# Oscillation Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities

- Ravi P. Agarwal
^{1}and - A. Zafer
^{2}Email author

**2009**:938706

**DOI: **10.1155/2009/938706

© R.P. Agarwal and A. Zafer. 2009

**Received: **21 January 2009

**Accepted: **19 April 2009

**Published: **2 June 2009

## Abstract

We obtain new oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the form , with , , where is a time scale interval with , the functions are right-dense continuous with , is the forward jump operator, , and . All results obtained are new even for and . In the special case when and our theorems reduce to (Y. G. Sun and J. S. W. Wong, Journal of Mathematical Analysis and Applications. 337 (2007), 549–560). Therefore, our results in particular extend most of the related existing literature from the continuous case to arbitrary time scale.

## 1. Introduction

Let be a time scale which is unbounded above and a fixed point. For some basic facts on time scale calculus and dynamic equations on time scales, one may consult the excellent texts by Bohner and Peterson [1, 2].

By a proper solution of (1.1) on we mean a function which is defined and nontrivial in any neighborhood of infinity and which satisfies (1.1) for all , where denotes the set of right-dense continuously differentiable functions from to . As usual, such a solution of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. The equation is called oscillatory if every proper solution is oscillatory.

where is the forward difference operator.

Further extensions of these results can be found in [30, 31], where the authors have studied some related super-half-linear differential equations with delay and advance arguments.

for which almost nothing is available in the literature.

## 2. Lemmas

We need the following preparatory lemmas. The first two lemmas are given by Wong and Sun as a single lemma [34, Lemma ] for . The proof for the case is exactly the same, in fact one only needs to replace the exponents by in their proof. Lemma 2.3 is the well-known Young inequality.

Lemma 2.1.

where is any positive number with

Lemma 2.2.

Lemma 2.3 (Young's Inequality).

## 3. The Main Results

The main results of this paper are contained in the following three theorems. The arguments used in the proofs have common features with the ones developed in [22, 30, 34].

Theorem 3.1.

then (1.1) is oscillatory.

Proof.

To arrive at a contradiction, let us suppose that is a nonoscillatory solution of (1.1). First, we assume that is positive for all , for some .

where stands for the inverse function. In our case, since , dynamic equation (3.15) has a unique solution satisfying . Clearly, the unique solution is . Therefore, on .

Using differential calculus, see [7], the result follows.

which of course contradicts (3.3). This completes the proof when is eventually positive. The proof when is eventually negative is analogous by repeating the arguments on the interval instead of .

A close look at the proof of Theorem 3.1 reveals that one cannot take . The following theorem is a substitute in that case.

Theorem 3.2.

then (1.1) with is oscillatory.

Proof.

The remainder of the proof is the same as that of Theorem 3.1.

As it is shown in [34] for the sublinear terms case, we can also remove the sign condition imposed on the coefficients of the sub-half-linear terms to obtain interval criterion which is applicable for the case when some or all of the functions , , are nonpositive. We should note that the sign condition on the coefficients of super-half-linear terms cannot be removed alternatively by the same approach. Furthermore, the function cannot take the value zero on intervals of interest in this case. We have the following theorem.

Theorem 3.3.

then (1.1) is oscillatory.

Proof.

The remainder of the proof is the same as that of Theorem 3.1, hence it is omitted.

## 4. Applications

To illustrate the usefulness of the results we state the corresponding theorems for the special cases , , and . One can easily provide similar results for other specific time scales of interest.

### 4.1. Differential Equations

where are continuous functions with , and . Let

Theorem 4.1.

then (4.1) is oscillatory.

Theorem 4.2.

then (4.1) with is oscillatory.

Theorem 4.3.

then (4.1) is oscillatory.

### 4.2. Difference Equations

where , with , and . Let , and

Theorem 4.4.

then (4.13) is oscillatory.

Theorem 4.5.

then (4.13) with is oscillatory.

Theorem 4.6.

then (4.13) is oscillatory.

### 4.3. -Difference Equations

where with , with , and . Let with , and

Theorem 4.7.

then (4.25) is oscillatory.

Theorem 4.8.

then (4.25) with is oscillatory.

Theorem 4.9.

then (4.25) is oscillatory.

## 5. Examples

Example 5.1.

Example 5.2.

Example 5.3.

## 6. Remarks

- (1)
Literature

Equation (1.1) has been studied by Sun and Wong [34] for the case and . Our results in Section 4.1 coincide with theirs when , and therefore the results can be considered as an extension from to . Since the results in [34] are linked to many well-known oscillation criteria in the literature, the interval oscillation criteria we have obtained provide further extensions of these to time scales.

The results in Sections 4.2 and 4.3 are all new for all values of the parameters. Although there are some results for difference equations in the special case , there is hardly any interval oscillation criteria for the -difference equations case.

- (2)
Generalization

- (3)
Forms Related to (1.1)

It is not difficult to see that time scale modifications of the previous arguments give rise to completely parallel results for the above dynamic equations. For an illustrative example we provide below the version of Theorem 3.1 for (6.4). The other theorems for (6.4), (6.5), and (6.6) can be easily obtained by employing arguments developed for (1.1) in this paper.

Theorem 6.1.

- (4)
An Open Problem

It is of theoretical and practical interest to obtain interval oscillation criteria when there are only sub-half-linear terms in (1.1), that is, when holds for all . Also, the open problems stated in [34] for the special case with naturally carry over for (1.1).

## Authors’ Affiliations

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