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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.

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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