=0)(k) x(n-i) +x(n-j+1)x(n-j)+1)/Sigma(i=0)(k)x(n-i), j=1,2,...,k, where n is an element of {0, 1,...}, k is an element of {1,2,...)and the initial values x(-k),x(-k+1),...,x(0) are positive real numbers. For these difference equations, we investigate the oscillatory behavior of the positive solutions and prove that the unique equilibrium (x) over bar = 1 is globally asymptotically stable. (C) 2004 Elsevier Inc. All fights reserved.