- Research Article
- Open Access

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

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

**2009**:938706

https://doi.org/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.

## Keywords

- Dynamic Equation
- Oscillation Criterion
- Jump Operator
- Infinite Interval
- Time Scale Modification

## 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

## References

- Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introduction with Applications*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar - Bohner M, Peterson A:
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.View ArticleMATHGoogle Scholar - Došlý O, Řehák P:
*Half-Linear Differential Equations, North-Holland Mathematics Studies*.*Volume 202*. Elsevier; North-Holland, Amsterdam, The Netherlands; 2005:xiv+517.Google Scholar - Agarwal RP, Grace SR:
*Oscillation Theory for Difference and Functional Differential Equations*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2002.Google Scholar - Agarwal RP, Grace SR, O'Regan D:
*Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2002:xiv+672.View ArticleMATHGoogle Scholar - Bohner M, Tisdell CC:
**Oscillation and nonoscillation of forced second order dynamic equations.***Pacific Journal of Mathematics*2007,**230**(1):59-71. 10.2140/pjm.2007.230.59MathSciNetView ArticleMATHGoogle Scholar - Došlý O, Marek D:
**Half-linear dynamic equations with mixed derivatives.***Electronic Journal of Differential Equations*2005,**2005**(90):1-18.Google Scholar - Erbe L, Peterson A, Saker SH:
**Hille-Kneser-type criteria for second-order linear dynamic equations.***Advances in Difference Equations*2006,**2006:**-18.Google Scholar - Erbe L, Peterson A, Saker SH:
**Oscillation criteria for second-order nonlinear delay dynamic equations.***Journal of Mathematical Analysis and Applications*2007,**333**(1):505-522. 10.1016/j.jmaa.2006.10.055MathSciNetView ArticleMATHGoogle Scholar - Kartsatos AG:
**Maintenance of oscillations under the effect of a periodic forcing term.***Proceedings of the American Mathematical Society*1972,**33:**377-383. 10.1090/S0002-9939-1972-0330622-0MathSciNetView ArticleMATHGoogle Scholar - Manojlović JV:
**Oscillation criteria for second-order half-linear differential equations.***Mathematical and Computer Modelling*1999,**30**(5-6):109-119. 10.1016/S0895-7177(99)00151-XMathSciNetView ArticleMATHGoogle Scholar - Řehák P:
**Half-linear dynamic equations on time scales: IVP and oscillatory properties.***Nonlinear Functional Analysis and Applications*2002,**7**(3):361-403.MathSciNetMATHGoogle Scholar - Saker SH:
**Oscillation criteria of second-order half-linear dynamic equations on time scales.***Journal of Computational and Applied Mathematics*2005,**177**(2):375-387. 10.1016/j.cam.2004.09.028MathSciNetView ArticleMATHGoogle Scholar - Sun YG, Agarwal RP:
**Forced oscillation of****th-order nonlinear differential equations.***Functional Differential Equations*2004,**11**(3-4):587-596.MathSciNetMATHGoogle Scholar - Sun YG, Saker SH:
**Forced oscillation of higher-order nonlinear differential equations.***Applied Mathematics and Computation*2006,**173**(2):1219-1226. 10.1016/j.amc.2005.04.065MathSciNetView ArticleMATHGoogle Scholar - Sun YG, Wong JSW:
**Note on forced oscillation of****th-order sublinear differential equations.***Journal of Mathematical Analysis and Applications*2004,**298**(1):114-119. 10.1016/j.jmaa.2004.03.076MathSciNetView ArticleMATHGoogle Scholar - Teufel H Jr.:
**Forced second order nonlinear oscillation.***Journal of Mathematical Analysis and Applications*1972,**40:**148-152. 10.1016/0022-247X(72)90037-6MathSciNetView ArticleGoogle Scholar - Wang Q-R, Yang Q-G:
**Interval criteria for oscillation of second-order half-linear differential equations.***Journal of Mathematical Analysis and Applications*2004,**291**(1):224-236. 10.1016/j.jmaa.2003.10.028MathSciNetView ArticleMATHGoogle Scholar - Wong JSW:
**Second order nonlinear forced oscillations.***SIAM Journal on Mathematical Analysis*1988,**19**(3):667-675. 10.1137/0519047MathSciNetView ArticleMATHGoogle Scholar - El-Sayed MA:
**An oscillation criterion for a forced second order linear differential equation.***Proceedings of the American Mathematical Society*1993,**118**(3):813-817.MathSciNetMATHGoogle Scholar - Anderson DR:
**Oscillation of second-order forced functional dynamic equations with oscillatory potentials.***Journal of Difference Equations and Applications*2007,**13**(5):407-421. 10.1080/10236190601116209MathSciNetView ArticleMATHGoogle Scholar - Anderson DR, Zafer A: Interval criteria for second-order super-half-linear functional dynamic equations with delay and advanced arguments. to appear in Journal of Difference Equations and ApplicationsGoogle Scholar
- Li W-T:
**Interval oscillation of second-order half-linear functional differential equations.***Applied Mathematics and Computation*2004,**155**(2):451-468. 10.1016/S0096-3003(03)00790-2MathSciNetView ArticleMATHGoogle Scholar - Li W-T, Cheng SS:
**An oscillation criterion for nonhomogenous half-linear differential equations.***Applied Mathematics Letters*2002,**15**(3):259-263. 10.1016/S0893-9659(01)00127-6MathSciNetView ArticleMATHGoogle Scholar - Nasr AH:
**Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential.***Proceedings of the American Mathematical Society*1998,**126**(1):123-125. 10.1090/S0002-9939-98-04354-8MathSciNetView ArticleMATHGoogle Scholar - Sun YG, Ou CH, Wong JSW:
**Interval oscillation theorems for a second-order linear differential equation.***Computers & Mathematics with Applications*2004,**48**(10-11):1693-1699. 10.1016/j.camwa.2003.08.012MathSciNetView ArticleMATHGoogle Scholar - Sun YG:
**A note Nasr's and Wong's papers.***Journal of Mathematical Analysis and Applications*2003,**286**(1):363-367. 10.1016/S0022-247X(03)00460-8MathSciNetView ArticleMATHGoogle Scholar - Wong JSW:
**Oscillation criteria for a forced second-order linear differential equation.***Journal of Mathematical Analysis and Applications*1999,**231**(1):235-240. 10.1006/jmaa.1998.6259MathSciNetView ArticleMATHGoogle Scholar - Yang Q:
**Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential.***Applied Mathematics and Computation*2003,**135**(1):49-64. 10.1016/S0096-3003(01)00307-1MathSciNetView ArticleMATHGoogle Scholar - Zafer A: Interval oscillation criteria for second order super-half-linear functional differential equations with delay and advanced arguments. to appear in Mathematische NachrichtenGoogle Scholar
- Güvenilir AF, Zafer A:
**Second-order oscillation of forced functional differential equations with oscillatory potentials.***Computers & Mathematics with Applications*2006,**51**(9-10):1395-1404. 10.1016/j.camwa.2006.02.002MathSciNetView ArticleMATHGoogle Scholar - Řehák P:
**Hardy inequality on time scales and its applications to half-linear dynamic equations.***Journal of Inequalities and Applications*2005,**2005**(7):495-507.MATHGoogle Scholar - Řehák P:
**On certain comparison theorems for half-linear dynamic equations on time scales.***Abstract and Applied Analysis*2004,**2004**(7):551-565. 10.1155/S1085337504306251View ArticleMATHGoogle Scholar - Sun YG, Wong JSW:
**Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities.***Journal of Mathematical Analysis and Applications*2007,**334**(1):549-560. 10.1016/j.jmaa.2006.07.109MathSciNetView ArticleMATHGoogle Scholar - Beckenbach EF, Bellman R:
*Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, N. F.*.*Volume 30*. Springer, Berlin, Germany; 1961:xii+198.Google Scholar

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