1. Introduction

The study of dynamic equations on time-scales, which goes back to its founder Hilger [1], is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time-scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations.

Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2], Bohner and Guseinov [3], and references cited therein. A book on the subject of time-scales, by Bohner and Peterson [4], summarizes and organizes much of the time-scale calculus; see also the book by Bohner and Peterson [5] for advances in dynamic equations on time-scales.

In the recent years, there has been increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of various equations on time-scales; we refer the reader to the papers [6–38]. To the best of our knowledge, it seems to have few oscillation results for the oscillation of third-order dynamic equations; see, for example, [14–16, 21, 35]. However, the paper which deals with the third-order delay dynamic equation is due to Hassan [21].

Hassan [21] considered the third-order nonlinear delay dynamic equations

(1.1)

where is required, and the author established some oscillation criteria for (1.1) which extended the results given in [16].

To the best of our knowledge, there are no results regarding the oscillation of the solutions of the following third-order nonlinear neutral delay dynamic equations on time-scales up to now:

(1.2)

We assume that is a quotient of odd positive integers, and are positive real-valued rd-continuous functions defined on such that the delay functions are rd-continuous functions such that and

As we are interested in oscillatory behavior, we assume throughout this paper that the given time-scale is unbounded above. We assume and it is convenient to assume We define the time-scale interval of the form by .

For the oscillation of neutral delay dynamic equations on time-scales, Mathsen et al. [26] considered the first-order neutral delay dynamic equations on time-scales

(1.3)

and established some new oscillation criteria of (1.3) which as a special case involve some well-known oscillation results for first-order neutral delay differential equations.

Agarwal et al. [7], Şahíner [28], Saker [31], Saker et al. [33], Wu et al. [34] studied the second-order nonlinear neutral delay dynamic equations on time-scales

(1.4)

by means of Riccati transformation technique, the authors established some oscillation criteria of (1.4).

Saker [32] investigated the second-order neutral Emden-Fowler delay dynamic equations on time-scales

(1.5)

and established some new oscillation for (1.5).

Our purpose in this paper is motivated by the question posed in [26]: What can be said about higher-order neutral dynamic equations on time-scales and the various generalizations? We refer the reader to the articles [23, 24] and we will consider the particular case when the order is 3, that is, (1.2). Set By a solution of (1.2), we mean a nontrivial real-valued function satisfying and and satisfying (1.2) for all

The paper is organized as follows. In Section 2, we apply a simple consequence of Keller's chain rule, devoted to the proof of the sufficient conditions which guarantee that every solution of (1.2) oscillates or converges to zero. In Section 3, some examples are considered to illustrate the main results.

2. Main Results

In this section we give some new oscillation criteria for (1.2). In order to prove our main results, we will use the formula

(2.1)

where is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller's chain rule (see Bohner and Peterson [4, Theorem ]).

Before stating our main results, we begin with the following lemmas which are crucial in the proofs of the main results.

For the sake of convenience, we denote: for Also, we assume that

there exists such that and

Lemma 2.1.

Assume that holds. Further, assume that is an eventually positive solution of (1.2). If

(2.2)

then there are only the following three cases for sufficiently large:

, , ,

or

,

Proof.

Let be an eventually positive solution of (1.2). Then there exists such that and for all From (1.2) we have

(2.3)

Hence is strictly decreasing on We claim that eventually. Assume not, then there exists such that

(2.4)

Then we can choose a negative and such that

(2.5)

Dividing by and integrating from to we have

(2.6)

Letting then by (2.2). Thus, there is a such that for

(2.7)

Integrating the previous inequality from to we obtain

(2.8)

Therefore, there exist and such that

(2.9)

We can choose some positive integer such that for Thus, we obtain

(2.10)

The above inequality implies that for sufficiently large which contradicts the fact that eventually. Hence we get

(2.11)

It follows from this that either or Since

(2.12)

which yields

(2.13)

If then there are two possible cases:

(1) eventually; or

(2) eventually.

If there exists a such that case (2) holds, then exists, and We claim that Otherwise, We can choose some positive integer such that for Thus, we obtain

(2.14)

which implies that and from the definition of we have which contradicts Now, we assert that is bounded. If it is not true, there exists with as such that

(2.15)

From

(2.16)

which implies that it contradicts that Therefore, we can assume that

(2.17)

By we get

(2.18)

which implies that so hence,

Assume that We claim that eventually. Otherwise, we have or By there exists we can choose some positive integer such that for and we obtain

(2.19)

which implies that and from the definition of we have which contradicts or Now, we have that here is finite. We assert that is bounded. If it is not true, there exists with as such that

(2.20)

From

(2.21)

which implies that it contradicts that Therefore, we can assume that

(2.22)

By we get

(2.23)

which implies that so hence, This completes the proof.

In [4, Section ] the Taylor monomials are defined recursively by

(2.24)

It follows from [4, Section ] that for any time-scale, but simple formulas in general do not hold for

Lemma 2.2 (see [15, Lemma ]).

Assume that satisfies case (i) of Lemma 2.1. Then

(2.25)

Lemma 2.3.

Assume that is a solution of (1.2) satisfying case (i) of Lemma 2.1. If

(2.26)

then satisfies eventually

(2.27)

Proof.

Let be a solution of (1.2) such that case of Lemma 2.1 holds for Define

(2.28)

Thus

(2.29)

We claim that eventually. Otherwise, there exists such that for Therefore,

(2.30)

which implies that is strictly increasing on Pick such that for Then we have

(2.31)

then for By Lemma 2.2, for any there exists such that

(2.32)

Hence there exists so that

(2.33)

By the definition of we have that

(2.34)

From (1.2), we obtain

(2.35)

Integrating both sides of (2.35) from to we get

(2.36)

which yields that

(2.37)

which contradicts (2.26). Hence and is nonincreasing. The proof is complete.

Lemma 2.4.

Assume that holds and is a solution of (1.2) which satisfies case (iii) of Lemma 2.1. If

(2.38)

where for then

Proof.

Let be a solution of (1.2) such that case of Lemma 2.1 holds for Then , Next we claim that Otherwise, there exists such that for all By the definition of we have that (2.35) holds. Integrating both sides of (2.35) from to we get

(2.39)

Integrating again from to we have

(2.40)

Integrating again from to we obtain

(2.41)

which contradicts (2.38), since by [23, Lemma ] and [3, Remark ], we get

(2.42)

Hence and completes the proof.

Theorem 2.5.

Assume that (2.2), (2.26), and (2.38) hold, Furthermore, assume that there exists a positive function such that for some and for all constants

(2.43)

where Then every solution of (1.2) oscillates or

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case Define the function by

(2.44)

Then Using the product rule, we have

(2.45)

By the quotient rule, we get

(2.46)

By the definition of and (1.2), we obtain (2.35). From (2.35) and (2.44), we have

(2.47)

from (2.25) and (2.27), for any we obtain

(2.48)

hence by (2.48), we have

(2.49)

In view of from (2.1) and of Lemma 2.1, we have

(2.50)

where By (2.49), we have

(2.51)

from we have by we have

(2.52)

so we get

(2.53)

by (2.44), we have

(2.54)

Therefore, we obtain

(2.55)

Integrating inequality (2.55) from to , we obtain

(2.56)

which yields

(2.57)

for all large which contradicts (2.43). If holds, from Lemma 2.1, then If case holds, by Lemma 2.4, then The proof is complete.

Remark 2.6.

From Theorem 2.5, we can obtain different conditions for oscillation of all solutions of (1.2) with different choices of .

For example, let Now Theorem 2.5 yields the following result.

Corollary 2.7.

Assume that (2.2), (2.26), and (2.38) hold, If

(2.58)

holds for some and for all constants then every solution of (1.2) is either oscillatory or

For example, let From Theorem 2.5, we have the following result which can be considered as the extension of the Leighton-Wintner Theorem.

Corollary 2.8.

Assume that (2.2), (2.26), and (2.38) hold, and If

(2.59)

then every solution of (1.2) is either oscillatory or

In the following theorem, we present a new Kamenev-type oscillation criteria for (1.2).

Theorem 2.9.

Assume that (2.2), (2.26), and (2.38) hold, Let and be as defined in Theorem 2.5. If for some and for all constants

(2.60)

where and

(2.61)

then every solution of (1.2) oscillates or

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case We proceed as in the proof of Theorem 2.5 to get (2.54) for all sufficiently large. Multiplying (2.54) by and integrating from to we have

(2.62)

Integration by parts, we obtain

(2.63)

Next, we show that if and then

(2.64)

If it is easy to see that (2.64) is an equality. If then we get

(2.65)

Using the inequality

(2.66)

we obtain for

(2.67)

and from this we see that (2.64) holds. From (2.62)–(2.64), we get

(2.68)

Thus

(2.69)

which implies that

(2.70)

This easily leads to a contradiction of (2.60). If holds, from Lemma 2.1, then If holds, by Lemma 2.4, then The proof is complete.

In the following theorem, we present a new Philos-type oscillation criteria for (1.2).

Theorem 2.10.

Assume that (2.2), (2.26), and (2.38) hold, Let and be as defined in Theorem 2.5. Furthermore, assume that there exist functions , , where such that

(2.71)

and has a nonpositive continuous -partial derivation with respect to the second variable and satisfies

(2.72)

If for some and for all constants

(2.73)

where

(2.74)

where then every solution of (1.2) oscillates or

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case We proceed as in the proof of Theorem 2.5 to get (2.54) for all sufficiently large. Multiplying both sides of (2.54), with replaced by by integrating with respect to from to we have

(2.75)

Integrating by parts and using (2.71) and (2.72), we obtain

(2.76)

Therefore, we get

(2.77)

This easily leads to a contradiction of (2.73). If case holds, from Lemma 2.1, then If case holds, by Lemma 2.4, then The proof is complete.

The following result can be considered as the extension of the Atkinson's theorem [39].

Theorem 2.11.

Assume that (2.2), (2.26), and (2.38) hold, If

(2.78)

then every solution of (1.2) is either oscillatory or

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case Define the function

(2.79)

Using the product rule, (2.25) and (2.27), for any we have that

(2.80)

By (1.2), we have that (2.35) holds, then from (2.80), we calculate

(2.81)

where the last inequality is true because due to (2.1) and because

(2.82)

Upon integration we arrive at

(2.83)

from This contradicts (2.78). If case holds, from Lemma 2.1, then If case holds, by Lemma 2.4, then The proof is complete.

Theorem 2.12.

Assume that (2.2), (2.26), and (2.38) hold, Furthermore, assume that there exists a positive function such that for some and for all constants

(2.84)

where is as defined as in Theorem 2.5. Then every solution of (1.2) is either oscillatory or

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume satisfies case Define the function as (2.44). We proceed as in the proof of Theorem 2.5 and we get (2.49). In view of from (2.1) and of Lemma 2.1, we have

(2.85)

from (2.27), there exists a constant such that so

(2.86)

By (2.49), we have

(2.87)

Therefore, we obtain

(2.88)

Integrating inequality (2.88) from to , we obtain

(2.89)

which yields

(2.90)

for all large which contradicts (2.84). If case holds, from Lemma 2.1, then If case holds, by Lemma 2.4, then The proof is complete.

Remark 2.13.

From Theorem 2.12, we can obtain different conditions for oscillation of all solutions of (1.2) with different choices of .

For example, let Now Theorem 2.12 yields the following results.

Corollary 2.14.

Assume that (2.2), (2.26), and (2.38) hold, If

(2.91)

holds for some and for all constants then every solution of (1.2) is either oscillatory or

For example, let From Theorem 2.12, we have the following result which can be considered as the extension of the Leighton-Wintner theorem.

Corollary 2.15.

Assume that (2.2), (2.26), and (2.38) hold, If (2.59) holds, then every solution of (1.2) is either oscillatory or

In the following theorem, we present a new Kamenev-type oscillation criteria for (1.2).

Theorem 2.16.

Assume that (2.2), (2.26), and (2.38) hold, Let and be as defined in Theorem 2.12. If for some and for all constants

(2.92)

where and

(2.93)

then every solution of (1.2) oscillates or

The proof is similar to that of Theorem 2.9 using inequality (2.88), so we omit the details.

In the following theorem, we present a new Philos-type oscillation criteria for (1.2).

Theorem 2.17.

Assume that (2.2), (2.26), and (2.38) hold, Let and be as defined in Theorem 2.12. Furthermore, assume that there exist functions , , where such that (2.71) holds, and has a nonpositive continuous -partial derivation with respect to the second variable and satisfies (2.72). If

(2.94)

holds for some and for all constants where

(2.95)

where Then every solution of (1.2) oscillates or

The proof is similar to that of the proof of Theorem 2.10 using inequality (2.88), so we omit the details.

The following result can be considered as the extension of the Belohorec's theorem [40].

Theorem 2.18.

Assume that (2.2), (2.26), and (2.38) hold If

(2.96)

then every solution of (1.2) is either oscillatory or satisfies

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case From and (2.1) we have

(2.97)

so

(2.98)

By (1.2), we have that (2.35) holds. Using (2.25) and (2.27), for any we obtain after dividing (2.35) by for all large

(2.99)

So,

(2.100)

Upon integration we arrive at

(2.101)

This contradicts (2.96). If case holds, from Lemma 2.1, then If case holds, by Lemma 2.4, then The proof is complete.

Remark 2.19.

One can easily see that the results obtained in [14–16, 21, 23, 24, 35] cannot be applied in (1.2), so our results are new.

3. Examples

In this section we give the following examples to illustrate our main results.

Example 3.1.

Consider the third-order neutral delay dynamic equations on time-scales

(3.1)

where is a quotient of odd positive integers,

Let . It is easy to see that (2.2), (2.26), and (2.38) hold. Also

(3.2)

Hence by Corollary 2.8, every solution of (3.1) is either oscillatory or

Example 3.2.

Consider the third-order neutral delay differential equation

(3.3)

Let . It is easy to see that all the conditions of Corollary 2.8 hold. Then by Corollary 2.8, every solution of (3.3) is either oscillatory or satisfies In fact, is a solution of (3.3).

Example 3.3.

Consider the third-order delay dynamic equation

(3.4)

where is a quotient of odd positive integers.

For , we have . Let . It is easy to see that (2.2) and (2.38) hold, and

(3.5)

Hence (2.26) holds. Also

(3.6)

so (2.78) holds. By Theorem 2.11, every solution of (3.4) is either oscillatory or satisfies .

Example 3.4.

Consider the third-order delay dynamic equation

(3.7)

where is a quotient of odd positive integers.

Let It is easy to see that (2.2), (2.26), and (2.38) hold. Also we have

(3.8)

Hence (2.96) holds. By Theorem 2.18, every solution of (3.7) is either oscillatory or satisfies