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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)]

Advances in Difference Equations20122012:57

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

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

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

Emden-Fowler type dynamic equations have some applications in the real world; see the background details introduced by Hilger [1]. Hence [2] studied a class of third-order Emden- Fowler neutral dynamic equations
r ( t ) ( x ( t ) - a ( t ) x ( τ ( t ) ) ) Δ 2 Δ + p ( t ) x γ ( δ ( t ) ) = 0
(1.1)

on a time scale with sup T = , 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 τ : T T and δ : T T 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 . 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 t 0 , T : = t 0 , 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 [47] 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 (A1)-(A4) hold and
( H ) t h e r e e x i s t s { c k } k 0 T s u c h t h a t lim k c k = a n d τ ( c k + 1 ) = c k .
Assume also that x is an eventually positive solution of (1.1). If
t 0 Δ t r ( t ) = ,
(1.2)

then there are only the following three cases for t t 1 , T , where t 1 t 0 , T sufficiently large:

Case (i). z ( t ) > 0 , z Δ ( t ) > 0 , z Δ 2 ( t ) > 0 , z Δ 3 ( t ) < 0 ;

Case (ii). z ( t ) > 0 , z Δ ( t ) > 0 , z Δ 2 ( t ) > 0 , z Δ 3 ( t ) < 0 , lim t x ( t ) = 0 ;

Case (iii). z ( t ) > 0 , z Δ ( t ) < 0 , z Δ 2 ( t ) > 0 , z Δ 3 ( t ) < 0 , lim t z ( t ) = l 0 , lim t x ( t ) = l / ( 1 - a ) 0 .

We note that there exists a mistake in the above statements. First, the case (ii) does not occur since zΔ> 0 and z Δ 2 > 0 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 [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), (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:

Case (1). z > 0 , z Δ > 0 , z Δ 2 > 0 , r z Δ 2 Δ < 0 ;

Case (2). z > 0 , z Δ < 0 , z Δ 2 > 0 , r z Δ 2 Δ < 0 ;

Case (3). z < 0 , z Δ < 0 , z Δ 2 > 0 , r z Δ 2 Δ < 0 .

Proof. Assume that x is an eventually positive solution of (1.1). Then, we have by (1.1) that ( r z Δ 2 ) Δ < 0 , and hence r z Δ 2 is decreasing and of one sign. The condition r z Δ 2 < 0 implies that there exist a t 1 t 0 , T and a constant M > 0 such that
r ( t ) z Δ 2 ( t ) M , f o r t [ t 1 , ) T ,
which yields
z Δ 2 ( t ) M r ( t ) , f o r t [ t 1 , ) T .
Integrating from t1 to t and letting t → ∞, we have by (1.2) that
lim t z Δ ( t ) = - .
Hence there exist a t 2 t 1 , T and a constant M1> 0 such that
z Δ ( t ) M 1 , f o r t [ t 2 , ) T .
Integrating the above inequality from t2 to t and letting t → ∞, we have
lim t z ( t ) = - ,
which yields z < 0 eventually. Then, we get
z < 0 , z Δ < 0 , z Δ 2 < 0 , ( r z Δ 2 ) Δ < 0 .
(2.1)
From (2.1) we have that limt→∞z (t) = -∞. Next we claim that x is bounded and (2.1) does not occur. If not, there exists a sequence t m m t 0 , T with t m → ∞ as m → ∞ such that
x ( t m ) = max { x ( s ) : t 0 s t m } a n d lim m x ( t m ) = .
It follows from τ (t) ≤ t that
z ( t m ) = x ( t m ) - a ( t m ) x ( τ ( t m ) ) ( 1 - a 0 ) x ( t m ) ,

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 z Δ 2 > 0 , then z > 0. Thus, for z Δ 2 > 0 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.

Proof. Assume that (3) holds. Then limt→∞z (t) ≤ 0. Next we claim that x is bounded. Similar as in the proof of Lemma 2.1, we have that limm→∞z (t m ) = ∞ which contradicts the fact that limt→∞z (t) ≤ 0. Thus, x is bounded. Hence we can suppose that lim supt→∞x (t) = x1, where 0 ≤ x1< ∞. Then, there exists a sequence t k k t 0 , T with t k → ∞ as k→ ∞ such that limk→∞x (t k ) = x1. Next we show that limt→∞x (t) = 0. If not, then x1> 0. Pick ε = x1(1 - a0)/(2a0), we find that x(τ(t k )) < x1 + ε eventually. Moreover,
0 = lim k z ( t k ) lim k ( x ( t k ) - a 0 ( x 1 + ε ) ) = x 1 ( 1 - a 0 ) 2 > 0 .

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, 47], one can renew those results of [2] 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.

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

(1)
School of Information and Control Engineering, Weifang University, Weifang, Shandong, 261061, P. R. China
(2)
School of Control Science and Engineering, Shandong University, Jinan, Shandong, 250061, P. R. China
(3)
School of Mathematical Sciences, University of Jinan, Jinan, Shandong, 250022, P. R. China

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