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Oscillation theorems for second order advanced neutral differential equations. (English) Zbl 1265.34232
The differential equation $$ \biggl (r(t)\Bigl [x(t)+p(t)x\bigl (\tau (t)\bigr)\Bigr ]'\biggr)^{\hskip -2pt{\prime }}+q(t)x\bigl (\sigma (t)\bigr)=0 \tag 1$$ is considered, where $q: [t_0,+\infty)\to (0,+\infty)$ is a continuous function, and $r$, $p$, $\tau $, $\sigma : [t_0,+\infty)\to (0,+\infty)$ are continuously differentiable functions such that $$ \tau '(t)\geq \tau _0 >0, \ \sigma '(t)\geq 0, \ \sigma (t)\geq t, \ \sigma \bigl (\tau (t)\bigr)=\tau \bigl (\sigma (t)\bigr), $$ $$ 0\leq p(t)\leq p_0 <+\infty , \ R(t)=\int \limits _{t_0}^t \frac {{\operatorname {d}}s}{r(s)}\to +\infty \quad \text{as } t\to +\infty, $$ $$ \int \limits _{t_0}^{+\infty }Q(t)\,{\operatorname {d}}t<+\infty , \quad\text{where } Q(t)=\min \Bigl \{q(t),q\bigl (\tau (t)\bigr)\Bigr \}. $$ It is proved that, if $\tau (t)\geq t$ and if one of the differential inequalities $$ w'(t)-\frac {\tau _0}{\tau _0+p_0}Q_1(t)w\bigl (\sigma (t)\bigr)\geq 0 $$ and $$ w'(t)-\frac {\tau _0}{\tau _0+p_0}Q_2(t)w\bigl (\sigma (t)\bigr)\geq 0,$$ where $$ Q_1(t)=\frac {1}{r(t)}\int \limits _t^{+\infty }Q(s)\, {\operatorname {d}}s, \;\;\; Q_2(t)=Q(t)\bigl (R(t)-R(t_1)\bigr), \qquad t_1\geq t_0, $$ has no positive solution, then every solution of (1) is oscillatory. An analogous result is obtained for the case when $\tau (t)\leq t$.
Reviewer: Nino Partsvania (Tbilisi)

34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations