On a recursive sequence

The aim in this paper is to investigate the dynamics of difference equation y_n+1=(py_n+y_n−k)/(qy_n+y_n−k), n=0,1,2,… where k∈{1,2,3,…}, the initial conditions y_−k, … ,y₋₁,y₀ and the parameters p and q are non‐negative.

The paper studies characteristics such as the character of semicycles, periodicity and the global stability of the above mentioned difference equation.

In particular, the 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.

The global behaviour of the solutions of equation y_n+1=(py_n+y_n−k)/(qy_n+y_n−k), n=0,1,2,… were investigated providing valuable conclusions on practical data.