Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time ScalesView this Special Issue
Oscillation for a Nonlinear Dynamic System with a Forced Term on Time Scales
We consider a class of two-dimensional nonlinear dynamic system with a forced term on a time scale and obtain sufficient conditions for all solutions of the system to be oscillatory. Our results not only unify the oscillation of two-dimensional differential systems and difference systems but also improve the oscillation results that have been established by Saker, 2005, since our results are not restricted to the case where for all and . Some examples are given to illustrate the results.
Let be a time scale, that is, a nonempty closed subset of , which is unbounded above. This paper is concerned with the two-dimensional dynamic system on . We assume that and it is convenient to let and define the time scale interval . For system (1), we assume the following. , , and . are nondecreasing functions with sign property and , for all ..
The problem of oscillation and nonoscillation of second-order dynamic equations on time scales has become an important research field due to its tremendous potential for various applications. We refer the reader to the recent papers [1–3] and the references therein. It is an interesting problem to extend oscillation criteria for second-order dynamic equations to the case of two-dimensional dynamic systems.
The system (1) includes two-dimensional linear and nonlinear differential and difference systems, which were investigated in the literature; see, for example, [4, 5] and the references therein. As a special case of (1), when , system (1) can be reduced to whose oscillation and nonoscillation results have been obtained by some authors; see, for example, [6–8] and the references therein. When , for all and , system (1) can be reduced to a single dynamic equation whose oscillatory behavior has been investigated; see, for example, [9, 10] and the references cited therein.
However, to the best of our knowledge, there are few results dealing with the oscillation of the solutions of forced dynamic systems on time scales up to now. Motivated by [4, 5, 11], we will consider the oscillation property of system (1) and establish some oscillation criteria for system (1) in this paper. Our results not only unify the oscillation of two-dimensional differential systems and difference systems but also improve the oscillation results that had been established by Saker , since our results are not restricted to the case where , for all and .
The remainder of this paper is organized as follows. Section 2 contains some basic definitions and the necessary results about time scales. In Section 3, we present some useful lemmas. In Section 4, we present and prove the main results. Examples are given to illustrate the applicability of the obtained results.
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 .
Lemma 1. Assume that are differentiable at and . Then, is differentiable at and
Lemma 2. If and , then
Lemma 3 (chain rule). Assume that is continuously differentiable and is delta differentiable; then is differentiable and
3. Some Basic Lemmas
A solution of (1) is said to be continuable if it exists on the entire interval . A continuable nontrivial solution is said to be oscillatory if , are both oscillatory. A component (or ) of a solution is said to be oscillatory if and only if (or ) is neither eventually positive nor eventually negative. Notice that if , the oscillation of follows from that of . Furthermore, we observe that the substitutions , transform (1) into the system where , , and , . The functions and are subject to the conditions imposed on and . Therefore, we restrict our discussion only to the case where is positive. In order to prove our results, we need the following lemmas.
Lemma 4. Suppose that and hold. If is a nonoscillatory solution of system (1), then the component is also nonoscillatory.
Proof. Assume that is a solution of (1) and is oscillatory, but is nonoscillatory. Without loss of generality, we let on . In view of the first equation of system (1) and and , we have on . Thus, or for all large on , which leads to a contradiction.
Lemma 5. Suppose that conditions and hold, and let denote a nonoscillatory solution of the system (1) on interval , , with for all ; moreover, let . If there exists a positive constant such that where the function is defined as then
Proof. From the second equation of (1) and Lemma 3, we obtain
By (10) and (11), we have
it follows from that and , for all .
Putting then In view of , we have which implies that since where satisfies Using nonlinear version of comparison theorem on time scales [13, Corollary 6.12], we have Therefore, By Lemmas 1 and 3, we obtain Then, we get , . Hence, The proof is completed.
4. Main Results
For simplicity, we list the conditions used in the main results as
For every and sufficiently small ,
Proof. Suppose that system (1) has a nonoscillatory solution on . By Lemma 4, we know that is nonoscillatory on . Without loss of generality, we may assume that , for all . In view of and (26), there exist and , such that , for . By , we have where is a finite positive constant. In view of (27) and (30), there exists a sufficiently large, such that (10) is satisfied for all . Applying Lemma 5, we obtain Since is nondecreasing, we have Integrating the above inequality from to , we get as , which is a contradiction. The proof is complete.
Example 7. Consider the system where .
Let , , , and . Since The system is oscillatory by Theorem 6. In fact, is such an oscillatory solution.
Proof. Suppose that system (1) has a nonoscillatory solution on . By Lemma 4, we know that is nonoscillatory on . Without loss of generality, we may assume that for all . In view of and (26), there exist and such that , for .
As seen in the proof of Lemma 5, we have Note that Otherwise, (11) is valid for some positive number . Then, by Lemma 5, we have , for all . Hence, holds, and its subsequent contradiction holds as before. It now follows where We now show that . Indeed, if , then (28), (30), and (39), respectively, imply that By (43), (44), and (45), we have Then, by Lemma 5, let ; we have , for all . Hence, holds, and its subsequent contradiction holds as before. In view of (41) and , we have for all large , where . For the sake of convenience, let for all large ; then and, in view of (29), Thus, by (36), we have however, which is contrary to (37). The proof is completed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project is supported by NSF of Shandong (ZR2013AL011).