Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time ScalesView this Special Issue
Oscillation Results for Second-Order Nonlinear Damped Dynamic Equations on Time Scales
This paper is concerned with second-order nonlinear damped dynamic equations on time scales of the following more general form . New oscillation results are given to handle some cases not covered by known criteria. An illustrative example is also presented.
Let denote the set of real numbers and a time scale, that is, a nonempty closed subset of with the topology and ordering inherited from . The theory of time scales was introduced by Hilger in his Ph.D. thesis  in 1988, and for a comprehensive treatment of the subject, see . Much recent attention has been concerned with the oscillation of dynamic equations on time scales; see, for example, [1–15]. In , Došlý and Hilger studied the second-order dynamic equation The authors gave a necessary and sufficient condition for the oscillation of all solutions of (1) on time scales. In [7, 8], Del Medico and Kong used the Riccati transformation as and obtained some sufficient conditions for oscillation of (1). In , Wang considered the nonlinear second-order damped differential equation and established new oscillation criteria. In , Tiryaki and Zafer considered the second-order nonlinear differential equation with nonlinear damping and gave interval oscillation criteria of (4). In , Huang and Wang considered the second-order nonlinear dynamic equation The authors gave some new oscillation criteria of (5) and extended the results in [7, 8]. In , Qiu and Wang studied the second-order nonlinear dynamic equation By employing the Riccati transformation as where , , the authors established interval oscillation criteria for (6). And in , Qiu and Wang obtained some new Kamenev-type oscillation criteria for dynamic equations of the following more general form: by using the transformation
In this paper, we consider second-order nonlinear damped dynamic equations of the form on a time scale . We will employ functions of the form and a generalized Riccati transformation as (7) and (9) which was used in [14, 15] and derive oscillation criteria for (10) in Section 2. An example is presented to demonstrate the obtained results in the final section.
Definition 1. A solution of (10) is said to have a generalized zero at if , and it is said to be nonoscillatory on if there exists such that for all . Otherwise, it is oscillatory. Equation (10) is said to be oscillatory if all solutions of (10) are oscillatory.
2. Main Results
In this section, we establish some oscillation criteria for (10). Our work is based on the application of the Riccati transformation. Throughout this paper we will assume that and (C1);(C2);(C3), and there exist and such that and for all ;(C4)for , , , above, we always have ;(C5).
Now, we give the first theorem.
Theorem 2. Assume that (C1)–(C5) hold and that there exists a function such that . Also, suppose that is a solution of (10) satisfying for with . For , define where , , and for . Then, satisfies where
Proof. By (C3) we see that and are both positive or both negative or both zero. When , which implies that , it follows that
When , which implies that , it follows that
When , which implies that and , it follows that
Hence, we always have so (12) holds. Then differentiating (11) and using (10), it follows that so (13) holds. Theorem 2 is proved.
Remark 3. In Theorem 2, the condition ensures that the coefficient of in (13) is always negative. The condition is obvious and easy to be fulfilled. For example, when for all , we have ; by (C4) we see that
Let and . For any function : , denote by the partial derivatives of with respect to . For , denote by the space of functions which are integrable on any compact subset of . Define These function classes will be used throughout this paper. Now, we are in a position to give the second theorem.
Theorem 4. Assume that (C1)–(C5) hold and that there exists a function such that . Also, suppose that there exist and such that and for any , where is defined as before, and Then, (10) is oscillatory.
Proof. Assume that (10) is not oscillatory. Without loss of generality we may assume that there exists such that for . Let be defined by (11). Then by Theorem 2, (12) and (13) hold.
For simplicity in the following, we let , , and and omit the arguments in the integrals. For , .
Multiplying (13), where is replaced by , by and integrating it with respect to from to with and , we obtain where , , are defined as before.
Noting that , by the integration by parts formula we have
Since on , from (12) we see that, for ,
Since on , we see that . For , , from and (C4), we have
For , , and , from (27) we have
For , , and , from (27) we have Therefore, for all , , we have Then, from (25), (26), and (30) we obtain that, for and , Hence, which contradicts (22) and completes the proof.
Corollary 6. Assume that (C1)–(C5) hold and that there exists a function such that . Also, suppose that there exists such that, for any , Then, (10) is oscillatory.
Corollary 7. Assume that (C1)–(C5) hold and that there exists a function such that . Also, suppose that there exist and such that, for any , where Then, (37) is oscillatory.
In this section, we will give an example to demonstrate Corollary 7.
Example 1. Consider the equations
where , , , in (40), and in (41), so we have both , . Letting , we have (1), ,
That is, (38) holds. By Corollary 7 we see that (40) and (41) are oscillatory. Consider (2), ,
That is, (38) holds. By Corollary 7 we see that (40) and (41) are oscillatory.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project was supported by the NNSF of China (no. 11271379).
S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten [Ph.D. thesis], Universität Würzburg, 1988.
M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.
A. Del Medico and Q. Kong, “Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain,” Journal of Mathematical Analysis and Applications, vol. 294, no. 2, pp. 621–643, 2004.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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