Advances in Mathematical Physics

Volume 2013 (2013), Article ID 871961, 7 pages

http://dx.doi.org/10.1155/2013/871961

Research Article

Oscillation of Two-Dimensional Neutral Delay Dynamic Systems

Xinli Zhang and Shanliang Zhu

College of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, China

Received 15 January 2013; Accepted 12 March 2013

Academic Editor: Changpin Li

Copyright © 2013 Xinli Zhang and Shanliang Zhu. 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 consider a class of nonlinear two-dimensional dynamic systems of the neutral type We obtain sufficient conditions for all solutions of the system to be oscillatory. Our oscillation results when improve the oscillation results for dynamic systems on time scales that have been established by Fu and Lin (2010), since our results do not restrict to the case where . Also, as a special case when , our results do not require to be a positive real sequence. Some examples are given to illustrate the main results.

1. Introduction

In this paper, we are concerned with oscillation of the two-dimensional nonlinear neutral dynamic systems

on time scales. Since we are interested in the oscillatory behavior of the solution of system (1) near infinity, we will assume throughout this paper that the time scales are unbounded. We assume that , and it is convenient to let , and define the time scale interval by . For system (1), we assume that

 , and ; , where , ;  and are real valued positive and rd-continuous functions defined on , and ;  are continuous, nondecreasing with for , . There exists continuous function such that for all , and for all . , where is a positive constant.

The theory of time scales, which has recently a lot of attention, was introduced by Hilger in his Ph.D. degree thesis in 1988 in order to unify continuous and discrete analysis (see [1]). Not only can this theory of the so-called “dynamic equations” unify the theories of differential equations and difference equations, but also extend these classical cases to cases “in between,” for example, to the so-called -difference equations and can be applied on other different types of time scales. Since Hilger formed the definition of derivatives and integrals on time scales, several authors have expounded on various aspects of the new theory; see the paper in [2] and the references cited therein. A book on the subject of time scales in [3] summarizes and organizes much of time scale calculus. The reader is referred to [3], Chapter 1, for the necessary time scale definitions and notations used throughout this paper.

Our main interest in this paper is to establish some oscillation results for system (1). We will relate our results to some earlier work for system (1). In the special case when , system (1) becomes the two-dimensional difference system

If is a positive real sequence, the oscillatory property of system (2) has been receiving attention. We refer the reader to the papers [4, 5] and the references cited therein. However, system (1) have been restricted to the case when in paper [4].

On the other hand, system (1) reduces to some important second-order dynamic equations in the particular case; for example,

We refer the reader to the recent papers [6–9] and the references cited therein. However, there are few works about oscillation of dynamic systems on time scales, motivated by [4] and the references cited therein, and in this paper, we investigate oscillatory properties for system (1). In Section 2, we present some basic definitions concerning the calculus on time scales. In Section 3, we discuss the case ; the case will be studied in Section 4. Examples are given in Section 5 to illustrate our theorems.

2. Preliminary

For completeness, we recall the following concepts and results concerning time scales that we will use in the sequel. More details can be found in [10–12].

The forward and backward jump operators are defined by

where and , where denotes the empty set. A point is called left-dense if and , right-dense if and , left-scattered if , and right-scattered if . A function is said to be rd-continuous if it is continuous at every right-dense point and if the left-sided limit exists at every left-dense point. The set of all such rd-continuous functions is denoted by . The graininess function for a time scale is defined by , and for any function , the notation denotes .

A function is called positively regressive (we write ) if it is rd-continuous function and satisfies for all . For a function , the (delta) derivative is defined by

if is continuous at and is right-scattered. If is not right-scattered, then the derivative is defined by

provided this limit exists. A function is said to be right-dense continuous if it is right continuous at each right-dense point and there exists a finite left limit at all left-dense points, and is said to be differentiable if its derivative exists. A useful formula is

Assume that are differentiable at and ; then, is differentiable at and

If and , then

Assume that is continuously differentiable and is delta differentiable. Then, is differentiable and

Hilger [1] showed that for to be rd-continuous and regressive, the solution of the initial value problem

is given by

where

3. The Case

In this section, we always assume that

For any , we define by

In the following, we will give some lemmas which are important in proving our first results.

Lemma 1. Suppose that ()–() and (14) hold, and is a solution of system (1) with eventually of one sign for . Then, is nonoscillatory, and there exists such that and are monotone for .

Proof. Assume that is a solution of system (1) and is nonoscillatory. Then, in view of   () and the hypothesis on , from the second equation of system (1), we have either or ≥0 for all . Thus is monotone and is eventually of one sign for all sufficiently large . Now, from the first equation of system (1), we can prove that is monotone and nonoscillatory for all sufficiently large . This completes the proof of the lemma.

Lemma 2. Suppose that and (14) hold. Let be a nonoscillatory solution of the inequality

defined for all sufficiently large . Then, is bounded.

Proof. Without loss of generality, we may assume that is an eventually positive solution of inequality (16), and the proof for the case eventually negative is similar. From (16), we have

for all sufficiently large . In view of (14), we have

Hence, is bounded.

We now establish some sufficient conditions for the oscillation of (1) by reducing our study to a first-order delay dynamic inequality where we apply the results of Zhang and Deng [12]. The main result from [12] is the following lemma.

Lemma 3. Assume that , and . If

then the inequality

cannot have an eventually positive solution, and the inequality

cannot have an eventually negative solution.

Now, we state and prove our main theorem.

Theorem 4. Assume that is bounded and with . Denote that . If there exists constant such that such that

then every solution of system (1) with bounded is oscillatory.

Proof. Let be a nonoscillatory solution of system (1) with bounded. Without loss of generality, we may assume that is eventually positive and bounded for all . From the second equation of system (1), we obtain for sufficiently large . In view of Lemma 1, we have two cases for sufficiently large :

(a) for ;(b) for .

Case (a). Because is negative and nonincreasing, there is a constant such that

Since and are bounded, defined by (15) is bounded. Integrating the first equation of system (1) from to and using (24), we have

From (25), we get , which contradicts the fact that is bounded. Case (a) cannot occur.

Case (b). We consider two possibilities.

(i) Let for be sufficiently large. Because is nondecreasing, there is a positive constant such that

for all sufficiently large . From (15) and the hypothesis , we obtain

for all sufficiently large . Integrating the second equation of system (1) from to , using (27), and then letting , we get

for all sufficiently large . From condition (22), we obtain

We claim that condition (22) implies

Otherwise, if , we can choose an integer so large that , which contradicts (29). From (9) and the monotonicity of , we have

From (26), (27), (31), and the second equation of system (1), we have

Combining the last inequality with (30), we have

for all sufficiently large . The last inequality together with (28) and the monotonicity of implies

and , for all sufficiently large , which contradicts (22). This case cannot occur.

(ii) Let for all sufficiently large . From (15), we have

where is sufficiently large and

In view of the hypothesis and the second equation of system (1), the last inequality implies

Integrating (37) from to , we have

Multiplying the last inequality by and then using the monotonicity of and the first equation of system (1), we have

By condition (23) and Lemma 3, the last inequality cannot have an eventually negative solution. This contradicts the assumption that eventually. The proof is complete.

Theorem 5. Let (16) hold. Assume that , , and there exists an constant such that and conditions (22) and (23) are satisfied. Then, all solutions of system (1) are oscillatory.

Proof. Let be a nonoscillatory solution of system (1). Without loss of generality, we may assume that is positive for . As in the proof of Theorem 4, we have two cases.

Case (a). Analogous to the proof of case (a) of Theorem 4, we can show that . By Lemma 2, is bounded, and, hence, is bounded, which is a contradiction. Hence, case (a) cannot occur.

Case (b). The proof of this case is similar to that of Theorem 4, and, hence, the details are omitted. The proof is now complete.

Remark 6. Theorems 4–5 include Theorems 5–6 in [4].

4. The Case

In this section, we always assume that

where is a positive constant.

Lemma 7. Suppose that ()–() and (40) hold, and is a nonoscillatory solution of system (1). Then, is eventually positive.

Proof. Without loss of generality, we may assume that , . Then, in view of () and the hypothesis on , we have for all from the second equation of system (1). We claim that

Otherwise, there exists such that

Now, from the first equation of system (1) and the monotonicity of , we have

Integrating the first equation of system (1) from to , we get

From (), (42), and the last inequality, we obtain . But in view of (15), we have . So, . This contradicts . This completes the proof of the lemma.

Theorem 8. Suppose that ()–() and (40) hold, and with . If

Then, every solution of system (1) is oscillatory.

Proof. Suppose that is a nonoscillatory solution of system (1). From Lemma 7, without loss of generality, we may assume that

Combining (15) with (40), we obtain that for . From the first equation of system (1), we get for all . So, is nondecreasing. There is a positive constant such that

From (15), we get

for . The last inequality together with () implies

Integrating the second equation of system (1) from to , using (49), and then letting , we obtain

From condition (45), we have

We claim that condition (45) implies

In fact, if , we can choose a constant so large that , which contradicts (51). From (9) and the monotonicity of , we have

From (49) and (53) and the second equation of system (1), we have

Combining the last inequality with (52), we have

The last inequality together with (28) and the monotonicity of implies

and ,, which contradicts (45). This case cannot occur. The proof is complete.

Remark 9. Theorem 8 improves Theorem 3.1 in [13], because condition (45) is weaker than condition (5) assumed in [13].

5. Some Examples

In this section, we present examples to illustrate the results obtained in the previous sections.

Example 10. Consider the system

where and , is a positive constant. Here, , , , , , . Choose , since

then, conditions (22) and (23) are

For , all the conditions of Theorem 4 are satisfied, and so all solutions of the system (57) are oscillatory. But the results [4] are not applicable.

Example 11. Consider the system

where and . Here, . Since

then condition (45) is

Condition (45) is satisfied. Hence, by Theorem 8, all solutions of system (60) are oscillatory.

Acknowledgment

This work is supported by the NSF of Shandong, China (BS2011DXD11).

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