Abstract
It is the purpose of this paper to give oscillation criteria for the third-order neutral dynamic equations with continuously distributed delay,
on a time scale , where γ is the quotient of odd positive integers. By using a generalized Riccati transformation and an integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.
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1 Introduction
We are concerned with the oscillatory behavior of third-order neutral dynamic equations with continuously distributed delay,
on an arbitrary time scale , where γ is a quotient of odd positive integers. Throughout this paper, we will assume the following hypotheses:
(H1) r and q are positive rd-continuous functions on and
(H2) , ;
(H3) is not a decreasing function for η and such that
(H4) is not decreasing function for ξ and such that
(H5) the function is assumed to satisfy and there exists a positive rd-continuous function on such that , for .
Define the function by
Furthermore, (1) is like the following:
A solution of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is non-oscillatory.
Much recent attention has been given to dynamic equations on time scales, or measure chains, and we refer the reader to the landmark paper of Hilger [1] for a comprehensive treatment of the subject. Since then, several authors have expounded various aspects of this new theory; see the survey paper by Agarwal et al. [2]. A book on the subject of time scales by Bohner and Peterson [3] also summarizes and organizes much of the time scale calculus. In the recent years, there has been increasing interest in obtaining sufficient conditions for the oscillation and non-oscillation of solutions of various equations on time scales; we refer the reader to the papers [4–19]. Candan [20] considered oscillation of second-order neutral dynamic equations with distributed deviating arguments of the form
where is a ratio of odd positive integers with and real-valued rd-continuous positive functions defined on . He established some new oscillation criteria and gave sufficient conditions to ensure that all solutions of nonlinear neutral dynamic equation are oscillatory on a time scale .
To the best of our knowledge, there is very little known about the oscillatory behavior of third-order dynamic equations. Erbe et al. [21] are concerned with the oscillatory behavior of solutions of the third-order linear dynamic equation
on an arbitrary time scale , where is a positive real-valued rd-continuous function defined on . Li et al. [22] considered third-order nonlinear delay dynamic equation
on a time scale , where is quotient of odd positive integers.
Erbe et al. [23, 24] established some sufficient conditions which guarantee that every solution of the third-order nonlinear dynamic equation
and the third-order dynamic equation
oscillate or converge to zero. Li et al. [25] considered the third-order delay dynamic equations
on a time scale , where is quotient of odd positive integers, a and r are positive rd-continuous functions on , and the so-called delay function satisfies , and as , is assumed to satisfy , for , and there exists a function p on such that , for .
Saker [26] considered the third-order nonlinear functional dynamic equations
on a time scale , where is quotient of odd positive integers. Recently Han et al. [27] and Grace et al. [28] considered the third-order neutral delay dynamic equation
on a time scale .
In this paper, we consider third-order neutral dynamic equation with continuously distributed delay on time scales which is not in literature. We obtain some conclusions which contribute to oscillation theory of third-order neutral dynamic equations.
2 Several lemmas
Before stating our main results, we begin with the following lemmas which play an important role in the proof of the main results. Throughout this paper, we let
and
where we have sufficiently large .
In order to prove our main results, we will use the formula
where is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller’s chain rule (see Bohner and Peterson [3]).
Lemma 2.1 Let be a positive solution of (1), is defined as in (3). Then has only one of the following two properties:
-
(I)
, , ,
-
(II)
, , ,
with , sufficiently large.
Proof Let be a positive solution of (1) on , so that , and
Then is a decreasing function and therefore eventually of one sign, so is either eventually positive or eventually negative on . We assert that on . Otherwise, assume that , then there exists a constant , such that
By integrating the last inequality from to t, we obtain
Let . Then from (H1), we have , and therefore eventually .
Since and , we have , which contradicts our assumption . Therefore, has only one of the two properties (I) and (II).
This completes the proof. □
Lemma 2.2 Let be an eventually positive solution of (1), correspondingly has the property (II). Assume that (2) and
hold. Then .
Proof Let be an eventually positive solution of (1). Since has the property (II), then there exists finite . We assert that . Assume that , then we have for all . Choosing and using (3) and (H2), we obtain
where . Using (H5) and (6), we find from (1) that
Note that has property (II) and (H4), and we have
where , . Integrating inequality (7) from t to ∞, we obtain
Using , we obtain
Integrating inequality (8) from t to ∞, we have
Integrating the last inequality from to ∞, we obtain
Because (7) and the last inequality contradict (5), we have . Since , . This completes the proof. □
Lemma 2.3 Assume that is a positive solution of (1), is defined as in (3) such that , , on , . Then
Proof Since is strictly decreasing on , we get for
Using the definition of , we obtain
□
Lemma 2.4 Assume that is a positive solution of (1), correspondingly has the property (I). Such that , , on , . Furthermore,
Then there exists a , sufficiently large, so that
is strictly decreasing, .
Proof Let . Hence . We claim there exists a such that , on . Assume not. Then on . Therefore,
which implies that is strictly increasing on . Pick so that , for . Then
so that , for . By (1), (3), and (H2), we obtain
Using (11), (H4), and (H5), we have
where , .
Now by integrating both sides of last equation from to t, we have
This implies that
which contradicts (10). So on and consequently,
and we find that is strictly decreasing on . The proof is now complete. □
3 Main results
In this section we give some new oscillation criteria for (1).
Theorem 3.1 Assume that (2), (5), and (10) hold. Furthermore, assume that there exists a positive function , for all sufficiently large , there is a such that
Then every solution of (1) is either oscillatory or tends to zero.
Proof Assume (1) has a non-oscillatory solution on . We may assume without loss of generality that , ; , and , for all . is defined as in (3). We suppose that . We shall consider only this case, since the proof when is eventually negative is similar. Therefore Lemma 2.1 and Lemma 2.2, we get
and either for or . Let on .
By (11) and (12), we have
where , .
Define the function by the Riccati substitution
Then
From (1), the definition of and using the fact is strictly decreasing for , , it follows that
Now we consider the following two cases: and . In the first case . Using the Keller chain rule (see [3]), we have
in view of (16), Lemma 2.2, Lemma 2.3, and (9), we have
In the second case . Applying the Keller chain rule, we have
in the view of (18), Lemma 2.2, Lemma 2.3, and (9), we have
By (17), (19), and the definition of and , we have, for ,
where . Define and by
Then using the inequality [15]
which yields
From this last inequality and (20), we find
Integrating both sides from T to t, we get
which contradicts assumption (13). This completes the proof of Theorem 3.1. □
Remark 3.1 From Theorem 3.1, we can obtain different conditions for oscillation of (1) with different choices of .
Remark 3.2 The conclusion of Theorem 3.1 remains intact if assumption (13) is replaced by the two conditions
For example, let . Now Theorem 3.1 yields the following results.
Corollary 3.1 Assume that (H1)-(H5), (5), and (10) hold. If
holds, then every solution (1) is either oscillatory or .
For example, let . Now Theorem 3.1 yields the following results.
Corollary 3.2 Assume that (H1)-(H5), (5), and (10) hold. If
then every solution (1) is either oscillatory or .
Theorem 3.2 Assume that (2), (5), and (10) hold. Furthermore, suppose that there exist functions , where such that
and H has a nonpositive continuous Δ-partial derivative with respect to the second variable and satisfies
and for all sufficiently large , there is a such that
where ρ is a positive Δ-differentiable function and
Then every solution of (1) is either oscillatory or tends to zero.
Proof Suppose that is a non-oscillatory solution of (1) and is defined as in (3). Without loss of generality, we may assume that there is a sufficiently large so that the conclusions of Lemma 2.1 hold and (24) holds for . If case (1) of Lemma 2.1 holds then proceeding as in the proof of Theorem 3.1, we see that (20) holds for . Multiplying both sides of (20) by and integrating from T to , we get
Integrating by parts and using , we obtain
It then follows from (26) that
It then follows from (24) that
Therefore, as in Theorem 3.1, by letting
Then using the inequality [15]
We have
Then for we have
and this implies that
for all large T, which contradicts (25). This completes the proof of Theorem 3.2. □
Remark 3.3 The conclusion of Theorem 3.2 remains intact if assumption (25) is replaced by the two conditions
Remark 3.4 Define w as (14), we also get
similar to the proofs of Theorem 3.1, we can obtain different results. We leave the details to the reader.
Example 3.1 Consider the following third-order neutral dynamic equation :
where , , , , , , , .
It is clear that condition (2), (5), and (10) hold. Therefore, by Theorem 3.1, picking , we have
Hence, by Theorem 3.1 every solution of (27) is oscillatory or tends to zero if .
Example 3.2 Consider the following third-order neutral dynamic equation :
where , , , , , , .
It is clear that condition (2), (5), and (10) hold. Therefore, by Theorem 3.1, picking , we have
Hence, by Theorem 3.1 every solution of (28) is oscillatory or tends to zero if .
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Şenel, M.T., Utku, N. Oscillation criteria for third-order neutral dynamic equations with continuously distributed delay. Adv Differ Equ 2014, 220 (2014). https://doi.org/10.1186/1687-1847-2014-220
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DOI: https://doi.org/10.1186/1687-1847-2014-220