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On oscillation of second-order neutral dynamic equations
Advances in Difference Equations volume 2013, Article number: 340 (2013)
a new oscillation criterion for a half-linear second-order neutral dynamic equation
is presented. An interesting example is provided to show that the delayed function plays an important role in the oscillatory behavior.
MSC:34K11, 34N05, 39A10.
This work is concerned with the oscillatory behavior of solutions to a second-order half-linear neutral delay dynamic equation
on a time scale , where . In what follows, we assume that is a quotient of odd positive integers, , , , , r, p, and q are real-valued rd-continuous functions defined on such that , , , and for .
By a solution of (1.1) we mean a nontrivial real-valued function x which has the properties and , and satisfying (1.1) for all . Our attention is restricted to those solutions of (1.1) which exist on some half-line and satisfy for any . A solution x of (1.1) is said to be 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 are oscillatory.
The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in order to unify continuous and discrete analysis (see Hilger ). Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al.  and the references cited therein. The books on the subject of time scales, by Bohner and Peterson [3, 4], summarize and organize much of time scale calculus and some applications of dynamic equations on time scales.
In recent years, there has been much research activity concerning the oscillatory and nonoscillatory behavior of solutions to various classes of differential, difference, and dynamic equations. We refer the reader to [5–28] and the references therein. If , then , , , , and equation (1.1) becomes a second-order neutral delay differential equation
If , then , , , , and equation (1.1) reduces to a second-order neutral delay difference equation
If where , then , , , , and equation (1.1) becomes a second-order neutral delay difference equation
In what follows, we present some background details that motivate the contents of this note. Jiang and Li  studied oscillatory properties of equation (1.2) in the case . Saker  and Sun and Saker  considered oscillation of equation (1.3) and established some criteria under the assumption that . To the best of our knowledge, there are few results for oscillation of equation (1.3) in the case where . Agarwal et al. , Saker , and Tripathy  studied oscillatory behavior of (1.1) in the case where . In particular, Tripathy  established some oscillation tests for (1.1) assuming that .
The purpose of this paper is to obtain a new sufficient condition for oscillation of (1.1). The result obtained is essentially new for equations (1.2)-(1.4). This paper is organized as follows. In Section 2, we use the Riccati transformation technique to prove the main results. In Section 3, we apply our criterion in equations (1.2)-(1.4) to establish some new oscillation criteria.
2 Oscillation results
To prove the main results, we use the formula
which is a simple consequence of Keller’s chain rule (see [, Theorem 1.90]). We also need the following lemma.
Lemma 2.1 (See [, Lemma 1])
Assume and . If , , then
In what follows, all functional inequalities are assumed to be satisfied for all sufficiently large t.
Theorem 2.2 Let . Suppose also that
Then equation (1.1) is oscillatory.
Proof Suppose to the contrary that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that there exists such that for all , where . Set
for . Hence, is of one sign. On the other hand, we have by (1.1) and [, Theorem 1.93] that
Thus, we get by (2.2) that
If , then by (2.5) there exist a constant and such that for ,
Integrating the latter inequality from to t, we have
which contradicts (2.3).
Assume now that . Define a Riccati substitution
Then . Noticing that is decreasing, we have
Dividing the latter inequality by and integrating the resulting inequality from t to l, we get
Note that . Letting in the above inequality, we obtain , which implies that
Then we have by (2.6) that
Similarly, we define another Riccati transformation
Then . Noting that is decreasing, we have , and so . From (2.7), we have
Differentiating (2.6), we get
By virtue of (2.1), we obtain
Using (2.11) in (2.10), we have
By (2.6) and the latter inequality, we find
due to . Differentiating (2.8), we obtain
Using (2.11) in (2.13), we have
From (2.8), (2.14), and the fact that , we get
Thus, by (2.12) and (2.15), we have
Using and (2.5) yields
Multiplying the latter inequality by and integrating the resulting inequality on ( sufficiently large), we obtain
Hence, by (2.1) and , we get
Similarly, we have
Therefore, we obtain by (2.16) that
Using the inequality
where is a constant, we get
Similarly, we have
Therefore, (2.17) implies that
due to (2.7) and (2.9), which contradicts (2.4). The proof is complete. □
Remark 2.3 Theorem 2.2 complements the results obtained in  since it can be applied in the case where .
Due to Theorem 2.2, we present the following results for oscillation of equations (1.2)-(1.4).
Corollary 3.1 Assume and let
where Q is defined as in Theorem 2.2 and
Then equation (1.2) oscillates.
Corollary 3.2 Assume and let
where Q is defined as in Theorem 2.2 and
Then equation (1.3) oscillates.
Corollary 3.3 Assume , , and let
where Q is defined as in Theorem 2.2 and
Then equation (1.4) oscillates.
Example 3.4 For , consider a second-order neutral delay differential equation
It is not difficult to verify that all conditions of Theorem 2.2 are satisfied. Hence, equation (3.1) is oscillatory. However, the equation
has a nonoscillatory solution . Therefore, the delayed argument plays an important role in oscillation of equation (1.1).
Remark 3.5 It would be of interest to find another method to study the equation
where or .
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The authors are grateful to reviewers for their comments and suggestions. This research is supported by Natural Science Foundation of Shandong Province (Z2007F08) and also by Weifang University Research Funds for Doctors (2012BS25).
The authors declare that they have no competing interests.
All authors contributed equally to this paper and also read and approved the final manuscript.
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Ji, T., Tang, S. & Thandapani, E. On oscillation of second-order neutral dynamic equations. Adv Differ Equ 2013, 340 (2013). https://doi.org/10.1186/1687-1847-2013-340
- neutral dynamic equation
- time scale