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Interval oscillation of a general class of second-order nonlinear differential equations with nonlinear damping. (English) Zbl 1064.34021

The authors are concerned with the oscillatory behavior of the second-order nonlinear differential equation with a nonlinear damping term

${\left[r\left(t\right){k}_{1}\left(x,{x}^{\text{'}}\right)\right]}^{\text{'}}+p\left(t\right){k}_{2}\left(x,{x}^{\text{'}}\right){x}^{\text{'}}+q\left(t\right)f\left(x\right)=0,\phantom{\rule{2.em}{0ex}}t\ge {t}_{0}\ge 0,\phantom{\rule{2.em}{0ex}}\left(\mathrm{b}\right)$

with $p,q:\left[{t}_{0},\infty \right)\to ℝ,$ $r:\left[{t}_{0},\infty \right)\to \left(0,\infty \right),$ $f:ℝ\to ℝ,$ ${k}_{1},{k}_{2}:{ℝ}^{2}\to ℝ·$ It is also assumed that

${k}_{1}^{2}\left(u,v\right)\le {\alpha }_{1}{k}_{1}\left(u,v\right),$

for some ${\alpha }_{1}>0$ and for all $\left(u,v\right)\in {ℝ}^{2}·$ Two cases are considered:

(a) $f\left(x\right)$ is differentiable, $xf\left(x\right)\ne 0$ and ${f}^{\text{'}}\left(x\right)\ge {\mu }_{1}$ for some ${\mu }_{1}>0$ and all $x\ne 0,$ and

$vf\left(u\right){k}_{2}\left(u,v\right)\ge {\alpha }_{2}{k}_{1}^{2}\left(u,v\right)\phantom{\rule{2.em}{0ex}}\left(\mathrm{b}1\right)$

for some ${\alpha }_{2}>0$ and for all $\left(u,v\right)\in {ℝ}^{2};$

(b) $f\left(x\right)$ is not necessarily differentiable, $f\left(x\right)/x\ge {\mu }_{2}$ for some ${\mu }_{2}>0$ and all $x\ne 0,$ and

$vu{k}_{2}\left(u,v\right)\ge {\alpha }_{3}{k}_{1}^{2}\left(u,v\right)\phantom{\rule{2.em}{0ex}}\left(\mathrm{b}2\right)$

for some ${\alpha }_{1}>0$ and for all $\left(u,v\right)\in {ℝ}^{2}·$ Using standard integral averaging technique, several interval oscillation criteria are obtained which require information on the behavior of the coefficients in equation (b) on a sequence of intervals $\left({a}_{n},{b}_{n}\right)$ such that ${a}_{n}\to \infty$ as $n\to \infty$. Unfortunately, rather specific assumptions (b1) and (b2) significantly restrict possible the applicability of the theorems.

The statement of the fundamental Lemma 1.1 should be corrected as follows: “If there exists an interval $\left(a,b\right)\subset \left[{t}_{0},\infty \right)$ such that (1.2) holds, then, for all $c\in \left(a,b\right),$ (1.3) is satisfied for every $H\in 𝒫$” instead of the incorrect formulation “If there exist an interval $\left(a,b\right)\subset \left[{t}_{0},\infty \right)$ and a $c\in \left(a,b\right)$ such that (1.2) holds, then (1.3) is satisfied for every $H\in 𝒫·$” The statement of Theorem 3.1 should be corrected by adding the phrase “and there exists a $c\in \left(a,b\right)$ such that (3.1) holds”.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory