Dichotomy of a perturbed Lyness difference equation

Abstract

We investigate in this paper the perturbed Lyness difference equation bxn+2xn=α+βxn+1+γxn2,n=0,1,2,…, where α,β,b are arbitrary positive real numbers and γ∈[0,∞) and the initial values x1,x0>0, which is a generalization of the Lyness difference equation xn+2xn=a+xn+1 extensively studied. It is known that for the Lyness difference equation, i.e., the perturbed Lyness difference equation with γ=0, all its solutions are periodic or strictly oscillatory. However, one here finds that this perturbed Lyness difference equation possesses the following dichotomy: for 0<γ<b, all of its solutions are globally asymptotically stable; for γ⩾b, all the sequences generated by it converge to +∞. We hence find that there exists the essential difference for the properties of solutions between the unperturbed Lyness difference equation and the perturbed Lyness difference equation.