Abstract
We study oscillatory properties of a class of secondorder nonlinear neutral functional differential equations with distributed deviating arguments. On the basis of less restrictive assumptions imposed on the neutral coefficient, some new criteria are presented. Three examples are provided to illustrate these results.
MSC: 34C10, 34K11.
Keywords:
oscillation; neutral differential equation; secondorder equation; distributed deviating argument1 Introduction
This paper is concerned with oscillation of the secondorder nonlinear functional differential equation
where , is a constant, and . Throughout, we assume that the following hypotheses hold:
(H_{2}) and is not eventually zero on any , ;
(H_{5}) is nondecreasing and the integral of (1.1) is taken in the sense of RiemannStieltijes.
By a solution of (1.1), we mean a function for some , which has the properties that , , and satisfies (1.1) on . We restrict our attention to those solutions x of (1.1) which exist on and satisfy for any . A solution x of (1.1) is termed oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions oscillate.
As is well known, neutral differential equations have a great number of applications in electric networks. For instance, they are frequently used in the study of distributed networks containing lossless transmission lines, which rise in high speed computers, where the lossless transmission lines are used to interconnect switching circuits; see [1]. Hence, there has been much research activity concerning oscillatory and nonoscillatory behavior of solutions to different classes of neutral differential equations, we refer the reader to [230] and the references cited therein.
In the following, we present some background details that motivate our research. Recently, Baculíková and Lacková [6], Džurina and Hudáková [12], Li et al.[15,18], and Sun et al.[22] established some oscillation criteria for the secondorder halflinear neutral differential equation
Baculíková and Džurina [4,5] and Li et al.[17] investigated oscillatory behavior of a secondorder neutral differential equation
where
Ye and Xu [26] and Yu and Fu [27] considered oscillation of the secondorder differential equation
Assuming , Thandapani and Piramanantham [23], Wang [24], Xu and Weng [25], and Zhao and Meng [30] studied oscillation of an equation
As yet, there are few results regarding the study of oscillatory properties of (1.1) under the conditions or . Thereinto, Li and Thandapani [19] obtained several oscillation results for (1.1) in the case where (1.2) holds, , and
In the subsequent sections, we shall utilize the Riccati substitution technique and some inequalities to establish several new oscillation criteria for (1.1) assuming that (1.3) holds or
All functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough.
2 Main results
In what follows, we use the following notation for the convenience of the reader:
where h, ρ, and η will be specified later.
Theorem 2.1Assume (H_{1})(H_{5}), (1.3), and let, , , andfor. Suppose further that there exists a realvalued functionsuch thatforand. If there exists a realvalued functionsuch that
then (1.1) is oscillatory.
Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a such that , , and for all and . Then . Applying (1.1), one has, for all sufficiently large t,
Using the inequality (see [[5], Lemma 1])
the definition of z, , and , we conclude that
By virtue of (1.1), we get
Thus, is nonincreasing. Now we have two possible cases for the sign of : (i) eventually, or (ii) eventually.
(i) Assume that for . Then we have by (2.3)
which yields
Then we obtain due to (1.3), which is a contradiction.
(ii) Assume that for . It follows from (2.2) and that
We define a Riccati substitution
Then . From (2.3) and , we have
Differentiating (2.5), we get
Therefore, by (2.5), (2.6), and (2.7), we see that
Similarly, we introduce another Riccati transformation:
Then . From (2.3) and , we obtain
Differentiating (2.9), we have
Therefore, by (2.9), (2.10), and (2.11), we find
Combining (2.8) and (2.12), we get
It follows from (2.4) that
Integrating the latter inequality from to t, we obtain
Define
Using the inequality
we get
On the other hand, define
Then we have by (2.14)
Thus, from (2.13), we get
which contradicts (2.1). This completes the proof. □
Assuming (1.2), where and are constants, we obtain the following result.
Theorem 2.2Suppose (H_{1})(H_{5}), (1.2), (1.3), and let, , , andfor. If there exists a realvalued functionsuch that
then (1.1) is oscillatory.
Proof As above, let x be an eventually positive solution of (1.1). Proceeding as in the proof of Theorem 2.1, we have , (2.3), and (2.4) for all sufficiently large t. Using (1.2), (2.3), and (2.4), we obtain
The remainder of the proof is similar to that of Theorem 2.1, and hence it is omitted. □
Theorem 2.3Suppose we have (H_{1})(H_{5}), (1.3), and letandfor. Assume also that there exists a realvalued functionsuch thatforand. If there exists a realvalued functionsuch that
then (1.1) is oscillatory.
Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a such that , , and for all and . As in the proof of Theorem 2.1, we obtain (2.3) and (2.4). In view of (2.3), is nonincreasing. Now we have two possible cases for the sign of : (i) eventually, or (ii) eventually.
(i) Suppose that for . Then, with a proof similar to the proof of case (i) in Theorem 2.1, we obtain a contradiction.
(ii) Suppose that for . We define a Riccati substitution
Then . From (2.3) and , we have
Differentiating (2.18), we obtain
Therefore, by (2.18), (2.19), and (2.20), we see that
Similarly, we introduce another Riccati substitution:
Then . Differentiating (2.22), we have
Therefore, by (2.22) and (2.23), we get
Combining (2.21) and (2.24), we have
It follows from (2.4) and that
Integrating the latter inequality from to t, we obtain
Define
Using inequality (2.14), we have
On the other hand, define
Then, by (2.14), we obtain
Thus, from (2.25), we get
which contradicts (2.17). This completes the proof. □
Assuming we have (1.2), where and are constants, we get the following result.
Theorem 2.4Suppose we have (H_{1})(H_{5}), (1.2), (1.3), and letandfor. If there exists a realvalued functionsuch that
then (1.1) is oscillatory.
Proof Assume again that x is an eventually positive solution of (1.1). As in the proof of Theorem 2.1, we have , (2.3), and (2.4) for all sufficiently large t. By virtue of (1.2), (2.3), and (2.4), we have (2.16) for all sufficiently large t. The rest of the proof is similar to that of Theorem 2.3, and so it is omitted. □
In the following, we present some oscillation criteria for (1.1) in the case where (1.4) holds.
Theorem 2.5Suppose we have (H_{1})(H_{5}), (1.2), (1.4), and let, , for, andfor. Assume further that there exists a realvalued functionsuch that (2.15) is satisfied. If there exists a realvalued functionsuch that, , for, and
then (1.1) is oscillatory.
Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a such that , , and for all and . Then . As in the proof of Theorem 2.1, we get (2.2). By virtue of (1.1), we have (2.3). Thus, is nonincreasing. Now we have two possible cases for the sign of : (i) eventually, or (ii) eventually.
(i) Suppose that for . Then, by the proof of Theorem 2.2, we obtain a contradiction to (2.15).
(ii) Suppose that for . It follows from (2.2), (2.3), and that
We define the function u by
Then . Noting that is nondecreasing, we get
Integrating this inequality from to l, we obtain
That is,
Thus, we get by (2.29)
Similarly, we define another function v by
Then . Noting that is nondecreasing and , we get
Thus, . Hence, by (2.30), we see that
Differentiating (2.29), we obtain
By (2.3) and , we have , and so
Similarly, we see that
Combining (2.33) and (2.34), we get
Using (2.28), (2.35), and , we obtain
Multiplying (2.36) by and integrating the resulting inequality from to t, we have
Set
Using inequality (2.14), we get
Similarly, we set
Then we have by (2.14)
Thus, from (2.30) and (2.32), we find
which contradicts (2.27). This completes the proof. □
With a proof similar to the proof of Theorems 2.4 and 2.5, we obtain the following result.
Theorem 2.6Suppose we have (H_{1})(H_{5}), (1.2), (1.4), and let, for, andfor. Assume also that there exists a realvalued functionsuch that (2.26) is satisfied. If there exists a realvalued functionsuch that, , for, and (2.27) holds, then (1.1) is oscillatory.
3 Applications and discussion
In this section, we provide three examples to illustrate the main results.
Example 3.1 Consider the secondorder neutral functional differential equation
Let , , , , , , , , , and . Then , , for , and . Moreover, letting , then
Hence, by Theorem 2.2, (3.1) is oscillatory. As a matter of fact, one such solution is .
Example 3.2 Consider the secondorder neutral functional differential equation
where is a constant. Let , , , , , , , , , and . Then , for , , and for . Further, setting ,
and
Therefore, we have
Hence, (3.2) is oscillatory due to Theorem 2.3.
Example 3.3 Consider the secondorder neutral functional differential equation
where , and β are positive constants. Let , , , , , , , , , and . Then , , for , , for , and . Further,
and
Hence, by Theorem 2.6, (3.3) is oscillatory when .
Remark 3.1 In this paper, we establish some new oscillation theorems for (1.1) in the case where p is finite or infinite on . The criteria obtained extend the results in [22] and improve those reported in [19]. Similar results can be presented under the assumption that . In this case, using [[5], Lemma 2], one has to replace with and proceed as above. It would be interesting to find another method to investigate (1.1) in the case where .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. They all read and approved the final version of the manuscript.
Acknowledgements
The authors express their sincere gratitude to the anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details.
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