Neutral Emden-Fowler Delay Dynamic Equations on Time Scales

Advances in Difference Equations Volume 2010, Article ID 586312,23pages doi:10.1155/2010/586312

Research Article

Oscillation Behavior of Third-Order

Neutral Emden-Fowler Delay Dynamic Equations on Time Scales

Zhenlai Han,

Tongxing Li,

Shurong Sun,

and Chenghui Zhang

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 Shurong Sun,sshrong@163.com

Received 14 September 2009; Revised 28 November 2009; Accepted 10 December 2009 Academic Editor: Leonid Berezansky

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.

We will establish some oscillation criteria for the third-order Emden-Fowler neutral delay dynamic equationsrtxt−atxτt^ΔΔΔptx^γδt 0 on a time scaleT, whereγ > 0 is a quotient of odd positive integers withr,a,andpreal-valued positive rd-continuous functions defined onT. To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales, so this paper initiates the study. Some examples are considered to illustrate the main results.

1. Introduction

The study of dynamic equations on time-scales, which goes back to its founder Hilger 1 , is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time-scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations.

Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al.2 , Bohner and Guseinov3 , and references cited therein. A book

on the subject of time-scales, by Bohner and Peterson4 , summarizes and organizes much of the time-scale calculus; see also the book by Bohner and Peterson5 for advances in dynamic equations on time-scales.

In the recent years, there has been increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of various equations on time-scales; we refer the reader to the papers6–38 . To the best of our knowledge, it seems to have few oscillation results for the oscillation of third-order dynamic equations; see, for example,14–

16,21,35 . However, the paper which deals with the third-order delay dynamic equation is due to Hassan21 .

Hassan21 considered the third-order nonlinear delay dynamic equations ct

atx^Δt^ΔγΔ

ft, xτt 0, t∈T, 1.1

whereτσt στtis required, and the author established some oscillation criteria for 1.1which extended the results given in16 .

To the best of our knowledge, there are no results regarding the oscillation of the solutions of the following third-order nonlinear neutral delay dynamic equations on time- scales up to now:

rtxt−atxτt^ΔΔ_Δ

ptx^γδt 0, t∈T. 1.2

We assume thatγ > 0 is a quotient of odd positive integers, r, aandp are positive real-valued rd-continuous functions defined on T such that r^Δt ≥ 0, 0 < at ≤ a0 <

1, lim_{t→ ∞}at a < 1,the delay functions τ : T → T, δ : T → T are rd-continuous functions such thatτt≤t, δt≤t,and lim_{t→ ∞}τt lim_{t→ ∞}δt ∞.

As we are interested in oscillatory behavior, we assume throughout this paper that the given time-scaleTis unbounded above. We assumet0 ∈ Tand it is convenient to assume t0>0.We define the time-scale interval of the formt0,∞_Tbyt0,∞_T t0,∞∩T.

For the oscillation of neutral delay dynamic equations on time-scales, Mathsen et al.

26 considered the first-order neutral delay dynamic equations on time-scales yt−rtyτt_Δ

ptyδt 0, t∈T, 1.3

and established some new oscillation criteria of1.3which as a special case involve some well-known oscillation results for first-order neutral delay differential equations.

Agarwal et al.7 , S¸ah´ıner28 , Saker31 , Saker et al.33 , Wu et al.34 studied the second-order nonlinear neutral delay dynamic equations on time-scales

rtyt ptyτt^Δγ_Δ

f

t, yδt

0, t∈T, 1.4

by means of Riccati transformation technique, the authors established some oscillation criteria of1.4.

Saker 32 investigated the second-order neutral Emden-Fowler delay dynamic equations on time-scales

atyt rtyτt^{Δ Δ}pty^γδt 0, t∈T, 1.5

and established some new oscillation for1.5.

Our purpose in this paper is motivated by the question posed in 26 : What can be said about higher-order neutral dynamic equations on time-scales and the various generalizations? We refer the reader to the articles23,24 and we will consider the particular case when the order is 3, that is, 1.2. Set t−1 : min_t∈t0,∞T{τt, δt}. By a solution of 1.2, we mean a nontrivial real-valued functionx∈Crdt₋₁,∞_T,Rsatisfyingx−ax◦τ ∈ C²_rdt0,∞_T,Randrx−ax◦τ^ΔΔ∈C¹_rdt0,∞_T,R,and satisfying1.2for allt∈t0,∞_T. The paper is organized as follows. In Section 2, we apply a simple consequence of Keller’s chain rule, devoted to the proof of the sufficient conditions which guarantee that every solution of 1.2 oscillates or converges to zero. In Section 3, some examples are considered to illustrate the main results.

2. Main Results

In this section we give some new oscillation criteria for1.2. In order to prove our main results, we will use the formula

xt^γ_Δ γ

₁

0

hx^σt 1−hxt ^γ−1x^Δtdh, 2.1

where x is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller’s chain rulesee Bohner and Peterson4, Theorem 1.90 .

Before stating our main results, we begin with the following lemmas which are crucial in the proofs of the main results.

For the sake of convenience, we denote:zt xt−atxτt,fort∈t0,∞_T.Also, we assume that

Hthere exists{ck}_k∈N0⊂Tsuch that limk→ ∞ck∞andτc_k1 ck.

Lemma 2.1. Assume thatHholds. Further, assume thatxis an eventually positive solution of 1.2. If

_∞

t0

Δt

rt ∞, 2.2

then there are only the following three cases fort≥t1sufficiently large:

izt>0,z^Δt>0,z^ΔΔt>0,z^ΔΔΔt<0,

iizt<0, z^Δt>0, z^ΔΔt>0, z^ΔΔΔt<0, lim_{t→ ∞}xt 0, or

iiizt > 0, z^Δt < 0, z^ΔΔt > 0, z^ΔΔΔt < 0, lim_{t→ ∞}zt l ≥ 0, lim_{t→ ∞}xt l/1−a≥0.

Proof. Letx be an eventually positive solution of 1.2. Then there exists t1 ≥ t0 such that xt>0, xτt>0,andxδt>0 for allt≥t1.From1.2we have

rtz^ΔΔt_Δ

−ptx^γδt<0, t≥t1. 2.3

Hence rtz^ΔΔt is strictly decreasing on t1,∞_T. We claim that z^ΔΔt > 0 eventually.

Assume not, then there existst2≥t1such that

rtz^ΔΔt<0, t≥t2. 2.4

Then we can choose a negativecandt3≥t2such that

rtz^ΔΔt≤c <0, t≥t3. 2.5

Dividing byrtand integrating fromt3tot,we have

z^Δt≤z^Δt3 c _t

t3

Δs

rs. 2.6

Lettingt → ∞,thenz^Δt → −∞by2.2. Thus, there is at4≥t3such that fort≥t4,

z^Δt≤z^Δt4<0. 2.7

Integrating the previous inequality fromt4tot,we obtain

zt−zt4≤z^Δt4t−t4. 2.8

Therefore, there existd >0 andt5≥t4such that

xt≤ −datxτt≤ −da0xτt, t≥t5. 2.9

We can choose some positive integerk0such thatck≥t5,fork≥k0.Thus, we obtain xck≤ −da0xτck −da0xck−1≤ −d−a0da²₀xτck−1

−d−a0da²₀xck−2≤ · · · ≤ −d−a0d− · · · −a^k−k₀ ⁰⁻¹da^k−k₀ ⁰xτck01 −d−a0d− · · · −a^k−k₀ ⁰⁻¹da^k−k₀ ⁰xck0.

2.10

The above inequality implies thatxck<0 for sufficiently largek,which contradicts the fact thatxt>0 eventually. Hence we get

z^ΔΔt>0. 2.11

It follows from this that eitherz^Δt>0 orz^Δt<0.Sincer^Δt≥0, rtz^ΔΔt_Δ

r^Δtz^ΔΔt r^σtz^ΔΔΔt<0, 2.12

which yields

z^ΔΔΔt<0. 2.13

Ifz^Δt>0,then there are two possible cases:

1zt>0,eventually; or 2zt<0,eventually.

If there exists a t6 ≥ t1 such that case 2 holds, then lim_{t→ ∞}zt exists, and lim_{t→ ∞}zt b ≤ 0. We claim that limt→ ∞zt 0.Otherwise, lim_{t→ ∞}zt b < 0.We can choose some positive integerk0such thatck≥t6,fork≥k0.Thus, we obtain

xck≤a0xτck a0xck−1≤a²₀xτck−1

a²₀xck−2≤ · · · ≤a^k−k₀ ⁰xτck01 a^k−k₀ ⁰xck0, 2.14

which implies that lim_{k→ ∞}xck 0,and from the definition ofzt,we have lim_{k→ ∞}zck

0,which contradicts lim_{t→ ∞}zt<0.Now, we assert thatxis bounded. If it is not true, there exists{sk}_k∈N⊂t6,∞_Twithsk → ∞ask → ∞such that

xsk sup

t0≤s≤sk

xs, lim

k→ ∞xsk ∞. 2.15

Fromτt≤t

zsk xsk−askxτsk≥1−a0xsk, 2.16

which implies that limk→ ∞zsk ∞,it contradicts that lim_{t→ ∞}zt 0.Therefore, we can assume that

lim sup

t→ ∞ xt x1, lim inf

t→ ∞ xt x2. 2.17

By 0≤a <1,we get

x1−ax1≤0≤x2−ax2, 2.18

which implies thatx1 ≤x2,sox1x2,hence, lim_{t→ ∞}xt 0.

Assume that z^Δt < 0. We claim that zt ≥ 0 eventually. Otherwise, we have lim_{t→ ∞}zt<0 or lim_{t→ ∞}zt −∞.ByH,there existst7≥t1,we can choose some positive integerk0such thatck≥t7fork≥k0,and we obtain

xck≤a0xτck a0xc_k−1≤a²₀xτc_k−1

a²₀xc_k−2≤ · · · ≤a^k−k₀ ⁰xτck01 a^k−k₀ ⁰xck0, 2.19

which implies that limk→ ∞xck 0,and from the definition ofz,we have limk→ ∞zck 0, which contradicts lim_{t→ ∞}zt<0 or lim_{t→ ∞}zt −∞.Now, we have that lim_{t→ ∞}zt l≥0, herelis finite. We assert thatxis bounded. If it is not true, there exists{sk}_k∈N⊂t6,∞_Twith sk → ∞ask → ∞such that

xsk sup

t0≤s≤sk

xs, lim

k→ ∞xsk ∞. 2.20

Fromτt≤t

zsk xsk−askxτsk≥1−a0xsk, 2.21

which implies that lim_{k→ ∞}zsk ∞,it contradicts that lim_{t→ ∞}zt l ≥ 0.Therefore, we can assume that

lim sup

t→ ∞ xt x1∗, lim inf

t→ ∞ xt x2∗. 2.22

By 0≤a <1,we get

x1∗−ax1∗≤l≤x2∗−ax2∗, 2.23

which implies thatx1∗≤x2∗,sox1∗ x2∗,hence, lim_{t→ ∞}xt l/1−a≥0.This completes the proof.

In4, Section 1.6 the Taylor monomials{hnt, s}^∞_n0are defined recursively by

h0t, s 1, hn1t, s _t

s

hnτ, sΔτ, t, s∈T, n≥1. 2.24

It follows from4, Section 1.6 thath1t, s t−sfor any time-scale, but simple formulas in general do not hold forn≥2.

Lemma 2.2see15, Lemma 4 . Assume thatzsatisfies case (i) ofLemma 2.1. Then

lim inf

t→ ∞

tzt

h2t, t0z^Δt ≥1. 2.25

Lemma 2.3. Assume thatxis a solution of1.2satisfying case (i) ofLemma 2.1. If _∞

t0

pth2δt, t0^γΔt∞, 2.26

thenzsatisfies eventually

z^Δt≥tz^ΔΔt, z^Δt

t is nonincreasing. 2.27

Proof. Letxbe a solution of1.2such that caseiofLemma 2.1holds fort≥t1.Define

Zt z^Δt−tz^ΔΔt. 2.28

Thus

Z^Δt −σtz^ΔΔΔt>0. 2.29

We claim thatZt>0 eventually. Otherwise, there existst2≥t1such thatZt<0 fort≥t2. Therefore,

z^Δt t

_Δ

−Zt

tσt >0, t≥t2, 2.30

which implies thatz^Δt/tis strictly increasing ont2,∞_T.Pickt3≥t2such thatδt≥δt3≥ t2,fort≥t3.Then we have

z^Δδt

δt ≥ z^Δδt3

δt3 P >0, 2.31

thenz^Δδt≥P δtfort≥t3.ByLemma 2.2, for any 0< k <1,there existst4 ≥t3such that zt

z^Δt ≥kh2t, t0

t , t≥t4. 2.32

Hence there existst5≥t4so that zδt≥kh2δt, t0

δt z^Δδt≥P kh2δt, t0

δt δt P kh2δt, t0, t≥t5. 2.33

By the definition ofz,we have that

xt≥zt. 2.34

From1.2, we obtain

rtz^ΔΔt_Δ

ptz^γδt≤0. 2.35

Integrating both sides of2.35fromt5tot,we get

rtz^ΔΔt−rt5z^ΔΔt5 P k^γ _t

t5

psh2δs, t0^γΔs≤0, 2.36

which yields that

rt5z^ΔΔt5≥P k^γ _t

t5

psh2δs, t0^γΔs, 2.37

which contradicts2.26. HenceZt>0 andz^Δt/tis nonincreasing. The proof is complete.

Lemma 2.4. Assume thatHholds andxis a solution of1.2which satisfies case (iii) ofLemma 2.1.

If

_∞

t0

psR^σsΔs∞, 2.38

whereRt:_t

t0σu/ruΔufort∈t0,∞_T,then lim_{t→ ∞}xt 0.

Proof. Let xbe a solution of1.2 such that caseiii of Lemma 2.1holds for t ≥ t1.Then lim_{t→ ∞}zt l ≥ 0, lim_{t→ ∞}xt l/1−a ≥0.Next we claim thatl 0.Otherwise, there existst2 ≥t1such thatzδt≥l >0 for allt≥t2.By the definition ofz,we have that2.35 holds. Integrating both sides of2.35fromtto∞,we get

z^ΔΔt≥ 1 rt

_∞

t

psz^γδsΔs. 2.39

Integrating again fromtto∞,we have

−z^Δt≥ _∞

t

1 ru

_∞

u

psz^γδsΔsΔu. 2.40

Integrating again fromt2to∞,we obtain zt1≥

_∞

t2

_∞

v

1 ru

_∞

u

psz^γδsΔsΔuΔv≥l^γ _∞

t2

_∞

v

1 ru

_∞

u

psΔsΔuΔv, 2.41

which contradicts2.38, since by23, Lemma 1 and3, Remark 4.7 , we get _∞

t0

_∞

v

1 ru

_∞

u

psΔsΔuΔv

_∞

t0

_∞

v

_∞

u

1

rupsΔsΔuΔv

_∞

t0

_∞

v

_σs

v

1

rupsΔuΔsΔv

_∞

t0

_σs

t0

_σs

v

1

rupsΔuΔvΔs

_∞

t0

ps _σs

t0

_σs

v

1

ruΔuΔvΔs _∞

t0

ps _σs

t0

_σu

t0

1

ruΔvΔuΔs

_∞

t0

ps _σs

t0

1 ru

_σu

t0

ΔvΔuΔs

_∞

t0

ps _σs

t0

σu−t0

ru ΔuΔs

_∞

t0

ps _σs

t0

σu

ruΔuΔs

_∞

t0

psR^σsΔs.

2.42

Hence lim_{t→ ∞}xt 0 and completes the proof.

Theorem 2.5. Assume thatH,2.2,2.26, and2.38hold, γ ≥ 1.Furthermore, assume that there exists a positive functionη∈C_rd¹ t0,∞_T,Rsuch that for some 0< k <1 and for all constants M >0

lim sup

t→ ∞

_t

t0

ηspsζs− rs

η^Δs2

4kγM^γ−1ηs

Δs∞, 2.43

whereζt: h2δt, t0/t^γ.Then every solutionxof 1.2oscillates or limt→ ∞xt 0.

Proof. Suppose that 1.2 has a nonoscillatory solution x.We may assume without loss of generality thatxt>0, xτt>0,andxδt>0 for allt∈t1,∞_T, t1 ∈t0,∞_T.Then by Lemma 2.1,zsatisfies three cases. Assume thatzsatisfies casei.Define the functionωby

ωt ηtrtz^ΔΔt

z^Δt_γ , t∈t1,∞_T. 2.44

Thenωt>0.Using the product rule, we have

ω^Δt

rtz^ΔΔt_σ ηt

z^Δtγ

_Δ

rtz^ΔΔt_Δ ηt

z^Δtγ. 2.45

By the quotient rule, we get

ω^Δt

rtz^ΔΔt_ση^Δt z^Δt_γ

−ηt

z^Δt_γΔ z^Δtγ

z^Δσtγ

rtz^ΔΔt_Δ ηt

z^Δtγ. 2.46

By the definition ofzand1.2, we obtain2.35. From2.35and2.44, we have

ω^Δt≤ η^Δt

η^σtω^σt−ηtptz^γδt z^Δt_γ −ηt

rtz^ΔΔtσ

z^Δtγ_Δ z^Δt_γ

z^Δσt_γ , 2.47

from2.25and2.27, for any 0< k <1,we obtain z^γδt

z^Δt_γ z^γδt z^Δδt_γ

z^Δδtγ

z^Δt_γ ≥

k^1/γh2δt, t0 δt

_γδt t

_γ k

h2δt, t0 t

_γ , 2.48

hence by2.48, we have

ω^Δt≤ η^Δt

η^σtω^σt−kηtptζt−ηt

rtz^ΔΔt_σ

z^Δt_γΔ z^Δt_γ

z^Δσt_γ . 2.49

In view ofγ ≥1,from2.1andiofLemma 2.1, we have

z^Δt_γΔ γ

₁

0

hz^Δσt 1−hz^Δt ^γ−1z^ΔΔtdh

≥γ

z^Δt_γ−1

z^ΔΔt≥γM^γ−1z^ΔΔt,

2.50

whereMz^Δt1.By2.49, we have

ω^Δt≤ η^Δt

η^σtω^σt−kηtptζt−γM^γ−1ηt

rtz^ΔΔtσ 2

z^Δt_γ

z^Δσt_γ z^ΔΔt

r^σtz^ΔΔσt, 2.51

fromi,we havez^Δt≤z^Δσt,byrtz^ΔΔt^Δ<0,we have

z^ΔΔt≥ r^σt

rt z^ΔΔσt, 2.52

so we get

ω^Δt≤ η^Δt

η^σtω^σt−kηtptζt−γM^γ−1ηt

rtz^ΔΔtσ 2

rt

z^Δσt_γ2, 2.53 by2.44, we have

ω^Δt≤ η^Δt

η^σtω^σt−kηtptζt−γM^γ−1 ηt

rt

η^σt2ω^σt². 2.54 Therefore, we obtain

ω^Δt≤ −kηtptζt rt η^Δt2

4γM^γ−1ηt. 2.55

Integrating inequality2.55fromt1tot, we obtain

−ωt1≤ωt−ωt1≤ − _t

t1

kηspsζs−rs

η^Δs2

4γM^γ−1ηs

Δs, 2.56

which yields

_t

t1

kηspsζs−rs

η^Δs2

4γM^γ−1ηs

Δs≤ωt1 2.57

for all larget,which contradicts2.43. Ifiiholds, fromLemma 2.1, then limt→ ∞xt 0.If caseiiiholds, byLemma 2.4, then lim_{t→ ∞}xt 0.The proof is complete.

Remark 2.6. From Theorem 2.5, we can obtain different conditions for oscillation of all solutions of1.2with different choices ofη.

For example, letηt t.NowTheorem 2.5yields the following result.

Corollary 2.7. Assume thatH,2.2,2.26, and2.38hold,γ≥1.If

lim sup

t→ ∞

_t

t0

sps

h2δs, t0 s

γ

− rs 4kγM^γ−1s

Δs∞ 2.58

holds for some 0 < k < 1 and for all constantsM > 0, then every solution xof 1.2 is either oscillatory or lim_{t→ ∞}xt 0.

For example, letηt 1.FromTheorem 2.5, we have the following result which can be considered as the extension of the Leighton-Wintner Theorem.

Corollary 2.8. Assume thatH,2.2,2.26, and2.38hold, andγ ≥1.If

lim sup

t→ ∞

_t

t0

ps

h2δs, t0 s

γ

Δs∞, 2.59

then every solutionxof1.2is either oscillatory or lim_{t→ ∞}xt 0.

In the following theorem, we present a new Kamenev-type oscillation criteria for1.2.

Theorem 2.9. Assume thatH,2.2,2.26, and2.38hold,γ ≥1.Letζandηbe as defined in Theorem 2.5. If for some 0< k <1 and for all constantsM >0

lim sup

t→ ∞

1 t^m

_t

t0

t−s^mηspsζs− rsB²t, s η^σs₂ 4kγM^γ−1ηst−s^m

Δs∞, 2.60

wherem >1,and

Bt, s t−s^mη^Δs

η^σs −mt−σs^m−1, t≥σs≥t0, 2.61 then every solutionxof1.2oscillates or lim_{t→ ∞}xt 0.

Proof. Suppose that 1.2 has a nonoscillatory solution x.We may assume without loss of generality thatxt>0, xτt>0,andxδt>0 for allt∈t1,∞_T, t1∈t0,∞_T.Then by Lemma 2.1,zsatisfies three cases. Assume thatzsatisfies casei.We proceed as in the proof ofTheorem 2.5to get2.54for allt≥t1sufficiently large. Multiplying2.54byt−s^mand integrating fromt1tot,we have

_t

t1

t−s^mkηspsζsΔs≤ − _t

t1

t−s^mω^ΔsΔs _t

t1

t−s^mη^Δs

η^σsω^σsΔs

− _t

t1

t−s^mγM^γ−1ηs

rs

η^σs₂ ω^σs²Δs.

2.62

Integration by parts, we obtain

− _t

t1

t−s^mω^ΔsΔs−t−s^mωs|^t_{t1 t}

t1

t−s^m_Δs

ω^σsΔs. 2.63

Next, we show that ift≥σsandm≥1,then t−s^m_Δs

≤ −mt−σs^m−1. 2.64

Ifμs 0,it is easy to see that2.64is an equality. Ifμs>0,then we get t−s^m_Δs

1 μs

t−σs^m−t−s^m

− 1 σs−s

t−s^m−t−σs^m

. 2.65

Using the inequality

x^m−y^m≥my^m−1 x−y

, x≥y >0, m≥1, 2.66

we obtain fort≥σs

t−s^m−t−σs^m

≥mt−σs^m−1σs−s, 2.67 and from this we see that2.64holds. From2.62–2.64, we get

_t

t1

t−s^mkηspsζsΔs

≤t−t1^mωt1

_t

t1

t−s^mη^Δs

η^σs −mt−σs^m−1

ω^σsΔs

− _t

t1

t−s^mγM^γ−1ηs

rs

η^σs₂ ω^σs²Δs.

2.68

Thus _t

t1

t−s^mηspsζs− rsB²t, s η^σs₂ 4kγM^γ−1ηst−s^m

Δs≤ 1

kωt1t−t1^m, 2.69 which implies that

1 t^m

_t

t1

t−s^mηspsζs− rsB²t, s η^σs₂ 4kγM^γ−1ηst−s^m

Δs≤ 1

kωt1 t−t1

t m

. 2.70

This easily leads to a contradiction of2.60. Ifiiholds, fromLemma 2.1, then lim_{t→ ∞}xt 0.Ifiiiholds, byLemma 2.4, then lim_{t→ ∞}xt 0.The proof is complete.

In the following theorem, we present a new Philos-type oscillation criteria for1.2.

Theorem 2.10. Assume thatH,2.2,2.26, and2.38hold,γ≥1.Letζandηbe as defined in Theorem 2.5. Furthermore, assume that there exist functionsH,h∈CrdD,R, whereD≡ {t, s: t≥s≥t0}such that

Ht, t 0, t≥t0, Ht, s>0, t > s≥t0, 2.71 andHhas a nonpositive continuousΔ-partial derivationH^Δst, swith respect to the second variable and satisfies

H^Δst, s Ht, sη^Δs

η^σs −ht, s η^σs

Ht, s. 2.72

If for some 0< k <1 and for all constantsM >0 lim sup

t→ ∞

1 Ht, t0

_t

t0

Kt, sΔs∞, 2.73

where

Kt, s Ht, sηspsζs−rsh−t, s²

4kγM^γ−1ηs, 2.74

whereh−t, s max{0,−ht, s},then every solutionxof 1.2oscillates or limt→ ∞xt 0.

Proof. Suppose that 1.2 has a nonoscillatory solution x.We may assume without loss of generality thatxt>0, xτt>0,andxδt>0 for allt∈t1,∞_T, t₁∈t0,∞_T.Then by Lemma 2.1,zsatisfies three cases. Assume thatzsatisfies casei.We proceed as in the proof ofTheorem 2.5to get2.54for allt ≥ t1sufficiently large. Multiplying both sides of2.54, withtreplaced bys,byHt, s,integrating with respect tosfromt1tot,we have

_t

t1

kHt, sηspsζsΔs

≤ − _t

t1

Ht, sω^ΔsΔs _t

t1

Ht, sη^Δs

η^σsω^σsΔs− _t

t1

Ht, s γM^γ−1ηs

rs

η^σs₂ω^σs²Δs.

2.75

Integrating by parts and using2.71and2.72, we obtain _t

t1

kHt, sηspsζsΔs

≤Ht, t1ωt1

_t

t1

H^Δst, sω^σsΔs _t

t1

Ht, sη^Δs

η^σsω^σsΔs

− _t

t1

Ht, s γM^γ−1ηs rs

η^σs₂ω^σs²Δs

≤Ht, t1ωt1 _t

t1

−ht, s η^σs

Ht, sω^σs−Ht, s γM^γ−1ηs

rs

η^σs₂ω^σs²

Δs

≤Ht, t1ωt1 _t

t1

h−t, s η^σs

Ht, sω^σs−Ht, s γM^γ−1ηs

rs

η^σs₂ω^σs²

Δs

≤Ht, t1ωt1

_t

t1

rsh−t, s² 4γM^γ−1ηs Δs.

2.76

Therefore, we get _t

t1

Ht, sηspsζs−rsh−t, s² 4kγM^γ−1ηs

Δs≤ 1

kHt, t1ωt1. 2.77

This easily leads to a contradiction of 2.73. If case ii holds, from Lemma 2.1, then lim_{t→ ∞}xt 0. If case iii holds, by Lemma 2.4, then lim_{t→ ∞}xt 0. The proof is complete.

The following result can be considered as the extension of the Atkinson’s theorem39 . Theorem 2.11. Assume thatH,2.2,2.26, and2.38hold,γ >1.If

lim sup

t→ ∞

_t

t0

ps

rsσs

h2δs, t0 σs

γ

Δs∞, 2.78

then every solutionxof1.2is either oscillatory or lim_{t→ ∞}xt 0.

Proof. Suppose that 1.2 has a nonoscillatory solution x.We may assume without loss of generality thatxt>0, xτt>0 andxδt>0 for allt∈t1,∞_T, t₁∈t0,∞_T.Then by Lemma 2.1,zsatisfies three cases. Assume thatzsatisfies casei.Define the functionω

ωt trtz^ΔΔt

z^Δtγ , t∈t1,∞_T. 2.79

Using the product rule,2.25and2.27, for any 0< k <1,we have that

z^γδt

z^Δσtγ z^γδt z^Δδtγ

z^Δδtγ

z^Δσtγ ≥

k^1/γh2δt, t0 δt

γδt σt

γ

k

h2δt, t0 σt

γ

. 2.80

By1.2, we have that2.35holds, then from2.80, we calculate

ω^Δt

rtz^ΔΔt σt

rtz^ΔΔt_Δ

z^Δσt_−γ

trtz^ΔΔt

z^Δt_−γΔ

≤rtz^ΔΔt

z^Δσt_−γ

−σtpt

zδt z^Δσt

γ

trtz^ΔΔt

z^Δt_−γΔ

≤rt

z^Δt_1−γΔ

1−γ −kσtpt

h2δt, t0 σt

γ

,

2.81

where the last inequality is true becausez^Δt^−γΔ≤0 due to2.1and because z^Δt_1−γΔ

1−γ¹

0

hz^Δσt 1−hz^Δt ^−γz^ΔΔtdh

≤

1−γ¹

0

hz^Δσt 1−hz^Δσt ^−γz^ΔΔtdh

1−γ

z^Δσt_−γ z^ΔΔt.

2.82

Upon integration we arrive at

_t

t1

kσsps

rs

h2δs, t0 σs

γ

Δs≤ _t

t1

z^Δs_1−γΔ 1−γ Δs−

_t

t1

ω^Δs

rs Δs

z^Δt_1−γ 1−γ −

z^Δt1_1−γ 1−γ −

_t

t1

ω^Δs

rs Δs

≤

z^Δt1_1−γ

γ−1 ωt1 rt1 −ωt

rt

_t

t1

ω^σs 1

rs _Δ

Δs

≤

z^Δt1_1−γ

γ−1 ωt1 rt1

2.83

fromr^Δt≥0.This contradicts2.78. If caseiiholds, fromLemma 2.1, then lim_{t→ ∞}xt 0.If caseiiiholds, byLemma 2.4, then lim_{t→ ∞}xt 0.The proof is complete.

Theorem 2.12. Assume thatH,2.2,2.26, and2.38hold,γ ≤ 1.Furthermore, assume that there exists a positive functionη∈C_rd¹ t0,∞_T,Rsuch that for some 0< k <1 and for all constants L >0

lim sup

t→ ∞

_t

t0

ηspsζs− rs

η^Δs2

4kγLσs^γ−1ηs

Δs∞, 2.84

where ζ is as defined as in Theorem 2.5. Then every solution x of 1.2 is either oscillatory or lim_{t→ ∞}xt 0.

Proof. Suppose that 1.2 has a nonoscillatory solution x.We may assume without loss of generality thatxt>0, xτt>0 andxδt>0 for allt1t∈t1,∞_T, ∈t0,∞_T.Then by Lemma 2.1,zsatisfies three cases. Assumezsatisfies casei.Define the functionωas2.44.

We proceed as in the proof ofTheorem 2.5and we get2.49. In view ofγ≤1,from2.1and iofLemma 2.1, we have

z^Δt_γΔ γ

₁

0

hz^Δσt 1−hz^Δt ^γ−1z^ΔΔtdh

≥γ

z^Δσt_γ−1 z^ΔΔt,

2.85

from2.27, there exists a constantL >0 such thatz^Δt≤Lt,so

z^ΔtγΔ

≥γLσt^γ−1z^ΔΔt. 2.86 By2.49, we have

ω^Δt≤ η^Δt

η^σtω^σt−kηtptζt−γLσt^γ−1 ηt

rt

η^σt₂ω^σt². 2.87 Therefore, we obtain

ω^Δt≤ −kηtptζt rt η^Δt2

4γLσt^γ−1ηt. 2.88

Integrating inequality2.88fromt1tot, we obtain

−ωt1≤ωt−ωt1≤ − _t

t1

kηspsζs− rs

η^Δs₂ 4γLσs^γ−1ηs

Δs, 2.89

which yields

_t

t1

kηspsζs− rs

η^Δs2

4γLσs^γ−1ηs

Δs≤ωt1 2.90

for all larget,which contradicts2.84. If caseiiholds, fromLemma 2.1, then lim_{t→ ∞}xt 0.If caseiiiholds, byLemma 2.4, then lim_{t→ ∞}xt 0.The proof is complete.

Remark 2.13. From Theorem 2.12, we can obtain different conditions for oscillation of all solutions of1.2with different choices ofη.

For example, letηt t.NowTheorem 2.12yields the following results.

Corollary 2.14. Assume thatH,2.2,2.26, and2.38hold,γ≤1.If

lim sup

t→ ∞

_t

t0

sps

h2δs, t0 s

γ

− rs

4kγLσs^γ−1s

Δs∞ 2.91

holds for some 0< k <1 and for all constantsL >0,then every solutionxof1.2is either oscillatory or lim_{t→ ∞}xt 0.

For example, letηt 1.FromTheorem 2.12, we have the following result which can be considered as the extension of the Leighton-Wintner theorem.

Corollary 2.15. Assume thatH,2.2,2.26, and2.38hold,γ ≤1.If 2.59holds, then every solutionxof 1.2is either oscillatory or lim_{t→ ∞}xt 0.

In the following theorem, we present a new Kamenev-type oscillation criteria for1.2.

Theorem 2.16. Assume thatH,2.2,2.26, and2.38hold,γ≤1.Letζandηbe as defined in Theorem 2.12. If for some 0< k <1 and for all constantsL >0

lim sup

t→ ∞

1 t^m

_t

t0

t−s^mηspsζs− rsB²t, s η^σs2

4kγLσs^γ−1ηst−s^m

Δs∞, 2.92

wherem >1,and

Bt, s t−s^mη^Δs

η^σs −mt−σs^m−1, t≥σs≥t0, 2.93 then every solutionxof1.2oscillates or lim_{t→ ∞}xt 0.

The proof is similar to that of Theorem 2.9using inequality 2.88, so we omit the details.

In the following theorem, we present a new Philos-type oscillation criteria for1.2.

Theorem 2.17. Assume thatH,2.2,2.26, and2.38hold,γ≤1.Letζandηbe as defined in Theorem 2.12. Furthermore, assume that there exist functionsH,h∈CrdD,R, whereD≡ {t, s: t≥s≥t0}such that2.71holds, andHhas a nonpositive continuousΔ-partial derivationH^Δst, s with respect to the second variable and satisfies2.72. If

lim sup

t→ ∞

1 Ht, t0

_t

t0

Kt, sΔs∞ 2.94

holds for some 0< k <1 and for all constantsL >0,where

Kt, s Ht, sηspsζs− rsh−t, s²

4kγLσs^γ−1ηs, 2.95

whereh−t, s max{0,−ht, s}.Then every solutionxof1.2oscillates or lim_{t→ ∞}xt 0.

The proof is similar to that of the proof ofTheorem 2.10using inequality2.88, so we omit the details.

The following result can be considered as the extension of the Belohorec’s theorem 40 .

Theorem 2.18. Assume thatH,2.2,2.26, and2.38holdγ <1.If

lim sup

t→ ∞

_t

t0

ps

r^γδsh2δs, t0^γΔs∞, 2.96

then every solutionxof1.2is either oscillatory or satisfies lim_{t→ ∞}xt 0.

Proof. Suppose that 1.2 has a nonoscillatory solution x.We may assume without loss of generality thatxt>0, xτt>0,andxδt>0 for allt1t∈t1,∞_T, ∈t0,∞_T.Then by Lemma 2.1,zsatisfies three cases. Assume thatzsatisfies casei.Fromiand2.1we have

rtz^ΔΔt_1−γΔ

1−γ¹

0

hrtz^ΔΔt^σ 1−hrtz^ΔΔt ^−γ

rtz^ΔΔt_Δ dh

≤

1−γ¹

0

hrtz^ΔΔt 1−hrtz^ΔΔt ^−γ

rtz^ΔΔt_Δ dh

1−γ

rtz^ΔΔt_−γ

rtz^ΔΔt_Δ ,

2.97

so

rtz^ΔΔt_−γ

rtz^ΔΔt_Δ

≥

rtz^ΔΔt_1−γΔ

1−γ . 2.98

By1.2, we have that2.35holds. Using2.25and2.27, for any 0< k <1,we obtain after dividing2.35byrtz^ΔΔt^γfor all larget

0≥

rtz^ΔΔt_Δ

ptz^γδt rtz^ΔΔt_γ

rtz^ΔΔt_−γ

rtz^ΔΔt_Δ pt

zδt rtz^ΔΔt

γ

≥

rtz^ΔΔt_1−γΔ

1−γ pt

r^γδt

zδt z^Δδt

z^Δδt z^ΔΔδt

γ

≥

rtz^ΔΔt_1−γΔ

1−γ pt

r^γδt

kh2δt, t0

δt δt

_γ

rtz^ΔΔt_1−γΔ

1−γ k^γ pt

r^γδth2δt, t0^γ.

2.99