I. P. Stavroulakis

Oscillations of Functional Differential Equations

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 $\US{N\TO\INFTY}{\LMS}\SUM\LIMITS_{I="n-k}^np_i<1$." FOR WHEN $\US{N\TO\!\INFTY}{\LMF}\SUM\LIMITS_{I="n\!-k}^{n-1}p_i\!\!\leq\!\!\Big(\frac{k}{k+1}\Big)^{k+1}$" P~\!(S\!) AND \LEQ\FRAC{1}{E}$ $\US{T\TO\!\INFTY}{\LMS}\INT_{T\!-\TAU}^TP(S)DS\!<1$, $(*)'$

Mathematics Subject Classification: 34k15, 34K25.

Key words and phrases: Delay differential equation, delay difference equation, oscillation.