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On oscillation of differential and difference equations with non-monotone delays. (English) Zbl 1256.39013
The authors study the delay differential equation $x'(t)+ p(t) x(h(t))= 0,\qquad t\geq 0,\tag{1}$ with $\begin{gathered} (\forall t\geq 0)((p(t)\geq 0)\wedge (h(t)\leq t)),\\ \lim_{t\to+\infty} h(t)=+\infty,\qquad \liminf_{t\to+\infty}\, \int^t_{h(t)} p(u)\,du> e^{-1},\end{gathered}$ and the difference equation $\Delta x(n)+ p(n) x(h(n))= 0,\qquad n\geq 0,\tag{2}$
$\begin{gathered} (\forall n\geq 0)((\Delta x(n)= x(n+1)- x(n))\wedge (p(n)\geq 0)\wedge (h(n)\leq n)),\\ \lim_{n\to+\infty}h(n)= +\infty.\end{gathered}$ The inequalities $\limsup_{t\to+\infty}\, \int^t_{h(t)} p(u)\,du> 1,\qquad\limsup_{n\to+\infty}\, \sum^n_{j= h(n)} p(j)> 1$ do not imply oscillations of (1) and (2).
The authors prove that there are not constants $$A$$ and $$B$$ such that $\limsup_{t\to+\infty}\, \int^t_{h(t)} p(u)\,du> A,\qquad \limsup_{n\to+\infty}\, \sum^n_{j= h(n)} p(j)> B$ imply oscillation of (1) and (2).
The authors present new sufficient oscillation conditions.

##### MSC:
 39A21 Oscillation theory for difference equations 39A10 Additive difference equations 39A12 Discrete version of topics in analysis 34K11 Oscillation theory of functional-differential equations
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