Global stability of a max-type difference equation

Abstract

We show that every positive solution to the difference equationxn = max fenced(frac(A1, xn - p1α1), frac(A2, xn - p2α2), ..., frac(Ak, xn - pkαk)), n ∈ N0,where pi, i = 1, ..., k are natural numbers such that 1 ≤ p1 < ⋯ < pk, k ∈ N, Ai > 0, αi ∈ (- 1, 1), i = 1, ..., k, converges to max1 ≤ i ≤ k fenced(Aifrac(1, αi + 1)). This result improves and complements the main result in our recent note: S. Stević, Global stability of a difference equation with maximum, Appl. Math. Comput. 210 (2009) 525-529, since it also considers the case when αi ∈ (- 1, 0], i{dotless} = 1, ..., k.