Global behavior of the difference equation xn+1=xn-l+11+a0xn+a1xn-1+⋯+alxn-l+xn-l+1

In this paper, we find the asymptotic behavior of solutions of the third order difference equationxn+1=xn-21+pxn+qxn-1+xn-2,n=0,1,2,…for all admissible non-negative values of the parameters p, q where the initial conditions x−2, x−1, x0 are positive. We show that the solutions do not exhibit a periodic attitude for all parameters of the above mentioned difference equation. It is worth to mention that this difference equation was an open problem introduced by Kulenovic and Ladas. Note that we also generalize and extend the above mentioned equation and we investigate the same arguments as the third difference equation for the zero equilibrium point of the higher case.