Asymptotic behavior of solutions of a first-order impulsive neutral differential equation in Euler form (Q533484)

Let \(0<\alpha \), \(\beta >1\), \(b_{k}\), \(k=1,2,\dots,\) are constants, \(0<t_{0}<t_{k}<t_{k+1}\) with \(t_{k}\rightarrow \infty \) as \(t\rightarrow \infty \). Also, assume \(P(t) \geq 0\) and \(Q, P\in PC([t_{0},\infty ),{\mathbb R})\), where \(PC([t_{0},\infty ),{\mathbb R})\) denotes the set of all functions \(g: [t_{0},\infty )\rightarrow {\mathbb R}\) such that \(g\) is continuous for \(t_{k}\leq t<t_{k+1}\) and \(g(t)\rightarrow g(t^{-}_{k})\), as \(t\rightarrow t^{-}_{k}\). The following neutral differential equation with impulsive perturbation is studied: \[ \begin{aligned} & \frac{d}{dt}\;[x(t)-Q(t)x(\alpha t)]+ \frac{P(t)}{t}\;x(\beta t) = 0 , \quad t\geq t_{0}>0 , \; t\neq t_{k} \\ & x(t_{k}) = b_{k}x(t^{-}_{k})+(1-b_{k}) \int^{t_{k}}_{\beta t_{k}} \frac{P(s/ \beta )}{t}\;x(s)\, ds ,\quad k = 1,2,\dots . \end{aligned} \] Sufficient conditions are obtained for every solution of the considered equation to tend to a constant as \(t\rightarrow \infty \).