Electron. J. Diff. Eqns., Vol. 2004(2004), No. 59, pp. 1-30.

Damped second order linear differential equation with deviating arguments: Sharp results in oscillation properties

Leonid Berezansky & Yury Domshlak

Abstract:
This article presents a new approach for investigating the oscillation properties of second order linear differential equations with a damped term containing a deviating argument
$$ x''(t)-[P(t)x(r(t))]'+Q(t)x(l(t))=0,\quad r(t)\leq t. $$
To study this equation, a specially adapted version of Sturmian Comparison Method is developed and the following results are obtained:
(a) A comprehensive description of all critical (threshold) states with respect to its oscillation properties for a linear autonomous delay differential equation
$$ y''(t)-py'(t-\tau)+qy(t-\sigma)=0, \quad \tau>0,\;\infty<\sigma<\infty.$$
(b) Two versions of Sturm-Like Comparison Theorems. Based on these Theorems, sharp conditions under which all solutions are oscillatory for specific realizations of $P(t), r(t)$ and $l(t)$ are obtained. These conditions are formulated as the unimprovable analogues of the classical Knezer Theorem which is well-known for ordinary differential equations ($P(t)=0$, $l(t)=t$).
(c) Upper bounds for intervals, where any solution has at least one zero.

Submitted March 23, 2004. Published April 19, 2004.
Math Subject Classifications: 34K11
Key Words: Linear differential equation with deviating arguments, second order, damping term, oscillation, Sturmian comparison method

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Leonid Berezansky
Department of Mathematics
Ben-Gurion University of the Negev
Beer-Sheva 84105, Israel
e-mail: brznsky@cs.bgu.ac.il
Yury Domshlak
Department of Mathematics
Ben-Gurion University of the Negev
Beer-Sheva 84105, Israel
e-mail: domshlak@cs.bgu.ac.il

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