Oscillation criteria for certain third-order delay dynamic equations
© Yang; licensee Springer 2013
Received: 3 February 2013
Accepted: 6 April 2013
Published: 19 June 2013
This paper is concerned with the oscillatory behavior of a certain class of third-order nonlinear variable delay neutral functional dynamic equations,
on a time scale T with , where , , . By using the generalized Riccati transformation and a lot of inequality techniques, some new oscillation criteria for the equations are established, results are presented that not only complement and improve those related results in the literature, but also improve some known results for a third-order delay dynamic equation with a neutral term. Further, the main results improve some related results for third-order neutral differential equations. Some examples are given to illustrate the importance of our results.
MSC:34K11, 34C10, 39A10.
where , , . Throughout this article, we assume that:
(H1): T is an arbitrary time scale with , and with , we define the time scale interval by . , i.e., are rd-continuous functions. are continuous functions with () and ().
(H2): are delay functions with , and , .
(H3): , , and , and .
(H4): There exist constants and , such that (), ().
(H5): , .
We recall that a solution of equation (1.1) is said to be oscillatory on if it is neither eventually positive nor eventually negative; otherwise, the solution is said to be nonoscillatory. Equation (1.1) is said to be oscillatory if all of its solutions are oscillatory. Our attention is restricted to those solutions of (1.1) where is not eventually identically zero.
Equations of this type arise in a number of important applications such as problems in biological population dynamics, in neural network, in quantum theory, in computer science and in control theory. Hence, it is important and useful to study the oscillatory properties of solutions of equation (1.1). Recently, there has been an increasing interest in studying the oscillatory behavior of first and second-order dynamic equations on time scales (see [1–7]). However, there are very few results regarding the oscillation of third-order equations. Among these papers dealing with the subject, we refer in particular to [8–17], the monographs [1, 2] and the references therein. Our concern is especially motivated by several recent papers such as [9–13].
and obtained the result that every solution of equation (1.6) oscillates or converges to zero.
Therefore, this topic is fairly new for dynamic equations on time scales. The purpose of this article is to obtain new oscillation criteria for the oscillation of (1.1), these criteria can improve the restriction of the conditions for the equation, which promote some existing results. We should note that many of our results of this article are new for the corresponding third-order nonlinear differential and difference equations. In fact, the obtained results extend, unify and correlate many of the existing results in the literature.
We shall employ the following lemmas.
Lemma 2.1 
Lemma 2.2 
, where , .
, , , .
Then, for every , there exists a constant , , such that ().
Lemma 2.3 
Suppose that a and b are nonnegative real numbers, then for all , where the equality holds if and only if .
Lemma 2.4 
, , , ,
, , , .
which contradicts with . So, , this implies that or . This completes the proof. □
holds, then .
which contradicts with . So , in view of , hence .
which contradicts with . So, , further, . This completes the proof. □
Due to the above reasons, in the next section, we assume that either (2.4) or (2.5) holds.
Lemma 2.7  (Hölder’s inequality)
Let and . For rd-continuous functions , we have , where and .
3 Main results
In this section, we establish some sufficient conditions which guarantee that every solution of (1.1) either oscillates on or converges as .
where , then every solution of equation (1.1) is either oscillatory or .
Proof Suppose that equation (1.1) has a nonoscillatory solution on . We may assume without loss of generality that and , for all , . Then, by Lemma 2.5, we see that satisfies either case (i) or case (ii).
Taking limsup on both sides of the above inequality as , we obtain a contradiction to condition (3.1).
If case (ii) in Lemma 2.5 holds, then by Lemma 2.6, we have . This completes the proof. □
Remark 3.1 From Theorem 3.1, we can obtain different conditions for oscillation of all solutions of (1.1) with different choices of . For example, (where M is a constant) or (). Then we have the following results respectively.
Corollary 3.2 If , then every solution of equation (1.1) is either oscillatory or .
Corollary 3.3 If , then every solution of equation (1.1) is either oscillatory or .
for some constant , where the function is defined as in Theorem 3.1, then every solution of equation (1.1) is either oscillatory or .
contradicting (3.12). This completes the proof. □
Remark 3.2 From Theorem 3.4, we can obtain different conditions for oscillation of all solutions of (1.1) with different choices of . For example, (where M is a constant) or (), then we have the following results, respectively.
for some constant , then every solution of equation (1.1) is either oscillatory or .
for some constant , then every solution of equation (1.1) is either oscillatory or .
Remark 3.3 Clearly, Kamenev-type oscillation criteria for second-order linear differential equation was extended to third-order nonlinear variable delay dynamic equations on time scales. One can easily see that the recent results cannot be applied in equation (1.1), so our results are new ones.
If (3.12) does not hold, then we have the following result.
for some constant , where , is defined as in Theorem 3.1, then every solution of equation (1.1) is either oscillatory or .
which contradicts (3.17). This completes the proof. □
Obviously, our results in this paper not only extend and improve some known results, and show some results of [3–8, 10, 14, 15] to be special examples of our results, but also unify the oscillation of the third-order nonlinear variable delay differential equations and the third-order nonlinear variable delay difference equations with a nonlinear neutral term. The theorems in this paper are new even for the cases and .
4 Applications and examples
In this section, we give some examples to illustrate our main results.
It is easy to verify that all conditions of Corollary 3.3 are satisfied. Hence, every solution of equation (4.1) is oscillatory or tends to zero as . For example, it is not difficult to verify that is a solution of equation (4.1). The important point to note here is that the recent results due to [9–13, 15] do not apply to equation (4.1) for the condition (1.7) or (1.8) can be a restrictive condition.
and so conditions of Corollary 3.6 are satisfied as well. Altogether, by Corollary 3.6, we have that every solution of equation (4.2) is oscillatory or tends to zero as . But the results in [9–16] are inapplicable for equation (4.2).
This work was supported by the Natural Science Foundation of Hunan Province (12JJ6006) and Hunan Province Science and Technology Project (2012FJ3107) and Scientific Research Fund of Hunan Provincial Education Department (09A082).
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