Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 6847956, 9 pages
http://dx.doi.org/10.1155/2016/6847956

Research Article

Comparison Criteria for Nonlinear Functional Dynamic Equations of Higher Order

Taher S. Hassan1,2 and Said R. Grace³

¹Department of Mathematics, Faculty of Science, University of Hail, Hail 2440, Saudi Arabia
²Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
³Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt

Received 9 February 2016; Accepted 22 May 2016

Academic Editor: Yuriy V. Rogovchenko

Copyright © 2016 Taher S. Hassan and Said R. Grace. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We will consider the higher order functional dynamic equations with mixed nonlinearities of the form , on an above-unbounded time scale , where , ,   with  , and . The function is a rd-continuous function such that for . The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.

1. Introduction

In this paper, we consider comparison criteria for higher order nonlinear dynamic equation with mixed nonlinearities of the form

on an above-unbounded time scale , where

(i) is an integer, and , , with , , and ;(ii) for .

Without loss of generality we assume . For and , we denote by the space of right-dense continuous functions from to and by the set of functions in with right-dense continuous -derivatives. Throughout this paper we make the following assumptions:

(iii) and , are constants and for , such that

(iv) such that , on .(v) rd-continuous function such that , , and we let be a nondecreasing function on .

Recall that the knowledge and understanding of time scales and time scale notation are assumed. For an excellent introduction to the calculus on time scales, see [1–3]. By a solution of (1) we mean a nontrivial real-valued function for some such that , and satisfies (1) on , where is the space of right-dense continuous functions. An extendable solution of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise it is said to be nonoscillatory.

In the last few years, there has been an increasing interest in obtaining sufficient conditions for the oscillation/nonoscillation of solutions of different classes of dynamic equations; we refer the reader to [4–13] and the references cited therein. Special cases of (1) have been studied by many authors. When and for , and , Grace et al. [14] established some oscillation criteria for higher order nonlinear dynamic equation of the form

where is the ratio of positive odd integers. In paper by Grace [15], some new criteria for the oscillation of the even order dynamic equation

where and are the ratios of positive odd integers, were given. Recently, Hassan and Kong [16] obtained asymptotics and oscillation criteria for the th-order half-linear dynamic equation with deviating argument

and Grace and Hassan [17] establish oscillation criteria for more general higher order dynamic equation

The purpose of this paper is to derive comparison criteria for higher order nonlinear dynamic equation with mixed nonlinearities (1).

2. Preliminaries

We will employ the following lemmas. Consider the inequality

where and are positive real-valued, rd-continuous functions on and satisfies condition (2), is a rd-continuous function and as , and and are positive real numbers.

Now, we present the following lemma.

Lemma 1. If inequality (8) has an eventually positive solution, then the equation

has also an eventually positive solution.

Proof. Let be an eventually positive solution of inequality (8). It is easy to see that eventually. Let be sufficiently large so that , , and for . Then, in view of

there is such that , for . Inequality (8) becomes

Integrating (11) from to and letting , we have

where

Now, we define a sequence of successive approximations as follows:

It is easy to show that

Then, the sequence is nonincreasing and bounded for each . This means that we may define . Since

we find that

By Lebesgue’s dominated convergence theorem on time scale, one can easily find

Therefore

where

Then

Equation (19) then gives

Hence (9) has a positive solution . This completes the proof.

The second one is cited from [18, 19].

Lemma 2. Assume that (3) holds. Then there exists an -tuple with satisfying

The next lemma is cited from [17] and improves the well-known lemma of Kiguradze.

Lemma 3. Assume that (2) holds. If (1) has an eventually positive solution , then there exists an integer with being odd such that

eventually, and

eventually.

3. Main Results

In the following main theorem, we will use the following notations: for and for and, for any , define , , and , , by the following recurrence formulas:

with and , provided the improper integrals involved are convergent.

Theorem 4. Assume that for sufficiently large the first-order dynamic equation

is oscillatory, where

for every number with being odd. Then

(1)if is even, every solution of (1) is oscillatory,(2)if is odd and

then every solution of (1) either is oscillatory or tends to zero eventually.

Proof. Assume that (1) has a nonoscillatory solution on . Then, without loss of generality, and , for and . By Lemma 3, there exists an integer , , with being odd such that (24) and (25) hold for .
(I) When , from (1), we get

From (23), we have

Using the arithmetic-geometric mean inequality (see [20, Page 17]), we have

Then for ,

This together with (30) shows that

Integrating above inequality from to and then using the fact that is strictly increasing and is nondecreasing, we get

and by (25) we see that . Hence by taking limits as we have

which implies

Integrating above inequality (37) from to and letting , we get

Continuing this process times, we find

Also, from (24) and (25), we get

It follows that

Then for ,

Analogously, we have

Then for

From (39) and (44), we get

Let ; we get

or

In view of Corollary 2.3.5 in [21], there exists a positive solution of (27) which contradicts the assumption of the theorem.
(II) When (in this case is odd), therefore

Since eventually, then . Assume that . Then for sufficiently large , we have for and . It follows that

where . Then from (1), we have

Integrating from to , we get

And by (48) we see that . Hence by taking limits as we have

which implies

Again integrating above inequality from to and then taking , we get

which implies

Continuing this process () times, we find

Integrating above inequality to , we get

Hence by (29), we have , which contradicts the fact that eventually. This shows that This completes the proof.

Theorem 5. Assume that for sufficiently large the second-order dynamic equation

is oscillatory, where

for every integer number with being odd. Then

(1)if is even, then every solution of (1) is oscillatory,(2)if is odd and (29) holds, then every solution of (1) either is oscillatory or tends to zero eventually.

Proof. Assume that (1) has a nonoscillatory solution on . Then, without loss of generality, and on . By Lemma 3, there exists an integer , , with being odd such that (24) and (25) hold for .
(I) When , as seen in the proof of Theorem 4, we obtain, for ,

Hence, we have

Since for ,

we have

It follows from (62) that we have, for ,

Continuing this process, we have

Then for

From (60) and (67), we get

Set ; we have, for ,

or

In view of Lemma 1, there exists a positive solution of (58) which contradicts the assumption of the theorem.
(II) When , as shown in the proof of Theorem 4, we show that if (29) holds, then . This completes the proof.

Remark 6. The conclusion of Theorems 4 and 5 remains intact if assumption (29) is replaced by one of the following conditions:

Theorem 7. Assume that

And for sufficiently large , the first-order dynamic equation

is oscillatory, where

Then

(1)if is even, every solution of (1) is oscillatory,(2)if is odd, then every solution of (1) either is oscillatory or tends to zero eventually.

Theorem 8. Assume that (72) holds and, for sufficiently large , the second-order dynamic equation

is oscillatory, where

Then

(i)if is even, every solution of (1) is oscillatory,(ii)if is odd, every solution of (1) either is oscillatory or tends to zero eventually.

Proof of Theorems 7 and 8. Assume that (1) has a nonoscillatory solution on . Then, without loss of generality, it is sufficiently large, such that and , for and . By Lemma 3, there exists an integer , , with being odd such that (24) and (25) hold for .
(I) When , we claim that (72) implies that . In fact, if , then for

Since on , then for . Then there exists such that for and . It follows that

where . Then from (1), we have

Integrating from to , we get

And by (48) we see that . Hence by taking limits as we have

which implies

Integrating above inequality from to and then taking , we get

which implies

Again, integrating above inequality from to and noting that eventually, we get

Then by (72), we have , which contradicts the fact that on . This shows that if (72) holds, then . The rest of proof of (I) is similar to proof (I) of Theorems 4 and 5, respectively, with and hence can be omitted.
(II) When (in this case is odd), therefore

Since eventually, then . Assume that . Then for sufficiently large , we have for and . It follows that

where . Then from (1), we have

Integrating from to , we get

And using (48) we see that . Hence by taking limits as we have

which implies

Integrating above inequality from to and then taking , we get

which implies

Again, integrating above inequality from to and noting that eventually, we get

Then by (72), we have , which contradicts the fact that on . This shows that . This completes the proof.

Remark 9. The conclusion of Theorems 7 and 8 remains intact if assumption (72) is replaced by one of the following conditions:

Theorem 10. Assume that

Then

(1)if is even, every solution of (1) is oscillatory,(2)if is odd, then every solution of (1) either is oscillatory or tends to zero eventually.

Proof. Assume that (1) has a nonoscillatory solution on . Then, without loss of generality, it is sufficiently large, such that and , for and . By Lemma 3, there exists an integer , , with being odd such that (24) and (25) hold for .
(I) When , this implies that is strictly increasing on . Then for sufficiently large , we have for . It follows that

where . Equation (1) becomes

Replacing by in (98), integrating from to , we obtain

Hence by (96), we have , which contradicts the fact that eventually.
(II) When (in this case is odd), therefore

This implies that is strictly decreasing on . Then . Assume that . Then for sufficiently large , we have for . It follows that

where . As is case (I), we get a contradiction with the fact that eventually. This shows that . This completes the proof.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

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