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**Some oscillation problems for a second-order linear delay differential equation.**
*(English)*
Zbl 0915.34064

The following scalar delay differential equation is considered
\[
\ddot x(t)+\sum_{k=1}^m a_k(t) x(g_k(t))=0,\quad g_k(t)\leq t .
\]
Nonoscillation and properties related to nonoscillation are studied. The main result is that under some natural assumptions for a delay differential equation the following four assertions are equivalent: nonoscillation of solutions to this equation and the corresponding differential inequality, positiveness of the fundamental function, and the existence of a nonnegative solution to a generalized Riccati inequality.

The equivalence of the oscillation properties with a differential inequality is applied to obtain new explicit conditions for nonoscillation and oscillation and to prove some known results in a different way.

A generalized Riccati inequality is applied to compare oscillation properties of two equations without comparing their solutions. These results can be regarded as a natural generalization of the Sturm comparison theorem for a second-order ordinary differential equation. By applying the positiveness of the fundamental function the positive solutions to two nonoscillatory equations are compared. Conditions on the initial function are given implying the positiveness of the corresponding solution.

The equivalence of the oscillation properties with a differential inequality is applied to obtain new explicit conditions for nonoscillation and oscillation and to prove some known results in a different way.

A generalized Riccati inequality is applied to compare oscillation properties of two equations without comparing their solutions. These results can be regarded as a natural generalization of the Sturm comparison theorem for a second-order ordinary differential equation. By applying the positiveness of the fundamental function the positive solutions to two nonoscillatory equations are compared. Conditions on the initial function are given implying the positiveness of the corresponding solution.

Reviewer: I.Ginchev (Varna)

### MSC:

34K11 | Oscillation theory of functional-differential equations |

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

### Keywords:

scalar linear delay differential equation; oscillations; fundamental function; nonnegative solutions; Riccati inequality; Sturm comparison theorem
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\textit{L. Berezansky} and \textit{E. Braverman}, J. Math. Anal. Appl. 220, No. 2, 719--740 (1998; Zbl 0915.34064)

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