Stability and periodic character of a third order difference equation

In this paper, we consider the third order difference equation yn+1=αyn−2β+γynkyn−1kyn−2k,n=0,1,2,… where the initial conditions y−2,y−1,y0y−2,y−1,y0 and the parameters α,βα,β and γγ are positive real numbers and k≥2k≥2 is a fixed integer. We investigate the stability, the periodic character and the boundedness nature of solutions of the above mentioned difference equation.