- Open Access
Corrigendum to "Oscillation behavior of third-order neutral Emden-Fowler delay xdynamic equations on time scales" [Adv. Difference Equ., 2010, 1-23 (2010)]
© Ji et al; licensee Springer. 2012
- Received: 24 October 2011
- Accepted: 9 May 2012
- Published: 9 May 2012
In this article, we revise results obtained by Han et al.
Mathematics Subject Classification 2000: 34K11; 39A10.
- neutral dynamic equation
- time scale
on a time scale with sup , where the authors assume the following hypotheses hold.
(A1) γ > 0 is the quotient of odd positive integers;
(A2) r and p are positive real-valued rd-continuous functions defined on such that rΔ(t) ≥ 0;
(A3) a is a positive real-valued rd-continuous function defined on such that 0 < a(t) ≤ a0< 1 and limt→∞a(t) = a1;
(A4) the functions and are rd-continuous functions such that τ(t) ≤ t, δ (t) ≤ t, and limt→∞τ(t) = limt→∞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 . For some concepts related to the notion of time scales; see . 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),  obtained various oscillation theorems by using some lemmas, one of which we present below for the convenience of the reader.
then there are only the following three cases for , where sufficiently large:
We note that there exists a mistake in the above statements. First, the case (ii) does not occur since zΔ> 0 and imply that limt→∞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 .
Now we use notation z as in Lemma 1.1 and present the following new lemmas.
Lemma 2.1. Let (1.2), (A1), (A2), and (A4) hold with (A3) replaced by (A 3*) a is a positive real-valued rd-continuous function defined on
such that 0 < a(t) ≤ a0< 1. Suppose that x is an eventually positive solution of (1.1). Then there are only the following three cases eventually:
which implies that limm→∞z (t m ) = ∞, this contradicts the fact that limt→∞z (t) = - ∞. Hence x is bounded, and so (2.1) does not hold.
If zΔ> 0 and, then z > 0. Thus, for only the cases (1), (2), and (3) may occur. The proof is complete. □
Lemma 2.2. Let 0 < a(t) ≤ a0< 1. If case (3) holds, then limt→∞x (t) = 0.
This is a contradiction. The proof is complete. □
In this article, we establish Lemmas 2.1 and 2.2 which improve Lemma 1.1 used in . Using these lemmas and methods given in [2, 4–7], one can renew those results of  and present some other new results. In particular, new results only require that 0 < a(t) ≤ a0< 1 rather than (H), 0 < a(t) ≤ a0< 1, and limt→∞a(t) = a1. The details are left to the reader.
To achieve new results, we are forced to require that 0 < a(t) ≤ a0< 1. The question regarding the oscillatory properties of (1.1) without this assumption remains open at the moment.
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).
- 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
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.