On the upper bounds for the distance between zeros of solutions of a first-order linear neutral differential equation with several delays

This work is devoted to studying the distribution of zeros of a first-order neutral differential equation with several delays

$${\left[y(t)+a(t)y\left(t-\sigma \right)\right]}^{\prime }+\sum _{j=1}^n{b}_j(t)y\left(t-{\mu }_j\right)=0,t\ge t_0.$$

New estimations for the upper bounds of the distance between successive zeros are obtained. The properties of a positive solution of a first-order differential inequality with several delays in a closed interval are studied, and many results are established. We apply these results to a first-order neutral differential equation with several delays and also to a first-order differential equation with several delays. Our results for the differential equation with several delays not only provide new estimations but also improve many previous ones. Also, the results are formulated in a general way such that they can be applied to any functional differential equation for which studying the distance between zeros is equivalent to studying this property for a first-order differential inequality with several delays. Further, new estimations of the upper bounds for certain equations are given. Finally, a comparison with all previous results is shown at the end of this paper.