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New nonoscillation and oscillation criteria are derived for scalar delay differential equations

It is well known that a scalar linear equation with delay

In [

In this theorem for

Further results on the critical case for (

The main results of this paper include the following.

Let

Let one assume that

In this paper we obtain new nonoscillation and oscillation sufficient conditions for (

We consider a scalar delay differential equation

Along with (

A function absolutely continuous on each interval

Let

One will say that (

To formulate a comparison result, consider the following equation:

Let (

Suppose that all solutions of (

Let exist

A nonoscillatory solution of (

Equation (

Our first result is a simple consequence of Theorem

Theorems

For every integer

Let for

(a) For the proof we will use a transformation applied to delay equations for the first time in [

Now we want to compare Theorem

Let (

The above statement is corroborated by Lemma

To apply Theorem

To compare Theorem

We start with the following question: for what functions

Let

Consider first the equation

For sufficiently large

We consider general equation (

(a) Let an integer

(b) Let an integer

Let the assumptions of case (a) be valid. Then, by Theorem

Let the assumptions of case (b) be valid. Suppose, on the contrary, that (

We show that equation of type (

Now we consider (

Let

(a) If there exists a

(b) If there exists a

Let the assumptions of case (a) be valid. By Theorem

Compare (

The proof of part (b) can be carried out in a way similar to that of the proof of part (a) and, therefore, it is omitted.

Consider the equation of the type of (

Similarly, one can show (by Theorem

In [

Let (

Let

The aim of the following theorems is to obtain nonoscillation conditions for (

Let

There exist

By Lemma

Let

There exist

Consider (

It is easy to generalize Theorem

Let

The following statement generalizes Theorem

Let

The proof of Theorem

In conclusion we note that there exist numerous results on nonoscillation for various classes of delay differential equations in a noncritical case. We refer, for example, to monographs [

J. Baštinec was supported by the Grant 201/10/1032 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00216 30529 and by the Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology. L. Berezansky was partially supported by grant 25/5 “Systematic support of international academic staff at Faculty of Electrical Engineering and Communication, Brno University of Technology” (Ministry of Education, Youth and Sports of the Czech Republic) and by the Grant 201/10/1032 of the Czech Grant Agency (Prague). J. Diblík was supported by the Grant 201/08/0469 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00216 30519 and by the Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology. Z. Šmarda was supported by the Council of Czech Government Grant MSM 00216 30503 and MSM 00216 30529, and by the Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology.