On oscillation of differential and difference equations with non-monotone delays (Q426440)

The authors study the delay differential equation NEWLINE\[NEWLINEx'(t)+ p(t) x(h(t))= 0,\qquad t\geq 0,\tag{1}NEWLINE\]NEWLINE with NEWLINE\[NEWLINE\begin{gathered} (\forall t\geq 0)((p(t)\geq 0)\wedge (h(t)\leq t)),\\ \lim_{t\to+\infty} h(t)=+\infty,\qquad \liminf_{t\to+\infty}\, \int^t_{h(t)} p(u)\,du> e^{-1},\end{gathered}NEWLINE\]NEWLINE and the difference equation NEWLINE\[NEWLINE\Delta x(n)+ p(n) x(h(n))= 0,\qquad n\geq 0,\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\begin{gathered} (\forall n\geq 0)((\Delta x(n)= x(n+1)- x(n))\wedge (p(n)\geq 0)\wedge (h(n)\leq n)),\\ \lim_{n\to+\infty}h(n)= +\infty.\end{gathered}NEWLINE\]NEWLINE The inequalities NEWLINE\[NEWLINE\limsup_{t\to+\infty}\, \int^t_{h(t)} p(u)\,du> 1,\qquad\limsup_{n\to+\infty}\, \sum^n_{j= h(n)} p(j)> 1NEWLINE\]NEWLINE do not imply oscillations of (1) and (2).NEWLINENEWLINE The authors prove that there are not constants \(A\) and \(B\) such that NEWLINE\[NEWLINE\limsup_{t\to+\infty}\, \int^t_{h(t)} p(u)\,du> A,\qquad \limsup_{n\to+\infty}\, \sum^n_{j= h(n)} p(j)> BNEWLINE\]NEWLINE imply oscillation of (1) and (2).NEWLINENEWLINE The authors present new sufficient oscillation conditions.