I. P. Stavroulakis
abstract:
Consider the first order delay differential equation
\begin{equation}
x'(t)+p(t)x(t-\tau)=0,\;\;\;\tau>0,\;\;\;t\geq t_0,\tag{$*$}
\end{equation}
and its discrete analogue
\begin{equation}
x_{n+1}-x_n+p_nx_{n-k}=0,\;\;\;k\in Z^+,\;\;\;n=0,1,2,\dots.\tag*{$(*)'$}
\end{equation}
Oscillation criteria are established for $(*)$ in the case where
$0\!<\!\us{t\to\!\infty}{\lmf}\!\int_{t\!-\tau}^t p~\!(s\!)
\leq\frac{1}{e}$ and
$\us{t\to\!\infty}{\lms}\int_{t\!-\tau}^tp(s)ds\!<1$,
and for $(*)'$ when
$\us{n\to\!\infty}{\lmf}\sum\limits_{i=n\!-k}^{n-1}p_i\!\!\leq\!\!\Big(\frac{k}{k+1}\Big)^{k+1}$
and $\us{n\to\infty}{\lms}\sum\limits_{i=n-k}^np_i<1$.
Mathematics Subject Classification: 34k15, 34K25.
Key words and phrases: Delay differential equation, delay difference equation, oscillation.