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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 $$\left[ r(t)k_{1}(x,x^{\prime})\right] ^{\prime}+p(t)k_{2}(x,x^{\prime })x^{\prime}+q(t)f(x)=0,\qquad t\geq t_{0}\geq0,\tag{b}$$ with $p,q:[t_{0},\infty)\to\Bbb{R},$ $r:[t_{0},\infty )\to(0,\infty),$ $f:\Bbb{R}\to\Bbb{R},$ $k_{1} ,k_{2}:\Bbb{R}^{2}\to\Bbb{R}.$ It is also assumed that $$ k_{1}^{2}(u,v)\leq\alpha_{1}k_{1}(u,v), $$ for some $\alpha_{1}>0$ and for all $(u,v)\in\Bbb{R}^{2}.$ Two cases are considered: (a) $f(x)$ is differentiable, $xf(x)\neq0$ and $f^{\prime} (x)\geq\mu_{1}$ for some $\mu_{1}>0$ and all $x\neq0,$ and $$vf(u)k_{2}(u,v)\geq\alpha_{2}k_{1}^{2}(u,v)\tag{b1}$$ for some $\alpha_{2}>0$ and for all $(u,v)\in\Bbb{R}^{2};$ (b) $f(x)$ is not necessarily differentiable, $f(x)/x\geq\mu_{2}$ for some $\mu_{2}>0$ and all $x\neq0,$ and $$vuk_{2}(u,v)\geq\alpha_{3}k_{1}^{2}(u,v)\tag{b2} $$ for some $\alpha_{1}>0$ and for all $(u,v)\in\Bbb{R}^{2}.$ 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 $(a_{n},b_{n})$ 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 $(a,b)\subset\lbrack t_{0},\infty)$ such that (1.2) holds, then, for all $c\in(a,b),$ (1.3) is satisfied for every $H\in\cal{P}$” instead of the incorrect formulation “If there exist an interval $(a,b)\subset\lbrack t_{0},\infty)$ and a $c\in(a,b)$ such that (1.2) holds, then (1.3) is satisfied for every $H\in\cal{P}.$” The statement of Theorem 3.1 should be corrected by adding the phrase “and there exists a $c\in(a,b)$ such that (3.1) holds”.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
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References:
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