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# Corrigendum to "Oscillation behavior of third-order neutral Emden-Fowler delay xdynamic equations on time scales" [Adv. Difference Equ., 2010, 1-23 (2010)]

- Tao Ji
^{1}, - Shuhong Tang
^{1}and - Tongxing Li
^{2, 3}Email author

**2012**:57

https://doi.org/10.1186/1687-1847-2012-57

© Ji et al; licensee Springer. 2012

**Received:**24 October 2011**Accepted:**9 May 2012**Published:**9 May 2012

## Abstract

In this article, we revise results obtained by Han et al.

**Mathematics Subject Classification 2000:** 34K11; 39A10.

## Keywords

- oscillation
- third-order
- neutral dynamic equation
- time scale

## 1. Introduction

on a time scale with sup $\mathbb{T}=\infty $, where the authors assume the following hypotheses hold.

(*A*_{1}) γ *>* 0 is the quotient of odd positive integers;

(*A*_{2}) *r* and *p* are positive real-valued rd-continuous functions defined on
such that *r*^{Δ}(*t*) ≥ 0;

(*A*_{3}) *a* is a positive real-valued rd-continuous function defined on
such that 0 *< a*(*t*) ≤ *a*_{0}*<* 1 and lim_{t→∞}*a*(*t*) = *a*_{1};

(*A*_{4}) the functions $\tau :\mathbb{T}\to \mathbb{T}$ and $\delta :\mathbb{T}\to \mathbb{T}$ are rd-continuous functions such that *τ*(*t*) ≤ *t*, *δ* (*t*) ≤ *t*, and lim_{t→∞}*τ*(*t*) = lim_{t→∞}*δ*(*t*) = ∞.

A time scale
is an arbitrary nonempty closed subset of the real numbers ℝ. Since we are interested in oscillatory behavior, we suppose that the time scale under consideration is not bounded above and is a time scale interval of the form ${\left[{t}_{0},\infty \right)}_{T}:=\left[{t}_{0},\infty \right)\cap T$. For some concepts related to the notion of time scales; see [3]. Regarding the oscillation properties of (1.1) with *a*(*t*) = 0, Saker [4–7] established some types of criteria, e.g., Hille-Nehari-type and Philos-type.

To establish oscillation criteria for (1.1), [2] obtained various oscillation theorems by using some lemmas, one of which we present below for the convenience of the reader.

*Lemma*1.1. (See [2, Lemma 2.1]). Let

*z*(

*t*) :=

*x*(

*t*) -

*a*(

*t*)

*x*(

*τ*(

*t*)). Assume that (

*A*

_{1})-(

*A*

_{4}) hold and

*x*is an eventually positive solution of (1.1). If

then there are only the following three cases for $t\in {\left[{t}_{1},\infty \right)}_{T}$, where ${t}_{1}\in {\left[{t}_{0},\infty \right)}_{T}$ sufficiently large:

Case (*i*). $z\left(t\right)>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{z}^{\Delta}\left(t\right)>0,{z}^{{\Delta}^{2}}\left(t\right)>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{z}^{{\Delta}^{3}}\left(t\right)<0;$

Case (*ii*). $z\left(t\right)>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{z}^{\Delta}\left(t\right)>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{{z}^{\Delta}}^{{\phantom{\rule{0.1em}{0ex}}}^{2}}\left(t\right)>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{{z}^{\Delta}}^{{\phantom{\rule{0.1em}{0ex}}}^{3}}\left(t\right)<0,\underset{t\to \infty}{\text{lim}}x\left(t\right)=0;$

Case (*iii*). $z\left(t\right)>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{z}^{\Delta}\left(t\right)<0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{{z}^{\Delta}}^{{\phantom{\rule{0.1em}{0ex}}}^{2}}\left(t\right)>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{{z}^{\Delta}}^{{\phantom{\rule{0.1em}{0ex}}}^{3}}\left(t\right)<0,\underset{t\to \infty}{\text{lim}}z\left(t\right)=l\ge 0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\underset{t\to \infty}{\text{lim}}x\left(t\right)=l/\left(1-a\right)\ge 0.$

We note that there exists a mistake in the above statements. First, the case (*ii*) does not occur since *z*^{Δ}*>* 0 and ${z}^{{\Delta}^{2}}>0$ imply that lim_{t→∞}*z* (*t*) = ∞, and so *z >* 0 eventually. Second, the restrictive assumption (*H*) can be omitted. Hence the purpose of this article is to revise the related results in [2].

## 2. Revised results

Now we use notation *z* as in Lemma 1.1 and present the following new lemmas.

*Lemma* 2.1. Let (1.2), (*A*_{1}), (*A*_{2}), and (*A*_{4}) hold with (*A*_{3}) replaced by (*A* 3*) *a* is a positive real-valued rd-continuous function defined on

such that 0 *< a*(*t*) ≤ *a*_{0}*<* 1. Suppose that *x* is an eventually positive solution of (1.1). Then there are only the following three cases eventually:

Case (1). $z>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{z}^{\Delta}>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{{z}^{\Delta}}^{{\phantom{\rule{0.1em}{0ex}}}^{2}}>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\left(r{{z}^{\Delta}}^{{\phantom{\rule{0.1em}{0ex}}}^{2}}\right)}^{\Delta}<0;$

Case (2). $z>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{z}^{\Delta}<0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{{z}^{\Delta}}^{{\phantom{\rule{0.1em}{0ex}}}^{2}}>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\left(r{{z}^{\Delta}}^{{\phantom{\rule{0.1em}{0ex}}}^{2}}\right)}^{\Delta}<0;$

Case (3). $z<0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{z}^{\Delta}<0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{{z}^{\Delta}}^{{\phantom{\rule{0.1em}{0ex}}}^{2}}>0,\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{\left(r{{z}^{\Delta}}^{{\phantom{\rule{0.1em}{0ex}}}^{2}}\right)}^{\Delta}<0.$

*Proof*. Assume that

*x*is an eventually positive solution of (1.1). Then, we have by (1.1) that ${\left(r{z}^{{\Delta}^{2}}\right)}^{\Delta}<0$, and hence $r{z}^{{\Delta}^{2}}$ is decreasing and of one sign. The condition $r{z}^{{\Delta}^{2}}<0$ implies that there exist a ${t}_{1}\in {\left[{t}_{0},\infty \right)}_{T}$ and a constant

*M >*0 such that

*t*

_{1}to

*t*and letting

*t*→ ∞, we have by (1.2) that

*M*

_{1}

*>*0 such that

*t*

_{2}to

*t*and letting

*t*→ ∞, we have

*z <*0 eventually. Then, we get

_{t→∞}

*z*(

*t*) = -∞. Next we claim that

*x*is bounded and (2.1) does not occur. If not, there exists a sequence ${\left\{{t}_{m}\right\}}_{m\in \mathbb{N}}\in {\left[{t}_{0},\infty \right)}_{T}$ with

*t*

_{ m }→ ∞ as

*m*→ ∞ such that

*τ*(

*t*) ≤

*t*that

which implies that lim_{m→∞}*z* (*t*_{
m
} ) = ∞, this contradicts the fact that lim_{t→∞}*z* (*t*) = - ∞. Hence *x* is bounded, and so (2.1) does not hold.

If *z*^{Δ}*>* 0 and${z}^{{\Delta}^{2}}>0$, then *z >* 0. Thus, for ${z}^{{\Delta}^{2}}>0$ only the cases (1), (2), and (3) may occur. The proof is complete. □

*Lemma* 2.2. Let 0 *< a*(*t*) ≤ *a*_{0}*<* 1. If case (3) holds, then lim_{t→∞}*x* (*t*) = 0.

*Proof*. Assume that (3) holds. Then lim

_{t→∞}

*z*(

*t*) ≤ 0. Next we claim that

*x*is bounded. Similar as in the proof of Lemma 2.1, we have that lim

_{m→∞}

*z*(

*t*

_{ m }) = ∞ which contradicts the fact that lim

_{t→∞}

*z*(

*t*) ≤ 0. Thus,

*x*is bounded. Hence we can suppose that lim sup

_{t→∞}

*x*(

*t*) =

*x*

_{1}, where 0 ≤

*x*

_{1}

*< ∞*. Then, there exists a sequence ${\left\{{t}_{k}\right\}}_{k\in \mathbb{N}}\in {\left[{t}_{0},\infty \right)}_{T}$ with

*t*

_{ k }→ ∞ as

*k*→ ∞ such that lim

_{k→∞}

*x*(

*t*

_{ k }) =

*x*

_{1}. Next we show that lim

_{t→∞}

*x*(

*t*) = 0. If not, then

*x*

_{1}

*>*0. Pick

*ε*=

*x*

_{1}(1 -

*a*

_{0})/(2

*a*

_{0}), we find that

*x*(

*τ*(

*t*

_{ k }))

*< x*

_{1}+

*ε*eventually. Moreover,

This is a contradiction. The proof is complete. □

## 3. Discussions

In this article, we establish Lemmas 2.1 and 2.2 which improve Lemma 1.1 used in [2]. Using these lemmas and methods given in [2, 4–7], one can renew those results of [2] and present some other new results. In particular, new results only require that 0 *< a*(*t*) ≤ *a*_{0}*<* 1 rather than (*H*), 0 *< a*(*t*) ≤ *a*_{0}*<* 1, and lim_{t→∞}*a*(*t*) = *a*_{1}. The details are left to the reader.

To achieve new results, we are forced to require that 0 *< a*(*t*) ≤ *a*_{0}*<* 1. The question regarding the oscillatory properties of (1.1) without this assumption remains open at the moment.

## Declarations

### Acknowledgements

The authors express their sincere gratitude to the Editor and anonymous referees for careful reading of the original manuscript and useful comments that helped to improve presentation of results and accentuate important details. T. Li would like to express their gratitude to Professors Ravi P. Agarwal and Martin Bohner for their kindly guidance. This research is supported by Natural Science Foundation of Shandong Province (Z2007F08) and also by Weifang University Research Funds for Doctors (2012BS25).

## Authors’ Affiliations

## References

- Hilger S: Analysis on measure chains-a unified approach to continuous and discrete calculus.
*Results Math*1990, 18: 18–56.MathSciNetView ArticleGoogle Scholar - Han Z, Li T, Sun S, Zhang C: Oscillation behavior of third-order neutral Emden-Fowler delay dynamic equations on time scales.
*Adv Difference Equ*2010, 2010: 1–23.View ArticleGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introduction with Applications.*Birkhäuser, Boston; 2001.View ArticleGoogle Scholar - Kubiaczyk I, Saker SH: Asymptotic properties of third order functional dynamic equations on time scales.
*Ann Pol Math*2011, 100: 203–222. 10.4064/ap100-3-1MathSciNetView ArticleGoogle Scholar - Saker SH: Oscillation of third-order functional dynamic equations on time scales.
*Sci China Math*2011, 54: 2597–2614. 10.1007/s11425-011-4304-8MathSciNetView ArticleGoogle Scholar - Saker SH: On oscillation of a certain class of third-order nonlinear functional dynamic equations on time scales.
*Bull Math Soc Sci Math Roumanie Tome*2011, 54: 365–389.MathSciNetGoogle Scholar - Saker SH:
*Oscillation Theory of Dynamic Equations on Time Scales: Second and Third Orders.*Lambert Academic Publishing, Germany; 2010.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.