Advances in Difference Equations Volume 2010, Article ID 450264,24pages doi:10.1155/2010/450264
Research Article
Oscillatory Behavior of Quasilinear Neutral Delay Dynamic Equations on Time Scales
Zhenlai Han,
1, 2Shurong Sun,
1, 3Tongxing Li,
1and Chenghui Zhang
21School 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,[email protected] Received 6 December 2009; Accepted 4 March 2010
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.
By means of the averaging technique and the generalized Riccati transformation technique, we establish some oscillation criteria for the second-order quasilinear neutral delay dynamic equations rt|xΔt|γ−1xΔtΔ q1t|yδ1t|α−1yδ1t q2t|yδ2t|β−1yδ2t 0, t ∈ t0,∞T, wherext yt ptyτt, and the time scale interval ist0,∞T : t0,∞∩T. Our results in this paper not only extend the results given by Agarwal et al.2005but also unify the oscillation of the second-order neutral delay differential equations and the second-order neutral delay difference equations.
1. Introduction
The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. Thesis in 1988 in order to unify continuous and discrete analysis see Hilger 1. Several authors have expounded on various aspects of this new theory and references cited therein. A book on the subject of time scale, by Bohner and Peterson 2, summarizes and organizes much of the time scale calculus; we refer also the last book by Bohner and Peterson3for advances in dynamic equations on time scales.
A time scaleTis an arbitrary closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales existsee Bohner and Peterson 2.
In the last few years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales which attempts to harmonize the oscillation theory for the continuous and the discrete to include them in one comprehensive theory and to eliminate obscurity form both, for instance, the papers4–20 and the reference cited therein.
For oscillation of delay dynamic equations on time scales, see recently papers21–32.
However, there are very few results dealing with the oscillation of the solutions of neutral delay dynamic equations on time scales; we refer the reader to33–44.
Agarwal et al.33and Saker37consider the second-order nonlinear neutral delay dynamic equations on time scales:
rt
xt ptxt−τΔγΔ
ft, xt−δ 0 for t∈T, 1.1
where 0 ≤ pt < 1,γ ≥ 0 is a quotient of odd positive integer,τ, δ are positive constants, r, p∈CrdT,R,∞
t01/rt1/γΔt ∞, f ∈CT×R,Rsuch thatuft, u>0 for all nonzero u, and there exists a nonnegative functionqtdefined onTsatisfing|ft, u| ≥qt|uγ|.
Agwo 35 examines the oscillation of the second-order nonlinear neutral delay dynamic equations:
xt−rxτtΔΔH
t, xh1t, xΔh2t
0 for t∈T. 1.2
Li et al. 36 discuss the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation of the form
xt ptxτ0tΔΔ
q1txτ1t−q2txτ2t et fort∈T. 1.3
Saker et al.38,39, Sah´ıner40, and Wu et al.43consider the second-order neutral delay and mixed-type dynamic equations on time scales:
rt
xt ptxτtΔγΔ
ft, xδt 0 for t∈T, 1.4
where 0≤pt<1,γis a quotient of odd positive integer,r, p∈CrdT,R,∞
t01/rt1/γΔt
∞, τ, δ ∈ CrdT,T, τt ≤ t,limt→ ∞τt ∞,limt→ ∞δt ∞,f ∈ CT×R,Rsuch that uft, u>0 for all nonzerou, and there exists a nonnegative functionqdefined onTsatisfing
|ft, u| ≥qt|uγ|.
Zhu and Wang 44 study existence of nonoscillatory solutions to neutral dynamic equations on time scales:
xt ptx gtΔ
ft, xht 0 fort∈T. 1.5
Recently, Tripathy42has established some new oscillation criteria for second-order nonlinear delay dynamic equations of the form
rtxt ptxt−τΔγΔ
qt|xt−δ|γsgnxt−δ 0, fort∈T, 1.6
where 0≤pt,γ≥0 is a quotient of odd positive integer,τ, δare positive constants,r, p, q∈ CrdT,R,and∞
t01/rt1/γΔt ∞.
To the best of our knowledge, there are no results regarding the oscillation of the solutions of the following second-order nonlinear neutral delay dynamic equations on time scales up to now:
rtxΔtγ−1xΔt Δ
q1tyδ1tα−1yδ1t
q2tyδ2tβ−1yδ2t 0 fort∈t0,∞T,
1.7
wherext yt ptyτt, and the time scale interval ist0,∞T: t0,∞∩T.
In what follows we assume the following:
A1α, β,andγare positive constants with 0< α < γ < β;
A2r, q1, q2∈Crdt0,∞T,R,R 0,∞,∞
t01/rt1/γΔt ∞, rΔt≥0;
A3p∈Crdt0,∞T,R,and−1< p0≤pt<1, p0constant;
A4τ, δ1, δ2 ∈ Crdt0,∞T,T, τt ≤ t, δ1t ≤ t, δ2t ≤ t, for t ∈ t0,∞T, and limt→ ∞τt ∞,limt→ ∞δ1t ∞,limt→ ∞δ2t ∞.
To develop the qualitative theory of delay dynamic equations on time scales, in this paper, by using the averaging technique and the generalized Riccati transformation, we consider the second-order nonlinear neutral delay dynamic equation on time scales1.7and establish several oscillation criteria. Our results in this paper not only extend the results given but also unify the oscillation of the second-order quasilinear delay differential equation and the second-order quasilinear delay difference equation. Applications to equations to which previously known criteria for oscillation are not applicable are given.
By a solution of 1.7, we mean a nontrivial real-valued function y ∈ C1rdty,∞T, ty∈t0,∞T, which has the propertyrt|xΔt|γ−1xΔt∈C1rdty,∞Tand satisfying1.7for t∈ty,∞T. Our attention is restricted to those solutionsyof1.7which exist on some half linety,∞Twith sup{|yt|:t≥ t1}>0 for anyt1 ∈ty,∞T. A solutionyof1.7is called oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory. Equation1.7is called oscillatory if all solutions are oscillatory.
Equation1.7includes many other special important equations; for example, ifq1t≡ 0,1.7is the prototype of a wide class of nonlinear dynamic equations called Emden-Fowler neutral delay superlinear dynamic equation:
rtxΔtγ−1xΔt Δ
q2tyδ2tβ−1yδ2t 0 fort∈t0,∞T, 1.8 wherext yt ptyτt.
Ifq2t≡0,1.7is the prototype of nonlinear dynamic equations called Emden-Fowler neutral delay sublinear dynamic equation:
rt|xΔt|γ−1xΔtΔ
q1tyδ1tα−1yδ1t 0 fort∈t0,∞T, 1.9 wherext yt ptyτt.
We note that if γ 1, then 1.7 becomes second-order nonlinear delay dynamic equation on time scales:
rt
yt ptyτtΔΔ
q1tyδ1tα−1yδ1t
q2tyδ2tβ−1yδ2t 0 for t∈t0,∞T.
1.10
Ifpt≡0, then1.7becomes second-order nonlinear delay dynamic equation on time scales:
rtyΔtγ−1yΔt Δ
q1tyδ1tα−1yδ1t
q2tyδ2tβ−1yδ2t 0 fort∈t0,∞T.
1.11
Ifγ 1, pt≡ 0, then1.7becomes second-order nonlinear delay dynamic equation on time scales:
rtyΔtΔ
q1tyδ1tα−1yδ1t q2tyδ2tβ−1yδ2t 0 fort∈t0,∞T.
1.12
Ifγ 1, rt≡ 1, pt≡ 0, then1.7becomes second-order nonlinear delay dynamic equation on time scales:
yΔΔt q1tyδ1tα−1yδ1t q2tyδ2tβ−1yδ2t 0 fort∈t0,∞T.
1.13
It is interesting to study1.7because the continuous version and its special cases have several physical applications, see1and whentis a discrete variable, and include its special cases also, are important in applications.
The paper is organized as follows: In the next section we present the basic definitions and apply a simple consequence of Keller’s chain rule, Young’s inequality:
|ab| ≤ 1 p|a|p 1
q|b|q, a, b∈R, p >1, q >1, 1 p 1
q 1, 1.14
and the inequality
λABλ−1−Aλ≤λ−1Bλ, λ≥1, 1.15
whereAandBare nonnegative constants, devoted to the proof of the sufficient conditions for oscillation of all solutions of1.7. In Section3, we present some corollaries to illustrate our main results.
2. Main Results
In this section we shall give some oscillation criteria for 1.7 under the cases when 0 ≤ pt < 1 and−1 < p0 ≤ pt < 0. It will be convenient to make the following notations in the remainder of this paper. Define
θ min
β−α β−γ,β−α
γ−α
, δt min
t≥t0{δ1t, δ2t}, γ
⎧⎪
⎨
⎪⎩
γ2, γ≥1, γ, 0< γ <1,
Q1t θ q1t
1−pδ1tαβ−γ/β−α q2t
1−pδ2tβγ−α/β−αδt σt
γ ,
Q2t θ
q1tαβ−γ/β−α
q2tβγ−α/β−αδt σt
γ .
2.1
We define the function spaceRas follows:H ∈Rprovided thatHis defined fort0 ≤ s ≤ σt, t, s ∈ t0,∞T, Ht, s ≥ 0, Hσt, t 0, HΔsσt, s −ht, sHσt, σs,and ht, sis rd-continuous function and nonnegative. For given functionρ, η∈C1rdt0,∞T,R, we set
λt, s hσt, s−ρΔs ρs , Θit, s
Qis−ηΔsρσs
ρs λt, sηs, i 1,2.
2.2
In order to prove our main results, we will use the formula xtγΔ
γxΔt 1
0
hxσt 1−hxtγ−1dh, 2.3
wherextis delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller’s chain rulesee Bohner and Peterson2, Theorem 1.90.
Also, we assume the conditionH−1 < p0 ≤pt<0 and limt→ ∞pt p >−1, and there exists{ck}k≥0such that limk→ ∞ck ∞andτck1 ck.
Lemma 2.1. Assume 0 ≤ pt < 1. If ytis an eventually positive solution of 1.7, then, there exists at∗∈t0,∞Tsuch thatxt>0, xΔt≥0 fort∈t∗,∞T. Moreover,
rt|xΔt|γ−1xΔtΔ
≤ −q1t
1−pδ1t
xδ1tα
− q2t
1−pδ2t
xδ2tβ
<0, t∈t∗,∞T.
2.4
Lemma 2.2. Assume 0≤pt<1, ∞
t0
q2t
1−pδ2tδ2tβΔt ∞. 2.5
Ifytis an eventually positive solution of 1.7, then
xΔΔt<0, xt≥txΔt, xt
t is strictly decreasing. 2.6
The proof of Lemmas2.1and2.2is similar to that of Saker et al.39, Lemma 2.1; so it is omitted.
Lemma 2.3. Assume that the conditionHholds:
∞
t0
q2tδ2tβΔt ∞. 2.7
Ifytis an eventually positive solution of 1.7, then
rtxΔtγ−1xΔt Δ
<−q1txδ1tα−q2txδ2tβ<0, t∈t∗,∞T, 2.8
xΔΔt<0, xt≥txΔt, xt
t is strictly decreasing, 2.9
or limt→ ∞yt 0.
Proof. Sinceytis an eventually positive solution of1.7, there exists a numbert1∈t0,∞T such thatyt > 0,yτt > 0,yδ1t > 0 andyδ2t > 0 for allt ∈ t1,∞T. In view of 1.7, we have
rtxΔtγ−1xΔt Δ
−q1t
yδ1tα
−q2t
yδ2tβ
, t∈t1,∞T. 2.10
By−1< p0≤pt<0, thenyt> xt, we get
rtxΔtγ−1xΔt Δ
<−q1txδ1tα−q2txδ2tβ<0, t∈t1,∞T. 2.11
Lett∗ ≥ t1, then2.8holds, andrt|xΔt|γ−1 xΔtis an eventually decreasing function. It follows that
t→ ∞limrtxΔtγ−1xΔt l≥ −∞. 2.12 We prove thatrt|xΔt|γ−1xΔtis eventually positive.
Otherwise, there exists at2 ∈ t1,∞T such thatrt2|xΔt2|γ−1xΔt2 c < 0, then we have rt|xΔt|γ−1xΔt ≤ rt2|xΔt2|γ−1xΔt2 c fort ∈ t2,∞T, and hencexΔt ≤
−−c1/γ1/rt1/γ,which implies that
xt≤xt2−−c1/γ t
t2
1 rs
1/γ
Δs−→ −∞ ast−→ ∞. 2.13
Therefore, there existsd >0,andt3≥t2such that
yt≤ −d−ptyτt≤ −dp0yτt, t≥t3. 2.14
We can choose some positive integerk0such thatck≥t3,fork≥k0.Thus, we obtain yck≤ −dp0yτck −dp0yck−1≤ −d−r0dp02yτck−1
−d−p0dp20xck−2≤ · · · ≤ −d−p0d− · · · −pk−k0 0−1dpk−k0 0yτck01
−d−p0d− · · · −pk−k0 0−1dpk−k0 0yck0.
2.15
The above inequality implies thatyck<0 for sufficiently largek,which contradicts the fact thatyt > 0 eventually. HencexΔt > 0 eventually. Consequently, there are two possible cases:
ixt>0,eventually;
iixt<0,eventually.
If Caseiholds, we can get
zt≥tzΔt>0, zΔΔt<0, zt
t is nonincreasing. 2.16
Actually, byrtxΔtγΔ rΔtxΔtγrσtxΔtγΔ <0, rΔt≥0, we can easily verify thatxΔtγΔ<0.Using2.3, we get
xΔtγΔ
γxΔΔt 1
0
h
xΔσ
1−hxΔγ−1
dh. 2.17
From1
0hxΔσ 1−hxΔγ−1dh >0, we have thatxΔΔtis eventually negative.
LetXt : xt−txΔt, sinceXΔt xΔt−xΔt σtxΔΔt −σtxΔΔt is eventually positive, soXtis eventually increasing. Therefore,Xtis either eventually positive or eventually negative. IfXtis eventually negative, then there is at3 ∈ t∗,∞T such thatXt<0 fort∈t3,∞T. So,
xt t
Δ
txΔt−xt
tσt −Xt
tσt >0, t∈t3,∞T, 2.18 which implies that xt/t is strictly increasing for t ∈ t3,∞T. Pickt4 ∈ t3,∞T so that δit≥δit∗fort∈t4,∞T, i 1,2. Then
xδit
δit ≥ xδit∗
δit∗ : di>0, 2.19
so thatxδit≥diδitfort∈t4,∞T. By2.8, we have
rt xΔtγ
−rt4
xΔt4γ
<− t
t4
q1sxδ1sαq2sxδ2sβ
Δs, 2.20
which implies that
rt4
xΔt4γ
> rt xΔtγ
t
t4
q1sxδ1sαq2sxδ2sβ Δs
≥d1α
t
t4
q1sδ1sαΔsd2β
t
t4
q2sδ2sβΔs,
2.21
which contradicts2.7. Hence, without loss of generality, there is at∗≥t0such thatXt>0, that is,xt> txΔtfort≥t∗. Consequently,
xt t
Δ
txΔt−xt
tσt −Xt
tσt <0, t≥t∗, 2.22
and we have thatxt/tis strictly decreasing fort≥t∗.
If there exists at4 ≥t1such that Caseiiholds, then limt→ ∞xtexists, limt→ ∞xt l ≤ 0,and we claim that limt→ ∞xt 0.Otherwise, limt→ ∞xt < 0.We can choose some positive integerk0such thatck≥t4,fork≥k0.Thus, we obtain
yck≤p0yτck p0yck−1≤p20yτck−1
p20yck−2≤ · · · ≤pk−k0 0yτck01 pk−k0 0yck0, 2.23 which implies that limk→ ∞yck 0,and limk→ ∞xck 0,which contradicts limt→ ∞xt
<0.
Now, we assert thatytis bounded. If it is not true, there exists{tk}withtk → ∞as k → ∞such that
ytk sup
t0≤s≤tk
ys, lim
k→ ∞ytk ∞. 2.24
Fromτt≤tand
xtk ytk ptkyτtk≥ 1−p0
ytk, 2.25
which implies that limk→ ∞xtk ∞,it contradicts the existence of limt→ ∞xt.Therefore, we can assume that
lim sup
t→ ∞ yt y1, lim inf
t→ ∞ yt y2. 2.26
By−1< p <0,we get
y1py1≤0≤y2py2, 2.27
thusy1 ≤y2,andy1 y2.Hence, limt→ ∞yt 0.The proof is complete.
Theorem 2.4. Assume that2.5holds, 0≤pt<1,
t→ ∞limsup
⎡
⎣t 1
rt ∞
t
q2s
1−pδ2sβ δ2s
s β
Δs 1/γ⎤
⎦ ∞. 2.28
Then1.7is oscillatory ont0,∞T.
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is eventually positive. We shall consider only this case, since the proof whenytis eventually negative is similar. In view of Lemmas 2.1and 2.2, there exists a t∗ ∈ t0,∞T such that xt>0, xΔt≥0, xΔΔt<0, xt≥txΔt, andxt/tis strictly decreasing fort∈t∗,∞T.
From2.4we have forT ≥t, T, t∈t∗,∞T, T
t
q2s
1−pδ2sβ
xδ2sβΔs≤ − T
t
rs
xΔsγΔ
Δs
rt
xΔtγ
−rT
xΔTγ
≤rt xΔtγ
,
2.29
and hence
1 rt
∞
t
q2s
1−pδ2sβ
xδ2sβΔs 1/γ
≤xΔt. 2.30
So,
xt≥txΔt≥t 1
rt ∞
t
q2s
1−pδ2sβ
xδ2sβΔs 1/γ
≥t 1
rt ∞
t
q2s
1−pδ2sβ δ2s
s xs
β Δs
1/γ
≥xβ/γtt 1
rt ∞
t
q2s
1−pδ2sβ δ2s
s β
Δs 1/γ
.
2.31
So,
t 1
rt ∞
t
q2s
1−pδ2sβ δ2s
s β
Δs 1/γ
≤ 1
xt
β/γ−1
. 2.32
Now note thatβ/γ >1 imply
t 1
rt ∞
t
q2s
1−pδ2sβ δ2s
s β
Δs 1/γ
≤ 1
xt∗
β/γ−1
. 2.33
This contradicts2.28. The proof is complete.
Remark 2.5. Theorem2.4includes results of Agarwal et al.21, Theorem 4.4and Han et al.
25, Theorem 3.1.
Theorem 2.6. Assume that the conditionHand2.7hold:
lim sup
t→ ∞
⎡
⎣t 1
rt ∞
t
q2s δ2s
s β
Δs 1/γ⎤
⎦ ∞. 2.34
Then every solution of 1.7either oscillates or tends to zero ast → ∞.
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is eventually positive. In view of Lemma2.3, either limt→ ∞yt 0 or there exists at∗∈t0,∞T such thatxt> 0, xΔt≥ 0, xΔΔt< 0, xt≥ txΔt,andxt/tis strictly decreasing for t∈t∗,∞T.
Then from2.8, we have forT ≥t, T, t∈t∗,∞T, T
t
q2sxδ2sβΔs≤ − T
t
rs
xΔsγΔ
Δs
rt
xΔtγ
−rT
xΔTγ
≤rt xΔtγ
,
2.35
Since the rest of the proof is similar to Theorem 2.4, so we omit the detail. The proof is complete.
Theorem 2.7. Assume that2.5holds. 0≤pt<1, letρ, η∈Crd1 t0,∞T,R,
lim sup
t→ ∞
t
t0
⎡
⎣ρσs
Q1s−ηΔ
−ρΔη− rs γ1γ1
ρΔs
γ1 ρσsγ
σs s
γ⎤
⎦Δs ∞, 2.36
then1.7is oscillatory ont0,∞T, whereρΔs max{ρΔs,0}.
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is eventually positive. In view of Lemmas 2.1and 2.2, there exists at∗ ∈ t0,∞T such that xt>0, xΔt≥0, xΔΔt< 0, xt≥ txΔt, andxt/tis strictlydecreasing fort∈t∗,∞T. Define the functionωtby
ωt ρt
⎡
⎣rtxΔtγ−1xΔt
xγt ηt
⎤
⎦, t∈t∗,∞T. 2.37
We get
ωΔt ρΔt
ρt ωt ρσt
⎡
⎢⎣ rt
xΔtγΔ
xσtγ −rt
xΔtγxγtΔ
xγtxσtγ ηΔt
⎤
⎥⎦. 2.38
Ifγ≥1, by2.3, we get
xtγΔ
γxΔt 1
0
hxσt 1−hxtγ−1dh≥γxtγ−1xΔt, 2.39
So, from2.4we have
ωΔt≤ ρΔt
ρt ωt−ρσtq1t
1−pδ1tαxδ1tα xσtγ
−ρσtq2t
1−pδ2tβxδ2tβ
xσtγ −γρσtrt
xΔtγ1
xtxσtγ ρσtηΔt.
2.40
By Young’s inequality1.14, we obtain that β−γ
β−αq1t
1−pδ1tαxδ1tα
xσtγ γ−α β−αq2t
1−pδ2tβxδ2tβ xσtγ
≥ q1t
1−pδ1tαxδ1tα xσtγ
β−γ/β−α q2t
1−pδ2tβxδ2tβ xσtγ
γ−α/β−α
q1t
1−pδ1tαβ−γ/β−α q2t
1−pδ2tβγ−α/β−α
×
xδ1tα xσtγ
β−γ/β−α
xδ2tβ xσtγ
γ−α/β−α
≥ q1t
1−pδ1tαβ−γ/β−α q2t
1−pδ2tβγ−α/β−αxδt xσt
γ .
2.41
Fromxt/tbeing strictly decreasing,xδt/xσt≥ δt/σt, xt/xσt≥ t/σt, by 2.37and2.40, we get that
ωΔt≤ ρΔt
ρt ωt−μρσt
q1t
1−pδ1tαβ−γ/β−α q2t
1−pδ2tβγ−α/β−α
× δt
σt γ
− γρσt 1 rt1/γ
t σt
γωt
ρt −ηt
γ1/γ
ρσtηΔt, 2.42
that is,
ωΔt≤ −ρσt
Q1t−ηΔt
ρΔt
ρt ωt−γρσt 1
rt1/γ t
σt γ
ωt ρt −ηt
γ1/γ. 2.43
So,
ωΔt≤ −ρσt
Q1t−ηΔt
ρΔtηt ρΔt
ωt
ρt −ηt
−γρσt 1 rt1/γ
t σt
γ ωt
ρt −ηt γ1/γ.
2.44
Using the inequality1.15we have
ωΔt≤ −ρσt
Q1t−ηΔt
ρΔtηt rt γ1γ1
ρΔt
γ1 ρσtγ
σt t
γ2
. 2.45
Integrating the inequality above fromt∗totwe obtain ωt−ωt∗
≤ − t
t∗
⎛
⎝ρσs
Q1s−ηΔs
−ρΔsηs− rs γ1γ1
ρΔs
γ1 ρσsγ
σs s
γ2⎞
⎠Δs.
2.46
Therefore, t
t∗
⎛
⎝ρσs
Q1s−ηΔs
−ρΔsηs− rs γ1γ1
ρΔs
γ1 ρσsγ
σs s
γ2⎞
⎠Δs
≤ωt∗−ωt≤ωt∗,
2.47
which contradicts2.36.
If 0< γ <1, proceeding as the proof of above, we have2.37and2.38. By2.3, we get that
xtγΔ
γxΔt 1
0
hxσt 1−hxtγ−1dh≥γxσtγ−1xΔt. 2.48
So, from2.4we have
ωΔt≤ ρΔt
ρt ωt−ρσtq1t
1−pδ1tαxδ1tα xσtγ
−ρσtq2t
1−pδ2tβxδ2tβ
xσtγ −γρσtrt
xΔtγ1
xσtxγt ρσtηΔt.
2.49
Since the rest of the proof is similar to that of above, so we omit the detail. The proof is complete.
Theorem 2.8. Assume that the conditionHand2.7hold, letρ, η∈C1rdt0,∞T,R,
lim sup
t→ ∞
t
t0
⎡
⎣ρσs
Q2s−ηΔ
−ρΔη− rs γ1γ1
ρΔs
γ1 ρσsγ
σs s
γ⎤
⎦Δs ∞, 2.50
then every solution of 1.7 either oscillates or tends to zero as t → ∞, where ρΔs max{ρΔs,0}.
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is eventually positive. In view of Lemma2.3, either limt→ ∞yt 0 or there exists at∗∈t0,∞T such thatxt > 0, xΔt ≥ 0, xΔΔt < 0, xt ≥ txΔt,and xt/t is strictly decreasing for t∈t∗,∞T.
Since the rest of the proof is similar to Theorem2.7, so we omit the detail. The proof is complete.
Theorem 2.9. Assume that2.5holds. 0≤pt<1, letρ, η∈Crd1 t0,∞T,R,
lim sup
t→ ∞
1 Hσt, t0
× t
t0
Hσt, σsρs
Θ1t, s− rs γ1γ1
ρs ρσs
γσs s
γ
|λt, s|γ1
Δs ∞, 2.51
then1.7is oscillatory ont0,∞T.
Proof. We prove only caseγ≥1. The proof of case 0< γ <1 is similar. Proceeding as the proof of Theorem2.7, we have2.37and2.43. Replacingtin2.43bys, then multiplying2.43 byHσt, σs, and integrating fromT tot,t > T, t, T∈t∗,∞T, we have
t
T
Hσt, σsρσs
Q1s−ηΔs Δs
≤ − t
T
Hσt, σsωΔsΔs
t
T
Hσt, σsρΔsωs
ρsΔs
−γ t
T
Hσt, σsρσs 1
rs1/γ s
σs
γωs
ρs −ηs
γ1/γ Δs.
2.52
Integrating by parts and using the fact thatHσt, t 0, we get t
T
Hσt, σsωΔsΔs −Hσt, TωT−
t
T
HΔsσt, sωsΔs. 2.53
So, t
T
Hσt, σsρσt
Q1s−ηΔs
Δs≤Hσt, TωT
− t
T
hσt, sHσt, σsωsΔs
t
T
Hσt, σsρΔsωs
ρsΔs
−γ t
T
Hσt, σsρσs 1
rs1/γ s
σs
γωs
ρs −ηs
γ1/γ Δs,
2.54
that is, t
T
Hσt, σsρσs
Q1s−ηΔs
Δs≤Hσt, TωT
− t
T
hσt, s− ρΔs
ρs Hσt, σsωsΔs
−γ t
T
Hσt, σsρσs 1
rs1/γ s
σs
γωs
ρs −ηs
γ1/γ Δs.
2.55
Hence, t
T
Hσt, σsρtΘ1t, sΔs≤Hσt, TωT
t
T
Hσt, σsρs|λt, s|
ωs
ρs −ηs Δs
−γ t
T
Hσt, σsρσs 1
rs1/γ s
σs γ
ωs
ρs −ηs
γ1/γΔs.
2.56
Using the inequality1.15we have
1
Hσt, T
× t
T
Hσt, σsρs
Θ1t, s− rs γ1γ1
ρs ρσs
γσs s
γ2
|λt, s|γ1
Δs≤ωT. 2.57
SetT t∗, so,
lim sup
t→ ∞
1 Hσt, t∗
× t
t∗
Hσt, σsρs
Θ1t, s− rs γ1γ1
ρs ρσs
γσs s
γ2
|λt, s|γ1
Δs≤ωt∗. 2.58
This contradicts2.51and finishes the proof.
Theorem 2.10. Assume that the conditionHand2.7hold, letρ, η∈C1rdt0,∞T,R,
lim sup
t→ ∞
1 Hσt, t0
× t
t0
Hσt, σsρs
Θ2t, s− rs γ1γ1
ρs ρσs
γσs s
γ
|λt, s|γ1
Δs ∞, 2.59
then every solution of 1.7either oscillates or tends to zero ast → ∞.
Proof. Suppose that 1.7 has a nonoscillatory solution yt. We may assume that yt is eventually positive. In view of Lemma2.3, either limt→ ∞yt 0 or there exists at∗∈t0,∞T such thatxt > 0, xΔt ≥ 0, xΔΔt < 0, xt ≥ txΔt,and xt/t is strictly decreasing for t∈t∗,∞T.
Since the rest of the proof is similar to Theorem2.9, so we omit the detail. The proof is complete.
Following the procedure of the proof of Theorem2.7, we can also prove the following theorem.
Theorem 2.11. Assume that2.5holds. 0≤pt<1, let H∈R, ρ, η∈C1rdt0,∞T,R,
0<inf
s≥t0
t→ ∞liminf Hσt, s Hσt, t0
≤ ∞, 2.60
lim sup
t→ ∞
1 Hσt, t0
t
t0
Hσt, σsρsrs
ρs ρσs
γσs s
γ
|λt, s|γ1Δs <∞, 2.61
there existsϕ∈Crdt0,∞T,Rsuch that
∞
t0
ρσs 1
rs1/γ s
σs
γϕs
ρs−ηs
γ1/γ
Δs ∞, ifγ≥1, 2.62
or
∞
t0
ρσs 1
rs1/γ s σs
ϕs
ρs−ηs
γ1/γ
Δs ∞, if 0< γ <1, 2.63
and for anyT∈t0,∞T,
lim sup
t→ ∞
1
Hσt, T
× t
T
Hσt, σsρs
Θ1t, s− rs γ1γ1
ρs ρσs
γσs s
γ
|λt, s|γ1
Δs≥ϕT, 2.64
and then1.7is oscillatory ont0,∞T, whereϕt max{ϕt,0}.
Proof. We prove only caseγ≥1. The proof of case 0< γ <1 is similar. Proceeding as the proof of Theorem2.9, we have2.56and2.57. So we have for allt > T, t, T∈t∗,∞T,
lim sup
t→ ∞
1
Hσt, T
× t
T
Hσt, σsρs
Θ1t, s− rs γ1γ1
ρs ρσs
γσs s
γ2
|λt, s|γ1
Δs≤ωT. 2.65
By2.64, we obtain
ϕT≤ωT, T∈t∗,∞T. 2.66
Define
Pt 1
Hσt, t∗ t
t∗
Hσt, σsρs
λt, s%%
%%ωs
ρs −ηs
Δs,
Qt γ
Hσt, t∗ t
t∗
Hσt, σsρσs s
σs γ
1 rs1/γ
ωs
ρs −ηs
γ1/γΔs.
2.67
Then, by2.56and2.64, we have that lim inf
t→ ∞ Qt−Pt≤ωt∗
−lim sup
t→ ∞
1 Hσt, t∗
t
t∗
Hσt, σsρsΘ1t, sΔs≤ωt∗−ϕt∗<∞. 2.68
We claim that t
t∗
ρσs s
σs γ
1 rs1/γ
ωs
ρs −ηs
γ1/γΔs <∞. 2.69
Suppose, to the contrary, that t
t∗
ρσs s
σs γ
1 rs1/γ
ωs
ρs −ηs
γ1/γΔs ∞. 2.70
By2.60, there exists a positive constantk1such that
s≥tinf0
lim inf
t→ ∞
Hσt, s Hσt, t0
≥k1. 2.71
Letk2 be an arbitrary positive number, then it follows from2.70 that there exists aT1 ∈ t∗,∞Tsuch that
t
t∗
ρσs s
σs γ
1 rs1/γ
ωs
ρs −ηs
γ1/γΔs≥ k2
γk1, t∈T1,∞T. 2.72
Hence,
Qt γ
Hσt, t∗
× t
t∗
Hσt, σs s
t∗
ρστ
τ στ
γ 1 rτ1/γ
ωτ
ρτ −ητ γ1/γΔτ
Δ Δs
γ Hσt, t∗
× t
t∗
−HΔsσt, ss
t∗
ρστ τ
στ
γ 1 rτ1/γ
ωτ
ρτ −ητ
γ1/γΔτ
Δs
≥ γ
Hσt, t∗
× t
T1
−HΔsσt, ss
t∗
ρστ τ
στ
γ 1 rτ1/γ
ωτ
ρτ −ητ
γ1/γΔτ
Δs
≥ k2 k1
1 Hσt, t∗
t
T1
−HΔsσt, s
Δs k2 k1
Hσt, T1 Hσt, t∗.
2.73
By 2.71, there exists a T2 ∈ T1,∞T such that Hσt, T1/Hσt, t∗ ≥ k1, t ∈ T2,∞T, which implies thatQt≥k2. Sincek2is arbitrary, then
tlim→ ∞Qt ∞. 2.74
In view of2.68, we consider a sequence{Tn}∞n 1⊂t∗,∞Twith limn→ ∞Tn ∞satisfying
nlim→ ∞QTn−PTn lim inf
t→ ∞ Qt−Pt<∞. 2.75
Then, there exists a constantMsuch thatQTn−PTn≤Mfor all sufficiently largen∈N.
Since2.70ensures that
n→ ∞limQTn ∞, lim
n→ ∞PTn ∞ 2.76
so,
PTn
QTn −1≥ −M QTn >−1
2 or PTn QTn ≥ 1
2 2.77
hold for all sufficiently largen∈N. Therefore,
nlim→ ∞
Pγ1Tn
QγTn ∞. 2.78
On the other hand, from the definition ofPwe can obtain, by H ¨older’s inequality,
PTn≤ 1
HσTn, t∗
× Tn
t∗
HσTn, σsρσs s
σs
γ 1 rs1/γ
ωs
ρs −ηs γ1/γΔs
γ/γ1
× Tn
t∗
HσTn, σsρs
ρs ρσs
γσs s
γ2
rs|λTn, s|γ1Δs
1/γ1
, 2.79
and, accordingly,
Pγ1Tn QγTn ≤ 1
γγ 1 HσTn, t∗
Tn
t∗
HσTn, σsρs
ρs ρσs
γσs s
γ2
rs|λTn, s|γ1Δs.
2.80
So, because of2.78, we get
nlim→ ∞
1 γγ
1 HσTn, t∗
Tn
t∗
HσTn, σsρs
ρs ρσs
γσs s
γ2
rs|λTn, s|γ1Δs ∞, 2.81
