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Oscillation of Two-Dimensional Neutral Delay Dynamic Systems
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.
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 ). 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  and the references cited therein. A book on the subject of time scales in  summarizes and organizes much of time scale calculus. The reader is referred to , 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 .
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  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.
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  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.
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 . The main result from  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.
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.
4. The Case
In this section, we always assume that where is a positive constant.
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.
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.
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  are not applicable.
This work is supported by the NSF of Shandong, China (BS2011DXD11).
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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.