Global Behavior of the Difference Equation xn+1=xn-1g(xn)‎

Abstract EN

We consider the following difference equation xn+1=xn-1g(xn), n=0,1,…, where initial values x-1,x0∈[0,+∞) and g:[0,+∞)→(0,1] is a strictly decreasing continuous surjective function.

We show the following.

(1) Every positive solution of this equation converges to a,0,a,0, …, or 0,a,0,a,… for some a∈[0,+∞).

(2) Assume a∈(0,+∞).

Then the set of initial conditions (x-1,x0)∈(0,+∞)×(0,+∞) such that the positive solutions of this equation converge to a,0,a,0,…, or 0,a,0,a,… is a unique strictly increasing continuous function or an empty set.