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We study the half-linear delay differential equation

In this paper we study oscillatory properties of the delay second-order half-linear differential equation

We suppose that

Under the solution of (

The solution of (

It is well known that the behavior of delay differential equations is very different from the behavior of ordinary differential equations. Among others, the Sturm theory fails and oscillatory solutions may coexist with nonoscillatory solutions.

In certain special cases, it is possible to compare asymptotics of (

In this paper we compare (

Let us recall the Riccati technique, which is one of the methods frequently used in oscillation theory of both (

The following lemma plays a crucial role in the qualitative theory of half-linear second order ordinary differential equations.

Denote

(

there is

there is

there is

As we show below, the assumptions used in the paper ensure that the positive solutions are eventually increasing and concave down. The main step when we compare the ordinary half-linear differential equation and its delay counterpart (

Suppose that

Note that the proof of Lemma

Equation (

As another particular example of a criterion which suffers from the presence of the constants

The above mentioned disadvantage has been removed for the linear delay equation

The aim of this paper is to derive a result analogical to the estimate from [

The proof of the following statement can be found in [

Let

The following lemma shows that under certain additional conditions we can utilize (

Suppose that (

Conditions (

We show that

Hence there exists

Suppose that conditions (

Suppose, by contradiction, that (

The oscillation criterion from Theorem

Note that a similar result like Theorem

Theorem

Consider the perturbed Euler type half-linear delay differential equation

We claim that the oscillation of (

In this section we use a slight modification of the estimates from the first part of the paper to derive similar results for the second order neutral differential equation

Similarly as for (

Similarly like for the delay equation, the positive solution is increasing and concave down. More precisely, the following lemma holds. For linear version of this lemma see [

Let

The proof is essentially the same as the proof of [

Without loss of generality we can suppose that

Suppose that there exists

If

Suppose that

According to Lemma

The following lemma is an alternative to Lemma

Suppose that (

Similarly like in Lemma

Suppose by contradiction that there exists

Now we can formulate the comparison theorem which relates neutral differential equations to ordinary second-order half-linear differential equations.

Suppose that (

Having proved important estimates in the preceding two lemmas, the proof of the theorem is a modification of the proof of Theorem

A version of Theorem

This research is supported by the Grant P201/10/1032 of the Czech Science Foundation.