The Scientific World Journal

Volume 2014 (2014), Article ID 768215, 10 pages

http://dx.doi.org/10.1155/2014/768215

Research Article

Nonoscillatory Solutions for System of Neutral Dynamic Equations on Time Scales

Zhanhe Chen,¹ Taixiang Sun,¹ Qi Wang,² and Hongjian Xi³

¹College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China

²College of Science, Guangxi University of Science and Technology, Liuzhou, Guangxi 545006, China

³College of Information and Statistics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, China

Received 30 December 2013; Accepted 2 February 2014; Published 16 March 2014

Academic Editors: T. S. Hassan, Z. Liu, S. Sun, and Z. Zheng

Copyright © 2014 Zhanhe Chen et al. 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 discuss nonoscillatory solutions to the -dimensional functional system of neutral type dynamic equations on time scales. We will establish some sufficient conditions for nonoscillatory solutions with the property .

1. Introduction

The theory of dynamic equations on time scales was introduced by its founder Hilger in his PHD thesis [1] in 1988. The study of dynamic equations on time scales is an area of mathematics that has recently received a lot of attention. It has been created in order to unify continuous and discrete analysis. In recent years there has been much research activity concerning the oscillation and nonoscillation of solutions of dynamic equations on time scales; we refer the reader to the papers [2–12]. In [13, 14], authors studied nonoscillatory solutions to the -dimensional functional differential systems of neutral type and obtained some sufficient conditions for nonoscillatory solutions with the property , . Using the idea and method of [13, 14], in this paper, we will study the nonoscillatory solutions for systems of neutral dynamic equations on time scales, which have the following form:

where natural number , , is a continuous, real-valued positive function defined on the time scale and through this paper we assume that

(a) is a continuous and increasing function with and for some ;(b), , are continuous and increasing functions; not identically equal to zero in any neighbourhood of infinity; , , hold for any ;(c) is a continuous and increasing function with ;(d) is a continuous function, and for , where is a positive constant.

2. Some Preliminary Results

A time scale is an arbitrary nonempty closed subset of the real numbers. Throughout this paper, as a matter of convenience, for any , , we denote the sets by , which is called a close interval in . Open intervals and half-open intervals and so forth are defined accordingly.

A function is defined for as

A vector function is a solution to the system (1) if there is a such that functions , , are continuously differentiable and satisfies (1) on . Let be the set of all solutions to the system (1) satisfying for any . A solution is called nonoscillatory if there exists a such that every component is different from zero for . Otherwise a solution is said to be oscillatory.

Since we restrict our attention to asymptotic properties of nonoscillatory solutions to the system (1), we suppose that the time scale under consideration is not bounded above; that is, it is a time scale interval of the form . On any time scale we define the forward operator and delta derivative as follows.

Definition 1 (see [15]). Let be a time scale. For any , we define the forward jump operator by

Definition 2 (see [15]). Let be a time scale. Assume is a function and let . The (delta) derivative of at is defined by

For some other preliminary concepts on time scales, one can refer to [15]. In the remainder of this section, we present some lemmas indispensable which will be used later.

Lemma 3. Let be a solution to (1) with on . Then is nonoscillatory and are monotone on for some .

Proof. Because on , using the conditions (b) and (d), does not change its sign and not identically equal to zero on for some . This implies that is monotone and on for some . Continue similarly as above and then conclude the desire results. The proof is complete.

Lemma 4. Assume that (a) holds and for . Let be a nonoscillatory solution to the functional inequality

where is a continuous function. If for , then is bounded on . Moreover, if   for , then .

Proof. We firstly claim that there exists a negative integer such that for some . In fact, if not the case, then there is such that for any . We need only to consider two cases: Case 1, for each . Then there exists satisfying , which implies , a contradiction. Case 2, . Then, by the monotonicity of , , also a contradiction.

Without loss of generality, we may assume that is a positive solution of the functional inequality on . Then for all .

If for , then , for any and each . For any , from the claim above, let for some and some . Then . Therefore, is bounded on .

If for , then for any and each . For any , from the claim above, let for some and some . Then , which implies that .

Similarly as Lemma 4, we can prove the following lemma.

Lemma 5. Assume that (a) holds and for . Let be a nonoscillatory solution to the functional inequality:

where is a continuous function. If for , then is bounded on . Moreover, if for , then .

Lemma 6. Assume that and are continuous functions and for all . If , then the following two statements hold.

(i)The functional inequality

has no eventually positive solution.(ii)The functional inequality

has no eventually negative solution.

Proof. We only prove (i) for the proof of (ii) is similar. Suppose that the functional inequality (7) has an eventually positive solution . Without loss of generality, we may assume that for any . Then, from (7), for any . It follows that is nondecreasing on .

Let be the linear extension of the function , then is continuous, is delta differentiable, and on . By [15, Theorem 1.87], for any , there is such that . Because for all , for all . Therefore,

By (7), we have

Integrating the inequality above from to on , we get, for ,

By the hypothesis, there exist and a constant such that holds for . So we have for ,

Here, to give the last inequality, we have used the inequality for all . From (7) and (9), we have

Integrating the inequality above from to on , we get, for ,

Continuing this progress, we conclude that, for each natural number ,

Since , it follows that for any , a contradiction. The proof is complete.

Let be a nonoscillatory solution to (1). It follows, from Lemma 3, that the function has to be eventually of constant sign. Hence, either

or

for sufficiently large .

Lemma 7. Let be a nonoscillatory solution to (1) on , and for . Then there exist with or and some , such that for

Proof. Putting for and , then and so we can pick . It is obvious that . Without loss of generality, we may assume that for . Note that for all . If , it is obvious that (18) holds. Next, we assume , then , for . We will show recursively that (18) holds.

We firstly show that for . Otherwise, for . Then, from (1) and (b), we have for . So, by (1) and (b),

holds for all , which contradicts the condition (b). Hence, , for .

We now show that for . Otherwise, for . Then, from (1) and (b), we have for . So, by (1) and (b),

holds for all , which contradicts the condition (b). This implies that , for .

Continuing this progress, we can prove that (18) holds. To complete the proof, it remains to show that . Suppose , then, from (1) and (d), we have on for . Note that and for all . It is easy to show that for all . But then,

holds for all , which contradicts the condition (b). This completes the proof.

Lemma 8. Let be a nonoscillatory solution to (1) on , and for . Then there exist with or and some , such that for , either

or

Proof. Without loss of generality, we may assume that for . Note that for all . We consider the following two cases.

(I) for . We will show recursively that (22) holds. We firstly show that for . Otherwise, for . Then, from (1) and (b), it is easy to see that for . So, by (1) and (b) again,

holds for all , which contradicts the condition (b). Hence, and for . Continuing this progress similarly, we conclude that (22) holds.

(II) for . Analogically as in the proof of Lemma 7, we can prove that (23) holds. This completes the proof.

For the sake of convenience, we denote by , , and the set of all nonoscillatory solutions to the system (1) satisfying (18), (22), and (23) correspondingly. Denote by the set of all nonoscillatory solutions to the system (1). From Lemmas 7 and 8, we have the classification of nonoscillatory solutions to the system (1) as follows:

(i) is odd and ,

(ii) is odd and ,

(iii) is even and ,

(iv) is even and ,

Remark 9. Assume that and for , where is a constant. Then for .

In fact, if for some . Let and suppose, without loss of generality, that for . Note that and for . It follows that for . On the other hand, for . By Lemma 5, which implies that , a contradiction.

Lemma 10. Let be a nonoscillatory solution to (1) on . Suppose that , and for . Then

Proof. Suppose that , , then there is a positive constant such that for . By (1) and (b), we have

It follows from (b) that . Obviously there is a positive constant such that for . Similarly as above, we can prove that . Continuing this progress, we conclude that (29) holds. Using (29), it is easy to see that (30) holds. The proof is complete.

3. Main Results and Proofs

We will now give the main results and their proofs. In the squeal, for the convenience of expressions, we define and by recursion formula as follows:

where , , are continuous functions and .

Remark 11. From the definitions of and , it is easy to show the following two properties:

(i)if for each , then

for , ;(ii)if for each and for some , then

for , .

Theorem 12. Assume that is odd and . If the following statements hold:

(1)there is a constant such that, for any ,

(2)for any ,

(3)there is a continuous function such that

(4)for each even with ,

(5)

then, for every nonoscillatory solution to (1), , .

Proof. Let be a nonoscillatory solution to (1). Without loss of generality, we may assume that for all . Because is odd and , the expression (26) holds. We consider the following five cases.

(I) on . In this case, we have

By Lemma 4, . From (1) and (41), we have is a positive and nonincreasing function, which implies that . It follows, from Lemma 10, that , .

(II) on . In this case, we have

From (1) and (42), there is a positive constant such that for . Integrating the first equation of (1) over , and using the inequality above, we get

By the condition (b), . The conditions (35), (36), and (42) imply that is bounded on , which arrives a contradiction as for . Therefore, .

(III) on . In this case, we have

Integrating the second equation of (1) from to , we obtain that, for and ,

Integrating the third equation of (1) from to , we get that, for and ,

Continuing this progress, we have that, for any and , ,

Combining inequalities above, we obtain that, for and ,

From (1), (b), (d), and (44), we have that, for and ,

Substituting (49) to (48), we have that, for and ,

Because of for ,

Using the monotonicity of , we have that, for and ,

Combining (50) and (52), we have that, for and ,

Multiplying the inequality above by and putting , we get that

By Lemma 6, the inequality above has no eventually negative solution, a contradiction. Hence, .

(IV) on , . In this case, we have that, for ,

Integrating the first equation of (1) from to , we get that, for and ,

Integrating the equation of (1), we obtain, for and , , that

Integrating the equation of (1), we obtain, for and , , that

Combining inequalities above, we obtain that, for and ,

Note that (52) holds if , , , and for . So we have that, for and , ,

Putting , then

Substituting (61) to (59), we have, on for some sufficiently large , that

So, for

This contradicts (39) and hence, .

(V) on . In this case, we have that, for ,

Similarly as in the case (IV) of the proof of this theorem, we can get, for some sufficiently large , that

Using the monotonicity of , we have, for , that

So, for

which gives a contradiction with (40). Thus . The proof is complete.

Theorem 13. Assume that is even, in the system (1) and conditions (35)–(39) hold. Then, for every nonoscillatory solution to (1), we have , .

Proof. Let be a nonoscillatory solution to (1) and the expression (27) holds. Without loss of generality, we may assume that for all . We consider the following five cases.

(I) on . Analogically as the case (I) in the proof of Theorem 12, we can show that , .

(II) on . Also analogically as in the proof of Theorem 12(II), we can prove that ;

(III) on . In this case, we have

We continue analogically as (III) in the proof of Theorem 12. We can get, for , that

From (1), (b), (d), and (68), we have that, for and ,

We substitute (70) with (69), we have that, for and ,

By the same way as in the proof of Theorem 12(III), we can obtain that .

(IV) on , . In this case, we have that, for ,

By the same way as in the proof of Theorem 12(IV), we can obtain, for and , that

From (1), (b), (d) and (72), we have, for and , that

Therefore, it holds that, for and ,

Similarly as in the proof of Theorem 12(IV), we can prove that .

(V) on . Similarly as in the proof of the case (IV) of this theorem, we can get, for and , that

And at the end to contradict to (39) for . Thus . This completes the proof.

Theorem 14. Let be odd, in the system (1). If (37) the following statements hold:

(1)there is a constant such that, for any ,

(2)for any ,

(3)the function in (37) satisfies that

(4)for each even with ,

(5)

Then, for every nonoscillatory solution to (1), , .

Proof. Let be a nonoscillatory solution to (1). Without loss of generality, we may assume that for all . By the expression (25) and Remark 9, we have . We consider the following four cases.

(I) on . In this case, we have that

Integrating the second equation of (1) from to , we obtain that, for and ,

Integrating the equation of (1) from to , we get, for and , that

Combining inequalities above, we obtain that, for and ,

From (1), (b), (d), and (44), we have that, for and ,

Substitute (86) with (71), and notice that the inequality holds for any . Then we have that, for and ,

Multiplying the inequality above by and putting , we get that

By Lemma 6, the inequality above has no eventually positive solution, a contradiction. Hence, .

(II) on , . In this case, we have that, for any ,

Integrating the first equation of (1) over , we get that

Similarly, by integrating the equations of (1) and the obtained inequalities institute to (90), we get, for , that

Integrating the equation of (1), we obtain, for and , , that

Combining inequalities above and (91), we obtain that, for and ,

Note that (86) and the inequality hold. So we have that, for and ,

Putting , using the monotonicity of , then we have, for some sufficiently large , that

So, for ,

This contradicts (80) and hence, .

(III) on . In this case, we have that, for ,

Analogically as derived at part (II) of the proof of this theorem, we have that

And putting , we can obtain that

which give a contradiction with (81), and so .

(IV) on . In this case, we have that

By Lemma 5, . From (100), we have . Then it follows, by Lemma 10, that , . The proof is complete.

Similarly to Theorem 14, we can prove the following theorem.

Theorem 15. Assume that is even, in the system (1) and conditions (77)–(80) hold. Then, for every nonoscillatory solution to (1), we have , .

Example 16. We consider a system on the time scale . In order to simplify calculations, we may suppose that . Let , , , , , , , , , , , , and ; that is to say, the system has the following form:

We calculate the conditions of Theorem 12 as follows:

It follows, from Theorem 12, that for every nonoscillatory solution to (101), , . In fact, functions , , , , and are particular components of such a kind of solutions.

Example 17. We also consider a system on the time scale and let . Let , , , , , , , , , , , and ; that is to say, the system has the following form:

We calculate the conditions of Theorem 13 as follows:

It follows, from Theorem 13, that for every nonoscillatory solution to (103), , . In fact, functions , , , and are particular components of such a kind of solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper has been supported by the NSF of China (11261005 and 11161029) and NSF of Guangxi (2012GXNSFDA276040) and the Scientific Research Foundation of Guangxi University (Grant no. XBZ10098).

References

S. Hilger, Ein Maßkettenkalkül vnit Anwendung auf Zentruvnsvnannigfaltigkeiten [Ph.D. thesis], Universitüt Würzburg, 1988.

R. P. Agarwal, D. O'Regan, and S. H. Saker, “Oscillation criteria for second-order nonlinear neutral delay dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 203–217, 2004.

L. Erbe, A. Peterson, and S. H. Saker, “Oscillation and asymptotic behavior of a third-order nonlinear dynamic equation,” Canadian Applied Mathematics Quarterly, vol. 14, pp. 124–147, 2006.

S. R. Grace, R. P. Agarwal, B. Kaymakçalan, and W. Sae-jie, “On the oscillation of certain second order nonlinear dynamic equations,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 273–286, 2009.

T. Sun, H. Xi, X. Peng, and W. Yu, “Nonoscillatory solutions for higher-order neutral dynamic equations on time scales,” Abstract and Applied Analysis, vol. 2010, Article ID 428963, 16 pages, 2010.

T. Sun, H. Xi, and X. Peng, “Asymptotic behavior of solutions of higher-order dynamic equations on time scales,” Advances in Difference Equations, vol. 2011, Article ID 237219, 14 pages, 2011.

T. Sun, H. Xi, and W. Yu, “Asymptotic behaviors of higher order nonlinear dynamic equations on time scales,” Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 177–192, 2011.

S. H. Saker, “Oscillation of third-order functional dynamic equations on time scales,” Science China Mathematics, vol. 54, no. 12, pp. 2597–2614, 2011.

C. Zhang, T. Li, R. P. Agarwal, and M. Bohner, “Oscillation results for fourth-order nonlinear dynamic equations,” Applied Mathematics Letters, vol. 25, pp. 2058–2065, 2012.

R. P. Agarwal, M. Bohner, T. Li, and C. Zhang, “Oscillation theorems for fourth-order half-linear delay dynamic equations with damping,” Mediterranean Journal of Mathematics, 2013.

M. Remili, “Oscillation criteria for second order nonlinear perturbed differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 79, pp. 1–17, 2013.

S. R. Grace, S. Sun, and Y. Wang, “On the oscillation of fourth order strongly superlinear and strongly sublinear dynamic equations,” Journal of Applied Mathematics and Computing, vol. 44, pp. 119–132, 2013.

E. S. Špániková and H. S. Šamajová, “Asymptotic properties of solutions to n-dimensional neutral differential systems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, pp. 2877–2885, 2009.

E. S. Špániková and H. S. Šamajová, “On asymptotic behaviour of solutions to n-dimensional systems of neutral differential equations,” Abstract and Applied Analysis, vol. 2011, Article ID 791323, 19 pages, 2011.

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.