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Academic research paper on topic "New oscillation results for second-order neutral delay dynamic equations"

﻿0 Advances in Difference Equations

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New oscillation results for second-order neutral delay dynamic equations

Chenghui Zhang1, Ravi P Agarwal2, Martin Bohner3* and Tongxing Li1

Correspondence: bohner@mst.edu 3Department of Mathematics and Statistics, Missouri S&T, Rolla, MO 65409-0020, USA Full list of author information is available at the end of the article

Abstract

This paper is concerned with oscillatory behavior of a certain class of second-order neutral delay dynamic equations

(r(t)[x(t) +p(t)x(T (t))]A)A + q(t)x(S(t)) = 0,

on a time scale T with supT = to, where 0 < p(t) < p0< to. Some new results are presented that not only complement and improve those related results in the literature, but also improve some known results for a second-order delay dynamic equation without a neutral term. Further, the main results improve some related results for second-order neutral differential equations. MSC: 34K11; 34N05; 39A10

Keywords: oscillation; neutral delay dynamic equation; second-order equation; time scale

ringer

1 Introduction

In this paper, we are concerned with oscillation of a class of second-order neutral delay dynamic equations,

(r(t)[x(t) +p(t)x(r (t))]A)A + q(t)x(S(t)) = 0, (1.1)

where t e [t0, to) t := [t0, to) n T, and

(H) r,p, q e Crd([t0, to) t , R), r(t) >0, 0 < p(t) < p0 < to, q(t) > 0; (H2) S e Crd([t0, to) t , T), S(t) < t, limt^TO S(t) = to, t o S = S o t;

(H3) t e Cld([t0, to)t ,T), t(t) < t, TA(t) > T0 > 0, t([t0, to)t) = [t(t0), to)T,where T0 is a constant.

Throughout this paper, we assume that solutions of (1.1) exist for any t e [t0, to) t. A solution x of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, we call it nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions oscillate.

A time scale T 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 [t0, to) t . For some concepts related to the notion of time scales, see [1, 2].

© 2012 Zhang et al.; licensee Springer. 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 originalworkis properly cited.

Recently, there has been an increasing interest in obtaining sufficient conditions for oscillatory or nonoscillatory behavior of different classes of differential equations and dynamic equations on time scales; we refer the reader to the papers [3-38]. In the following, we present some details that motivate the contents of this paper. Regarding oscillation of second-order neutral differential equations, Grammatikopoulos etal. [16] established that the condition

/ q(s)[l-p(s - 5)]ds = œ

ensures oscillation of the linear neutral differential equation

(x(t) + p(t)x(t - t))" + q(t)x(t - 5) = 0.

Later, Grace and Lalli [15] obtained that the conditions "œ dt

fœ dt

1Â = œ (1.2)

J to r(t)

(0 '(s))2r(s - 5)

ji (0(s)q(s)[l-p(s - 5)]

40 (s)

ds = œ

for some positive function 0 e C1([t0, œ), R) ensure oscillation of the linear neutral differential equation

(r(t)(x(t) + p(t)x(t - t )) ')' + q(t)x(t - 5) = 0.

Baculikova and Dzurina [8] established that the conditions

1<p1 <p(t) <p2 < œ, t'(t) > t0 > 0, t o 5 = 5 o t, 5(t)< t(t)< t,

I q(s)ds = œ

ensure oscillation of the linear neutral differential equation

(r(t)[x(t) + p(t)x(T (t))]')' + q(t)x(5(t)) = 0. (1.3)

Recently, Zhong et al. [38] improved this result, and they obtained that the conditions (1.2), p > 0, p =1, and

t "(e _mgm \ ds=»

j,0 \ 1+p(1 + n) V) (s)S'(s) )

for some constant e e (0,1) and for some positive function Q e C1([t0, R) guarantee oscillation of the linear neutral differential equation

(r(t)(x(t) + px(t - t))')' + q(t)x(S(t)) = 0.

Hasanbulli and Rogovchenko [20] used the standard integral averaging technique to obtain some new oscillation criteria for the second-order neutral delay differential equation

(r(t)(x(t) + p(t)x(t - t))')' + q(t)f(x(t),x{o(t))) = 0.

For oscillation of second-order dynamic equations on time scales, Erbe et al. [13] established a sufficient condition which ensures that the solution x of the delay dynamic equation

(r(t)xA(t))A + q(t)x( t (t)) = 0 (1.4)

is either oscillatory or satisfies limt^œ x(t) = 0 under the condition

fœ 1 ft

Ta q(s)AsAt = œ.

t0 r(t) t0

Zhang [36] obtained some oscillation results for (1.4) in the case where

fœ 1 ft ^ fœ Au

q(s) TT As At = œ. (.5)

t0 r(t) t0 s r(u)

Agarwal et al. [4], Saker [29], and Tripathy [33] considered the equation

(r(t)(x(t) + p(t)x(t - t))A)A + q(t)x(t - 5) = 0, (1.6)

and established some oscillation results for (1.6) provided that 0 < p(t) < 1 and fœ At

1a = œ. (()

t0 r(t)

In particular, Tripathy [33] obtained some oscillation criteria for (1.6) when (1.7) holds and 0 < p(t) < p0 < œ, and established that the condition

I min{q(s), q(s - t)} As = œ

ensures oscillation of (1.6).

The question regarding the study of oscillatory properties of (1.1) (including the case when T = R) has been solved by some recent papers; see [4, 8,17,19, 25, 29, 32, 33] etc. Based on the conditions t (t) < t and a (t) < t, they established some results. The ideas can be divided into two aspects, i.e., comparison methods and the Riccati transformation. In order to compare our results in Section 2 with those related subjects in [4, 8,17,19, 25, 29, 32, 33], we list their results as follows.

Theorem 1.1 (See [8]) Let (1.2) hold and t 1 be the inverse function of t . Assume (Hi)-(H3) for T = R and S(t) < t(t) < t. If

ft ( fS(s) du \ liminf / mini q(s), q( t (s))} / —— ds

t—TO Jt"1(S(t))\ Jt0 r(u)J

To + po T0e '

then (1.3) is oscillatory.

Theorem 1.2 (See [17, 32]) Let (1.2) hold. Assume H1MH3) for T = R andt > S(t) > t(t). If there exists a positive function a e C1([t0, to), R) such that

limsup I

t >to Jt0

t\ ■ t t\ i t W\ /1 po\ r(T (s))(a'(s))2

a(s) mi^q(s), q(t(s)} - ( 1 + -) 4Toa(s)

ds = to,

then (1.3) is oscillatory.

Theorem 1.3 (See [4, 29]) Let (1.7) hold. Assume (H1MH3) for t(t) = t - t, S(t) = t - S, and p0 = 1. If there exists a positive function a e C1d([t0, to) t , R) such that

r(s - S)(aA(s))2 -

limsup I

t—>to jt0

a(s)q(s)(1-p(s - S)) - :

4a (s)

as = to,

then (1.1) is oscillatory.

Theorem 1.4 (See [19, 25]) Let (1.7) hold. Assume (H1MH3) and t > S(t) > t (t). If there exists a positive function a e C1d([t0, to) t , R) such that

limsup I

t—TO Jt0

M •/ M I 1 M r(t (s))(aA(s))2 n

a(s) mi^q(s), q(T (s)) } - (1 + 4t0a(s)

As = to,

then (1.1) is oscillatory.

Theorem 1.5 (See [33]) Let (1.7) hold. Assume (H1)-(H3) for t(t) = t - t, S(t) = t - S, and S > t >0. If there exists a positive function a e C1d([t0, to) t , R) such that

(1+ p0)r(s - t)(aA(s))21

limsup I

t—TO Jt0

a(s) min{q(s), q(s - t)} - ■ then (1.1) is oscillatory.

4a (s)

As = to,

The natural question now is: Can one obtain new oscillation criteria for (1.1) that improve the results in [4,19, 25, 29, 33]? The aim of this paper is to give an affirmative answer to this question. As a special case when T = R, the obtained results improve those by [8,15, 17,32,38]. As a special case whenp(t) = 0, the obtained results improve those reported in [13,36].

2 Main results

In this section, we establish the main results. All functional inequalities considered in this section are assumed to hold eventually, that is, they are satisfied for all t large enough. For

our further references, let us denote

Q(t) := min{q(t),q(t(t))} and z(t) :=x(t) + p(t)x(t(t)).

Theorem 2.1 Assume (Hi)-(H3) and (1.7). If there exist functions n, a e Cjd([t0, <) T, I such that n(t) > 0, a(t) > 0, and

limsup /" (na (s)Q(s)

t—œ Jt2 V

Ç 5(s) Av Jt1 r(v) a (s) Av t1 r(v)

- E(s) ) As = œ

for all sufficiently large t1 and for some t2 > ti, where

Jt1 r(v)

E(s) := -i]a (s)

r(s)a2(s)

Jt1 r(v) ra (s) Av

- r(s)a(s) A

B2(s) 4A(s)

na (s)

ça (s) Av.

Jt1 r(v)

T0r(T(s))a2(s)- (Kt(s))a(s))z

ÇT (s) Av

' ÇTa (s) Av. Jt1 r(v)

,rs na(s) ft[ D, , nA(s) 2na(s)a(s) ^

A(s):= ^ -asr-;-, B(s):= —+

r(s)n2(s) fa(s) Av'

-'t1 r(v)

n(s) cA)'

C(s) :=

T0ÏÏ

T (s) Av_ t1 r(v)

r(T(s))n2(s) ÇTa(s) Av'

-'t1 r(v)

tfte« (1.1) is oscillatory.

nA(s) 2T0na(s)a(s) ft D(s) := —— +

•T (s) Av t1 r(v)

ÇTa (s) Av ' Jt1 r(v)

Proof Assume that (1.1) has a nonoscillatory solution x on [t0, œ) T. Without loss of generality, suppose that it is an eventually positive solution. From (1.1) and [1, Theorem 1.93], we obtain

(r( t (t)) za{t (t)))A + p0^ T (t))x{5{T (t))) = 0.

Combining (1.1) and (2.2), we are led to

(r(t)zA(t))A + p0(r(T(t))zA(T(t)))A + Q(t)z(S(t)) < 0.

From (1.1) and (1.7), we have

zA (t)>0 and (r(t)zA(t))A <0 for t e [t1, <) T, where t1 e [t0, <) T is large enough. Define the function w by - r(t)zA(t)

«(t) := n(t)

+ r(t)a(t)

t > t1.

Hence, we have «(t) > 0 for t e [t1, œ) T and

«A = nA

= —w + n" (ra)A + n" rZ

nA a( A " (rZA )AZ - r(zA)2

= — w + n" (ra)A + n"-

nA.....a, A a (rZA )A ..a r(zA)2

= — w + n" (ra) + n nz

nA a( A a (rzA)A

= — w + n" (ra)A + n"-

- r\~z) z".

izA\2 z

By virtue of (r(t)zA(t))A < 0, we have rt As

z(t) > zA(t)r(t)jt —, and so

f t As_ Jt1 r(s)

t1 r(s)

which implies that

, , ct As

z(t) Jt1 r(s)

z" (t) > f"(t) As •

Jt1 r(s)

On the other hand, we have by (2.4) that

+ a2-2—. rn

Putting (2.7) and (2.8) into (2.5), we have (rzA)A

wA < n"

f t As Jt1

ra2 f^L - (ra)A

f" (t) As_ Jt1 r(s)

(rzA)A

< n" ^ - n"

2n" a Jl

n f"(t) A.

Jt1 r(s) tt1

+ 2n" a Jt1 A 1

n Jt1 r(s) 2

W--—-7-:-W

t As_ 2 Jt1 r(s)

rn2 f" (t) As Jt1 r(s)

f" (t) As

Jt1 r(s)

- (ra)A

Now, define the function u by

.. ..r r(t (t))zA(t (t)) u(t) := n(t) -r—tt-+ r1

(t (t)) a(t)

, t > t1.

z(t (t))

Hence, we have by [1, Theorem 1.93] and (H3) that

(2.10)

uA = nA

(r o T)(zA o T)

+ (r o t)a

(r o t )(zA o t)

+ (r o t)a

= — u + n" ((r o t)a)A + n" n

(r o t)(zA o t)

= — u + n" ((r o t)a)A + n n

A " ((r o t)(zA o t))A(z o t) — (r o t)(zA o t)(zA o t)ta

(z o t)(z o t" )

A \ 2 zA o T \ z o t

nA " n , ) A " ((r o t)(zA o t))A " a / ,

= — u + n ((r o t)a) + n--n t (r o t)

n z o t" \z o T / z o T'

,, nA " u , )A " ((r o T)(zA o T))A

<— u + n ((r o t)a) + n -

n z o t"

- Ton" (r o T)

zA o T \ z o T

z o T z o T"

Note that (2.6) implies that

z(T (t)) A

z(T" (t))- ftr"(t) ^

Jtl r(s)

On the other hand, we have by (2.10) that

zA o T

_(r o t)n

_(r o t)n_

+ a2-2

(r o t)n

(2.13)

Putting (2.12) and (2.13) into (2.11), we have

. " ((r o T)(zA o T))A "

uA < n"-"--n"

z o T"

T0(r o t)a

f-r (t) As

A n " F(t) As n

nA 2Ton"a Jt1 r(s)

n n CT" (t) AS.

Jt1 r(s)

¡•r (t) As. 2 J h _r(s)

t) . f _

t1 r(s) 2

r" (t) As

r (t) As

r o T )

((r o T)(zA o T))A z" z" z o T"

(r o T )n2 fta (t) As it

t1 r(s) r(t) As

T0(r o t)a

2 J t1 r(s)

f r" (t) As

Jt1 r(s)

r o T )

ß2 4C

^ " ((r o T)(zA o T))A "

< n ----n

r0(r o t)a

rr (t) As t

•4 rs

j-r (t) _A±

2 J t1 r(s) rr" (t) As

r o T )

ß2 , x

+ —. (2.14) 4C

Recalling (2.9) and (2.14), we have by (2.3) and (2.6) that

, p0 A (rzA)A + f°((r o T)(zA o T))A «A + P° uA < n" --)-^---— + E

< -n" Q^ + E

fä(t) As.

< -n" Q iL*t + E.

— ' ^ r" (t) As Jt1 r(s)

Hence, we have

fS(s) Av

f"(s) Av I ' T0

Jt, r(v)

n" (s)Q(s)-£(sn As < W(t2)+ p0u(t2),

t1 r(v)

which contradicts (2.1). The proof is complete. □

Based on Theorem 2.1, we have the following corollary when n(t) = t and a(t) = 0.

Corollary 2.2 Assume H1MH3) and (1.7). If

t. rS(s) Av 1 / f"(s) Av fr"(s) A-w

limsup / ( " (s)Q(s) -—( + ^ r(t (s)) 1 () r(v) ) ) As = to

t—su Jt2\1 )Q( ^ A) 4"w v n A T021 (it;« ^

for all sufficiently large t1 and for some t2 > t1, then (1.1) is oscillatory.

When T = R,we have from Corollary 2.2 the following result for the neutral differential equation (1.3).

Corollary 2.3 Assume H1MH3) for T = R and (1.2). If

fS(s) dv_ , Jt1 r(v) 1

nm—- ^(r(s)+\$ *(s))))ds=TO

fs — Jt1 r(v)

for all sufficiently large t1 and for some t2 > t1, then (1.3) is oscillatory. Example 2.4 Consider the second-order neutral differential equation

x(t) + 1 x(t -1) + Yx(t) = 0 for t > 1, (2.15)

where y > 0 is a constant, r(t) = 1, p0 = 1/2, t0 = 1, q(t) = Q(t) = y/t2, t(t) = t - 1, and S(t) = t. Note that

t , fS(s) iv w x x

■mupiHw-¿KT02* <s))))ds

t1 r(v) t

f ds 3

limsup/ —= to , if y > -.

t— t2 s 8

Hence, by Corollary 2.3, (2.15) is oscillatory if y >3/8. Let 0 (t) = t andp = p(t) = 1/2. Then

(0 '(s))2r(s - S)"

ji (0(s)q(s) [1 -p(s - S)]

40 (s) and

ds = to , if y > 1/2

(TO(am ^ 1-£ (0;(s))2r(S(s)^ 1 + 2(1 +e)

0(s)q(sh-^-T- /.fl^w^ ds = to, if y > —2——

Jt0 \ 1+p(1 + e) 40(s)S'(s) J 4(1- e)

for some constant e e (0,1). Since

1 + 5(1 + e) 3

-2-> -,

4(1-e) 8

our result is better than [15, 38] in some cases.

Example 2.5 For t > 1, consider the second-order neutral delay differential equation

x(t)+pox( 22

+ ^ *( i) =0> (2.16)

where 0 < p0 < to and a > 0 is a constant. In [8], Baculikova and Dzurina obtained that the condition

1 + 2po

e ln 2

ensures oscillation of (2.16) (using Theorem 1.1). Letting n(t) = t2, an application of Theorem 2.1 yields that the condition

a > — f— + ^P^ for some constant k0 e (0,1) ko \8 4 J

guarantees oscillation of (2.16). For example, we can put a > 1/6 + V2p0/3 (by letting k0 = 3/4). Hence, our result improves that in [8] since

1 + 2p0 1 V2p0

e ln 2 > 6 + 3 .

Example 2.6 For t > 1, consider the second-order neutral delay differential equation

x(i)+p0x( 2

+ ax(t) = 0, (2.17)

where 0 < p0 < to and a > 0 is a constant. An application of Corollary 2.3 implies that the condition

a > - + po 4

guarantees oscillation of (2.17). However, applications of Theorem 1.2 and Theorem 1.4 (by letting a(t) = t) yield that

a > ^ + po

ensures oscillation of (2.17). Hence, our result is new. Note that Theorem 1.3 and Theorem 1.5 cannot be applied in (2.17).

In the following, we give an oscillation criterion for (1.1) when

< (X) . (.18)

to r(t)

Theorem 2.7 Assume (H1)-(H3) and (2.18). Suppose further that there exist two functions n, a e Cld([to, x ) T, R) such that n(t) > 0, a(t) > 0, and (2.1) holds for all sufficiently large t1 and for some t2 > t1. If there exists a positive function b e Cld([t0, x ) T, R) such that

b(t) -77^7 > 0, b(t) - 1 > 0, (2.19) r(t)R(t) r(r(t))R(r(t))

limsup/ (n"(sQWt;^^-E*(s))As = x, (2.20)

t^x Jt0\ R( T" (s)) /

^ r As

R(t):=i rs),

E.(s) := -n" (s)[r(s)b2(s) - (r(s)b(s))A]

4A.(s)

n"(s)[T0Kt(s))b2(s)-(r(t(s))b(s))A] -

4C.(s)

A.(s) := n" (s) B(s) := nA?) + 2n" (s)b(s)

r(s)n2(s)' ' n(s) n(s)

C.(s) := T0n" (s) D(s) := ^ + 2 T0n" (s)b(s)

r( T(s))n2(s)' ' n(s) n(s)

then (1.1) is oscillatory.

Proof Assume that (1.1) has a nonoscillatory solution x on [t0, x ) T. Without loss of generality, suppose that it is an eventually positive solution. By the proof of Theorem 2.1, we have (2.3). From (1.1), there exists ti e [t0,x)T such that zA(t) > 0 or zA(t) < 0 for t e [t1, x ) T. The proof of the case when zA(t) > 0 is the same as that of Theorem 2.1, and we can get a contradiction to (2.1). Now, we assume zA(t) < 0. Then we have

zA(s) < ^zA(t), s > t > t1.

Integrating this from t to x , we get

zAtl> 1 zA( T(t)) 1

z(t) > -r(t)R(t), z( T(t)) > -r( T(t))R(т(t)),

(R)A >0. (2.21)

Define the function w by - r(t)zA(t)

w(t) := n(t) Then w(t) > 0,

+ r(t)b(t)

, t > ti.

= — w + n" (rb)A + rfv~ ' n

(rzA)A _ „ /Z-\2z Z n r\z)zF'

+ b - 2—. rn

Note that z/z" > 1. By virtue of ( 2.22) and ( 2.23), we have

A , n (rzA)A ^ r i 2

wA < n"

(rzA)A

z o t" l Now, define the function u by / ,.r r(T (t))zA(T (t))

u(t):=n(t) -¡-7-77

L z(r (t))

Hence, we have u(t) > 0,

n" 2 w--- w

(r (t)) b(t)

t > ti.

A^" n ,, u.A n ((r o T)(zA o T))A uA <— u + n" ((r o t)b) + n" ■

- Ton" (r o t)

z o T"

A2 zA o T z o T

zA o ^ 2

(r o T)n

z o T z o T"

(r o T)n

+ b2 - 2

(r o T)n

Note that z(t (t))/z(Tn (t)) > 1. By virtue of (2.25) and (2.26), we have ((r o t)(zA o t))A •

uA < n"

z o T"

r nA 2T0n" b' nn

- n" [To(r o t)b2-((r o t)b) A]

Ton" 2 u -7-r"2 u

(r o T)n2

""" - n"[To(r o T)b2-((r o T)b)A] +

((r o T)(zA o T))A

Recalling (2.24) and (2.27), we have by (2.3) and (2.21) that

(2.22)

(2.24)

(2.26)

(2.27)

A po A n (rzA)A + f°((r o t)(zA o t))A z" „ R" wA + — uA < n"----+ E* < -n" Q-+ £••

z o T"

~R o t"

Hence, we have

f((sQ^TST^ -E-s) As < + -Jt A t a (s)) / ro

which contradicts (2.20). The proof is complete.

Example 2.8 Consider the second-order neutral differential equation

(t2(x(t) + p0x(t -1))')' + q0x(t) = 0 for t > 2,

(2.28)

where r(t) = t2, r0 = 1, q(t) = Q(t) = qo, r(t) = t - 1, and 5(t) = t. Note that R(t) = t"1. Let ^(t) = 1/t and a(t) = ¿(t) = 1/(t -1). Then

t , rS(s) Av

lim s up| (s)Q(s) - E(s) A

t^oo jt2

= lim sup I

t^oo Jt2

o (s) Av tl r(v)

qo + 2

= oo , if q0 >

(s + 1)2 s ' (s -l)2 4s(s -l)2

(s + 1)2 4s3

lim sup /"Y(s)Q(s) R(s) t^oo ,/t0 V

, , ,, £.(s)| As R( t o (s)) (V

= lim sup I

t^oo J 2

trqo(s -1) 2

= oo , if q0 >

(s + 1)2 (s -1)2 4s (s -1)2

(s + 1)2 4s3

Hence, by Theorem 2.7, (2.28) is oscillatory if q0 >(1 + p0)/4. Whenp0 = 0, q0 > 1/4 is a sharp condition for oscillation of the equation (t2x'(t))' + q0x(t) = 0. Note that the results of [13, 36] cannot give this result (see (1.5)), and hence our results improve those of [13, 36].

3 Discussions

In this paper, we have suggested some new oscillation criteria for second-order neutral delay dynamic equation (1.1) by employing the generalized Riccati substitution. To achieve these results, we are forced to require, similar as in [33], that TA(t) > r0>0,ro S = S o t, and r([t0, oo) T) = [ t(t0), oo ) T. It would be interesting to seek other methods for further study of oscillatory properties or asymptotic problems of equation (1.1) in the case where P(t) > 1.

During the past three decades, there have been many classical results regarding oscillatory behavior of equation (1.1) in the case where T = R, some of which provided that 0 < p(t) < 1; see, for example, [15,16]. Examples given in this paper reveal some advantages even when one applies the obtained criteria to the case where 0 < p(t) < 1.

These results show that the delay argument t plays an important role in oscillation of second-order neutral delay dynamic equations; see the details in Example 2.6 and differences between Corollary 2.3 and Theorem 1.2, Theorem 1.4. Let us go through Example 2.6. One can easily see some superiorities in comparison to those related results, e.g., Theorem 1.2 and Theorem 1.4.

As a special case whenp(t) = o, the established results improve those of [13, 36] in some sense, which is shown by Example 2.8.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Allauthors contributed equally to the writing of the paper. Allauthors read and approved the finalmanuscript. Author details

1 Schoolof ControlScience and Engineering, Shandong University, Jinan, Shandong 250061, P.R. China. 2Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA. 3Department of Mathematics and Statistics, Missouri S&T, Rolla, MO 65409-0020, USA.

Acknowledgements

This research is supported by NNSF of P.R. China (Grant Nos. 61034007, 51277116, 50977054). Received: 9 October 2012 Accepted: 3 December 2012 Published: 28 December 2012 References

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doi:10.1186/1687-1847-2012-227

Cite this article as: Zhang et al.: New oscillation results for second-order neutral delay dynamic equations. Advances in Difference Equations 2012 2012:227.

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