Necessary and sufficient condition for oscillation of nonlinear neutral first-order differential equations with several delays

Abstract

In this work, necessary and sufficient conditions for oscillations of the solutions of a class of nonlinear neutral first-order differential equations with several delays of the form $$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}[x(t)+r(t)x(t-\tau )]+ \sum _{i=1}^m \phi _i(t)H(x(t-\sigma _i))=0, \end{aligned}$$ are established under various ranges of r(t). Our main tools are Knaster–Tarski fixed point theorem and Banach’s fixed point theorem. Finally, some illustrating examples are presented to show that feasibility and effectiveness of main results.