Oscillation Criteria of Second-Order Dynamic Equations with Damping on Time Scales

and Applied Analysis 3 so we have

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of dynamic equations on time scales; for example, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein.In Došlý and Hilger [9], the authors considered the second-order dynamic equation as and gave necessary and sufficient conditions for the oscillation of all solutions on unbounded time scales.In Del Medico and Kong [7,8], the authors employed the following Riccati transformation: and gave sufficient conditions for Kamenev-type oscillation criteria of (6) on a measure chain.
In [14], S ¸enel had tried to establish Kamenev-type oscillation criteria for (1).However, it seemed that several mistakes had been made and the obtained theorems and corollaries are incorrect.In this paper, we will correct some mistakes in [14] and establish some Kamenev-type oscillation criteria for (1) by employing functions in some function classes and a similar generalized Riccati transformation as (13) and as used in [15,16] for nonlinear differential equations.Finally, two examples are included to show the significance of the results.

Preliminary Results
To establish Kamenev-type criteria for oscillation of (1), we give three lemmas in this section.
Remark 3. In [14, (A * )], the key condition that −()/() is regressive is missed; then the assumption may not be well presented.In this paper, the condition is added as (C6).

Main Results
In this section we establish Kamenev-type criteria for oscillation of (1).Our approach to oscillation problems of ( 1) is based largely on the application of the Riccati transformation.Firstly, we give some definitions.
These function classes will be used throughout this paper.Now, we are in a position to give our first theorem.
Multiplying (36), where  is replaced by , by , and integrating it with respect to  from  1 to  with  ∈ T and  ≥ ( 1 ), we obtain Noting that (, ) = 0, by the integration by parts formula, we have When 0 <  < 1, we have When  = 1, we have When  > 1, on the one hand, we have On the other hand, when  > 0, we also have Using the inequality let  = , and Then we have .
So, when  = 0, we get while when  > 0 we get Therefore, for all  > 0, by (52), we have which implies that which contradicts (49) and completes the proof.
When  = 0, (35) is simplified as Now we have the following theorem.
Proof.Assume that (1) is not oscillatory.Without loss of generality we may assume that there exists Let () be defined by (67).Then, by Lemma 7, we have where Φ 0 () is simplified as When  = 1, we have When  > 1, on the one hand, we have On the other hand, when  > 0, we also have Therefore, for all  > 0, we always have which contradicts (68) and completes the proof.

Examples
In this section, we will show the application of our oscillation criteria in two examples.We first give an example to demonstrate Theorem 9 (or Corollary 10).