Asymptotic behavior of solutions of nonlinear neutral delay differential equations

Abstract

This paper is concerned with the nonlinear neutral delay differential equation with positive and negative coefficients $$ [x(t)-c(t)x(t-\tau)]'+p(t)f(x(t-\delta))-q(t)f(x(t-\sigma))=0,\,\ t\geq t_0, $$ where $\tau\in(0,\infty)$, $\delta$ and $\sigma \in[0,\infty)$, $c(t)\in C([t_0,\infty), R)$, $p(t$) and $q(t)\in C([t_0,\infty), [0,\infty))$, $f\in C(R,R)$. Sufficient conditions are obtained under which every solution of the above equation is bounded and tends to a constant as $t\to\infty$. Our results extend and improve some known results.