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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 $$u^{(n)}(t)+p(t)u(\tau (t))=0,\tag 1$$ where $p:\bbfR_{+}\to\bbfR$ is a locally integrable function of constant sign and $\tau :\bbfR_{+}\to\bbfR$ is a continuous nondecreasing function such that $\tau (t)\to\infty$ as $t\to\infty$. Equation (1) is studied as an ordinary differential equation ($\tau (t)\equiv t$) as well as a functional-differential equation in both the delayed case ($\tau (t)\le t$) and the advanced case ($\tau (t)\ge t$). 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.

34K11Oscillation theory of functional-differential equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory