We are concerned with the oscillation of the forced second-order neutral nonlinear differential
equations with delayed argument in the form . No restriction is imposed on the potentials , , and to be nonnegative. Our methodology is somewhat different from those of previous authors.
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
,, is nondecreasing, for , and , for and ,, and are continuous functions defined on , , , is nondecreasing, , and .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:
where . The remaining argument is the same as that in Theorem 2.4 and this completes the proof.
Remark 2.7. Let , and . Theorems 2.4 and 2.6 reduce to Theorem of [8, 11].
We will use the function class to study the oscillatory of (1.1). We say that a function belongs to the function class , denoted by if , where , which satisfies , and for , and has the partial derivative on such that is locally integrable with respect to in .
We defined the operator by
and the function is defined by
It is easy to verify that is a linear operator and satisfies
Theorem 2.8. Suppose that, for any , there exist constants such that and satisfies (2.1). Assume that there exists a function , such that, for each and for some ,
for j=1, 2, then (1.1) is oscillatory, where the operator B is defined by (2.30), is defined by (2.31),
, and are positive constants satisfying () and () of Lemma 2.1.
Proof. We proceed as in Theorem 2.4. Assume to the contrary that there exists a solution of (1.1) such that , when , for some depending on the solution . From the proof of Theorem 2.4, we obtain (2.16) for all . Applying to (2.17), we have that
By (2.32) and above inequality, we have, for , that
That is,
Taking the super limit in the above inequality, we have that
which contradicts assumption (2.33). When is eventually negative, the proof follows the same argument using the interval instead of . This completes the proof.
Theorem 2.9. Suppose that, for any , there exist constants such that and satisfies (2.23). Assume that there exists a function such that, for each and for some ,
for j = 1, 2, then (1.1) is oscillatory, where the operator B is defined by (2.30), is defined by (2.31), and is defined by (2.25).
The proof of Theorem 2.9 can be completed by following the proofs of Theorems 2.6 and 2.8 with suitable changes, we omit it here.
3. Corollaries
As Theorems 2.4–2.9 are rather general, it is convenient for applications to derive a number of oscillation criteria with the appropriate choice of the functions and .
With an appropriate choice of the functions , one can derive from Theorems 2.4 and 2.6 a number of oscillation criteria for (1.1). For example, we consider the simple case ; hence ,. Let , where are given in Theorem 2.4. This determines . Choose , , and k being natural number. We obtain from Theorems 2.4 and 2.6 the following corollaries.
Corollary 3.1. Let be chosen to satisfy (), () of Lemma 2.1 for . If for any , there exist constants such that , then it holds that
and for , and , one has that
then all solutions of the equation
are oscillatory.
Corollary 3.2. Assume that, for any , there exist constants such that and
Also assume that, for , and that there exists some such that
where , then all solutions of (3.3) are oscillatory.
If we choose for and , then we have that
Thus by Theorems 2.8 and 2.9, we have two new oscillation results.
Corollary 3.3. Suppose that, for any , there exist constants such that and satisfies (2.1). For each and for some ,
for j = 1, 2, then (1.1) is oscillatory, where
, and are positive constants satisfying () and () of Lemma 2.1.
Corollary 3.4. Suppose that, for any , there exist constants such that and satisfies (2.23). Assume that there exists a function such that, for each and for some ,
for j = 1, 2, then (1.1) is oscillatory, where is defined by (2.25).
We say that a function belongs to the function class , if , where , which satisfies for and has partial derivatives and on such that
where are locally integrable with respect to and , respectively, in .
If we choose , for , then we have that
where are defined as the following:
Thus by Theorems 2.8 and 2.9, we also have the two following oscillation results.
Corollary 3.5. Suppose that, for any , there exist constants such that and satisfies (2.1). For each and for some ,
for j = 1, 2, then (1.1) is oscillatory, where are defined by (3.12),
, and are positive constants satisfying () and () of Lemma 2.1.
Corollary 3.6. Suppose that, for any , there exist constants such that and satisfies (2.23). Assume that there exists a function such that, for each and for some ,
for j = 1, 2, then (1.1) is oscillatory, where is defined by (3.12) and are defined by (2.25).
Acknowledgments
The aouthors thank the referees for their helpful comments to improve this paper. This research was supported by the NNSF of China (Grant no. 10771118) and NSF of Shandong (Grants no. ZR2009AM011 and no. ZR2009AQ010).