On the global behavior of a high-order rational difference equation

In this paper, we consider the (k+1)(k+1)-order rational difference equationyn+1=p+qyn+ryn−k1+yn−k,n=0,1,2,… where k∈{1,2,3,…}k∈{1,2,3,…}, and the initial conditions y−k,…,y−1,y0y−k,…,y−1,y0 and the parameters p, q and r are non-negative. We investigate the global stability, the periodic character and the boundedness nature of solutions of the above mentioned difference equation. In particular, our results solve the open problem introduced by Kulenovic and Ladas in their monograph [Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002].