Abstract and Applied Analysis

Volume 2014 (2014), Article ID 615374, 12 pages

http://dx.doi.org/10.1155/2014/615374

Research Article

Oscillation Theorems for Second-Order Half-Linear Neutral Delay Dynamic Equations with Damping on Time Scales

Quanxin Zhang and Shouhua Liu

Department of Mathematics, Binzhou University, Shandong 256603, China

Received 11 March 2014; Accepted 22 April 2014; Published 20 July 2014

Academic Editor: Tongxing Li

Copyright © 2014 Quanxin Zhang and Shouhua Liu. 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 establish the oscillation criteria of Philos type for second-order half-linear neutral delay dynamic equations with damping on time scales by the generalized Riccati transformation and inequality technique. Our results are new even in the continuous and the discrete cases.

1. Introduction

In reality, it is known that the movement in the vacuum or ideal state is rare, while the movement with damping and disturbance is extensive. In recent years, the study of the oscillation of the second-order dynamic equations with damping on time scales is emerging; see [1–7], for example. Besides, the study of the oscillation for the second-order linear and nonlinear or semilinear dynamic equations can be found in [8–23] and of the oscillation for the high-order dynamic equations can be found in [24–33]. Then, inspired by the above work, this paper will study the oscillatory behavior of all solutions of a more extensive second-order half-linear neutral delay dynamic equation with damping, which is given as follows:

where , , .

Here, we give the following hypotheses at first.

(H₁) is a time scale (i.e., a nonempty closed subset of the real numbers ) which is unbounded above and when with , we define the time scale interval of the form by .(H₂) are positive rd-continuous functions such that , where is defined as the set of all regressive and rd-continuous functions and is all positively regressive elements of .(H₃) is a strictly increasing and differentiable function such that

(H₄) is a continuous function such that, for some positive constant ,

The solution of (1) defines a nontrivial real-valued function satisfying (1) for . A solution of (1) is called oscillatory if it is neither eventually positive nor negative; otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory. Here, we pay attention to those solutions of (1) which are not the eventually identical zero.

The purpose of this paper is to establish the oscillation criteria of Philos [34] for (1). The two famous results of Philos [34] about oscillation of second-order linear differential equations are extended to (1), while it satisfies

Besides, two criteria of (1) about the fact that each solution is either oscillatory or converges to zero are obtained when

The paper is organized as follows. In Section 2, we present some basic definitions and results about the theory of calculus on time scales. In Section 3, we give some lemmas. Section 4 introduces the main results of this paper. We established four new oscillatory criteria when conditions (4) and (5) hold, respectively, for the solutions of (1) in this paper.

2. Some Preliminaries

On the time scale we define the forward and backward jump operators by

A point is said to be left-dense if it satisfies , right-dense if it satisfies , left-scattered if it satisfies , and right-scattered if it satisfies . The graininess function of the time scale is defined by . For a function , the (delta) derivative is defined by

if is continuous at and is right-scattered. If is right-dense, then the derivative is defined by

provided this limit exists. A function is said to be rd-continuous if it is continuous at each right-dense point and if there exists a finite left limit at every left-dense point. Denote by the set of rd-continuous functions , and denote by the set of functions which is -differentiable and the derivative is rd-continuous. The derivative of , the shift of , and the graininess function are related by the following formula:

We will make use of the following product and quotient rules for the derivative of the product and the quotient of two differentiable functions and :

For , the Cauchy integral of is defined by

The integration by parts formula reads

and the infinite integral is defined by

For more details, see [8, 9].

3. Several Lemmas

In this section, we present six lemmas that are needed in Section 4. The first lemma is well known, and it can be found in Chapter 2 of [8]. Lemma 2 is Theorem 1.93 of [8]; Lemma 3 is the simple corollary of Theorem 1.90 in [8]; Lemma 4 is Theorem 41 in [35]; and Lemma 5 is Theorem 3 in [36].

Lemma 1. If , that is, is rd-continuous, such that for all , then the initial value problem has a unique and positive solution on , denoted by . This “exponential function” satisfies the semigroup property .

Lemma 2. Assume that is strictly increasing and is a time scale. Let . If and exist on , where

then

Lemma 3. If is differentiable, then

Lemma 4. Assume that and are nonnegative real numbers; then

where the equality holds if and only if .

Lemma 5. Let and . Then for positive rd-continuous functions we have

where and .

Lemma 6. Assume that ()–() and (4) hold. Let be an eventually positive solution of (1). Then there exists such that

Proof. Suppose that is an eventually positive solution of (1). There exists such that and for . From the definition of , we get for , and at the same time for , from (1), we get

Hence, from Lemma 1 and (11) we obtain

for . So

is decreasing. By Lemma 1, is either eventually positive or eventually negative. Therefore, for arbitrary , we have

Otherwise, we assume that (24) is not satisfied; then there exits such that for all . Because (23) is decreasing, from Lemma 1 we have

for , where . By (25) and Lemma 1, we get

that is,

After integrating the two sides of inequality (27) from to , we have

Next, we find the limits of the two sides of (28) when . From (4), we get . Therefore, is eventually negative, which is contradictory to . So the inequality (24) holds.

From (24) and (21), it is obvious that the second inequality of (20) holds. This completes the proof.

4. Main Results

Firstly, the two famous results of Philos [24] about oscillation of second-order linear differential equations are extended to (1) when condition (4) is satisfied.

Theorem 7. Assume that ()–() and (4) hold. Let ,  be rd-continuous function, such that

and has a nonpositive continuous -partial derivative with respect to the second variable and satisfies (31). Let be a rd-continuous function and satisfies

If there exist a positive and differentiable function such that for and a real rd-continuous function such that

where , , and , then (1) is oscillatory on .

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists , such that and for all . By the definition of , it follows

Since it satisfies , there exists such that for all . Then if it satisfies , we have

By Lemma 6 and (), we obtain that

on (where is short hand for ), and

holds. Moreover, using Lemmas 3 and 6, it follows that

In Lemma 2, let , and is unbounded above by , so , and by ; using Lemma 2, we get

Thus

By the above inequality and the first inequality in (37), we obtain that

holds on . Now we define the function by

Then we have on , and

then we obtain

on , where . Thus, for every with , by (13), we get

where  . So using Lemma 4, let

Using the inequality (18), we have

where . Thus

From (46) and (50), we obtain

that is,

From condition (34), we have

By (46), we have

and from the above inequality, let , and denote that

meanwhile noting (54), we obtain

Now we assert that

holds. Suppose to the contrary that

By (31), there exists a constant such that

From (59), there exists a for arbitrary real number such that

for . By (13), we have

From (60), there exists such that for , so . Since is arbitrary, we have

Selecting a sequence with satisfying

then there exists a constant such that

for sufficiently large positive integer . From (63), we can easily see

and (65) implies that

From (65) and (66), we have

that is,

for sufficiently large positive integer , which together with (67) implies

On the other hand, by Lemma 5, we obtain

The above inequality shows that

Hence, (70) implies

which contradicts (32). Therefore (58) holds. Noting for , by using (58), we obtain

which is contradicting with (33). This completes the proof.

Remark 8. From Theorem 7, we can obtain different conditions for oscillation of all solutions of (1) with different choices of and . For example, or .

Theorem 9. Assume that ()–(), (4), (30)-(31), and (33) hold, where , , , and are defined in Theorem 7. Assume that

holds, where , . Then (1) is oscillatory on .

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists , such that and for all . So and there exists such that

for . Define the function by

We proceed as in the proof of Theorem 7 to obtain (46) and (50), so that

Hence, (76) implies

then we have

From the above inequality and (75), we have

Therefore, there exists a sequence with such that

Definitions of and are as in Theorem 7; from (46), and noting (81), we have

For the above sequence ,

We obtain (58) by using reductio ad absurdum. The rest of the proof is similar to that of Theorem 7 and hence is omitted. This completes the proof.

If (4) is not satisfied, that is, if condition (5) holds, we can obtain the following results.

Theorem 10. Assume that ()–(), (5), and (30)–(34) hold, where , , , and are defined in Theorem 7. Assume that

holds. Then every solution of (1) is either oscillatory or converges to zero on .

Proof. As the proof of Theorem 7, assume that (1) has a nonoscillatory solution on . Without loss of generality we may assume that there exists , such that and for all . So and there exists such that

for . In the proof of Lemma 6, we find that is either eventually positive or eventually negative. Thus, we will distinguish the following two cases:

(I)  for  ;(II)  for  .

Case (I). When is an eventually positive and the proof is similar to that of the proof of Theorem 7, we can obtain that (1) is oscillatory.

Case (II). When is an eventually negative, is decreasing and exists. Therefore, there exists such that

for . Define the function . Equations (1) and (89) yield

The inequality (90) is the assumed inequality of [8, Theorem 6.1] (see also [37, Lemma 1]). All assumptions of [8, Theorem 6.1], for example, , are satisfied as well. Hence the conclusion of [8, Theorem 6.1] holds; that is,

for all , and thus

for all . Assuming and using (87) in (92), we can get , and this is a contradiction to the fact that for . Thus ; that is, . Then, it follows from that holds. This completes the proof.

Using the same method as in the proofs of Theorems 9 and 10, we can easily obtain the following results.

Theorem 11. Assume that ()–(), (5), (30)-(31), (33), (75)-(76), and (87) hold, where , , , and are defined in Theorem 9. Then every solution of (1) is either oscillatory or converges to zero on .

Remark 12. The theorems in this paper are new even for the cases of and .

Example 13. Consider a second-order half-linear delay 2-difference equation with damping

where . Here, we have

Then is unbounded above, , and . Conditions () and () are clearly satisfied, () holds with , and () is satisfied as

Next, by [37, Lemma 2] and (), we obtain

so

Hence (4) is satisfied. Now let ; then

Since

let ; then condition (30) holds. We have

so condition (31) holds. Let as ; then for all , and

for all . Hence

We get

thus condition (32) holds. Let ; then

that is, condition (33) holds. Since

then

Moreover, (103) implies

Thus, when is enough large, we have

so (34) is satisfied. By Theorem 7, (93) is oscillatory on . Similarly, conditions (75) and (76) are satisfied as well. By Theorem 9, we can also obtain that (93) is oscillatory. But the other known results cannot be applied in (93).

Conflict of Interests

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

Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This work was supported by the Natural Science Foundation of Shandong Province of China under Grant no. ZR2013AM003, the Development Program in Science and Technology of Shandong Province of China under Grant no. 2010GWZ20401, and the Science Foundation of Binzhou University under Grant no. BZXYKJ0810.

References

S. H. Saker, R. P. Agarwal, and D. O'Regan, “Oscillation of second-order damped dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 1317–1337, 2007.

L. Erbe, T. S. Hassan, and A. Peterson, “Oscillation criteria for nonlinear damped dynamic equations on time scales,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 343–357, 2008.

W. Chen, Z. Han, S. Sun, and T. Li, “Oscillation behavior of a class of second-order dynamic equations with damping on time scales,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 907130, 15 pages, 2010.

Q. Zhang, “Oscillation of second-order half-linear delay dynamic equations with damping on time scales,” Journal of Computational and Applied Mathematics, vol. 235, no. 5, pp. 1180–1188, 2011.

Q. Zhang and F. Qiu, “Oscillation theorems for second-order half-linear delay dynamic equations with damping on time scales,” Computers & Mathematics with Applications, vol. 62, no. 11, pp. 4185–4193, 2011.

Y. B. Sun, Z. L. Han, S. R. Sun, and C. Zhang, “Oscillation of a class of second-order half-linear neutral delay dynamic equations with damping on time scales,” Acta Mathematicae Applicatae Sinica, vol. 36, no. 3, pp. 480–494, 2013 (Chinese).

Q. Zhang and L. Gao, “Oscillation of second-order nonlinear delay dynamic equations with damping on time scales,” Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 145–158, 2011.

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

M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.

L. Erbe, “Oscillation criteria for second order linear equations on a time scale,” The Canadian Applied Mathematics Quarterly, vol. 9, no. 4, pp. 345–375, 2001.

R. P. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, “Dynamic equations on time scales: a survey,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002.

L. Erbe, A. Peterson, and P. Řehák, “Comparison theorems for linear dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 275, no. 1, pp. 418–438, 2002.

M. Bohner and S. H. Saker, “Oscillation of second order nonlinear dynamic equations on time scales,” Rocky Mountain Journal of Mathematics, vol. 34, no. 4, pp. 1239–1254, 2004.

S. H. Saker, “Oscillation criteria of second-order half-linear dynamic equations on time scales,” Journal of Computational and Applied Mathematics, vol. 177, no. 2, pp. 375–387, 2005.

R. P. Agarwal, M. Bohner, and S. H. Saker, “Oscillation of second order delay dynamic equations,” The Canadian Applied Mathematics Quarterly, vol. 13, no. 1, pp. 1–17, 2005.

Y. Şahiner, “Oscillation of second-order delay differential equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5–7, pp. e1073–e1080, 2005.

L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for second-order nonlinear delay dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 505–522, 2007.

T. S. Hassan, “Oscillation criteria for half-linear dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 176–185, 2008.

S. Sun, Z. Han, and C. Zhang, “Oscillation of second-order delay dynamic equations on time scales,” Journal of Applied Mathematics and Computing, vol. 30, no. 1-2, pp. 459–468, 2009.

S. R. Grace, M. Bohner, and R. P. Agarwal, “On the oscillation of second-order half-linear dynamic equations,” Journal of Difference Equations and Applications, vol. 15, no. 5, pp. 451–460, 2009.

L. Erbe, T. S. Hassan, and A. Peterson, “Oscillation criteria for nonlinear functional neutral dynamic equations on time scales,” Journal of Difference Equations and Applications, vol. 15, no. 11-12, pp. 1097–1116, 2009.

D. R. Anderson and A. Zafer, “Nonlinear oscillation of second-order dynamic equations on time scales,” Applied Mathematics Letters, vol. 22, no. 10, pp. 1591–1597, 2009.

S. R. Grace, R. P. Agarwal, B. Kaymakçalan, and W. Sae-jie, “Oscillation theorems for second order nonlinear dynamic equations,” Journal of Applied Mathematics and Computing, vol. 32, no. 1, pp. 205–218, 2010.

Y. Qi and J. Yu, “Oscillation criteria for fourth-order nonlinear delay dynamic equations,” Electronic Journal of Differential Equations, vol. 2013, no. 79, pp. 1–17, 2013.

T. Sun, Q. He, H. Xi, and W. Yu, “Oscillation for higher order dynamic equations on time scales,” Abstract and Applied Analysis, vol. 2013, Article ID 268721, 8 pages, 2013.

X. Wu, T. Sun, H. Xi, and C. Chen, “Kamenev-type oscillation criteria for higher-order nonlinear dynamic equations on time scales,” Advances in Difference Equations, vol. 2013, article 248, 2013.

X. Wu, T. Sun, H. Xi, and C. Chen, “Oscillation criteria for fourth-order nonlinear dynamic equations on time scales,” Abstract and Applied Analysis, vol. 2013, Article ID 740568, 11 pages, 2013.

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.

T. Sun, W. Yu, and H. Xi, “Oscillatory behavior and comparison for higher order nonlinear dynamic equations on time scales,” Journal of Applied Mathematics & Informatics, vol. 30, no. 1-2, pp. 289–304, 2012.

Z. Zhang, W. Dong, Q. Li, and H. Liang, “Positive solutions for higher order nonlinear neutral dynamic equations on time scales,” Applied Mathematical Modelling, vol. 33, no. 5, pp. 2455–2463, 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. 2012, 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.

B. Karpuz and O. Öcalan, “Necessary and sufficient conditions on asymptotic behaviour of solutions of forced neutral delay dynamic equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3063–3071, 2009.

C. G. Philos, “Oscillation theorems for linear differential equations of second order,” Archiv der Mathematik, vol. 53, no. 5, pp. 482–492, 1989.

G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 2nd edition, 1988.

A. Tuna and S. Kutukcu, “Some integral inequalities on time scales,” Applied Mathematics and Mechanics, vol. 29, no. 1, pp. 23–29, 2008.

M. Bohner, “Some oscillation criteria for first order delay dynamic equations,” Far East Journal of Applied Mathematics, vol. 18, no. 3, pp. 289–304, 2005.