Dynamics of a higher-order rational difference equation

Our aim in this paper is to investigate the dynamics of difference equation,yn+1=p+qyn1+yn-k,n=0,1,2,…where the initial conditions y−k, … , y−1, y0 are non-negative, k ∈ {1, 2, 3, …}, and the parameters p and q are non-negative. We study characteristics such as the global stability, the boundedness of positive solutions and the character of semicycles of the above mentioned difference equation. In particular, our results solve the open problem introduced by Kulenovic and Ladas in their monograph [M.R.S. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002].