Abstract
By using a Riccati transformation and inequality, we present some new oscillation theorems for the second-order nonlinear dynamic equation with damping on time scales. An example illustrating the importance of our results is also included.
1. Introduction
The theory of time scales, which has recently received a lot of attraction, was introduced by Hilger in his Ph.D. Thesis in 1990 [1] in order to unify continuous and discrete analysis. The books on the subjects of time scale, that is, measure chain, by Bohner and Peterson [2, 3] summarize and organize much of time scale calculus.
We are concerned with second-order nonlinear dynamic equations with damping on a time scale here and are real-valued positive rd-continuous positive functions defined on , and is a quotient of odd positive integers. We assume that , , and define .
In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations [4–13]. However, there are few papers dealing with the oscillation of dynamic equations with damping term [14–17].
Saker [18] presented several oscillation criteria for the nonlinear second-order dynamic equation where and .
Hassan [19] studied the oscillation behavior of the second-order half-linear dynamic equation and obtained several new results.
Bohner et al. [20] established some oscillation criteria for the second-order nonlinear dynamic equation
Erbe et al. [16] considered the second-order nonlinear dynamic equations with damping and established some sufficient conditions for oscillation of (1.5).
Saker et al. [17] investigated the oscillation of second-order dynamic equations with damping term of the form and obtained some new oscillation criteria for (1.6).
Zafer [21] studied the second-order nonlinear dynamic equations on time scales and presented some oscillation and nonoscillation criteria. Obviously, (1.7) is the special situation of (1.1).
Note that in the special case when (1.1) becomes the second-order nonlinear damped differential equation and when , (1.1) becomes the second-order nonlinear damped difference equation where .
This paper is organized as follows: in Section 2, we give some preliminaries and lemmas. In Section 3, we will establish some oscillation criteria for (1.1). In Section 4, we give an example to illustrate the main results.
2. Preliminaries
It will be convenient to make the following notations:
Lemma 2.1. Assume that is -differentiable. Then from Keller's chain rule [2, Theorem ],
Lemma 2.2 (see [22]). If , , then attains its maximum value at , and .
Lemma 2.3. Suppose that is an eventually positive solution of equation (1.1), and Then there exists a , such that for ,
Proof. Pick such that on . From (1.1), we have So, we get Therefore, We claim that . If not, there exist and a constant such that hence Integrating the above inequality from to , we obtain which is a contradiction. Hence, Obviously, by (2.7) and (2.11), we can see that From (2.11) and (2.12), we have It follows from (2.13) that In view of (2.14) and , it is easy to get that
3. Main Results
In this section, we will give some new oscillation criteria for (1.1).
Theorem 3.1. Assume that (2.3) holds. Further, suppose that , and there exists a positive -differentiable function , such that for all sufficiently large , where , . Then every solution of (1.1) oscillates on .
Proof. Let be a nonoscillatory solution of (1.1) on . Without loss of generality, we assume , for . Consider the generalized Riccati substitution then , and by the product rule and then the quotient Using (1.1) and (3.2), we find If , from Lemma 2.1, we get hence In view of Lemma 2.3 and (3.2), we obtain If , from Lemma 2.1, we get hence In view of Lemma 2.3, we have Therefore, From Lemma 2.3, we get Integrating the above inequality from to , we have which leads to a contradiction to (3.1). This completes the proof.
Remark 3.2. From Theorem 3.1, we can obtain different conditions for oscillation of all solutions of (1.1) with different choice of .
Theorem 3.3. Assume that (2.3) holds. Further, suppose that , and there exist positive -differentiable functions and , such that for all sufficiently large , where , . Then every solution of (1.1) oscillates on .
Proof. Let be a nonoscillatory solution of (1.1) on . Without loss of generality, we assume , for . Consider the generalized Riccati substitution as in (3.2). Then , and by the product rule and then the quotient it follows from (1.1) and (3.2) that If , from Lemma 2.1, we get hence In view of Lemma 2.3, we see that If , from Lemma 2.1, we get So, In view of Lemma 2.3, we find Therefore, From Lemma 2.2, we obtain Integrating the above inequality from to , we get which leads to a contradiction to (3.14). This completes the proof.
Remark 3.4. From Theorem 3.3, we can obtain different conditions for oscillation of all solutions of (1.1) with different choice of and .
In the following, we will establish Kamenev-type oscillation criteria for (1.1).
Theorem 3.5. Assume that (2.3) holds. Further, suppose that , and there exists a positive -differentiable function , such that for and all sufficiently large , where , . Then every solution of (1.1) is oscillatory on .
Proof. We may assume that (1.1) has a nonoscillatory solution such that . Define by (3.2) as before, then we get (3.24). From (3.24), we have Thus Upon integration, we arrive at Note that (see Saker [11]); then using (3.28), we have Therefore, Hence, which contradicts (3.26). This completes the proof.
Theorem 3.6. Assume that (2.3) holds. Further, suppose that , and there exists a positive -differentiable function , such that for and all sufficiently large , where , . Then every solution of (1.1) oscillates on .
Proof. In view of Theorem 3.3, the proof is similar to that of [18, Theorem ].
In the following, we will establish the Philos-type oscillation criteria for (1.1).
Theorem 3.7. Assume that (2.3) holds. Further, suppose that , there exists a positive -differentiable function and , where such that has a continuous and nonpositive -partial derivative with respect to the second variable and satisfies and for sufficiently large , where Then every solution of (1.1) oscillates on .
Proof. Let be a nonoscillatory solution of (1.1) on . Without loss of generality, we assume , for . Define by (3.2) as before, then we have (3.11). From (3.11), we have
Thus,
Integrating the right side by parts, we have
and then by using (3.34) and (3.35), we arrive at
Define
By employing the inequality
we obtain
Therefore,
which contradicts (3.36). The proof is complete.
Theorem 3.8. Assume that (2.3) holds. Further, suppose that , there exists a positive -differentiable function and , where such that (3.28) holds, and has a continuous and nonpositive -partial derivative with respect to the second variable and satisfies If for sufficiently large where then every solution of (1.1) oscillates on .
Proof. In view of Theorem 3.3, the proof is similar to [16, Theorem ].
Theorem 3.9. Assume that (2.3) holds. Further, suppose that , and for all sufficiently large , Then every solution of (1.1) oscillates on .
Proof. Let be a nonoscillatory solution of (1.1) on . Without loss of generality, we assume , for . From (1.1) and Lemma 2.3, we get for , Letting , we obtain In view of Lemma 2.3, we obtain Thus which is a contradiction. This completes the proof.
4. Example
In this section, we will give an example to illustrate our results.
Example 4.1. Consider the second-order damped dynamic equation on time scales where Obviously, .
It is easy to see that (2.3) holds. For , one has and for , Hence, by Theorem 3.1, every solution of (4.1) is oscillatory.
Remark 4.2. It is easy to see that the results in [16–21] cannot be applied in (4.1), and to the best of our knowledge nothing is known regarding the oscillatory behavior of (1.1), so our results are new.
Acknowledgments
This research is supported by the Natural Science Foundation of China (11071143, 60904024), China Postdoctoral Science Foundation funded project (20080441126, 200902564), Shandong Postdoctoral funded project (200802018) and supported by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), also supported by University of Jinan Research Funds for Doctors (XBS0843).