Unbounded Solutions of the Difference Equationxn=xn−lxn−k−1

Abstract

The following difference equationxn=xn−lxn−k−1,n∈ℕ0, wherek,l∈ℕ,k<l,gcd(k,l)=1, and the initial valuesx-l,…,x-2,x-1are real numbers, has been investigated so far only for some particular values ofkandl. To get any general result on the equation is turned out as a not so easy problem. In this paper, we give the first result on the behaviour of solutions of the difference equation of general character, by describing the long-term behavior of the solutions of the equation for all values of parameterskandl, where the initial values satisfy the following conditionmin{x-l,…,x-2,x-1}.