Tianetal. Advances in Difference Equations (2015) 2015:267 DOI 10.1186/s13662-015-0604-6

0 Advances in Difference Equations

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Oscillation and asymptotic behavior of third-order neutral differential equations with distributed deviating arguments

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Yazhou Tian1,2*, Yuanli Cai1, Youliang Fu2 and Tongxing Li2

"Correspondence: tianyazhou369@163.com 1 Schoolof Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, P.R. China Full list of author information is available at the end of the article

Abstract

By employing a generalized Riccati transformation and integral averaging technique, two Philos-type criteria are obtained which ensure that every solution of a class of third-order neutral differential equations with distributed deviating arguments is either oscillatory or converges to zero. These results extend and improve related criteria reported in the literature. Two illustrative examples are provided.

MSC: 34K11

Keywords: oscillation; asymptotic behavior; third-order neutral differential equation; distributed deviating argument; generalized Riccati transformation

1 Introduction

Differential equations with distributed deviating arguments are often used for modeling various problems arising in the engineering and natural sciences. Therefore, analysis of qualitative properties of solutions to such equations is crucial for applications; see Wang [1]. On the basis of these background details, we investigate the oscillation and asymptotic behavior of a third-order neutral differential equation with distributed deviating arguments

r(t)( t > to,

n b a y « d

x(t) + J p(t, £)x(t(t, £)) d£ J +y q(t, £)f(x(a(t, £))) d£ = 0,

ft Spri

where a > 1 is the ratio of odd positive integers. Throughout, we suppose that the following assumptions hold.

(Ai) r(t) e C1([i0, to), (0, to)), r'(t) > 0, r-1/a(s) ds = to; (A2) p(t, f) e C([to, to) x [a, b], [0, to)), 0 < jbap(t, f) df < P <1;

(A3) t(t, f) e C([t0, to) x [a, b],R) is a nondecreasing function for f satisfying t (t, f) < t

and liminft^TO t(t, f) = to for f e [a,b]; (A4) q(t, f) e C([t0, to) x [c, d], [0, to));

(A5) a(t, f) e C([t0, to) x [c, d],R) is a nondecreasing function for f satisfying a(t, f) < t and liminft^TO a (t, f) = to for f e [c, d];

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(A6) f (x) e C(R, R) and there exists a positive constant K such that/(x)/xa > K for all x = 0. Define a new function z(t) by , b

z(t)=x(t) + p(t, %)x(t(t, %)) d%.

By a solution of (1.1) we mean a nontrivial function x(t) e C([Tx, to), R), Tx > t0, which has the properties z(t) e C2([TX, to),R) and r(t)(z"(t))a e C1([TX, to),R) for Tx > to. Our attention is restricted to those solutions of (1.1) which satisfy sup{|x(t) |: t > T} > 0 for any T > Tx .A solution x(t) of (1.1) is said to be oscillatory on [Tx, to) if it is neither eventually positive nor eventually negative. Otherwise it is called nonoscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory.

Recently, there has been much research activity concerning the oscillation and asymptotic properties of various classes of differential equations; see, e.g., [1-21] and the references cited therein. So far, there are few results dealing with the asymptotic behavior of third-order neutral differential equations with distributed deviating arguments, we refer the reader to [17,19]. The third-order neutral differential equation

and its special cases have been studied by Baculikova and Dzurina [5,6], Candan [7], Grace etal. [9], Jiang and Li [10], and Li etal. [14]. Using Riccati transformation, Zhang etal. [19] considered a class of third-order neutral differential equations

and they obtained several Philos-type (see [15]) criteria for (1.2), whereas §enel and Utku [17] studied (1.1).

In the study of oscillation of differential equations, there are two techniques which are used to reduce the higher-order equations to the first-order Riccati equations (or inequalities). One of them is the Riccati transformation technique which has been recently extended to dynamic equations on time scales; see, e.g., §enel and Utku [17]. The other one is termed the generalized Riccati transformation technique; we refer the reader to Li [11], Li et al. [12], Li and Saker [13], and the related references cited therein. In particular, Li [11] used the generalized Riccati substitution and established several oscillation criteria for a second-order ordinary differential equation

is oscillatory and showed that the results established by the Riccati transformation technique cannot be applied.

[r(t)( [x(t) +p(t)x( t (t))]")"]' + q(t/(x(* (t))) = 0

(r(t)x'(t))' + q(t)x(t) = 0.

Furthermore, he proved that the equation

In the special case when a = 1, (1.1) reduces to (1.2). Now the following question arises. Could we obtain new Philos-type oscillation criteria for (1.1) by using a generalized Riccati transformation which differs from that of [19]? Motivated by Li [11], Li et al. [12], and Li and Saker [13], our purpose in this paper is to give a positive answer to this question. In Section 2, four lemmas are given to prove the main results. In Section 3, we establish two Philos-type theorems for (1.1). In Section 4, two examples and some conclusions are presented to illustrate the main results. As customary, all functional inequalities considered in this paper are supposed to hold for all t large enough.

2 Some lemmas

Lemma 2.1 Suppose that conditions (Ai)-(A6) are satisfied and letx(t) be a positive solution of (1.1). Then z(t) has only one of the following two properties:

(I) z(t) > 0, z'(t) > 0, z"(t) > 0, z"'(t) < 0;

(II) z(t) > 0, z'(t) < 0, z"(t) > 0, z"'(t) < 0, for t > t1, where t1 > t0 is sufficiently large.

Proof Assume that x(t) is a positive solution of (1.1). Then there exists a t1 > t0 such that, for t > t1,

x(t) > 0, x(t(t,%))>0, % e [a, b], and x(a(t,%))>0, % e [c,d].

From (1.1) and the definition of z(t), we have z(t) > 0 and

q(t, % )f(x( a (t, %))) d% < 0.

Thus r(t)(z"(t))a is nonincreasing and of one sign. Therefore, z"(t) is also of one sign and so we have two possibilities: z"(t) < 0 or z"(t) > 0 for t > t2 > t1. We assert that z"(t) > 0 for t > t2. Otherwise, there exists a constant M >0 such that, for t > t2,

z"(t) < -Ma —11- < 0.

r a(t)

Integrating this inequality from t2 to t, we obtain f t 1

z'(t) < z'(t2) - Ma —— ds.

Jt2 r a (s)

Letting t —^ to and using (A1), we get limt—TO z'(t) = -to. Thus z!(t) < 0 eventually. But conditions z"(t) < 0 and z'(t) < 0 imply that z(t) < 0, which contradicts our assumption z(t) > 0. Hence, z"(t) > 0 for t > t2. Furthermore, we have, for t > t2,

[r(t)(z"(t))a]' = r'(t)(z"(t))a + ar(t)(z"(t))a-V"(t) < 0.

This yields z"'(t) < 0 for t > t2 due to condition (A1). Therefore, z(t) has only one of the two properties (I) and (II). This completes the proof. □

Lemma 2.2 Letx(t) be a positive solution of (1.1) and assume that corresponding z(t) has the property (II). If

pTO /*TOr 1 fTO f d

q(s' $) d$ ds

Jta Jv j(u) Ju Jc

du dv = to,

then limt^TO x(t) = 0.

Proof Let x(t) be a positive solution of (1.1). Since z(t) has the property (II), there exists a finite constant l > 0 such that limt^TO z(t) = l > 0. We prove that l = 0. Assume now that l >0. Then we have l + e > z(t) > l for all e >0. Choose 0 < e < l(1 - P)/P. It is easy to verify that

x(t) = z(t)- p(t, $)x(T(t, $)) d$

p b p b

> l - p(t, $)z(T(t, $)) d$ > l - z(t(t, a)) / p(t, $) d$

> l - P(l + e) = N(l + e) > Nz(t),

where N =(l - P(l + e))/(l + e)> 0. Using (A6) and (2.2), we conclude that

q(t, $)z> (t, $)) d$.

Noting that z(t) has the property (II) and using (A5), we have

q(t, $) d$ = -q1(t)z"(a1(t)),

where qx(t) = KNa fca q(t,$) d$ and oi(t) = a (t, d). Integrating inequality (2.3) from t to to, we obtain

q1(s)z"(a1(s)) ds.

By virtue of z(ai(t)) > l,

r 1 fTO 1" z"(t) > l rt)jt q1(s) ds .

Integrating inequality (2.4) from t to to, we have

fTO r 1 fTO ~

-z'(t) > l/ — q1(s) ds t r(u) u

Integrating the latter inequality from ti to to, we obtain

p TO p TO " 1 p TO

z(t1) > l — q1(s) ds

t1 v r(u) u

which contradicts (2.1). Hence l = 0 and limt—TO z(t) = 0. Then it follows from 0 < x(t) < z(t) that limt—TO x(t) = 0. The proof is complete. □

Lemma 2.3 ([5], Lemma 3) Assume that u(t) > 0, u'(t) > 0, and u"(t) < 0for t > t0. If a(t) e C([t0, to), [0, to)), a(t) < t, and limt—TO a(t) = to, then, for every j e (0,1), there exists aTj > t0 such that, for t > Tj,

u(a(t)) > u(t).

Lemma 2.4 Assume that u(t) > 0, u'(t) > 0, u"(t) > 0, and u"'(t) < 0 for t > t0. Then, for every y e (0,1), there exists aTY > t0 such that, for t > TY,

u(t) > 1 y tu'(t).

Proof The proof is similar to that of Bacullkova and Dzurina ([5], Lemma 4), and hence it is omitted. □

3 Main results

D ={(t,s) e R2: t > s > and D0 = {(t,s) e R2: t > s >

The function H(t, s) e C(D, R) is said to belong to the class X (denoted by H e X) if it satisfies

(i) H(t, t) = 0, t > t0, H(t, s) > 0, (t, s) e D0;

(ii) dH(t,s)/ds < 0, there exist p(t) e C1([t0, to), (0, to)), b(t) e C1([t0, to), [0, to)), and h(t, s) e C(D0, R) satisfying

= H (...)

P '(s) I lU1/^ —— + (a + l)o« (s)

+ h(t. s).

Theorem 3.1 Assume that conditions (Al)-(A6) and (2.l) are satisfied. If there exists a function H e X such that. for some j e (0. l) and y e (0. l).

limsup -

l ftïH{t s) (s)__l P(s)r(s)|h(t.s)|a+l

t^œ H(t. t0) Jt0

(a + l)«+l Ha (t. s)

ds = œ. (3.l)

where a2(t) = a (t. c) and

1 a22(t^ a ^d

q(t, %) d%

+ p (t)r(t)b1+ a(t)-p(t)(r(t)b(t))', (.2)

then every solution x(t) of (1.1) is either oscillatory or satisfies limt—TO x(t) = 0.

Proof Assume that (1.1) has a nonoscillatory solution x(t). Without loss of generality, we may assume that x(t) is an eventually positive solution of (1.1). By Lemma 2.1, we observe that z(t) satisfies either (I) or (II) for t > t1. We consider each of two cases separately.

Suppose first that z(t) has the property (I). Then we obtain f b

x(t) = z(t)- / p(t, $)x(T(t, $)) d$

/»b p b

> z(t)-/ p(t,$)z(t(t,$))d$ >z(t)-z(T(t,b)) / p(t,$)d$

> (V^bp(t, $) d^jz(t) > (1- P)z(t).

Using (A5), (A6), and (3.3), we have

q(t, $)xa(a(t, $)) d$

q(t, $)za (a (t, $)) d$

q(t, $) d$ = -q2(t)za(a2(t)),

where q2(t) = K(1 - P)a fc q(t, $) d$ and a2(t) = a(t, c). Define a generalized Riccati transformation «(t) by

«(t) = p(t)

r(t)(z"(t))a

(z'(t))a

Then we have «(t)> 0 and ' r(t)(z"(t))

+ r(t)b(t)

t > t1.

«'(t) = p'(t) =

P (t) P'(t)

+ r(t)b(t)

+ P(t)

(z'(t))

«(t) + p(t)(r(t)b(t))' + p (t)

r(t)(z"(t))a

(z'(t)Y r(t)(z"(t))a (z<(t))a

+ r(t)b(t)

(A (A( (AMY (A WW)" ]' . w. /Z"(tna+1 — «(t) + p(t)(r(t)b(t)) + p (t)—(z^t)^--ap (t)r(t^ —

'(t) /

By virtue of (3.5), we conclude that

^^ (2« r(i)b(i)

Z'(t) r 5(t)V P (t)

Combining (3.4), (3.6), and (3.7), we have

«'(t) < Pt)«(t) + P(t)(r(t)b(t))' - P(t)q2(t)iaZ^

aP (tv «(t) zaua

—- ^77- r(t)b(t)

r°(t)\ P(t)

Using Lemma 2.4, for every y e (0,1), there exists a TY > t1 such that, for t > TY z(a2(t)) > 2Ya2(t)Z(a2(t)).

From Lemma 2.3, for every j e (0,1), there exists a Tj > TY such that, for t > Tj,

1 ja2(t) z'(t) > tz'Mt)).

Define

A* = and B* = r(t)b(t).

Using the inequality (see [13])

(A*) 1+ a - (A* - B*)i+ a < (B*)

1+ 1|A* -1 B*

we have

«+ a(t) 1

p 1+ «(t) a

Using inequalities (3.8)-(3.11), for t > T > Tj, we have

A*B*> 0, a = — > 1, odd

1\(r(t)b(t))a

+ a(r(t)b(t^ a+ -ay-^tr^. (3.11)

« (t) < P (t)(r(t)b(t))'- p (t)q2(t)[ 2 PYaa~~~) - p(t)r(t)b1+ ¿(t)

P(t) !

—- + (a + 1)ba (t) p (t)

«(t) - ■

(p(t)r(t))

- «1+ a (t)

= (t) + A(t)«(t) -B(t)«1+ «(t),

(3.12)

where ^(t) is defined as in (3.2), A(t) = (p'(t)/p(t)) + (a + 1)b1/a (t), andB(t) = a/(p(t)r(t))1/a. Multiplying inequality (3.12) by H(t, s) and integrating the resulting inequality from T to t, we have

i H(t,s)^(s)ds < i H(t,s)(-«'(s)+A(s)w(s)-B(s)«1+ «(s))ds Jt Jt

ft/ d h (t, s)

/t \ ds

= H(t, T)«(T) + j ( + H(t, s)A(s) )«(s) ds

- j H(t,s)B(s)«1+ a (s) ds

= H(t, T)«(T)- i h(t,s)«(s) ds -i H(t,s)B(s)«1+ «(s) ds Jt Jt

< H(t, T)«(T) + j [|h(t,s)|«(s)-H(t,s)B(s)«1+ a(s)]ds. (3.13) Letting C = |h(t,s)|, D = H(t,s)B(s), and using the inequality (see [13])

C« - D«1+ a <

aa Ca+1 (a + 1)a+1 Da

-, D >0,

we obtain

ft H (t. s)* (s) ds < H (t. t)«(t) + f P (s)r(s)'h(t;s)|a+l ds.

Jt ~ ) ( ) Jt (a + l)a+l Ha (t. s)

l P(s)r(s)|h(t. s)|'

H (t. T ) Jt

H (t. s)* (s) —

(a + l)a+l Ha (t. s)

ds < «(T) (3.l4)

for all sufficiently large t. which contradicts (3.l).

Assume now that z(t) has the property (II). By Lemma 2.2. we have limt^œ x(t) = 0. The proof is complete. □

It may happen that assumption (3.l) in Theorem 3.l fails to hold. Consequently. Theorem 3.l cannot be applied. The following theorem provides a new oscillation criterion for (l.l).

Theorem 3.2 Let conditions (Ai)-(A6) and (2.l) be satisfied. Assume that there exists a function H e X such that

0 < inf jliminf H(t.s) < œ (3.l5)

s>t^ t^œ H (t. t0) J

1 f p(s)r(s)|h(t, s)|a+1

limsup—--- I --—--as < to (3.16)

t^ H (t, t0)Jt0 Ha (t, s) ( )

hold. If there exists a function q>(t) e C([t0, to), R) such that, for all T > t0,

^ -1 -I [ ] — limsup p a (s)r a (s)[<p+(s)J a ds = to (3.17)

t >TO Jt0

"mœ mr St

t I /11 Ml I M I hit. Nil

ds > <p(T). (3.l8)

'u WM l P(s)r(s)|h(t.s)|a H(t.s)* - (a + l)a+l Ha (t. s)

where *(t) is defined by (3.2) and <p+(t) = maxj^(t). 0}. then the conclusion of Theorem 3.l remains intact.

Proof Assuming that z(t) has the property (I) and proceeding as in the proof of Theorem 3.l. we have (3.l3) and (3.l4) for all t > T. Hence. byvirtue of (3.l4).

'H(l T) L

/T , , ,, l P(s)r(s)|h(t.s)|a+l

limsup ^ I H(t.s)*(s) — ■

ds < «(T) (3.l9)

(a + l)a+l Ha (t. s) for all t > T. Thus. by (3.l8) and (3.l9). we have

V(T) < «(T) (.20)

limsup H(t,s)f (s) ds > <p(T).

t—>to H (t, T ) Jt

From (3.20), we obtain

[ to i i i [ i a+1

I p-i1(s)r-5(s)«—(s) ds > / p-i1(s)r-5(s)[^(s)] a ds Jt Jt

and hence, by (3.17),

[ TO I

J p - ;1(s)r- «(s)«"+1(s) ds = to.

To complete the proof, we show that (3.23) does not hold. Let

u(t) = HOT) jT H(t,s)p 1 (s)r " (s)«a+ (s) ds

v(t) = H(rrjJT 'h(t,s)|«(s) ds

for all t > T .It follows from (3.13) and (3.21) that

liminf[u(t) - v(t)] < «(T) - limsup —-—— I H(t,s)^(s) ds

t—TO t—TO H(t, T) Jt

< «(T) - <p(T) < to. Now by (3.15), there exists a positive constant $1 satisfying

inH liminf H(t' s)\ > $1 > 0. s>t^ t—TO H(t, t0)\

Let $2 be an any positive constant. It follows from (3.23) that, for all t > T1.

f tii i I p-« (s)r-« (s)«~ (s) ds > Jt a$1

where T1 > T is sufficiently large. Therefore, for all t > T1,

u(t) = HjJT)fTH(t,s)d(jT p-«(Z)r-«(Z^(Z)d^

> h^ i: (-^)(/; p - ^ >r- ^ ) )

> $2 ft (- dH(t, s)) ds = $2H(t, T1) > $2H(t, T1)

t' T) Jt1

$1H(t, T) JtA ds ) $1H(t, T) " $1H(t, tc)

(3.21)

(3.22)

(3.25)

By (3.25), there exists a T2 > T1 such that H(t, TO/H(t, fe) > $1 for all t > T2. Thus u(t) > $2 for all t > T2. Since $2 is an arbitrary constant,

lim u(t) = to. (3.6)

t—TO

Consider now a sequence {ti}TO1 in (T, to) with limi—TO ti = to such that lim [u(ti) - vfe)l = limingu(t) - v(t)l < to.

i—TO t—TO L J

By virtue of (3.24), there exist a natural number N0 and a constant L >0 such that, for all i > N0,

u(ti) - v(ti) < L. (7)

It follows from (3.26) that

lim u(ti) = to. (3.8)

i—TO

Combining (3.27) and (3.28), we conclude that

lim v(ti) = to (3.29)

i—TO

and, for i large enough,

v(ti) 1 fl Q C\\

uutj > 2. (0)

From (3.29) and (3.30), we obtain

r va+1(ti)

lim aU, =TO.

i—TO ua (ti)

On the other hand, by Holder's inequality, we have

v(ti) < jH~TT) jT H(ti,s)p ;(s)r «(s)«""(s)ds

, p(s)r(s)|h(ti,s)|a+1 M -rr:-^ I -77—;—:-ds

1 p !(ti, T) Jt

aaH(ti, T) Jt Ha (ti, s) Therefore, for all i large enough,

va+1(ti) < 1 fP(s)r(s)|h(ti, s)|a+1 ds (3 32)

ua(ti) - aa$1H(ti, t0) Jt0 Ha (ti, s) From (3.31) and (3.32), we deduce that

1 ft> p(s)r(s)|h(ti,s)|a+i lim 7777—7 i -ttzt.—;-ds = to,

-i' t0) Jtr

i—TO H(ti, t0)Jt0 Ha (ti, s)

l ft P(s)r(s)|h(t.s)|a+l lim —-r I --rrrr.—;-ds = œ.

1.10) J t0

t^TO H(t, t0)J t0 Ha (t, s) which contradicts (3.16). Therefore, (3.23) cannot hold. By virtue of (3.22), we get

fœ —1 —1 r 1 a+l J p a (s)r a (s) [<p+(s)J a ds

which contradicts (3.17).

Suppose that z(t) has the property (II). By Lemma 2.2, we obtain limt^TO x(t) = 0. This completes the proof. □

4 Examples and conclusions

Example 4.1 For t > l. consider a third-order differential equation

X(t) + /2 31

l4?. xf£+i id?

+ ,l32q0i x3( L+i\ d? = 0.

where q0 > 0 is a constant. Let a = 3. a = l/2. b = l. c = 0. d = l. r(t) = t2. p(t. ?) = 4?/(3t2). t(t. ?) = (t + ?)/3. q(t. ?) = 32q0?/t5. and a(t. ?) = (t + ?)/2. Then

= a (t. 0) = 2

b l 4? l l

J p(t. ?) d? ^ 3^2 d? = 22 < 2 and a2(t) = a(t. It is not difficult to verify that

i -4- ds = œ and i i i i 32q°? d? ds

Jl s2 Jl Jv u Ju J0 s5 .

du dv = œ.

Therefore. the conditions (Al)-(A6) and (2.l) are satisfied. Furthermore. we choose K = l. P = l/2. p(t) = t. b(t) = 0. and H (t. s) = (t — s)4. Then h(t. s) = (t — s)3 (5 — ts—l).

* (t) = (l — 2 ) t

l\3 /¿Y(2)2\3 fl32q0?Ji_ q^jY)3

limsuP HTTTT f

t^œ H (t.t0) Jt0

tV H (t. s)* (s)— l P (s)r(s)|h(t.s)|a+n

= ^ (t—) l ="mr (T^/

(a + l)a+l Ha (t. s)

t r q^ (t — s) 4l--l— s3(5 —ts—l)4

trq0(fiY )3

s 256 (t4s—l — 4t3 + 6t2s — 4ts2 + s3)

/ l 4 , 5 3 75 2 l25 2 625 — I -: t4s—l--13 + -t2s--ts2 + -

t3 + -1 s -

64 l28

-ts +-s'

ds = œ ,

if q0 > 1/(3y)3 for some ¡3 e (0,1) and y e (0,1). Hence, by Theorem 3.1, every solution x(t) of (4.1) is either oscillatory or converges to zero as t ^to in the case where q0 > 1,000,000/(531,441) ^ 1.9 (by letting 3 = y = 9/10). Observe that the results reported in [19] cannot be applied to (4.1) since a = 3.

Example 4.2 For t > 1, consider a third-order differential equation

r1i£./ t + — h 3t2

x(t) + ) d—

+ f1 ——d— = o,

where q0 > 0 is a constant. Let a = 1, a = 1/2, b = 1, c = 0, d = 1, r(t) = 1,p(t, — ) = 4— /(3t2), r (t, — ) = (t + — )/3, q(t, — ) = 16qo— /t3, and a(t, — ) = (t + — )/2. Then

fb f1 4— 1 1

J p(t, — ) d— = J — d— = — < ^ and CT2(t) = a(t,

It is easy to verify that

çœ 1 çœ />œr />œ /> 1

I —rrds = œ and I I I I q(s, — ) ds A r(s) h Jv L Ju Jo

= a (t,0) = 2

du dv = œ.

Therefore, the conditions (Ai)-(A6) and (2.1) are satisfied. Furthermore, we choose K = 1, P =1/2, p(t) = t, b(t) = 1/t, and H (t, s) = (t - s)2. Then h(t, s) = (t - s)(5 - 3ts-1),

f (t) = 1-2 )t

1\ iPY(2)2\ f1 16qo—

d— + t - -1 - H .qoPY + 2-,

limsup HTT)

tnœ H (t, ^ Jt0

H (t, s)f (s)-

p (s)r(s)|h(t, s)|

(a + 1)

h a (t, s)

: lirnsup (t _11)2 ^ QqoPY + 2j(t-s)21 -4s(5-3ts1)2 ds

dim sup 1 i i1 qoPY +2 (t2s-1 - 2t + s) - 1 (25s -30t + 9t2s-1) tnœ (t -1) A L\2 / 4

if q0 >1/(23y )forsome3 e (0,1) and y e (0,1). Therefore, by Theorem 3.1,everysolution x(t) of (4.2) is either oscillatory or converges to zero as t ^to in the case where q0 > 50/81 ^ 0.62 (by letting ¡3 = y = 9/10).

Remark 4.1 With an appropriate choice of the function H, one can derive from Theorems 3.1 and 3.2 a number of oscillation criteria for (1.1). For example, consider a Kamenev-type function H(t, 5) by H(t, s) = (t - s)n-1, (t, s) e D, where n > 2 is an integer. The remainder of the details are left to the reader.

Remark 4.2 Theorems 3.1 and 3.2 reported in this paper reduce to ([19], Theorems 3.1 and 3.2), respectively, when letting a = 1 and b(t) = 0.

Remark 4.3 Note that Theorems 3.1 and 3.2 ensure that every solution x(t) to (1.1) is either oscillatory or satisfies limt^TO x(t) = 0 and, unfortunately, these results cannot distinguish solutions with different behaviors. Since the sign of the derivative z' (t) is not fixed, it is not easy to establish sufficient conditions which guarantee that all solutions to (1.1) are just oscillatory and do not satisfy limt^TO x(t) = 0. Neither is it possible to use the technique exploited in this paper for proving that all solutions of (1.1) satisfy limt^TO x(t) = 0. Hence, these two interesting problems are left for future research.

Remark 4.4 It would be interesting to find a different method to investigate (1.1) when

0 < a < 1. It would also be of interest to find another method to study (1.1) in the case where f^ r-1/a(s) ds < to.

Competing interests

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

All four authors contributed equally to this work. They all read and approved the final version of the manuscript. Author details

1 School of Electronic and Information Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, P.R. China. 2Qingdao Technological University, Feixian, Shandong 273400, P.R. China.

Acknowledgements

The authors are grateful to the editors and two anonymous referees for a very thorough reading of the manuscript and for pointing out several inaccuracies. This research was supported by NNSF of P.R. China (Grant Nos. 61174217,61374074, and 61473133) and NSF of Shandong Province (Grant No. JQ201119).

Received: 23 February 2015 Accepted: 11 August 2015 Published online: 28 August 2015 References

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