1. Introduction

In this paper, we study the oscillatory behavior of the forced neutral nonlinear functional differential equation of the form

where . In this paper, we assume that

No restriction is imposed on the potentials , and to be nonnegative. As usual, a solution of (1.1) is called oscillatory if it is defined on some ray with and has unbounded set of zeros. (1.1) is called oscillatory if all of its solutions on some ray are oscillatory.

In the last decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of second-order linear and nonlinear delay differential equations (see, for example, [1–22] and the references therein). Let us consider the familiar forced Emden-Fowler equation

When , (1.2) is known as the superlinear equation, and when , it is known as the sublinear equation. The oscillation of (1.2) has been the subject of much attention during the last 50 years; see the seminal book by Agarwal, et al. [23]. Here, we refer to the papers [1–3] and the references cited therein. In this case, one can usually establish oscillation criteria for more general nonlinear equations by using a technique introduced by Kartsatos [9] where it is additionally assumed that is the second derivative of an oscillatory function. This approach has been expressed in [5, 6]. Sun [4] has extended these results to delay differential equations of the form of (1.2), where and the potentials and are allowed to change sign. However, Sun [4] does not say anything else for the oscillation of equation (1.2) with . Later, employing the arguments in [4], Çakmak and Tiryaki [7] have established similar oscillation criteria for the equation of the form

where is assumed to satisfy certain growth conditions.

Very recently, Sun et al. [13, 14] obtained some new oscillation criteria for the equations in the form

where . He also established oscillation theorems when . When , this approach was initiated by Agarwal and Grace [1, pages 244–249] for higher-order equations and subsequently developed in papers of Ou and Wong [15], Q.Yang [18], X.Yang [19], as well as Sun and Agarwal [16, 17].

In [24], Xu and Meng studied the oscillation of the equation

by using the generalized Riccati technique and the function class .

The purpose of this paper is to give some new oscillation criteria for (1.1), which can be regarded as further investigation for the (1.1) including the papers of Sun and Wong [13], Xu and Meng [24]. These criteria do not assume that , and are of definite sign. Our methodology is somewhat different from those of previous authors, and the results we obtained are more general than those of Sun and Wong [13].

2. Main Results

We will need the following lemmas that have been proved in [13].

Lemma 2.1 (see [13]). Let , be n-tuple satisfying . Then there exists an n-tuple satisfying which also satisfies either or

Lemma 2.2 (see [13]). Let be positive real numbers. Then (i),(ii).

Remark 2.3. For a given set of exponents satisfying , Lemma 2.1 ensures the existence of an n-tuple such that either () and () hold or () and () hold. When n = 2 and , in the first case, we have that where can be any positive number satisfying . This will ensure that , and conditions () and () are satisfied. In the second case, we simply solve () and () and obtain

Theorem 2.4. Suppose that, for any , there exist constants such that , and Let , for j = 1, 2. Assume that there exists a positive, nondecreasing function such that, for some and for some , for j = 1, 2, then (1.1) is oscillatory, where , and are positive constants satisfying () and () of Lemma 2.1.

Proof. Assume to the contrary that there exists a solution of (1.1) such that , when , for some depending on the solution . Set
By assumption, we have that for , and from (2.4) it follows that It is not difficult to show that is eventually positive. In fact, first, we know that for sufficiently large , since is nontrivial. Second, if there exists an such that , then for , that is, , and hence, as , which contradicts the fact that . Without loss of generality; say , . Thus we have that Define It follows from (2.7) that satisfies the following differential equality: Using (2.6), we have that and by the condition , we have that By assumption, we can choose such that , for , and for . Recall the arithmetic-geometric mean inequality (see [25]) where and are chosen to satisfy () and () of Lemma 2.1 for the given . Now return to (2.10) and identify and in (2.11) to obtain From (1.1), we can easily obtain , for . Therefore, we have that, for , Noting that for , we get by (2.13) that that is, Integrating (2.15) from to , we obtain By using (2.16) in (2.12), we have that, for , Multiplying both sides of (2.17) by as given in the hypothesis of Theorem 2.4 and integrating (2.17) from to , we obtain Using the integration by parts formula, we have that where . Substituting (2.19) into (2.18), we obtain Then From the hypothesis of Theorem 2.4 and (2.21), we have that which contradicts (2.2). When is eventually negative, we can obtain similar contradiction using the interval instead of . This completes the proof.

Remark 2.5. Let , , and for . It is easy to see that Theorem 2.4 reduces to Theorem of [7].

In Theorem 2.6, we do not impose any restriction on signs of those coefficients corresponding to sublinear terms of (1.1), that is, for . If it is nonpositive, we can easily see that Theorem 2.4 is invalid. However, the following theorem is valid for this case.

Theorem 2.6. Suppose that, for any , there exist constants such that and Assume that there exists a positive, nondecreasing function such that, for some and for some , for j = 1, 2, then (1.1) is oscillatory, where with for , , and .

Proof. Assume to the contrary that there exists a solution of (1.1) such that , when , for some depending on the solution . When is eventually negative, the proof follows the same argument using the interval instead of . Apply the assumption of and , then (1.1) is rearranged as Noting the assumption (2.23) and applying Lemma 2.2(i) to the first summation term in (2.26), we get that Introduce the Riccati substitution as (2.4) and apply Lemma 2.2(ii) to each of the nonlinear terms in the last sum in (2.27). Here , and We can obtain from (2.27) the following Riccati inequality: