Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients. (Q1874596)

The authors establish sufficient conditions under which every solution of the neutral differential equation \[ [x(t)-P(t)x(t-\tau)]'+Q_{1}(t)x(t-\sigma_{1})-Q_{2}(t)x(t-\sigma_{2})=0, \quad t\geq t_{0}, \] with \(\tau,\sigma_{1},\sigma_{2} \in (0,\infty)\) and \(P,Q_{i}\in C([0,\infty):\mathbb{R})\), tends to zero as \(t\to \infty\). The results in this work relaxes the restrictions on the coefficients and delays in \textit{J. H. Shen} and \textit{J. S. Yu, } [J. Math. Anal. Appl. 195, No. 2, 517--526 (1995; Zbl 0844.34078)].