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Properties $A$ and $B$ of $n$th-order linear differential equations with deviating argument. (English) Zbl 0939.34062

The authors investigate oscillatory properties of solutions to the $n$th-order linear differential equation

${u}^{\left(n\right)}\left(t\right)+p\left(t\right)u\left(\tau \left(t\right)\right)=0,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $p:{ℝ}_{+}\to ℝ$ is a locally integrable function of constant sign and $\tau :{ℝ}_{+}\to ℝ$ is a continuous nondecreasing function such that $\tau \left(t\right)\to \infty$ as $t\to \infty$. Equation (1) is studied as an ordinary differential equation ($\tau \left(t\right)\equiv t$) as well as a functional-differential equation in both the delayed case ($\tau \left(t\right)\le t$) and the advanced case ($\tau \left(t\right)\ge t$).

In a series of statements some sufficient conditions for equation (1) to have property $A$ or property $B$ are established. Moreover, comparisons with known results are given.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory