The qualitative behavior of solutions of a nonlinear difference equation

This paper is concerned with the qualitative behavior of solutions to the difference equationxn+1=p+qxn-k1+xn,n=0,1,2,…,where the initial conditions x−k, …, x−1, x0 are non-negative, k ∈ {1, 2, 3, …}, and the parameters p, q are non-negative. We start by establishing the periodicity, the character of semicycles, the global stability, and the boundedness of the above mentioned equation. We also present solutions that have unbounded behavior. It is worth to mention that this difference equation is a special case of an open problem was introduced by M.R.S. Kulenovic and G. Ladas [Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002]. Several computational examples are given to support our theoretical discussions. The presented numerical tests represent different types of qualitative behavior of solutions to our nonlinear difference equation.