^{1}

The aim of this paper is to offer sufficient conditions for property
(B) and/or the oscillation of the third-order nonlinear functional differential
equation with mixed arguments

We are concerned with the oscillatory and certain asymptotic behavior of all solutions of the third-order functional differential equations

By a solution of (

Recently, (

We will discuss both cases

We will establish suitable comparison theorems that enable us to study properties of (

In the paper, we are motivated by an interesting result of Grace et al. [

All functional inequalities considered in this paper are assumed to hold eventually; that is, they are satisfied for all

The following results are elementary but useful in what comes next.

Assume that

If

Assume that

We may assume that

The following result presents a useful relationship between an existence of positive solutions of the advanced differential inequality and the corresponding advanced differential equation.

Suppose that _{2}), (H_{3}), and (H_{4}), respectively. If the first-order advanced differential inequality

Let

We start our main results with the classification of the possible nonoscillatory solutions of (

Let

Let

Now, we suppose that

It follows from Lemma

In the following results, we provide criteria for the elimination of Cases (I)–(III) of Lemma

Let us denote for our further references that

Let

Let

It follows from the proof of Theorem

Now, we are prepared to provide new criteria for property (B) of (

Let (

Let

Employing an additional condition on the function

Let

First note that (

Consider the third-order nonlinear differential equation with mixed arguments

Now, we turn our attention to the case when

Let

Let

The following result is obvious.

Let (

Now, we present easily verifiable criterion for property (B) of (

Let (

The proof is similar to the proof of Corollary

Theorems

It is useful to notice that if we apply the traditional approach to (

Consider the third-order nonlinear differential equation with mixed arguments

Now, we eliminate Case (I) of Lemma

Let

Let

Combining Theorem

Let (

Assume that (

Let

Conditions (

We consider once more the third-order differential equation (

The following results are obvious.

Let (

Let

We recall again the differential equation (

The following result is intended to exclude Case (III) of Lemma

Let

Let

The following results are immediate.

Let (

Let (

Let (

Let (

In this paper, we have presented new comparison theorems for deducing the property (B)/oscillation of (