On the recursive sequence x(n+1) = (alpha - beta x(n-k))/g(x(n), x(n-1), ... ,x(n-k+1))
This paper is devoted to investigating the asymptotic behavior of the recursive sequence x(n+1) = alpha - beta x(n-k)/(x(n), x(n-1), ... ,x(n-k+1)), n = 0, 1, ... where alpha >= 0 beta > 0 and g is continuous on R-k. We show that under certain conditions this equation has a unique positive (negative) equilibrium point which is a global attractor with some basin S subset of Rk+1. Also we establish the oscillation of all solutions with initial conditions {x(-i)}(i=0)(k) such that (x(0), x(-1),..., x(-k)) is an element of S. We apply these results to the recursive sequence x(n+1) = alpha - beta x(n-k)/gamma +(i=0)Sigma(k-1)(a(i)x(n-i)+/-b(i)x(n-i)(2)), n = 0, 1, ... where alpha, gamma, a(i), b(i) >= 0, i = 0, ... , k - 1, and beta > 0. (C) 2010 Elsevier Ltd. All rights reserved.
difference equations stability attractivity