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
(H_{1})
(H_{2})
(H_{3})
(H_{4})
(H_{5})
By a solution of (1.1), we mean a function
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
where
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
As yet, there are few results regarding the study of oscillatory properties of (1.1)
under the conditions
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
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
Using the inequality (see [[5], Lemma 1])
the definition of z,
By virtue of (1.1), we get
Thus,
(i) Assume that
which yields
Then we obtain
(ii) Assume that
We define a Riccati substitution
Then
Differentiating (2.5), we get
Therefore, by (2.5), (2.6), and (2.7), we see that
Similarly, we introduce another Riccati transformation:
Then
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
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
Theorem 2.2Suppose (H_{1})(H_{5}), (1.2), (1.3), and let
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
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 let
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
(i) Suppose that
(ii) Suppose that
Then
Differentiating (2.18), we obtain
Therefore, by (2.18), (2.19), and (2.20), we see that
Similarly, we introduce another Riccati substitution:
Then
Therefore, by (2.22) and (2.23), we get
Combining (2.21) and (2.24), we have
It follows from (2.4) and
Integrating the latter inequality from
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
Theorem 2.4Suppose we have (H_{1})(H_{5}), (1.2), (1.3), and let
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
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
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
(i) Suppose that
(ii) Suppose that
We define the function u by
Then
Integrating this inequality from
Letting
That is,
Thus, we get by (2.29)
Similarly, we define another function v by
Then
Thus,
Differentiating (2.29), we obtain
By (2.3) and
Similarly, we see that
Combining (2.33) and (2.34), we get
Using (2.28), (2.35), and
Multiplying (2.36) by
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
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
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
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
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
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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