The positive solutions of nonautonomous hyperlogistic delay difference equations
Consider the nonautonomous hyperlogistic delay difference equation Deltax(n) = p(n)x(n) (1 - x(n-kn))(r), n = 0, 1, 2, ..., where {p(n)} is a sequence of positive real numbers, {kn} a sequence of nonnegative integers such that {n - k(n)} is nondecreasing, and r a ratio of two odd integers. Our main results give sufficient conditions that guarantee every solution to be positive. The conditions under which, for all n greater than or equal to 1, either x(n) > I or 0 < x(n) < I 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]. (C) 2004 Elsevier Ltd. All rights reserved.
positive solution hyperlogistic delay difference equation oscillation global attractivity