Geodesic
Damped second order linear differential equation with deviating arguments: sharp results in oscillation properties
Electronic Journal of Differential Equations, Tome 2004 (2004).

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Summary: 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.
Classification : 34K11
Keywords: linear differential equation with deviating arguments, second order, damping term, oscillation 
@article{EJDE_2004__2004__a138,
     author = {Berezansky, Leonid and Domshlak, Yury},
     title = {Damped second order linear differential equation with deviating arguments: sharp results in oscillation properties},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2004},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a138/}
}
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Berezansky, Leonid; Domshlak, Yury. Damped second order linear differential equation with deviating arguments: sharp results in oscillation properties. Electronic Journal of Differential Equations, Tome 2004 (2004). http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a138/