Asymptotic behavior of solutions to neutral differential equations with positive and negative coefficients (Q2765011)

Neutral delay differential equations of the form NEWLINE\[NEWLINE [x(t)+\lambda c(t)x(t-\sigma)]'+p(t)x(t-\tau)-Q(t)x(t-\delta)=0 NEWLINE\]NEWLINE are considered when \(t\to \infty\), under the main conditions: \(\lambda\in\{-1,1\}\), \(\sigma>0\), \(\tau, \delta \geq 0\), \(c,p,Q\in C([t_0,\infty),\mathbb{R}^+)\), and if there exists a constant \(A>0\) such that \(Q(t+\delta-\tau)\leq Ap^*(t)\) with \(p^*(t)=p(t)-Q(t+\delta-\tau)\) for \(t\geq\max\{t_0,t_0+\tau-\delta\}\). Sufficient conditions are given for the validity of the statements: all oscillatory solutions are vanishing and all solutions are vanishing. Illustrative examples are considered, too.