Dynamics of a Rational Difference Equation

The main goal of the paper is to investigate boundedness, invariant intervals, semicycles, and global attractivity of all nonnegative solutions of the equation x(n+1) = (alpha + beta x(n) + gamma x(n-k))/(1 _ x(n-k)), n is an element of N-0, where the parameters alpha, beta, gamma is an element of [0, infinity), k >= 2 is an integer, and the initial conditions x(-k), ... , x(0) is an element of [0, infinity). It is shown that the unique positive equilibrium of the equation is globally asymptotically stable under the condition beta <= 1. The result partially solves the open problem proposed by Kulenovic and Ladas in work (2002).