- Research Article
- Open Access

# Oscillation of Second-Order Mixed-Nonlinear Delay Dynamic Equations

- M Ünal
^{1}Email author and - A Zafer
^{2}

**2010**:389109

https://doi.org/10.1155/2010/389109

© M. Ünal and A. Zafer. 2010

**Received:**19 January 2010**Accepted:**20 March 2010**Published:**30 March 2010

## Abstract

New oscillation criteria are established for second-order mixed-nonlinear delay dynamic equations on time scales by utilizing an interval averaging technique. No restriction is imposed on the coefficient functions and the forcing term to be nonnegative.

## Keywords

- Difference Equation
- Linear Differential Equation
- Force Term
- Differential Calculus
- Nonnegative Real Number

## 1. Introduction

In this paper we are concerned with oscillatory behavior of the second-order nonlinear delay dynamic equation of the form

on an arbitrary time scale , where

the functions , , : are right-dense continuous with nondecreasing; the delay functions are nondecreasing right-dense continuous and satisfy for with as .

We assume that the time scale is unbounded above, that is, and define the time scale interval by . It is also assumed that the reader is already familiar with the time scale calculus. A comprehensive treatment of calculus on time scales can be found in [1–3].

By a solution of (1.1) we mean a nontrivial real valued function such that and for all with , and that satisfies (1.1). A function is called an oscillatory solution of (1.1) if is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. Equation (1.1) is said to be oscillatory if and only if every solution of (1.1) is oscillatory.

Notice that when , (1.1) is reduced to the second-order nonlinear delay differential equation

while when , it becomes a delay difference equation

Another useful time scale is and is a real number , which leads to the quantum calculus. In this case, (1.1) is the -difference equation

Interval oscillation criteria are more natural in view of the Sturm comparison theory since it is stated on an interval rather than on infinite rays and hence it is necessary to establish more interval oscillation criteria for equations on arbitrary time scales as in . As far as we know when , an interval oscillation criterion for forced second-order linear differential equations was first established by El-Sayed [4]. In 2003, Sun [5] demonstrated nicely how the interval criteria method can be applied to delay differential equations of the form

where the potential and the forcing term may oscillate. Some of these interval oscillation criteria were recently extended to second-order dynamic equations in [6–10]. Further results on oscillatory and nonoscillatory behavior of the second order nonlinear dynamic equations on time scales can be found in [11–23], and the references cited therein.

Therefore, motivated by Sun and Meng's paper [24], using similar techniques introduced in [17] by Kong and an arithmetic-geometric mean inequality, we give oscillation criteria for second-order nonlinear delay dynamic equations of the form (1.1). Examples are considered to illustrate the results.

## 2. Main Results

We need the following lemmas in proving our results. The first two lemmas can be found in [25, Lemma ].

Lemma 2.1.

Lemma 2.2.

The next two lemmas are quite elementary via differential calculus; see [23, 25].

Lemma 2.3.

Lemma 2.4.

The last important lemma that we need is a special case of the one given in [6]. For completeness, we provide a proof.

Lemma 2.5.

Proof.

which completes the proof.

In what follows we say that a function belongs to if and only if is right-dense continuous function on having continuous -partial derivatives on , with for all and for all . Note that in case , the -partial derivatives become the usual partial derivatives of . The partial derivatives for the cases and will be explicitly given later.

Denoting the -partial derivatives and of with respect to and by and , respectively, the theorems below extend the results obtained in [5] to nonlinear delay dynamic equation on arbitrary time scales and coincide with them when is replaced by . Indeed, if we set , then it follows that

as in [5]. However, we prefer using instead of for simplicity.

Theorem 2.6.

then (1.1) is oscillatory.

Proof.

Suppose on the contrary that is a nonoscillatory solution of (1.1). First assume that and are positive for all for some . Choose sufficiently large so that . Let .

Define

Similarly, by following the above calculation step by step, that is, multiplying both sides of (2.32) this time by after taking into account that

This contradiction completes the proof when is eventually positive. The proof when is eventually negative is analogous by repeating the above arguments on the interval instead of .

Corollary 2.7.

then (1.3) is oscillatory.

Corollary 2.8.

then (1.4) is oscillatory.

Corollary 2.9.

then (1.5) is oscillatory.

Notice that Theorem 2.6 does not apply if there is no forcing term, that is, . In this case we have the following theorem.

Theorem 2.10.

then (1.1) with is oscillatory.

Proof.

where and , are as in Lemma 2.2.

Corollary 2.11.

then (1.3) with is oscillatory.

Corollary 2.12.

then (1.4) with is oscillatory.

Corollary 2.13.

then (1.5) with is oscillatory.

It is obvious that Theorem 2.6 is not applicable if the functions are nonpositive for . In this case the theorem below is valid.

Theorem 2.14.

then (1.1) is oscillatory.

Proof.

Suppose that (1.1) has a nonoscillatory solution. Without losss of generality, we may assume that and are eventually positive on when is sufficiently large. If is eventually negative, one may repeat the same proof step by step on the interval

The rest of the proof is the same as that of Theorem 2.6 and hence it is omitted.

Corollary 2.15.

then (1.3) is oscillatory.

Corollary 2.16.

then (1.4) is oscillatory.

Corollary 2.17.

then (1.5) is oscillatory.

## 3. Examples

In this section we give three examples when , and , in (1.1). That is, we consider

For simplicity we take , thus . Note that and by Lemma 2.2.

Example 3.1.

We calculate

Since all conditions of Corollary 2.11 are satisfied, we conclude that (3.2) is oscillatory when (3.4) holds.

Example 3.2.

Since all conditions of Corollary 2.12 are satisfied, we conclude that (3.5) is oscillatory if (3.7) holds.

Example 3.3.

We see that (2.67) holds for all and . Since all conditions of Corollary 2.12 are satisfied, we conclude that (3.8) is oscillatory if and are positive.

## Declarations

### Acknowledgments

The paper is supported in part by the Scientific and Research Council of Turkey (TUBITAK) under Contract 108T688. The authors would like to thank the referees for their valuable comments and suggestions.

## 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 (Eds):
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar - Lakshmikantham V, Sivasundaram S, Kaymakçalan B:
*Dynamic Systems on Measure Chains, Mathematics and Its Applications*.*Volume 370*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.View ArticleGoogle 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 - Sun YG:
**A note on 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 - Agarwal RP, Anderson DR, Zafer A:
**Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities.***Computers and Mathematics with Applications*2010,**59**(2):977-993. 10.1016/j.camwa.2009.09.010MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Zafer A:
**Oscillation criteria for second-order forced dynamic equations with mixed nonlinearities.***Advances in Difference Equations*2009,**2009:**-20.Google 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
- 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 - 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 - Bohner M, Saker SH:
**Oscillation of second order nonlinear dynamic equations on time scales.***The Rocky Mountain Journal of Mathematics*2004,**34**(4):1239-1254. 10.1216/rmjm/1181069797MathSciNetView ArticleMATHGoogle Scholar - Došlý O, Hilger S:
**A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales.***Journal of Computational and Applied Mathematics*2002,**141**(1-2):147-158. 10.1016/S0377-0427(01)00442-3MathSciNetView ArticleMATHGoogle 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 - Erbe L, Hassan TS, Peterson A:
**Oscillation of second order neutral delay differential equations.***Advances in Dynamical Systems and Applications*2008,**3**(1):53-71.MathSciNetGoogle Scholar - Huang M, Feng W: Oscillation for forced second-order nonlinear dynamic equations on time scales. Electronic Journal of Differential Equations 2006, (145):1-8. 10.1016/j.jde.2006.02.002Google Scholar
- Kong Q:
**Interval criteria for oscillation of second-order linear ordinary differential equations.***Journal of Mathematical Analysis and Applications*1999,**229**(1):258-270. 10.1006/jmaa.1998.6159MathSciNetView ArticleMATHGoogle Scholar - Del Medico A, Kong Q:
**Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain.***Journal of Mathematical Analysis and Applications*2004,**294**(2):621-643. 10.1016/j.jmaa.2004.02.040MathSciNetView ArticleMATHGoogle Scholar - Del Medico A, Kong Q:
**New Kamenev-type oscillation criteria for second-order differential equations on a measure chain.***Computers & Mathematics with Applications*2005,**50**(8-9):1211-1230. 10.1016/j.camwa.2005.07.002MathSciNetView ArticleMATHGoogle 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 - Şahiner Y:
**Oscillation of second-order delay differential equations on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2005,**63**(5–7):e1073-e1080.MATHGoogle Scholar - Saker SH:
**Oscillation of nonlinear dynamic equations on time scales.***Applied Mathematics and Computation*2004,**148**(1):81-91. 10.1016/S0096-3003(02)00829-9MathSciNetView ArticleMATHGoogle Scholar - Zafer A:
**Interval oscillation criteria for second order super-half linear functional differential equations with delay and advanced arguments.***Mathematische Nachrichten*2009,**282**(9):1334-1341. 10.1002/mana.200610801MathSciNetView ArticleMATHGoogle Scholar - Sun YG, Meng FW:
**Interval criteria for oscillation of second-order differential equations with mixed nonlinearities.***Applied Mathematics and Computation*2008,**198**(1):375-381. 10.1016/j.amc.2007.08.042MathSciNetView 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, 30*. Springer, Berlin, Germany; 1961:xii+198.Google Scholar

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