Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 59, pp. 1-30.

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

Authors: Leonid Berezansky (Ben-Gurion Univ. of the Negev, Israel)
 Yury Domshlak (Ben-Gurion Univ. of the Negev, Israel)

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