Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations

doi:10.1155/2010/763278

Research Article

Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations

Zhenlai Han,

Tongxing Li,

Shurong Sun,

and Weisong Chen

1School of Science, University of Jinan, Jinan, Shandong 250022, China

2School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

3Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA

Correspondence should be addressed to Zhenlai Han,hanzhenlai@163.com Received 10 December 2009; Revised 22 June 2010; Accepted 1 July 2010 Academic Editor: A ˘gacık Zafer

Copyrightq2010 Zhenlai Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Some sufficient conditions are established for the oscillation of second-order neutral differential equationxt ptxτtqtfxσt 0,t ≥t0, where 0≤pt≤p0<∞. The results complement and improve those of Grammatikopoulos et al. Ladas, A. Meimaridou, Oscillation of second-order neutral delay differential equations, Rat. Mat. 11985, Grace and Lalli1987, Ruan 1993, H. J. Li1996, H. J. Li1997, Xu and Xia2008.

1. Introduction

In recent years, the oscillatory behavior of differential equations has been the subject of intensive study; we refer to the articles 1–13 ; Especially, the study of the oscillation of neutral delay differential equations is of great interest in the last three decades; see for example 14–38 and references cited therein. Second-order neutral delay differential equations have applications in problems dealing with vibrating masses attached to an elastic bar and in some variational problemssee39 .

This paper is concerned with the oscillatory behavior of the second-order neutral delay differential equation

xt ptxτt

qtfxσt 0, t≥t0, 1.1

wherep, q∈Ct0,∞,R, f ∈CR,R.Throughout this paper, we assume that

a0≤pt≤p0 <∞,qt≥0,andqtis not identically zero on any ray of the form t_∗,∞for anyt_∗≥t₀,wherep₀is a constant;

bfu/u≥k >0,foru /0, kis a constant;

cτ, σ ∈ C¹t0,∞,R,τt ≤ t, σt ≤ t,τt≡ τ₀ >0,σt> 0, lim_{t→ ∞}σt ∞, τ◦σσ◦τ,whereτ0is a constant.

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 equationor inequality.

One of them is the Riccati transformation technique. The other one is called the generalized Riccati technique. This technique can introduce some new sufficient conditions for oscillation and can be applied to different equations which cannot be covered by the results established by the Riccati technique.

Philos 7 examined the oscillation of the second-order linear ordinary differential equation

xt ptxt 0 1.2

and used the class of functions as follows. Suppose there exist continuous functionsH, h : D ≡ {t, s : t ≥ s ≥ t0} → Rsuch thatHt, t 0,t ≥ t0,Ht, s > 0, t > s ≥ t0,andH has a continuous and nonpositive partial derivative onDwith respect to the second variable.

Moreover, leth:D → Rbe a continuous function with

∂Ht, s

∂s −ht, s

Ht, s, t, s∈D. 1.3

The author obtained that if

lim sup

t→ ∞

1 Ht, t0

_t

t0

Ht, sps−1 4h²t, s

ds∞, 1.4

then every solution of1.2oscillates. Li4 studied the equation rtxt

ptxt 0, 1.5

used the generalized Riccati substitution, and established some new sufficient conditions for oscillation. Li utilized the class of functions as in7 and proved that if there exists a positive functiong∈C¹t0,∞,Rsuch that

lim sup

t→ ∞

1 Ht, t0

_t

t0

asrsht, sds <∞, lim sup

t→ ∞

1 Ht, t0

_t

t0

as

Ht, sψs− 1

4rsh²t, s

ds∞,

1.6

where as exp{−_s

0gudu} and ψs {ps rsg²s− rsgs}, then every solution of1.5oscillates. Yan13 used Riccati technique to obtain necessary and sufficient

conditions for nonoscillation of1.5. Applying the results given in4,13 , every solution of the equation

1 txt

1

t³xt 0 1.7

is oscillatory.

An important tool in the study of oscillation is the integral averaging technique. Just as we can see, most oscillation results in1,3,5,7,11,12 involved the function classH.Say a functionHHt, sbelongs to a function classH,denoted byH∈ H,ifH∈CD,R∪ {0}, whereD{t, s:t₀ ≤s≤t <∞}andR 0,∞,which satisfies

Ht, t 0, Ht, s>0, fort > s, 1.8

and has partial derivatives∂H/∂tand∂H/∂sonDsuch that

∂Ht, s

∂t h1t, s

Ht, s, ∂Ht, s

∂s −h2t, s

Ht, s. 1.9

In10 , Sun defined another type of function classXand considered the oscillation of the second-order nonlinear damped differential equation

rtyt

ptyt qtf

yt

0. 1.10

Say a functionΦ Φt, s, lis said to belong toX,denoted byΦ∈X,ifΦ∈CE,R, whereE{t, s, l:t0≤l≤s≤t <∞},which satisfiesΦt, t, l 0, Φt, l, l 0, Φt, s, l>

0,forl < s < t,and has the partial derivative∂Φ/∂sonEsuch that∂Φ/∂sis locally integrable with respect tosinE.

In 8 , by employing a class of function H ∈ H and a generalized Riccati transformation technique, Rogovchenko and Tuncay studied the oscillation of 1.10. Let D {t, s : −∞ < s ≤ t < ∞}. Say a continuous functionHt, s, H : D → 0,∞ belongs to the classHif:

iHt, t 0 andHt, s>0 for−∞< s < t <∞;

iiHhas a continuous and nonpositive partial derivative∂H/∂ssatisfying, for some h∈L_locD,R,∂H/∂s−ht, s

Ht, s,wherehis nonnegative.

Meng and Xu 22 considered the even-order neutral differential equations with deviating arguments

xt ^m

i1

p_itxτit _n

^l

j1

q_jtfj

x σ_jt

0, 1.11

where _m

i1pit ≤ p,p ∈ 0,1.The authors introduced a class of functionsF^∗.Let D0 {t, s∈R²;t > s≥t₀}andD {t, s∈R²;t≥s ≥t₀}.The functionH ∈CD,Ris said to

belong to the classF^∗defined byH∈F^∗,for shortif J1Ht, t 0,t≥t0,Ht, s>0 fort, s∈D0;

J2H has a continuous and nonpositive partial derivative onD₀ with respect to the second variable;

J3there exists a nondecreasing functionρ∈C¹t0,∞,0,∞such that

ht, s ∂Ht, s

∂s Ht, sρs

ρs. 1.12

Xu and Meng31 studied the oscillation of the second-order neutral delay differential equation

rt

yt ptyσt

ⁿ

i1

q_itfi

yτit

0, 1.13

wherep∈ Ct0,∞,0,1; by using the function classX an operatorT·;l, t ,and a Riccati transformation of the form

ωt rtzt

zt, zt yt ptyσt, 1.14

the authors established some oscillation criteria for 1.13. In 31 , the operatorT·;l, t is defined by

T g;l, t

_t

l

Φ²t, s, lgsds, 1.15

fort≥s≥l≥t₀andg∈Ct0,∞,R.The functionϕϕt, s, lis defined by

∂Φt, s, l

∂s ϕt, s, lΦt, s, l. 1.16

It is easy to verify thatT·;l, t is a linear operator and that it satisfies

T g;l, t

−2T gϕ;l, t

, forgs∈C¹t0,∞,R. 1.17

In 2009, by using the function classX and defining a new operatorT·;l, t , Liu and Bai21 considered the oscillation of the second-order neutral delay differential equation

rt

xt ptxτt^m−1

xt ptxτt

qtfxσt 0, 1.18

where 0≤pt≤1.The authors defined the operatorT·;l, t by

T g;l, t

_t

lΦt, s, lgsds, 1.19

fort≥s≥l≥t₀andg∈Ct0,∞,R.The functionϕϕt, s, lis defined by

∂Φt, s, l

∂s ϕt, s, lΦt, s, l. 1.20

It is easy to see thatT·;l, t is a linear operator and that it satisfies

T g;l, t

−T gϕ;l, t

, forg∈C¹t0,∞,R. 1.21

Wang11 established some results for the oscillation of the second-order differential equation

rtyt f

t, yt, yt

0 1.22

by using the function classHand a generalized Riccati transformation of the form

ωt δtrt

yt g

ytρt

. 1.23

Long and Wang6 considered1.22; by using the function classXand the operatorT·;l, t which is defined in31 , the authors established some oscillation results for1.22.

In 1985, Grammatikopoulos et al. 16 obtained that if 0 ≤ pt ≤ 1, qt ≥ 0,and _∞

t0 qs1−ps−σ ds∞,then equation xt ptxt−τ

qtxt−σ 0 1.24

is oscillatory. Li 18 studied 1.1 when 0 ≤ pt ≤ 1, τt t −τ, σt t −σ and established some oscillation criteria for 1.1. In 15,19,25 , the authors established some

general oscillation criteria for second-order neutral delay differential equation at

xt ptxt−τ

qtfxt−σ 0, 1.25

where 0 ≤ pt ≤ 1.In 2002, Tanaka 27 studied the even-order neutral delay differential equation

xt htxt−τ ⁿf t, x

gt

0 1.26

where 0 ≤ μ ≤ ht ≤ λ < 1 or 1 < λ≤ ht ≤ μ.The author established some comparison theorems for the oscillation of1.26. Xu and Xia28 investigated the second-order neutral differential equation

xt ptxt−τ

qtfxt−σ 0, 1.27

and obtained that if 0 ≤ pt ≤ p₀ <∞, fx/x ≥k > 0,forx /0,andqt ≥ M >0,then 1.27is oscillatory. We note that the result given in28 fails to apply the casesqt γ/t,or qt γ/t²forγ >0.To the best of our knowledge nothing is known regarding the qualitative behavior of1.1whenpt>1,0< qt≤M.

Motivated by10,21 , for the sake of convenience, we give the following definitions.

Definition 1.1. Assume thatΦt, s, l∈X. The operator is defined byT_n·;l, t by

T_n g;l, t

_t

l

Φⁿt, s, lgsds, 1.28

forn≥1, t≥s≥l≥t0andg∈Ct0,∞,R.

Definition 1.2. The functionϕϕt, s, lis defined by

∂Φt, s, l

∂s ϕt, s, lΦt, s, l. 1.29

It is easy to verify that Tn·;l, t is a linear operator and that it satisfies

T_n g;l, t

−nTn

gϕ;l, t

, forg ∈C¹t0,∞,R. 1.30

In this paper, we obtain some new oscillation criteria for1.1. The paper is organized as follows. In the next section, we will use the generalized Riccati transformation technique to give some sufficient conditions for the oscillation of1.1, and we will give two examples to illustrate the main results. The key idea in the proofs makes use of the idea used in23 . The method used in this paper is different from that of27 .

2. Main Results

In this section, we give some new oscillation criteria for1.1. We start with the following oscillation result.

Theorem 2.1. Assume that σt ≤ τt fort ≥ t0.Further, suppose that there exists a function g∈C¹t0,∞,Rsuch that for someβ≥1 and someH∈H,one has

lim sup

t→ ∞

1 Ht, t0

_t

t0

Ht, sψs− 1p₀ τ₀

β

4σsush²t, s

ds∞, 2.1

whereψt : ut{kQt 1p0/τ0σtg²t−gt }, Qt : min{qt, qτt}, ut : exp{−2_t

t0σsgsds}.Then every solution of 1.1is oscillatory.

Proof. Letxbe a nonoscillatory solution of1.1. Without loss of generality, we assume that there existst1 ≥ t0 such thatxt > 0,xτt > 0, xσt > 0, for allt ≥ t1.Definezt xt ptxτtfort≥t₀,thenzt>0 fort≥t₁.From1.1, we have

zt≤ −kqtxσt≤0, t≥t₁. 2.2

It is obvious thatzt ≤ 0 andzt> 0 fort≥ t1 implyzt> 0 fort≥ t1.Using2.2and conditionb,there existst2≥t1such that fort≥t2,we get

0zt qtfxσt

zt qtfxσt p₀

zτt qτtfxστt

zt p₀zτt

qtfxσt p₀qτtfxστt

≥

zt p₀ τ₀zτt

k

qtxσt p₀qτtxτσt

≥

zt p0

τ0zτt

kQt

xσt p₀xτσt

≥

zt p₀ τ₀zτt

kQtzσt.

2.3

We introduce a generalized Riccati transformation

ωt ut

zt

zσtgt

. 2.4

Differentiating2.4from2.2, we havezσt ≥ zt.Thus, there existst3 ≥ t1 such that for allt≥t₃,

ωt≤ −2σtgtωt ut

zt

zσt−σt ωt

ut −gt

2

gt

ut zt

zσtut

−σtg²t gt

−σtω²t ut .

2.5

Similarly, we introduce another generalized Riccati transformation

νt ut

zτt

zσt gt

. 2.6

Differentiating2.6, note thatσt ≤ τt,by2.2, we havezσt≥ zτt,then for all sufficiently larget,one has

νt≤ −2σtgtνt ut

τ₀zτt zσt −σt

νt

ut−gt

₂ gt

τ0utzτt

zσt ut

−σtg²t gt

−σtν²t ut.

2.7

From2.5and2.7, we have

ωt p0

τ0νt

≤ ut zσt

zt p0

τ0zτt

1p0

τ0

ut

−σtg²t gt

−σtω²t ut −p₀

τ₀

σtν²t

ut .

2.8

By2.3and the above inequality, we obtain

ωt p₀ τ₀νt

≤ −ψt−σtω²t ut −p₀

τ₀

σtν²t

ut . 2.9

Multiplying2.9byHt, sand integrating fromT tot,we have, for anyβ ≥ 1 and for all t≥T ≥t₃,

_t

T

Ht, sψsds≤ − _t

T

Ht, sωsds− _t

T

Ht, sσsω²s

us ds

− p₀ τ0

_t

T

Ht, sνsds−p₀ τ0

_t

T

Ht, sσsν²s

us ds

−Ht, sωs|^t_T− _t

T

−∂Ht, s

∂s ωs Ht, sσsω²s us

ds

−p0

τ0Ht, sνs ^t

T

− p0

τ0

_t

T

−∂Ht, s

∂s νs Ht, sσsν²s us

ds

Ht, TωT−

_t

T

ht, s

Ht, sωs Ht, sσsω²s us

ds

p0

τ₀Ht, TνT−p0

τ_{0 t}

T

ht, s

Ht, sνs Ht, sσsν²s us

ds

Ht, TωT−

_t

T

⎡

⎣

Ht, sσs

βus ωs

βus 4σsht, s

⎤

⎦

2

ds

_t

T

βus

4σsh²t, sds− _t

T

β−1

σsHt, s

βus ω²sds

p₀

τ₀Ht, TνT−p₀ τ₀

_t

T

⎡

⎣

Ht, sσs

βus νs

βus 4σsht, s

⎤

⎦

2

ds

p₀ τ₀

_t

T

βus

4σsh²t, sds−p₀ τ₀

_t

T

β−1

σsHt, s

βus ν²sds.

2.10

From the above inequality and using monotonicity ofH,for allt≥t₃,we obtain

_t

t3

Ht, sψs− 1p0

τ₀ β

4σsush²t, s

ds≤Ht, t0|ωt3|p0

τ₀Ht, t0|νt3|, 2.11

and, for allt≥t3, _t

t0

Ht, sψs− 1p₀ τ₀

β

4σsush²t, s

ds

≤Ht, t0 _t3

t0

ψsds|ωt3|p₀ τ0|νt3|

.

2.12

By2.12,

lim sup

t→ ∞

1 Ht, t0

_t

t0

Ht, sψs− 1p₀ τ₀

β

4σsush²t, s

ds

≤ _t3

t0

ψsds|ωt3|p0

τ0|νt3|<∞,

2.13

which contradicts2.1. This completes the proof.

Remark 2.2. We note that it suffices to satisfy2.1inTheorem 2.1for anyβ≥1,which ensures a certain flexibility in applications. Obviously, if2.1is satisfied for someβ₀≥1,it well also hold for anyβ₁ > β₀.Parameterβintroduced inTheorem 2.1plays an important role in the results that follow, and it is particularly important in the sequel thatβ >1.

With an appropriate choice of the functionsHandh,one can derive fromTheorem 2.1 a number of oscillation criteria for1.1. For example, consider a Kamenev-type function Ht, sdefined by

Ht, s t−sⁿ⁻¹, t, s∈D, 2.14

wheren >2 is an integer. It is easy to see thatH∈H,and

ht, s n−1t−s^n−3/2, t, s∈D. 2.15

As a consequence ofTheorem 2.1, we have the following result.

Corollary 2.3. Suppose thatσt≤τtfort≥t0.Furthermore, assume that there exists a function g∈C¹t0,∞,Rsuch that for some integern >2 and someβ≥1,

lim sup

t→ ∞ t¹⁻ⁿ _t

t0

t−sⁿ⁻³

t−s²ψs− 1p₀ τ₀

βn−1² 4σs us

ds∞, 2.16

whereuandψare as inTheorem 2.1. Then every solution of 1.1is oscillatory.

For an application ofCorollary 2.3, we give the following example.

Example 2.4. Consider the second-order neutral differential equation

xt 3sintxt−τ γ

t²xt−σ 0, t≥1, 2.17

whereσ≥τ,γ >0.Letpt 3sint,qt γ/t², fx x, andgt −1/2t.Thenut t, ψt γ−5/4/t.Takek1, p04.ApplyingCorollary 2.3withn3,for anyβ≥1,

lim sup

t→ ∞ t¹⁻ⁿ _t

t0

t−sⁿ⁻³

ψst−s²−βn−1² 1p₀

4 us

ds

lim sup

t→ ∞

1 t²

_t

1

γ−5 4

t−s² s −5βs

ds∞,

2.18

forγ >5/4.Hence,2.17is oscillatory forγ >5/4.

Remark 2.5. Corollary 2.3can be applied to the second-order Euler differential equation

xt γ

t²xt 0, t≥1, 2.19

whereγ >0.Letpt 0, qt γ/t², fx x, andgt −1/2t.Thenut t,ψt γ−1/4/t.Takek1,p00.ApplyingCorollary 2.3withn3,for anyβ≥1,

lim sup

t→ ∞ t¹⁻ⁿ _t

t0

t−sⁿ⁻³

ψst−s²−βn−1² 1p0

4 us

ds

lim sup

t→ ∞

1 t²

_t

1

γ−1 4

t−s² s −βs

ds∞,

2.20

forγ >1/4.Hence,2.19is oscillatory forγ >1/4.

It may happen that assumption 2.1 is not satisfied, or it is not easy to verify, consequently, thatTheorem 2.1does not apply or is difficult to apply. The following results provide some essentially new oscillation criteria for1.1.

Theorem 2.6. Assume thatσt≤τtfort≥t0,and for someH∈H,

0<inf

s≥t0

lim inf

t→ ∞

Ht, s Ht, t0

≤ ∞. 2.21

Further, suppose that there exist functionsg ∈C¹t0,∞,Randm∈ Ct0,∞,Rsuch that for allT ≥t₀and for someβ >1,

lim sup

t→ ∞

1 Ht, T

_t

T

Ht, sψs− 1 p₀ τ0

β

4σsush²t, s

ds≥mT, 2.22

whereu,ψare as inTheorem 2.1. Suppose further that

lim sup

t→ ∞

_t

t0

σsm²s

us ds∞, 2.23

wheremt:max{mt,0}.Then every solution of 1.1is oscillatory.

Proof. We proceed as in the proof ofTheorem 2.1, assuming, without loss of generality, that there exists a solutionxof1.1such thatxt>0, xτt>0, andxσt>0,for allt≥t₁. We define the functionsωandνas inTheorem 2.1; we arrive at inequality2.10, which yields fort > T≥t₁,sufficiently large

1 Ht, T

_t

T

Ht, sψs− 1 p₀ τ₀

β

4σsush²t, s

ds

≤ωT− 1 Ht, T

_t

T

β−1

σsHt, s

βus ω²sds

p₀

τ₀νT− p₀ τ₀

1 Ht, T

_t

T

β−1

σsHt, s

βus ν²sds.

2.24

Therefore, fort > T≥t1,sufficiently large

lim sup

t→ ∞

1 Ht, T

_t

T

Ht, sψs− 1p0

τ0

β

4σsush²t, s

ds

≤ωT p₀

τ₀νT−lim inf

t→ ∞

1 Ht, T

_t

T

β−1

σsHt, s

βus ω²s p₀

τ₀ν²s

ds.

2.25

It follows from2.22that

ωT p₀

τ₀νT≥mT lim inf

t→ ∞

1 Ht, T

_t

T

β−1

σsHt, s

βus ω²s p₀

τ₀ν²s

ds, 2.26

for allT ≥t1and for anyβ >1.Consequently, for allT ≥t1,we obtain ωT p₀

τ₀νT≥mT, lim inf

t→ ∞

1 Ht, t1

_t

t1

Ht, sσs

us ω²s p₀

τ₀ν²s

ds≤ β

β−1 ωt1

p₀

τ₀νt1−mt1

<∞.

2.27

In order to prove that

_∞

t1

σs

ω²s p₀/τ₀

ν²s

us ds <∞, 2.28

suppose the contrary, that is, _∞

t1

σs

ω²s p₀/τ₀

ν²s

us ds∞. 2.29

Assumption2.21implies the existence of aρ >0 such that

s≥tinf0

lim inf

t→ ∞

Ht, s Ht, t0

> ρ. 2.30

By2.30, we have

lim inf

t→ ∞

Ht, s

Ht, t0 > ρ >0, 2.31

and there exists aT₂≥T₁such thatHt, T1/Ht, t0≥ρ,for allt≥T₂.On the other hand, by virtue of2.29, for any positive numberκ,there exists aT1≥t1such that, for allt≥T1,

_t

t1

σs

ω²s p0/τ0

ν²s

us ds≥ κ

ρ. 2.32

Using integration by parts, we conclude that, for allt≥T₁, 1

Ht, t1 _t

t1

Ht, sσs

us ω²s p0

τ0

ν²s

ds

1 Ht, t1

_t

t1

−∂Ht, s

∂s

_s

t1

σv

ω²v p₀/τ₀

ν²v

uv dv

ds

≥ κ ρ

1 Ht, t1

_t

T1

−∂Ht, s

∂s

ds κHt, T1 ρHt, t1.

2.33

It follows from2.33that, for allt≥T2, 1

Ht, t1 _t

t1

Ht, sσs

us ω²s p0

τ0

ν²s

ds≥κ. 2.34

Sinceκis an arbitrary positive constant, we get

lim inf

t→ ∞

1 Ht, t1

_t

t1

Ht, sσs

us ω²s p0

τ₀ν²s

ds∞, 2.35

which contradicts2.17. Consequently,2.28holds, so _∞

t1

σsω²s

us ds <∞,

_∞

t1

σsν²s

us ds <∞, 2.36

and, by virtue of2.27, _∞

t1

σsm²s

us ds

≤ _∞

t1

σsω²s p₀/τ₀2

σsν²s

2p₀/τ₀

σsωsνs

us ds

≤ _∞

t1

σsω²s p₀/τ₀₂

σsν²s p₀/τ₀

σs

ω²s ν²s

us ds <∞,

2.37

which contradicts2.23. This completes the proof.

Choosing H as in Corollary 2.3, it is easy to verify that condition2.21 is satisfied because, for anys≥t0,

t→ ∞lim Ht, s Ht, t0 lim

t→ ∞

t−sⁿ⁻¹

t−t₀ⁿ⁻¹ 1. 2.38

Consequently, we have the following result.

Corollary 2.7. Suppose thatσt ≤τtfort≥ t₀.Furthermore, assume that there exist functions g ∈C¹t0,∞,Randm∈Ct0,∞,Rsuch that for allT ≥t₀,for some integern >2 and some β≥1,

lim sup

t→ ∞ t¹⁻ⁿ _t

T

t−sⁿ⁻³

t−s²ψs− 1 p₀ τ₀

βn−1² 4σs us

ds≥mT, 2.39

whereuandψare as inTheorem 2.1. Suppose further that2.23holds, wheremis as inTheorem 2.6.

Then every solution of 1.1is oscillatory.

FromTheorem 2.6, we have the following result.

Theorem 2.8. Assume thatσt ≤ τtfort ≥ t₀.Further, suppose thatH ∈ Hsuch that2.21 holds, there exist functionsg∈C¹t0,∞,Randm∈Ct0,∞,Rsuch that for allT ≥t₀and for someβ >1,

lim inf

t→ ∞

1 Ht, T

_t

T

Ht, sψs− 1p₀ τ₀

β

4σsush²t, s

ds≥mT, 2.40

whereuandψare as inTheorem 2.1. Suppose further that2.23holds, wheremis as inTheorem 2.6.

Then every solution of 1.1is oscillatory.

Theorem 2.9. Assume thatσt≤τtfort≥t₀.Further, assume that there exists a functionΦ∈X, such that for eachl≥t₀,for somen≥1,

lim sup

t→ ∞ Tn

ψs−n²

4 1p0

τ₀

usϕ²s σs ;l, t

>0, 2.41

whereψ,uare defined as inTheorem 2.1, the operatorT_nis defined by1.28, andϕ ϕt, s, lis defined by1.29. Then every solution of1.1is oscillatory.

Proof. We proceed as in the proof ofTheorem 2.1, assuming, without loss of generality, that there exists a solutionxof1.1such thatxt>0,xτt>0, andxσt>0,for allt≥t₁. We define the functionsωandνas inTheorem 2.1; we arrive at inequality2.9. Applying Tn·;l, t to2.9, we get

Tn

ωs p0

τ0νs

;l, t

≤Tn

−ψs−σsω²s us −p0

τ0

σsν²s us ;l, t

. 2.42

By1.30and the above inequality, we obtain

T_n

ψs;l, t

≤T_n

nϕωs−σsω²s

us np₀

τ₀ϕνs−p₀ τ₀

σsν²s us ;l, t

. 2.43

Hence, from2.43we have

T_n

ψs;l, t

≤T_n 1p0

τ0

n²usϕ²s 4σs ;l, t

, 2.44

that is,

Tn

ψs− 1p₀ τ₀

n²usϕ²s 4σs ;l, t

≤0. 2.45

Taking the super limit in the above inequality, we get

lim sup

t→ ∞ Tn

ψs− 1p0

τ₀

n²usϕ²s 4σs ;l, t

≤0, 2.46

which contradicts2.41. This completes the proof.

If we choose

Φt, s, l ρst−s^αs−l^β 2.47

forα, β >1/2 andρ∈C¹t0,∞,0,∞,then we have

ϕt, s, l ρs

ρs βt− αβ

sαl

t−ss−l . 2.48

Thus byTheorem 2.9, we have the following oscillation result.

Corollary 2.10. Suppose thatσt≤τtfort≥t₀.Further, assume that for eachl≥t₀,there exist a functionρ∈C¹t0,∞,0,∞and two constantsα, β >1/2 such that for somen≥1,

lim sup

t→ ∞

_t

l

ρⁿst−s^nαs−l^nβ

×

⎡

⎣ψs− n²

4 1p0

τ0

us σs

ρs

ρs βt− αβ

sαl t−ss−l

₂⎤

⎦ds >0,

2.49

whereψ,uare as inTheorem 2.1. Then every solution of1.1is oscillatory.

If we choose

Φt, s, l

H₁s, lH2t, s, 2.50 whereH1, H2∈ H,then we have

ϕt, s, l 1 2

h¹₁ s, l

H1s, l − h²₂ t, s H2t, s

, 2.51

whereh¹₁ s, l,h²₂ t, sare defined as the following:

∂H1s, l

∂s h¹₁ s, l

H1s, l, ∂H2t, s

∂s −h²₂ t, s

H2t, s. 2.52

According toTheorem 2.9, we have the following oscillation result.

Corollary 2.11. Suppose thatσt≤τtfort≥t0.Further, assume that for eachl≥t0,there exist two functionsH₁, H₂∈ Hsuch that for somen≥1,

lim sup

t→ ∞

_t

l

H1s, lH2t, s n

×

⎡

⎣ψs−n²

16 1p0

τ₀ us

σs

h¹₁ s, l

H1s, l− h²₂ t, s H2t, s

2⎤

⎦ds >0,

2.53

whereψ,uare as inTheorem 2.1. Then every solution of1.1is oscillatory.

In the following, we give some new oscillation results for1.1whenσt ≥ τtfor t≥t₀.

Theorem 2.12. Assume that σt ≥ τt for t ≥ t0. Suppose that there exists a function g ∈ C¹t0,∞,Rsuch that for someβ≥1 and for someH∈H,one has

lim sup

t→ ∞

1 Ht, t0

_t

t0

Ht, sψs− 1 p₀ τ₀

β

4τ₀ush²t, s

ds∞, 2.54

whereψt ut{kQt 1p0/τ0τ0g²t−gt }, ut exp{−2τ0

_t

t0gsds},andQis as inTheorem 2.1. Then every solution of1.1is oscillatory.

Proof. Letxbe a nonoscillatory solution of1.1. Without loss of generality, we assume that there exists a solutionxof1.1such thatxt>0,xτt>0, andxσt>0,for allt≥t1. Proceeding as in the proof ofTheorem 2.1, we obtain2.2and2.3. In view of2.2, we have zt>0 fort≥t₁.We introduce a generalized Riccati transformation

ωt ut

zt

zτtgt

. 2.55

Differentiating2.55from2.2, we havezτt ≥zt.Thus, there existst₂ ≥ t₁such that for allt≥t₂,

ωt≤ −2τ0gtωt ut

zt zτt−τ₀

ωt ut −gt

₂ gt

ut zt

zτtut

−τ0g²t gt

−τ0ω²t ut .

2.56

Similarly, we introduce another generalized Riccati transformation

νt ut

zτt

zτt gt

. 2.57

Differentiating2.57, then for all sufficiently larget,one has

νt −2τ0gtνt ut

τ₀zτt zτt −τ₀

νt

ut−gt

₂ gt

τ0utzτt

zτt ut

−τ0g²t gt

−τ0ν²t ut .

2.58

From2.56and2.58, we have

ωt p0

τ₀νt

≤ ut zτt

zt p0

τ₀zτt

1p0

τ0

ut

−τ0g²t gt

−τ₀ω²t ut −p0

ν²t ut.

2.59

Note thatzt > 0,then we have zσt ≥ zτt.By 2.3and the above inequality, we obtain

ωt p0

τ0νt

≤ −ψt−τ0ω²t

ut −p0ν²t

ut. 2.60

Multiplying2.60byHt, sand integrating fromT tot,we have, for anyβ≥ 1 and for all t≥T ≥t₂,

_t

T

Ht, sψsds≤ − _t

T

Ht, sωsds− _t

T

Ht, sτ₀ω²s

us ds

−p₀ τ0

_t

T

Ht, sνsds− p₀ τ0

_t

T

Ht, sτ₀ν²s

us ds

−Ht, sωs|^t_T− _t

T

−∂Ht, s

∂s ωs Ht, sτ0ω²s us

ds

−p0

τ₀Ht, sνs ^t

T

−p0

τ_{0 t}

T

−∂Ht, s

∂s νs Ht, sτ0ν²s us

ds

Ht, TωT− _t

T

ht, s

Ht, sωs Ht, sτ0ω²s us

ds

p₀

τ₀Ht, TνT−p₀ τ₀

_t

T

ht, s

Ht, sνs Ht, sτ0ν²s us

ds

Ht, TωT− _t

T

⎡

⎣

Ht, sτ0

βus ωs

βus 4τ0 ht, s

⎤

⎦

2

ds

_t

T

βus

4τ0 h²t, sds− _t

T

β−1

τ₀Ht, s

βus ω²sds

p₀

τ₀Ht, TνT−p₀ τ₀

_t

T

⎡

⎣

Ht, sτ0

βus νs

βus 4τ₀ ht, s

⎤

⎦

2

ds

p₀ τ0

_t

T

βus 4τ0

h²t, sds− p₀ τ0

_t

T

β−1

τ0Ht, s

βus ν²sds.

2.61 The rest of the proof is similar to that ofTheorem 2.1, we omit the details. This completes the proof.

TakeHt, s t−sⁿ⁻¹, t, s ∈ D,wheren > 2 is an integer. As a consequence of Theorem 2.12, we have the following result.

Corollary 2.13. Suppose thatσt≥τtfort≥t₀.Furthermore, assume that there exists a function g∈C¹t0,∞,Rsuch that for some integern >2 and someβ≥1,

lim sup

t→ ∞ t¹⁻ⁿ _t

t0

t−sⁿ⁻³

t−s²ψs− 1p₀ τ₀

βn−1² 4τ₀ us

ds∞, 2.62

whereuandψare as inTheorem 2.12. Then every solution of1.1is oscillatory.

For an application ofCorollary 2.13, we give the following example.

Example 2.14. Consider the second-order neutral differential equation xt ptxλt

γ

t²fxσt 0, t≥1, 2.63

where τt λt, σt ≥ λt,σλt λσt, 0 < λ < 1, γ > 0, 0 ≤ pt ≤ p₀ < ∞, and fx/x ≥ k > 0, for x /0.Let τ0 λ,qt γ/t², and gt −1/2λt. Then ut t, ψt kγ −1p₀/λ/4λ/t.ApplyingCorollary 2.13withn3,for anyβ≥1,

lim sup

t→ ∞ t¹⁻ⁿ _t

t0

t−sⁿ⁻³

ψst−s²− 1p0

τ0

βn−1² 4τ0 us

ds

lim sup

t→ ∞

1 t²

_t

1

kγ− 1 p₀/λ 4λ

1

st−s²− β λ

1p0

λ

s

ds∞,

2.64

forγ >1p₀/λ/4kλ.Hence,2.63is oscillatory forγ >1p₀/λ/4kλ.

By2.61, similar to the proof ofTheorem 2.6, we have the following result.

Theorem 2.15. Assume thatσt≥τtfort≥t₀.Assume also thatH∈Hsuch that2.21holds.

Moreover, suppose that there exist functionsg∈C¹t0,∞,Randm∈Ct0,∞,Rsuch that for allT ≥t0and for someβ >1,

lim sup

t→ ∞

1 Ht, T

_t

T

Ht, sψs− 1p₀ τ₀

β

4τ₀ush²t, s

ds≥mT, 2.65

whereuandψare as inTheorem 2.12. Suppose further that

lim sup

t→ ∞

_t

t0

m²s

us ds∞, 2.66

wheremis defined as inTheorem 2.6. Then every solution of1.1is oscillatory.

ChoosingHt, s t−sⁿ⁻¹,t, s∈D,wheren >2 is an integer. ByTheorem 2.15, we have the following result.

Corollary 2.16. Suppose thatσt≥τtfort≥t₀.Furthermore, assume that there exist functions g ∈ C¹t0,∞,Rand m ∈ Ct0,∞,Rsuch that for allT ≥ t₀,some integern > 2 and some β≥1,

lim sup

t→ ∞ t¹⁻ⁿ _t

T

t−sⁿ⁻³

t−s²ψs− 1p0

τ0

βn−1² 4τ0 us

ds≥mT, 2.67

whereuandψ are as inTheorem 2.12. Suppose further that2.66holds, wheremis defined as in Theorem 2.6. Then every solution of 1.1is oscillatory.

FromTheorem 2.15, we have the following result.

Theorem 2.17. Assume thatσt≥τtfort≥t₀.Assume also thatH∈Hsuch that2.21holds.

Moreover, suppose that there exist functionsg∈C¹t0,∞,Randm∈Ct0,∞,Rsuch that for allT ≥t₀and for someβ >1,

lim inf

t→ ∞

1 Ht, T

_t

T

Ht, sψs− 1p₀ τ₀

β

4τ₀ush²t, s

ds≥mT, 2.68

where u and ψ are as in Theorem 2.12. Suppose further that 2.66 holds, where m is as in Theorem 2.6. Then every solution of 1.1is oscillatory.

Next, by2.60, similar to the proof ofTheorem 2.9, we have the following result.

Theorem 2.18. Assume that σt ≥ τt fort ≥ t₀. Further, assume that there exists a function Φ∈X,such that for eachl≥t₀,for somen≥1,

lim sup

t→ ∞ T_n

ψs−n²

4 1p₀ τ₀

usϕ²s τ₀ ;l, t

>0, 2.69

whereψ, uare defined as inTheorem 2.12, the operatorTnis defined by1.28, andϕϕt, s, lis defined by1.29. Then every solution of1.1is oscillatory.

If we choose Φt, s, l as 2.47, then from Theorem 2.18, we have the following oscillation result.

Corollary 2.19. Suppose thatσt≥τtfort≥t₀.Further, assume that for eachl≥t₀,there exist a functionρ∈C¹t0,∞,0,∞and two constantsα, β >1/2 such that for somen≥1,

lim sup

t→ ∞

_t

l

ρⁿst−s^nαs−l^nβ

×

⎡

⎣ψs−n²

4 1p₀ τ₀

us τ₀

ρs

ρs βt− αβ

sαl t−ss−l

2⎤

⎦ds >0,

2.70

whereψ,uare as inTheorem 2.12. Then every solution of 1.1is oscillatory.

If we choose Φt, s, l as 2.50, then from Theorem 2.18, we have the following oscillation result.

Corollary 2.20. Suppose thatσt≥τtfort≥t₀.Further, assume that for eachl≥t₀,there exist two functionsH1, H2∈ Hsuch that for somen≥1,

lim sup

t→ ∞

_t

l

H₁s, lH2t, s _n

×

⎡

⎣ψs−n²

16 1p₀ τ₀

us τ₀

h¹₁ s, l

H1s, l − h²₂ t, s H2t, s

2⎤

⎦ds >0,

2.71

whereψ,uare as inTheorem 2.12. Then every solution of 1.1is oscillatory.

Remark 2.21. The results of this paper can be extended to the more general equation of the form

rt

xt ptxτt

qtfxσt 0. 2.72

The statement and the formulation of the results are left to the interested reader.

Remark 2.22. One can easily see that the results obtained in15,16,18,19,25,28 cannot be applied to2.17,2.63, so our results are new.