The Distribution of Zeros of Solutions of Differential Equations with a Variable Delay

Abstract

This paper is concerned with the distribution of zeros of solutions of the first order linear differential equations with a variable delay of the form

$$\begin{aligned} x'(t)+P(t)x\left( \tau (t)\right) =0 , \quad t\ge {t}_{0},\nonumber \end{aligned}$$

where P, $\tau \in C([{t}_{0},\;\infty ),[0,\;\infty ))$, $\tau (t)\le t$, $\tau (t)$ is nondecreasing, and $\lim \limits _{t\rightarrow +\infty }\tau (t)=+\infty$. By introducing a class of new series, we are able to derive sharper upper bounds on the distance between zeros of solutions of the above delay differential equations. Some examples and a table are given to support our accomplishment.