New Oscillation Results of Second-Order Damped Dynamic Equations with p-Laplacian on Time Scales

In 1988, the theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in order to unify continuous and discrete analysis; see also [2]. In recent years, there has been much research activity concerning the oscillation of solutions of dynamic equations on time scales; for example, see [3–19] and the references therein. Došlý and Hilger [10] considered the second-order dynamic equation

The rest of this paper is organized as follows.In Section 2, we give some basic lemmas.In Section 3, we derive new oscillation criteria for (12).In Section 4, two examples are included to show the significance of the results.

Basic Lemmas
Preliminaries about time scale calculus can be found in [3,4,6,7] and hence we omit them here.Note that we have the following properties for some typical time scales, respectively: (2) (3) Definition 1.A solution  of ( 12) is said to have a generalized zero at  * ∈ T if ( * )(( * )) ≤ 0, and it is said to be nonoscillatory on T if there exists  0 ∈ T such that ()(()) > 0 for all  >  0 .Otherwise, it is oscillatory.Equation ( 12) is said to be oscillatory if all solutions of (12) are oscillatory.

Main Results
In this section, we will derive new oscillation criteria of (12).Our approach to oscillation results of ( 12) is based on the application of the generalized Riccati transformation.Firstly, we give some definitions.
Proof.Assume that ( 12) is not oscillatory.Without loss of generality we may assume there exists Let () be defined by (38).Then by Lemma 5, (39) holds.
Multiplying (39), where  is replaced by , by , and integrating it with respect to  from  1 to  ∈ [ ( 1 ) , ∞) T , we obtain Noting that (, ) = 0, by the integration by parts formula we have Since by ( 54) we have ) Δ. (56) Hence which contradicts (52) and completes the proof.
When  = 0, (38) is simplified as Now we have the following theorem.
Proof.Assume that ( 12) is not oscillatory.Without loss of generality we may assume there exists Let () be defined by (58).Then by Lemma 5, we have which implies that So we obtain Letting  be replaced by , and integrating (62) with respect to  from  1 to  ∈ [ ( 1 ) , ∞) T , we obtain which contradicts (59) and completes the proof.
When  ≥ 0, it is easy to obtain −/ ≤ 0, so we have Then, Theorems 6 and 7 can be simplified as the following corollaries, respectively.
Remark 10.Compared to the theorems and corollaries in [15], the conclusions of Theorems 6 and 7 and Corollaries 8 and 9 are much simpler, and the proofs of Theorems 6 and 7 are more convenient.Furthermore, () may change sign in Theorems 6 and 7, for the employment of the function  −/ (,  1 ).
Remark 11.Consider the equations as the following form:

Examples
In this section, we will show the application of our oscillation results in two examples.Firstly, we give an example to demonstrate Theorem 6. (i) T = N.In this case, there exists Hence lim sup That is, (52) holds.By Theorem 6 we see that (72) is oscillatory.