Abstract
The purpose of this article is to give oscillation criteria for the third-order neutral dynamic equation where is a ratio of odd positive integers with , , and are positive real-valued rd-continuous functions defined on . We give new results for the third-order neutral dynamic equations and an example to illustrate the importance of our results.
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1 Introduction
In the present article, we are concerned with oscillations of the third-order nonlinear neutral dynamic equation
on a time scale . Throughout this paper it is assumed that is a ratio of odd positive integers, and are rd-continuous functions such that , , and is rd-continuous, , and are positive real valued rd-continuous functions defined on , is increasing. We define the time scale interval by . Furthermore, is a continuous function such that for all and there exists a rd-continuous positive function defined on such that .
We use throughout this paper the following notations for convenience and for shortening the equations:
A nontrivial function is said to be a solution of (1) if , and for and satisfies equation (1) for . A solution of (1) which is nontrivial for all large t is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory.
Recently, there has been many important research activity on the oscillatory behavior of dynamic equations. For example, on second-order dynamic equations, Saker [1], and Agarwal et al. [2], Saker [3], Hassan [4] and Candan [5, 6] considered the following nonlinear dynamic equations:
and
respectively, and they gave sufficient conditions which guarantee that every solution of the equation oscillates. Moreover, there are also some papers on third-order dynamic equations. For instance, Erbe et al. [7] considered the third-order nonlinear dynamic equation
Later, Erbe et al. [8] considered the third-order nonlinear dynamic equation
by giving Hille and Nehari type criteria. Then, Hassan [9] studied the third-order nonlinear dynamic equation
Lastly, Wang and Xu [10] studied asymptotic properties of a certain third-order dynamic equation,
As we see from all the above, our equation, a neutral dynamic equation, is more general than other third-order dynamic equations and therefore it is very important. For some other important articles on oscillations of second-order nonlinear neutral delay dynamic equation on time scales and oscillations of third-order neutral differential equations, we refer the reader to the papers [11, 12], and [13], respectively. We give [14, 15] as references for books on the time scale calculus.
2 Main results
Lemma 1 Assume that y is an eventually positive solution of (1) and
Then, there is a such that either
or
Proof Assume that for and therefore and for . Consequently, , eventually. Using (2) in (1) and the fact that , we obtain
Hence, we conclude that is a strictly decreasing function on . We claim that on . If not, then there exists a such that on . Then, there exist a negative constant c and such that
and it follows that
Integrating (5) from to t and using (3), we obtain
which implies that as . Therefore, there exists a such that
Dividing both sides of (6) by and integrating from to t, we obtain
Hence, we see from (3) that as , which contradicts the fact that , and therefore for . As a result of for it follows that on or on , which completes the proof. □
Lemma 2 Let y be an eventually positive solution of (1). Assume that Case (i) of Lemma 1 holds. Then, there exists a such that
where and
where .
Proof Since is strictly decreasing on , we have
it follows that
or
Similarly, integrating (8) from to t, we obtain
or
This completes the proof. □
Lemma 3 Let y be an eventually positive solution of (1). Assume that Case (ii) of Lemma 1 holds. If
then .
Proof Since Case (ii) of Lemma 1 is satisfied,
We claim that . Assume that . Then for any , we have for sufficiently large . Choose . On the other hand, since
we have
where . Then,
Substituting (10) into (4), we obtain
Integrating (11) from t to ∞, we get
or using ,
Integrating (12) from t to ∞ and dividing both sides by , we have
Integrating (13) from to ∞, we obtain
which contradicts (9) and therefore . By making use of , we conclude that . □
Theorem 2.1 Assume that . Furthermore, suppose that (3), (9), and
where , hold. Then, every solution of (1) is either oscillatory on or .
Proof Assume that (1) has a nonoscillatory solution; without loss of generality we may suppose that for and therefore and for . In the case when is negative the proof is similar. As we see from Lemma 1 we have two cases to consider. First we assume that satisfies Case (i) in Lemma 1. Then, by using (2) we see that
or
Substituting (15) into (4), we obtain
Furthermore, using Pötzche’s chain rule, we find
Define the function
It is easy to see that . Taking the derivative of , we see that
Substituting (16) into (19) and using (17), respectively, we have
Using (7) in (20) and the fact that is strictly decreasing, we obtain from (18)
Finally, integrating (21) from to t, we get
and consequently
which contradicts (14). When Case (ii) holds, we can conclude from Lemma 3 that . □
Theorem 2.2 Suppose that (3), (9) hold and . Furthermore, assume that there exists a positive rd-continuous △-differentiable function such that
where and . Then, every solution of (1) is either oscillatory on or .
Proof Suppose to the contrary that is nonoscillatory solution of (1). We assume that for , then and for . We first consider that satisfies Case (i) in Lemma 1. We proceed as in the proof of Theorem 2.1, and we obtain (16). Let us define the function
It is clear that . Taking the derivative of , we see that
Now using (16) in (24), we obtain
Substituting (17) into (25), we obtain
By using (7) into (26), we obtain
where . Let
and
By making use of the inequality
in (27), we have
Integrating both sides of (29) from to t then yields
which contradicts (22).
When Case (ii) holds, we can conclude from Lemma 3 that . □
Let and . The function is said to belong to class ℜ if , , on and H has a continuous △-partial derivative on with respect to the second variable.
Theorem 2.3 Assume that (3) and (9) hold and . Furthermore, is defined as in Theorem 2.2 and such that
where . Then every solution of (1) is either oscillatory on or .
Proof Assume that is a nonoscillatory solution of (1). Define as in (23). We proceed as in the proof of Theorem 2.2 to obtain (27). Multiplying both sides of (27) by , integrating with respect to s from to t, we get
where . Integrating by parts yields by (31)
Let
and
Then, using the inequality (28) in (32), we have
or
which contradicts (30) and completes the proof.
When Case (ii) holds, we can conclude from Lemma 3 that . □
Example 2.4 Consider the following third-order neutral nonlinear dynamic equation:
where , , , , and . We can verify that all conditions of Theorem 2.1 are satisfied, therefore every solution of (33) is oscillatory or . In fact, is a solution of (33).
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Candan, T. Asymptotic properties of solutions of third-order nonlinear neutral dynamic equations. Adv Differ Equ 2014, 35 (2014). https://doi.org/10.1186/1687-1847-2014-35
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DOI: https://doi.org/10.1186/1687-1847-2014-35