### Differential equations with several deviating arguments: Sturmian comparison method in oscillation theory. II.

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We consider preservation of exponential stability for the scalar nonoscillatory linear equation with several delays $$\dot{x}\left(t\right)+\sum _{k=1}^{m}{a}_{k}\left(t\right)x\left({h}_{k}\left(t\right)\right)=0,\phantom{\rule{1.0em}{0ex}}{a}_{k}\left(t\right)\ge 0$$ under the addition of new terms and a delay perturbation. We assume that the original equation has a positive fundamental function; our method is based on Bohl-Perron type theorems. Explicit stability conditions are obtained.

The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type $$\Delta x\left(n\right)+\sum _{k=-p}^{q}{a}_{k}\left(n\right)x(n+k)=0,\phantom{\rule{1.0em}{0ex}}n>{n}_{0},$$ where $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ is the difference operator and $\left\{{a}_{k}\left(n\right)\right\}$ are sequences of real numbers for $k=-p,...,q$, and $p>0$, $q\ge 0$. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.

We present a review of known stability tests and new explicit exponential stability conditions for the linear scalar neutral equation with two delays $$\dot{x}\left(t\right)-a\left(t\right)\dot{x}\left(g\left(t\right)\right)+b\left(t\right)x\left(h\left(t\right)\right)=0,$$ where $$\left|a\right(t\left)\right|<1,\phantom{\rule{1.0em}{0ex}}b\left(t\right)\ge 0,\phantom{\rule{1.0em}{0ex}}h\left(t\right)\le t,\phantom{\rule{1.0em}{0ex}}g\left(t\right)\le t,$$ and for its generalizations, including equations with more than two delays, integro-differential equations and equations with a distributed delay.

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