Global stability of a difference equation with maximum

Abstract

We prove that every positive solution to the difference equationxn = max fenced(frac(A1, xn - 1α1), frac(A2, xn - 2α2), ..., frac(Ak, xn - kαk)), n ∈ N0,where k ∈ N, Ai > 0, αi ∈ (0, 1), i = 1, ..., k, converges to the following quantity max fenced(A1frac(1, α1 + 1), ..., Akfrac(1, αk + 1)), confirming a quite recent conjecture of interest. We also prove another result on global convergence which concerns some cases when not all αi, i = 1, ..., k belong to the interval (0, 1).