Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1064.34021
Tiryaki, A.; Zafer, A.
Interval oscillation of a general class of second-order nonlinear differential equations with nonlinear damping. (English)
[J] Nonlinear Anal., Theory Methods Appl. 60, No. 1, A, 49-63 (2005). ISSN 0362-546X

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".
[Svitlana P. Rogovchenko (Famagusta)]

MSC 2000:: *34C10 Qualitative theory of oscillations of ODE: Zeros, etc.

Keywords: oscillation; interval oscillation criteria; integral averaging technique; nonlinear differential equations; damping

Cited in: Zbl 1102.34022

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]