Long-term Behavior of a Cyclic Max-type System of Difference Equations

We study the long-term behavior of positive solutions of the cyclic system of difference equations x (i) n+1 = max n α, (x (i+1) n ) p (x (i+2) n−1 ) q o , i = 1, . . . , k, n ∈ N0, where k ∈ N, min{α, p, q} > 0 and where we regard that x1 n = x (i2) n when i1 ≡ i2 (mod k). We determine the set of parameters α, p and q in (0,∞)3 for which all such solutions are bounded. In the other cases we show that the system has unbounded solutions. For the case p = q we give some sufficient conditions which guaranty the convergence of all positive solutions. The main results in this paper generalize and complement some recent ones.