Abstract

A class of nonlinear neutral delay differential equations is considered. Some new oscillation criteria of all solutions are derived. The obtained results generalize and extend some of well known previous results in the literature.

1. Introduction and Preliminaries

Consider the nonlinear neutral delay differential equation of the form where

Let . By a solution of we mean a function for some such that is continuously differentiable for and such that is satisfied for . Let be a given initial point and let be a given initial function. Then, one can show by using the method of steps that has a unique solution on satisfying the initial function

As usual, a solution of is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory if it is either eventually positive or eventually negative. Equation 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 .

In the case where , , and and are constants, Karpuz and Öcalan [1] improved the result of Ladas and Sficas [2] holding , , and conditions for oscillation. Also, the case including continuous functions as coefficients has been studied by many authors; see, for example, Kubiaczyk and Saker [3], Karpuz and Öcalan [1], Ahmed at al. [4], Chuanxi et al. [5], and Yu et al. [6]. In particular, Chen et al. [7] succeeded in getting some oscillation theorems for (3) which involve joint behaviour of and using the condition

Some further results on the oscillation for neutral delay differential equations can be found in the excellent paper of Saker and Kubiaczyk [8] and the recent paper of Ahmed et al. [9]. See also Shen and Debnath [10] and Wang [11].

Li [12] extended results of Chen et al. [7] for in the case when and introduced some new oscillation criteria under the hypothesis

Kubiaczyk et al. [13] have given some several sufficient conditions for oscillation of all solutions depending on the functions and when is allowed to oscillate, while Zhou [14] has established some new sufficient conditions for oscillation depending on an additional constant .

Here, in this paper, we continue in this direction of finding some sufficient conditions for to oscillate in the case when . To this end, let us site the next two results which will enable us to complete the proofs of our main results.

Lemma 1 (see [8]). Assume that there exists such that Let be an eventually positive solution of . Let Then

Theorem 2 (see [15]). Assume that ; then every solution of (3) oscillates if

2. Main Results

Theorem 3. Assume that condition (6) holds and either or Then all solutions of oscillate if and only if the corresponding differential inequality has no eventually positive solution.

Proof. The sufficiency is obvious. To prove necessity, assume that is an eventually positive solution of (12). We plan to show that has a nonoscillatory solution. Set as in (7). Then, from we have Integrating the last equation from to , and using Lemma 1, we have That is, which leads to Let be fixed so that (16) holds for all . Set and consider the set of functions Define a mapping on as It is easy to see, by using (16), that maps into itself. Moreover, for any we have for .
Next, define the sequence in as follows: Therefore, by using (16) and a simple induction, we can easily see that Set Then from Lebesgue’s Dominated Convergence Theorem, it follows that satisfies Again set Then and satisfies, for , This implies that where Clearly, is continuous on . To show that is positive for all , assume that there exists such that for and . Then and by (26) we obtain Then This is a contradiction with (10) or (11). Therefore, is positive on . Furthermore, it is easy to see that is a positive solution of , which implies that the inequality (12) having no eventually positive solution is a necessary condition for the oscillation of all solutions of . The proof is complete.

Remark 4. Theorem 3 is an extent of Theorem 2.1 due to Lalli and Zhang [16], Theorem 1 due to Chen et al. [7], and Theorem 1 due to Li [12].

Now we give an application of Theorem 3.

Theorem 5. Consider with . Suppose that condition (6) holds with where Then all solutions of are oscillatory.

Proof. Suppose that has an eventually positive solution . Set as in (7). Then, by Lemma 6, we have From (7), we have Hence, from , (31), and (35), respectively, we have That is, or where In view of Theorem 3, we have that the equation has an eventually positive solution. On the other hand, in view of Theorem 2, condition (30) implies that (40) cannot have an eventually positive solution. This is a contradiction. The proof is complete.

Lemma 6. Suppose that Let be an eventually positive solution of and defined by (7). Then

Proof. From and (7), we have Therefore, if (44) does not hold, then we have eventually that ; that is, which together with (41) and (42) yields Let be such that and also such that (16) holds for . Define Then, for . Set , and we have For convenience, we denote where is the greatest integer parts of . Then from (7), (41), and (42), we obtain Thus, we have But is decreasing, and for . Therefore, from (53), we get Substituting in , we obtain By Holder’s inequality, we have Then, from (55), we get Hence, Integrating (58) from to , we have Also we have Now, by noting that we see that Thus, we have eventually that exists. Then, from condition (43), we have Letting in (59), we obtain a contradiction with (64). The proof is complete.

Remark 7. Lemma 6 extends Lemma 2 in Li [12] where .

Theorem 8. Suppose that condition (6) holds with where is defined as in Theorem 3 and Then, all solutions of are oscillatory.

Proof. Suppose that has an eventually positive solution . Set as in (7). Then by Lemma 6 we have From (7), we have Hence, from and (69), we have Applying Holder’s inequality, we obtain which implies that where we have used condition (66) to obtain the last inequality. This implies that is a positive solution of the inequality that is, where As we see, (74) satisfies all conditions of Lemma 6; hence, eventually. On the other hand, since (74) satisfies all conditions of Lemma 6, then eventually, which is a contradiction. The proof is complete.