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# Oscillation of Second-Order Sublinear Dynamic Equations with Damping on Isolated Time Scales

*Advances in Difference Equations*
**volume 2010**, Article number: 103065 (2010)

## Abstract

This paper concerns the oscillation of solutions to the second sublinear dynamic equation with damping , on an isolated time scale which is unbounded above. In , *α* is the quotient of odd positive integers. As an application, we get the difference equation , where , , and is any real number, is oscillatory.

## 1. Introduction

During the past years, there has been an increasing interest in studying the oscillation of solution of second-order damped dynamic equations on time scale which attempts to harmonize the oscillation theory for continuousness and discreteness, to include them in one comprehensive theory, and to eliminate obscurity from both. We refer the readers to the papers [1–4] and the references cited therein.

In [5], Bohner et al. consider the second-order nonlinear dynamic equation with damping

where and are real-valued, right-dense continuous functions on a time scale , with . is continuously differentiable and satisfies and for . When , where , is the quotient of odd positive integers, (1.1) is the second-order sublinear dynamic equation with damping

When , (1.2) is the second-order sublinear dynamic equation

When , (1.3) is the second-order sublinear difference equation

In [6], under the assumption of being an isolated time scale, we prove that, when is allowed to take on negative values, is sufficient for the oscillation of the dynamic equation (1.3). As an application, we get that, when is allowed to take on negative values, is sufficient for the oscillation of the dynamic equation (1.4), which improves a result of Hooker and Patula [7, Theorem 4.1] and Mingarelli [8].

In this paper, we extend the result of [6] to dynamic equation (1.1). As an application, we get that the difference equation with damping

where , , , and is any real number, is oscillatory.

For completeness (see [9, 10] for elementary results for the time scale calculus), we recall some basic results for dynamic equations and the calculus on time scales. Let be a time scale (i.e., a closed nonempty subset of ) with . The forward jump operator is defined by

and the backward jump operator is defined by

where , where denotes the empty set. If , we say is right scattered, while, if , we say is left scattered. If , we say is right dense, while, if and , we say is left-dense. Given a time scale interval in the notation denotes the interval in case and denotes the interval in case . The graininess function *μ* for a time scale is defined by , and for any function the notation denotes . We say that is differentiable at provided

exists when (here, by , it is understood that approaches in the time scale) and when is continuous at and

Note that if , then the delta derivative is just the standard derivative, and when the delta derivative is just the forward difference operator. Hence, our results contain the discrete and continuous cases as special cases and generalize these results to arbitrary time scales (e.g., the time scale which is very important in quantum theory [11]).

## 2. Lemmas

We will need the following second mean value theorem (see [10, Theorem 5.45]).

Lemma 2.1.

Let be a bounded function that is integrable on . Let and be the infimum and supremum, respectively, of the function on . Suppose that is nonincreasing with on . Then, there is some number with such that

Lemmas 2.2 and 2.4 give two lower bounds of definite integrals on time scale, respectively.

Lemma 2.2.

Assume that , where . If there exists a real number such that , for all , then, for , , one has

Remark 2.3.

It is easy to know that, when , and, when , .

Proof.

For , using Theorem 1.75 of [9], we have

We consider the two cases and . First, if , then we have that

On the other hand, if , then

which implies that

From (2.3)–(2.6) and the additivity of the integral, we have

Lemma 2.4.

Assume that , where , with . Then, for , , one has

Proof.

For , using Theorem 1.75 of [9], we have

Setting , we have that

We consider the two possible cases and . First, if , we have that

On the other hand, if , then

which implies that

Hence, from (2.10)–(2.13), we have that

From (2.9), (2.10), (2.14), and the additivity of the integral, we have

## 3. Main Theorem

Theorem 3.1.

Assume that , where . Suppose that

(i)there exists a real number such that , for all ;

(ii)there exists a function such that for ,

Then, (1.1) is oscillatory.

Proof.

For the sake of contradiction, assume that (1.1) is nonoscillatory. Then, without loss of generality, there is a solution of (1.1) and a with , for all . Making the substitution in (1.1) and noticing that

we get that

Multiplying both sides of (3.3) by , integrating from to , and using an integration by parts formula, we get

Next, using the quotient rule and then Pötzsche's chain rule [9, Theorem 1.90] gives

where we used the fact that . Using this last inequality in (3.4), we get

Note that

Let us define , . Then, we get from (3.6) and (3.7) that

since , for . So the first term of (3.8) is nonnegative. From (3.8), we get that

From and , using the second mean value theorem [10, Theorem 5.45] and Lemmas 2.2 and 2.4, we get that

From and , the fifth term of (3.9) is nonnegative. From (3.9), and (3.10), we get that

Since , from (3.11), there exists such that, for , we have

Dividing both sides of this last inequality by and integrating from to , we get, using inequality (2.11) in [12], that

Since , we get , for large , which is a contradiction. Thus, (1.1) is oscillatory.

When , , and , it is easy to get that , . So we have the following corollary (see Corollary 2.4 of [6]). Corollary 3.2 shows that, with no sign assumption on , the condition

is sufficient for the oscillation of the difference equation (1.4).

Corollary 3.2.

Assume that . If

then (1.4) is oscillatory.

By using the idea in Theorem 3.1, we can also consider the differential equation

where . It is easy to get the following.

Theorem 3.3.

Suppose that there exists a function such that

Then, the differential equation (3.16) is oscillatory.

Example 3.4.

Consider the sublinear difference equation

where , , , and is any real number.

Take , . We have

Let . Then, we have , for large . So is concave for large . Therefore, we have

for large . That means . It is easy to get that and is nonincreasing for large . So from Theorem 3.1, (3.18) is oscillatory.

Example 3.5.

Let , , and consider the -difference equation

where , , , is any real number. Take . We have

and is nonincreasing. So from Theorem 3.1, (3.21) is oscillatory.

Example 3.6.

Let , and consider the differential equation

where , , , and is any real number.

Take . It is easy to know that

So from Theorem 3.3, (3.23) is oscillatory.

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## Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 10971232).

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Lin, Q., Jia, B. Oscillation of Second-Order Sublinear Dynamic Equations with Damping on Isolated Time Scales.
*Adv Differ Equ* **2010, **103065 (2010) doi:10.1155/2010/103065

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### Keywords

- Dynamic Equation
- Difference Equation
- Jump Operator
- Part Formula
- Closed Nonempty Subset