ON THE GLOBAL ASYMPTOTIC STABILITY OF THE DIFFERENCE EQUATION x_(n+1)= x_(n-1)x_(x-3)+x_(n-1)~2+a/x_(n-1)~2x_(n-3)+x_(n-1)+a

摘要

We investigate the dynamical behavior of the following fourth-order rational difference equation x_(n+1)=x_(n-1)x_(n-3)+x_(n-1)~2+a/x_(n-1)~2x_(n-3)+x_(n-1)+a,n=0,1,2... where α ∈ [0,∞) and the initial values x_(-3), x_(-2), x_(-1), x_0∈ (0, ∞). We find that the successive lengths of positive and negative semicycles of nontrivial solutions of the above equation occur periodically. We also show that the positive equilibrium of the equation is globally asymptotically stable.