The positive solutions of nonautonomous hyperlogistic delay difference equations

Consider the nonautonomous hyperlogistic delay difference equation Δxn = pnxn (1 − xnkn)r, N = 0,1,2,…, where {pn} is a sequence of positive real numbers, {kn} a sequence of nonnegative integers such that {n-kn} is nondecreasing, and r a ratio of two odd integers. Our main results given sufficient conditions that guarantee every solution to be positive. The conditions under which, for all n ≥ 1, either xn > 1 or 0 < xn < 1 are given, respectively. For the asymptotic behavior, we give sufficient conditions that guarantee every positive solution to converge to the equilibrium x = 1 of the model, or to oscillate about 1. The results improve and generalize some recent results established by Chen and Yu [1] and Zhou and Zhang [2]. Some remarks and examples illustrate our theorems. The methods used here are different from those in [3].