TY - JOUR

T1 - Damped second order linear differential equation with deviating arguments

T2 - Sharp results in oscillation properties

AU - Berezansky, Leonid

AU - Domshlak, Yury

PY - 2004/4/19

Y1 - 2004/4/19

N2 - 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, r(t) ≤ 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 - r) + qy(t - σ) = 0, r > 0, ∞ < σ < ∞. (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.

AB - 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, r(t) ≤ 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 - r) + qy(t - σ) = 0, r > 0, ∞ < σ < ∞. (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.

KW - Damping term

KW - Linear differential equation with deviating arguments

KW - Oscillation

KW - Second order

KW - Sturmian comparison method

UR - http://www.scopus.com/inward/record.url?scp=3042531114&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:3042531114

VL - 2004

SP - 1

EP - 30

JO - Electronic Journal of Differential Equations

JF - Electronic Journal of Differential Equations

SN - 1072-6691

ER -