Neutral delay differential equations with positive and negative coefficients

We studied the asymptotic behavior and the oscillatory properties of solutions of the neutral delay differential equation (NDDE) ;We proved that every solution of Eq. (1) oscillates if and only if the characteristic equation ;Necessary and sufficient conditions for the oscillation of all solutions were also obtained for the neutral differential equation with mixed arguments;Finally we note that our methods allowed us to establish easily verifiable sufficient conditions for the oscillation of all solutions of Eq. (1) as well as for Eq. (2).;The proofs of our results make use of classical results from analysis and differential equations together with some techniques developed by Ladas and his coworkers.