Oscillation criteria for differential equations with several retarded arguments

Abstract

Consider the first-order linear differential equation with several retarded arguments $$ x^{\prime}(t)+\sum\limits_{i=1}^{m}p_{i}(t)x(\tau_{i}(t))=0,\;\;\;t\geq t_{0}, $$ where the functions $p_{i},\tau_{i}\in C([t_{0,}\infty),\mathbb{R}^{+}),$ for every $i=1,2,\ldots,m,$ $\tau_{i}(t)\leq t$ \ for $t\geq t_{0}$ and $% \lim_{t\rightarrow \infty}\tau_{i}(t)=\infty $. The state of the art on the oscillation of all solutions to these equations are established especially in the case of non-monotone arguments. Examples illustrating the results are given.