Asymptotic behavior of impulsive neutral delay differential equations with positive and negative coefficients of Euler form

In this paper, we are concerned with asymptotic properties of solutions for a class of neutral delay differential equations with forced term, positive and negative coefficients of Euler form, and constant impulsive jumps of the form {[x(t)−C(t)g(x(τ(t)))]′+P(t)tf(x(αt))−Q(t)tf(x(βt))=h(t),t≥t0>0,t≠tk,x(tk+)−x(tk)=αk,k∈Z+.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} [x(t)-C(t)g(x(\tau(t)))]'+ \frac{P(t)}{t}f(x(\alpha t))-\frac {Q(t)}{t}f(x(\beta t))=h(t),\quad t\geq t_{0}>0, t\neq t_{k},\\ x(t_{k}^{+})-x(t_{k})=\alpha_{k},\quad k\in{\mathbb{Z}_{+}}. \end{cases} $$\end{document} By constructing auxiliary functions and applying the technique of considering asymptotic properties of nonoscillatory and oscillatory solutions we establish some sufficient conditions to guarantee that every solution of the system tends to zero as t→+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\to+\infty$\end{document}.