Dynamics of a non-linear difference equation

Our aim in this paper is to investigate the dynamics of difference equationyn+1=yn+pyn-kyn+q,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 periodicity, invariant interval, the character of semicycles, the global stability, and the boundedness of positive 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].