Unbounded solutions of the difference equation xn = x n-Lx-k-1

Abstract

The following difference equation xn = x n - l x n - k- 1, n € ℕ0, where k, l ,€ ℕk < l, gcd (k, l) = 1, and the initial values x - l, , x - 2, x - 1 are real numbers, has been investigated so far only for some particular values of k and l. 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 parameters k and l, where the initial values satisfy the following condition min { x - l, , x - 2, x - 1 }. Copyright