The Scientific World Journal

Volume 2014 (2014), Article ID 124310, 6 pages

http://dx.doi.org/10.1155/2014/124310

Research Article

Oscillations for Neutral Functional Differential Equations

Fatima N. Ahmed, Rokiah R. Ahmad, Ummul K. S. Din, and Mohd S. M. Noorani

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Malaysia

Received 30 May 2014; Revised 4 August 2014; Accepted 5 August 2014; Published 18 August 2014

Academic Editor: Taher S. Hassan

Copyright © 2014 Fatima N. Ahmed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We will consider a class of neutral functional differential equations. Some infinite integral conditions for the oscillation of all solutions are derived. Our results extend and improve some of the previous results in the literature.

1. Introduction

During the past few decades, neutral differential equations have been studied extensively and the oscillatory theory for these equations is well developed; see [1–19] and the references cited therein. In fact, the developments of oscillation theory for the neutral differential equations began in 1986 with the appearance of the paper of Ladas and Sficas [15]. A survey of the most significant efforts in this theory can be found in the excellent monographs of Győri and Ladas [12] and Agarwal et al. [1].

Consider the first-order neutral differential equations of the form

where

There are numerous numbers of oscillation criteria obtained for oscillation of all solutions of (1). In particular, many various sufficient conditions for oscillation are established in [3–5, 9–15, 18, 19]. In reviewing the literature, (1) is much studied in the case when

which has been considered as an essential condition for the oscillation.

However, Yu et al. [19] considered (1) when , in the case when (3) does not hold (in this case (1) is said to have integrally small coefficients).

In [9], Gopalsamy et al. studied (1) when , and proved that every solution of (1) is oscillatory if

In [5], some finite integral conditions for oscillation of all solutions of (1) when are given under less restrictive hypothesis on . See also Grammatikopoulos et al. [10], Ladas and Sficas [15], and Al-Amri [4].

Recently, Ahmed et al. [2, 3] investigated the oscillation behaviour of (1) and obtained some new oscillation results. Additional results on the oscillation behaviour of (1) can also be found in the articles of Kulenović et al. [14], Kubiaczyk and Saker [13], and Greaf et al. [11].

In [18], infinite integral conditions for oscillation of all solutions of (1) in the case when are obtained when the coefficient takes some different ranges. A primary purpose of this paper is to further study the oscillation of solutions of (1). Our results extend and generalize some of the relevant results in [1–19].

Define the functions and as follows:

If is an eventually positive solution of the equation

then and are also solutions of (7). Furthermore, is a differentiable solution, while is twice differentiable. (see Győri and Ladas [12]).

As usual, a solution of (1) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory if it is either eventually positive or eventually negative. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

In the sequel, unless otherwise specified, when we write a functional inequality, we assume that it holds for all sufficiently large .

2. Auxiliary Lemmas

To specify the proofs of our main results, we need the following essential lemmas.

Lemma 1 (see [12]). Assume that (3) holds. Let be an eventually positive solution of (7). Then

(a) is a decreasing function and either

or

(b)The following statements are equivalent:

(i)Equation (8) holds;(ii) ; (iii) ;(iv) .

(c)The following statements are equivalent:

(i)Equation (9) holds;(ii) ; (iii) ;(iv) .

Lemma 2 (see [16]). Assume that

If is an eventually positive solution of the delay differential equation

then, for the same ,

Lemma 3 (see [16]). If the equation

has an eventually positive solution, then one has eventually that

Lemma 4 (see [12]). Assume that

Then the differential inequality

has an eventually positive solution if and only if the equation

has an eventually positive solution.

3. Main Results

In this section, we establish some infinite integral conditions for all solutions of (1) to oscillate. We assume that condition (3) holds.

Theorem 5. Let conditions (2) and (3) hold with ,

Then every solution of (1) is oscillatory.

Proof. Assume that (1) has a nonoscillatory solution on . Then, without loss of generality, there is a , sufficiently large, so that , and on . Set to be defined as in (5). Then by Lemma 1, it follows that

As , it follows from (1) that

Dividing the last inequality by , we obtain

Let

This implies that . Substituting in (22) yields

So by Lemma 4, we have that the delay differential equation

has an eventually positive solution as well. Let

Then is positive and continuous, and there exists such that , and

Furthermore, satisfies the generalized characteristic equation

where

Let

Therefore

Applying the inequality (cf. Erbe et al. [8, page 32]),

to (31), we have

or

Then, for , we have

By interchanging the order of integration, we get

Hence

Then

From (35) and (38), we find that

However, using Lemma 3, it follows that

eventually. Therefore, from (40) in (39), we get

That is,

which implies by condition (19) that

On the other hand, from Lemma 2, we have

This is a contradiction with (43). The proof is complete.

Example 6. Consider the equation

where

Observe that

Then

All conditions of Theorem 5 are satisfied. Then all solutions of (45) oscillate.

Theorem 7. Let conditions (2) and (3) hold with , , . Assume further that is - periodic;

Then every solution of (1) is oscillatory.

Proof. Assume that (1) has a nonoscillatory solution on . Then, without loss of generality, there is a , sufficiently large, so that , and on . Let and be defined as in (5) and (6). It is easily seen, by direct substituting, that and are also solutions of (1) when and are constants; that is

By Lemma 1, we have that is decreasing and . Also, we have indeed that

Then

Using (54) in (52) implies that

As , we have . Then

In view of the periodicity of , (56) implies that

As is positive solution, so by Lemma 4, the delay differential equation

has an eventually positive solution as well. Let

Then is positive and continuous, and there exists such that , and

Furthermore, satisfies the generalized characteristic equation

where

Let

Therefore

Applying the inequality (32) to (64), we have

or

Then, for , we have

By interchanging the order of integration, we get

Hence

Then

From (67) and (70), we find that

However, using Lemma 3, it follows that

eventually. Therefore, from (72) in (71), we get

or

which implies by condition (50) that

On the other hand, from Lemma 2, we have

This is a contradiction with (75). The proof is complete.

Example 8. Consider the equation

where

Observe that

Also,

Then all conditions of Theorem 7 are satisfied and therefore all solutions of (77) oscillate.

Remark 9. Theorems 5 and 7 generalize and extend Theorems 3.1 and 3.2 of Saker and Elabbasy [18], respectively, and Theorem 6.4.3 in Gyori and Ladas [12], where . See also the results of Ahmed et al. [2] and Kubiaczyk and Saker [13].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research has been completed with the support of these Grants: FRGS/2/2013/SG04/UKM/02/3 and DIP-2012-31.

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