On the oscillation of the solutions to delay and difference equations

Abstract

Consider the first-order linear delay differential equation
\begin{equation*}
x^{\prime }(t)+p(t)x(\tau (t))=0,\;\;\;t\geq t_{0},\eqno(1)
\end{equation*}%
where $p,\tau \in C([t_{0},\infty )$, $\mathbb{R}^{+}$), $\tau (t)$ is
nondecreasing, $\tau (t)}\tau (t)=\infty $, and the (discrete analogue) difference equation
\begin{equation*}
\Delta x(n)+p(n)x(\tau (n))=0\text{, \ \ }n=0,1,2,...\text{,}\eqno%
(1)^{\prime }
\end{equation*}%
where $\Delta x(n)=x(n+1)-x(n),$ $p(n)$ is a sequence of nonnegative real
numbers and $\tau (n)$ is a nondecreasing sequence of integers such that $%
\tau (n)\leq n-1$ for all $n\geq 0$ and $\lim_{n\rightarrow \infty }\tau
(n)=\infty .$ Optimal conditions for the oscillation of all solutions to the
above equations are presented.