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We study necessary and sufficient conditions for the oscillation of the
third-order nonlinear ordinary differential equation with damping term and deviating argument

The aim of this paper is to investigate the third order nonlinear functional differential equation with deviating argument

The following assumptions will be made.

Throughout the paper we assume that the operator

It is well known, see, for example, [

Asymptotic properties of equations of type (

Recently, oscillation criteria for (

have been presented in [

If the differential operator

where

For

Let

Let for some

In our previous paper [

Motivated by [

The role of the deviating argument

A function

The following theorem shows the possible types of nonoscillatory solutions for (

Any nonoscillatory solution

Without loss of generality suppose that there exists a solution

O. Boruvka [

Theorem

The following lemma is similar to [

Any solution

In view of Theorem

Now let us prove (

Let

By contradiction, assume

In view of Theorem

Nonoscillatory solution

Our main result here deals with the existence of solutions of Type II.

Assume

We prove the existence of solutions of (

Let

Let us show that

Now we prove the continuity of

Theorem

In this section we give a sufficient condition for oscillation of (

Assume

To prove the first assertion, it is sufficient to show that (

Consider the function

Now let

Theorem

Applying Theorems

Assume (

From Theorem

Assume that

In addition, if

By Theorem

If

Finally, if

Let

Assume that

We conclude this section with the following result on the continuability of solutions of (

Assume

Let

Now consider these two cases. (a) Let

(b) Let

Theorem

In this section we study Property A for (

We start with the following result on the boundedness of nonoscillatory solutions, which extends [

Assume that

Without loss of generality, assume that

The next result describes the asymptotic properties of nonoscillatory solutions and will be used later.

Assume (

Theorem

By Theorem

In order to complete the proof, define for

Using the previous results, we obtain a sufficient condition for property A.

Assume (

Let

Suppose assumptions of Theorem

The assertion follows from Theorem

The first and third authors are supported by the Research Project 0021622409 of the Ministry of Education of the Czech Republic and Grant 201/08/0469 of the Czech Grant Agency. The fourth author is supported by the Research Project PRIN07-Area 01, no. 37 of the Italian Ministry of Education.