### Applications of Langenhop inequality to difference equations: Lower bounds and oscillation.

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The paper is concerned with oscillation properties of $n$-th order neutral differential equations of the form $${[x\left(t\right)+cx\left(\tau \left(t\right)\right)]}^{\left(n\right)}+q\left(t\right)f\left(x\left(\sigma \right(t\left)\right)\right)=0,\phantom{\rule{1.0em}{0ex}}t\ge {t}_{0}>0,$$ where $c$ is a real number with $\left|c\right|\ne 1$, $q\in C([{t}_{0},\infty ),\mathbb{R})$, $f\in C(\mathbb{R},\mathbb{R})$, $\tau ,\sigma \in C([{t}_{0},\infty ),{\mathbb{R}}_{+})$ with $\tau \left(t\right)<t$ and ${lim}_{t\to \infty}\tau \left(t\right)={lim}_{t\to \infty}\sigma \left(t\right)=\infty $. Sufficient conditions are established for the existence of positive solutions and for oscillation of bounded solutions of the above equation. Combination of these conditions provides necessary and sufficient conditions for oscillation of bounded solutions of the equation. Furthermore, the results are generalized to equations in which...

One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor's Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality...

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