New Oscillatory Behavior of Second-Order Nonlinear Dynamic Equations with Damping on Time Scales

Abstract

We establish four new oscillation criteria of Grace-type for the second-order nonlinear dynamic equations with damping. These criteria extend known criteria for corresponding dynamic equations. Our results are new even in the continuous and the discrete cases.

1. Introduction

In this paper, we deal with the oscillatory behavior of all solutions of the second-order nonlinear dynamic equation with damping: Our aim is to give some oscillatory criteria of Grace-type of (1). Here, we give the following hypotheses: (H₁) is a time scale (i.e., a nonempty closed subset of the real numbers ) which is unbounded above and with ; we define the time scale interval of the form by ; (H₂) is the ratio of two positive and odd integers; (H₃)  are positive and rd-continuous functions such that ; (H₄) is a continuous function such that, for some positive constant ,

By a solution of (1), we mean a nontrivial real-valued function satisfying (1) for . We recall that a solution of (1) is said to be oscillatory on in case it is neither eventually positive nor eventually negative; otherwise, the solution is said to be nonoscillatory. Equation (1) is said to be oscillatory in case all of its solutions are oscillatory. Our attention is restricted on those solutions of (1) in which is not the eventually identical zero.

In 2009, Grace et al. [1] considered the second-order half-linear dynamic equations: and obtained some oscillatory criteria of Grace-type of (3). In recent years, there have been numerous researches and many research activities concerning the oscillation and nonoscillation of solutions of (3) and its special cases; we refer the reader to the papers [2–8].

In this paper, we will establish two sufficient conditions for oscillation of all solutions of (1) by use of the generalized Riccati transformation and the inequality technique, under the condition that Moreover, if condition (4) does not hold, that is, holds, two sufficient conditions are obtained for oscillation or convergence to zero of (1).

In order to prove the main results of this paper, we will use the following rules: For more details about differential and integral theory on the time scale, see [9, 10].

2. The Main Results

In order to prove the main results of this paper, we first give the following two lemmas.

Lemma 1 (see [9, Theorem 1.90]). Assume that is delta-differentiable and eventually positive or eventually negative; then

Lemma 2. Assume that , , and (4) hold. Let be an eventually position solution of (1). Then there exists such that
Similar to the proof of Lemma 3.3 in paper [8], we can give the proof of Lemma 2; thus, the proof is omitted here.

Theorem 3. Assume that (H₁)–(H₄) and (4) hold. If there exists a positive nondecreasing -differentiable function such that, for every , where then (1) is oscillatory on .

Proof. Suppose, to the contrary, that is a nonoscillatory solutions of (1) on . Without loss of generality, we may assume that for all , . We shall consider only this case, since the proof when is eventually negative is similar. By Lemma 2, we obtain on that Define the function by Then, we have on ; by (1), (6), (7), and (12), we obtain Now and thus Using (17) in (15), we have Integrating (18) from to , we obtain which gives a contradiction by (10). This completes the proof.

Theorem 4. Assume that (H₁)–(H₄) and (4) hold. If there exists a positive -differentiable function such that, for every , where has been defined in (11), then (1) is oscillatory on .

Proof. Suppose, to the contrary, that is a nonoscillatory solution of (1) on . We may assume, without loss of generality, that for all , . When is eventually negative, the proof is similar. Proceeding as in the proof of Theorem 3, we obtain (14). Using (8) and (9), we have Using (12) and (21) in (14), we have By , we have Substituting (23) in (22), we obtain That is, on , Now using inequality (17) as the form, that is, this implies on that Using (27) in (25), we have on that Integrating both sides of this inequality from to , taking the of the resulting inequality as and applying condition (20), we obtain a contradiction to the fact that for . This completes the proof.

Now, when condition (5) holds, using the same method of proof of Theorem 3.3 and Theorem 3.4 in paper [8], we can obtain the following theorems.

Theorem 5. Assume that (H₁)–(H₄) and (5) hold. If there exists a positive -differentiable function such that (10) holds, where has been defined in (11), and then every solution of (1) is either oscillatory or converges to zero on .

Theorem 6. Assume that (H₁)–(H₄), (5), and (29) hold. If there exists a positive -differentiable function such that (20) holds, where has been defined in (11), then every solution of (1) is either oscillatory or converges to zero on .

Remark 7. Our results in this paper extend some known results and make some results of [1] special cases of our results. The theorems in this paper are new even for the cases and .

Example 8. Consider the second-order nonlinear delay 2-difference equations: Here, The conditions (H₁) and (H₂) are clearly satisfied, (H₄) holds with , and (H₃) is satisfied as Next, by [11, Lemma 2], using (H₃), we have for , so that, as , Hence, (4) is satisfied. Now let ; for all , then, as , so (10) is satisfied as well. Altogether, by Theorem 3, (30) is oscillatory.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors sincerely thank the reviewers and editors for their valuable suggestions and useful comments that have led to the present improved version of the original paper. This work was supported by the Natural Science Foundation of Shandong Province of China under Grant no. ZR2013AM003 and the Development Program in Science and Technology of Shandong Province of China under Grant no. 2010GWZ20401.