# Neutral delay differential equations with positive and negative coefficients

#### Abstract

We studied the asymptotic behavior and the oscillatory properties of solutions of the neutral delay differential equation (NDDE) $$\rm{d\over dt} \lbrack y(t) + py(t - \tau)\rbrack + q\sb1y(t - \sigma\sb1) + q\sb2y(t - \sigma\sb2) = 0\eqno(1)$$where the coefficients and delays are such that p, q$\sb1$, q$\sb2$, $\tau \in$ $\IR$ and $\sigma\sb1$, $\sigma\sb2$ $\in$ (0,$\sp{\infty}$).^ We proved that every solution of Eq. (1) oscillates if and only if the characteristic equation $\lambda$ + $\lambda$pe$\sp{-\lambda\tau}$ + q$\sb1$e$\sp{-\lambda\sigma\sb1}$ + q$\sb2$e$\sp{-\lambda\sigma\sb2}$ = 0 has no real roots. Furthermore we proved that every bounded solution of Eq. (1) oscillates if and only if the characteristic equation has no roots in $(-\infty{,}0\rbrack.$^ Necessary and sufficient conditions for the oscillation of all solutions were also obtained for the neutral differential equation with mixed arguments$$\rm{d\over dt}\lbrack y(t) + py(t - \tau)\rbrack + q\sb1y(t - \sigma\sb1) + q\sb2y(t + \sigma\sb2) = 0\eqno(2)$$where p, q$\sb1$, q$\sb2$, $\tau$, $\sigma\sb1$, $\sigma\sb2 \in$ (0,$\infty$).^ 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. ^

#### Subject Area

Mathematics

#### Recommended Citation

Kevin J Farrell,
"Neutral delay differential equations with positive and negative coefficients"
(1988).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI8913020.

http://digitalcommons.uri.edu/dissertations/AAI8913020