M. K. Grammatikopoulos,  R. Koplatadze,  I. P. Stavroulakis

On the Oscillation of Solutions of First Order Differential
Equations with Retarded Arguments

abstract:
For the differential equation
$$ u^{\prime}(t)+\sum_{i=1}^{m}p_{i}(t)u(\tau_{i}(t))=0, $$
where $p_{i}\in L_{loc}(R_{+};R_{+})$, $\tau_{i}\in C(R_{+};R_{+})$, $\tau_{i}(t)\leq t $
for $t\in R^{+},$ $\lim\limits_{t\rightarrow+\infty}\tau_{i}(t)=+\infty$ $(i=1,\dots,m),$
optimal integral conditions for the oscillation of all solutions are established.