Abstract

By employing a generalized Riccati technique and functions in some function classes for integral averaging, we derive new oscillation criteria of second-order damped dynamic equation with -Laplacian on time scales of the form , where the coefficient function may change sign. Two examples are given to demonstrate the obtained results.

1. Introduction

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 and they gave some necessary and sufficient conditions for the oscillation of all solutions as (1) on unbounded time scales. Erbe et al. [11] studied the second-order nonlinear dynamic equation By means of generalized exponential functions and generalized Riccati transformation techniques, some oscillation criteria were given for (2). Del Medico and Kong [8, 9] employed the Riccati transformation and gave sufficient conditions for Kamenev-type oscillation criteria of (1) on a measure chain. Wang [18] considered the second-order nonlinear damped differential equation used the generalized Riccati transformations, and established new oscillation criteria of (3). Saker et al. [16] studied the second-order damped dynamic equation with damping and they gave some oscillation criteria. Agarwal et al. [5] studied the second-order half-linear dynamic equation and established Philos-type oscillation criteria. Huang and Wang [13] considered the second-order nonlinear dynamic equation The authors extended the results in [8, 9], and established some new Kamenev-type oscillation criteria. Qiu and Wang [14] studied the dynamic equations of a more general form and established some similar oscillation criteria.

Şenel [17] discussed the oscillation of second-order damped dynamic equations of the following form: on a time scale satisfying and , where . However, it seemed that several mistakes had been made and the obtained theorems and corollaries were incorrect. Qiu and Wang [15] corrected some mistakes in [17] and established some Kamenev-type oscillation criteria of (8) by employing functions in some function classes and a similar generalized Riccati transformation as used in [13, 18, 19].

Hassan and Kong [12] considered the second-order dynamic equation with -Laplacian and damping where , . The authors used the notations where , by which (9) can be written into the form and established some oscillation criteria of (9).

In this paper, we will consider the second-order damped dynamic equation with -Laplacian on a time scale satisfying and , where the coefficient function may change sign. Throughout this paper, we will assume that(C1), , ;(C2), is positively regressive, which means , and (C3), and for any , (C4), and there exists a function such that .

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.

2. 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:(1), (2), (3),

Definition 1. A solution of (12) 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  (12) is said to be oscillatory if all solutions of (12) are oscillatory.

For simplicity, throughout this paper, we denote , where , and are denoted similarly.

To establish new oscillation criteria of (12), we give three lemmas in this section.

Lemma 2. Assume that (C1)–(C4) hold, and there exists a sufficiently large such that is a solution of (12) satisfying for . Then for , we have

Proof. Let such that is a solution of (12) satisfying for ; then we also have . It is obvious that (12) is equivalent to which is simplified as By (12) and (C4), we have
We claim that for . Assume the contrary; that is, there exists such that . Also, for we have Integrating (22) from to , we obtain That is, which implies that Integrating (25) from to and letting , by (C2) we obtain which contradicts . Hence for , and . By (21), we obtain
Lemma 2 is proved.

Lemma 3. Assume that (C1)–(C4) hold and is a solution of (12) satisfying for with . Then for , if , we have and if , we have where

Proof. Since is a solution of (12) satisfying for with , by Lemma 2 we have From (21) it follows that Integrating (32) from to with , we obtain Hence when , we have
Since , , for , we get and we obtain
Lemma 3 is proved.

Remark 4. In Lemma 3, when and for any , the integral in must be convergent, which means

Lemma 5. Assume that (C1)–(C4) hold and is a solution of (12) satisfying for with . For , define where , . Then u(t) satisfies where

Proof. Without loss of generality we may assume there exists such that for ; then Lemmas 2 and 3 hold. Let be defined by (38). Then, differentiating (38) and using (12), it follows that Using the fact that , we obtain
When , using the Pötzsche chain rule, we have and it follows that By Lemmas 2 and 3, for , we obtain So (42) becomes
When , we have and it follows that By Lemmas 2 and 3, for , we obtain So (42) becomes
By (46) and (50), (39) holds. Lemma 5 is proved.