# Oscillation Theorems for Second-Order Damped Nonlinear Differential Equations

Authors: Hui-Zeng Qin; Yongsheng Ren
Journal: International Journal of Differential Equations
 (2009)

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## Abstract

We present new oscillation criteria for the differential equation of the form[r(t)U(t)]′+p(t)k2(x(t),x′(t))|x(t)|νU(t)+q(t)ϕ(x(g1(t)),x′(g2(t)))f(x(t))=0, where U(t)=k1(x(t),x′(t))|x′(t)|α−1x′(t), α≤β,ν=(β−α)/(α+1). Our research is different from most known ones in the sense that H function is not employed in our results, though Riccati's substitution and its generalized forms are used. Our criteria which  are established under quite general assumptions are an extension for previous results. In particular, by taking β=α, the above-mentioned equation can be reduced into the various types of equations concerned by people currently.


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## Body

## 1. Introduction

The existence of the oscillatory solutions of the nonlinear  differential equation with damping,(1.1)[r(t)x′(t)]′+p(t)x′(t)+q(t)g(x(t))=0,t≥t0,
has received considerable  attention from researchers for a long time.People previously focused  on the casesr(t)>0,p(t)≥0,q(t)>0. In recent years, people concerned that r(t)>0,p(t),q(t) may change sign for t∈[t0,∞), regarding work in this area can be seen in literature [1–4].Recently, Li [5] has extended (1.1) to more general equations of the form(1.2)(r(t)x′(t))′+p(t)x′(t)+q(t)g(x(t))f(x′(t))=0,t≥t0.
Yamaoka [6] has studied the following class of particular equation:(1.3)(|x′(t)|p-2x′(t))′+2p-1t|x′(t)|p-2x′(t)+q(t)g(x(t))=0,t≥t0,p>1.
Tiryaki and Zafer [1] and other authors [2, 7] have considered the following equation of the form(1.4)(r(t)h(x(t))x′(t))′+p(t)x′(t)+q(t)g(x(t))=0.
Zheng [8] has discussed the oscillation problem for the following equation:(1.5)(r(t)h(x(t))ψ(x′(t)))′+p(t)ψ(x′(t))+q(t)g(x(t))=0.It is worth noting that (1.1), (1.2), and (1.3) can transform into an undamping equation. For example, the equation(1.6)(r(t)|x′(t)|α-1x′(t))′+p(t)|x′(t)|α-1x′(t)+q(t)g(x(t))=0
can transform into the undamping equation(1.7)(r̃(t)|x′(t)|α-1x′(t))′+q̃(t)g(x(t))f(x′(t))=0,t≥t0,
where r̃(t)=r(t)e∫Tt‍(p(s)/r(s))ds,q̃(t)=q(t)e∫Tt‍(p(s)/r(s))ds. Although (1.4) and (1.5) can not be transformed into the undamping equation, but from the conditions 0<c≤h(x)≤c1<∞ given by [1, 8, 9], if h(t) is changed into c or c1, the above-mentioned equations consistent with (1.6). This shows that under the above conditions, there is no essential difference between (1.4), (1.5), and the undamping equation. We note that the condition ∫t0∞‍(1/r(s))1/αds=∞ must be used for (1.5); however, at this point the condition ∫t0∞‍(1/r̃(s))1/αds=∞ cannot be guaranteed.We have removed the condition∫t0∞‍(1/r(s))1/αds=∞, considered the oscillation problem for the following equation:
(1.8)[r(t)χ(x′(t))]′+q(t)ϕ(x(g1(t)),x′(g2(t)))f(x(t))=0,
applied the results to the above-mentioned equation, and obtained a very good result.In this paper, we consider the oscillatory behavior of the following differential equation of the form:(1.9)[r(t)U(t)]′+p(t)k2(x(t),x′(t))|x(t)|(β-α)/(α+1)U(t)+q(t)ϕ(x(g1(t)),x′(g2(t)))f(x(t))=0,
where U(t)=k1(x(t),x′(t))|x′(t)|α-1x′(t).Today, Riccati transformation, and its generalized forms are one of the most effective method in the oscillatory theory of nonlinear differential equations. Most obvious merits of Riccati's approach is thatq(t) may change sign in (1.8). For getting the more general results [10, 11], a lot of authors have introduced to a class of Y function(1.10)Y={Φ∈C1(E,R)∣Φ(t,t,l)=Φ(t,l,l)=0,Φ(t,s,l)≠0,l<s<t},E={(t,s,l)∣t0≤l≤s≤t<∞},
where ∂Φ/∂s exists on E and is integral with respect to s. By using this method, peoples have obtained some general results, but its shortcoming is that the property of q(t) can be weakened as t→∞. We use the method similar to [4], that is, replace the above-mentioned function Φ(t,s,l) with ρ∈C1([t0,∞),R+). Perhaps the reason that people like to use this method is that integrating by parts with respect to s on [l,t] can employ Φ(t,t,l)=Φ(t,l,l)=0.For (1.9), we make the following assumptions:(A)
xf(x)>0,x≠0,f′(x)|f(x)|(1-β)/β≥C1,α,β>0,ϕ∈C(R2,R+),0<nϕ≤ϕ(x,y)≤Nϕ;(B)
0<C0≤k1(u,v)≤C2,0≤C3≤k2(u,v)≤C4;(C)
r∈C1(I,R+),gi∈C(I,R+),0≤gi′(t),i=1,2,I=[t0,∞),p,q∈C([t0,∞),R).In the paper, a solution of (1.9) is called oscillatory if it has zeros unbounded set. If the solutions are oscillatory, (1.9) is called to be oscillatory equation.


## 2. Main Theorem

We establish some lemmas which are useful in our discussions.Lemma 2.1.
Leta,b,γ,λ>0, then
(2.1)at-bt1+1/γ≤γγ(γ+1)γ+1aγ+1bγ,(2.2)at-λ+btγ≥(λ+γ)λ-λ/(λ+γ)γ-γ/(λ+γ)aγ/(λ+γ)bλ/(λ+γ).Lemma2.1 can easily be proved by using the extremum of one variable function. For the sake of convenience, we denote(2.3)q-(t)=min{q(t),0},q+(t)=max{q(t),0},q̃(t)=nϕq+(s)+Nϕq-(s),p1(t)=C4p+(t)+C3p-(t),p2(t)=C3p+(t)+C4p-(t).Theorem 2.2.
Assume thatα=β holds and there exists ρ∈C1([t0,∞),R+),∀T∈[t0,∞), such that
(2.4)∫Tt‍(q̃(s)-C2ααr(s)C1α(α+1)α+1|ρ′(s)ρ(s)|α+1)e∫t0s‍(pi(τ)/r(τ))dτρ(s)ds→∞,t→∞,i=1,2,(2.5)limt→∞Q(t)=∫Tt‍q̃(s)e∫t0s‍(p1(τ)/r(τ))dτds>0.
If any one of the following two conditions holds, then the solution x=x(t) of (1.9) is oscillatory.
(1)(2.6)R(t)=∫t0t‍r-1/α(s)e-1/α∫t0s(‍p1(τ)/r(τ))dτds→∞,t→∞.
(2)R(∞)<∞, and
(2.7)∫Tt‍(Q(s)r-1(s)e-∫t0s(‍p1(τ)/r(τ))dτ)1/αds=∫Tt‍(r-1(s)∫Ts‍q̃(ς)e-∫ςs(‍p1(τ)/r(τ))dτdς)1/αds→∞,t→∞,(2.8)limt→∞Q1/α(t)(R(∞)-R(t))≥αC1C21/α.Proof.
Letx=x(t) be a nonoscillatory solution of (1.9). Then, there exists T≥t0 such that x=x(t)≠0,t∈(T,∞). Without loss of generality, we may assume that x=x(t)≥0,t∈(T,∞).
Define the Riccati Transformation by(2.9)W(t)=ρ(t)r(t)k1(x(t),x′(t))|x′(t)|α-1x′(t)f(x(t)),t≥T.
From conditions (A) and (B), we have
(2.10)|W(t)|1/α|x(t)|C01/α(ρ(t)r(t))1/α≥|x′(t)|≥|W(t)|1/α|x(t)|C21/α(ρ(t)r(t))1/α,|f(x)|≥(C1α)α|x|α,t≥T.
Differentiating W(t), and applying (1.9) and (2.10), we have
(2.11)W′(t)=ρ′(t)ρ(t)W(t)-p(t)k2(x(t),x′(t))r(t)W(t)-ρ(t)q(t)|f(x(t))||x(t)|αϕ(x(g1(t)),x′(g2(t)))-αW(t)x′(t)x(t)≤{(ρ′(t)ρ(t)-p1(t)r(t))W(t)-(C1α)αρ(t)q̃(t)-α|W(t)|(α+1)/αC21/α(ρ(t)r(t))1/α,W(t)<0,(ρ′(t)ρ(t)-p2(t)r(t))W(t)-(C1α)αρ(t)q̃(t)-α|W(t)|(α+1)/αC21/α(ρ(t)r(t))1/α,W(t)>0.
By (2.1), we have
(2.12)W′(t)≤{-((C1α)αρ(t)q̃(t)-C2r(t)(α+1)α+1(ρ′(t)ρ(t)-p1(t)r(t))α+1)ρ(t),W(t)<0,-((C1α)αρ(t)q̃(t)-C2r(t)(α+1)α+1(ρ′(t)ρ(t)-p2(t)r(t))α+1)ρ(t),W(t)>0.
Integrating the above inequality from T to t≥T, we have
(2.13)W(t)≤W(T)-∫Tt‍((C1α)αq̃(s)-C2r(s)(α+1)α+1(ρ′(s)ρ(s)-pi(s)r(s))α+1)ρ(s),i=1,2.
Condition (2.4) shows that limt→∞W(t)=-∞. Without loss of generality, we may assume that W(t)<0,t≥T, by applying (2.11), we have
(2.14)q̃(t)e∫t0t(‍p1(s)/r(s))ds+C1C2-1/αr-1/α(t)e-(1/α)∫t0t(‍p1(s)/r(s))dsZ(α+1)/α(t)≤Z′(t),Z(t)=-W(t)ρ(t)e∫t0t‍(p1(s)/r(s))ds>0,t≥T.
Integrating the above inequality fromT to t≥T, we obtain
(2.15)Z(T)+Q(t)+C1C2-1/α∫Tt‍r-1/α(τ)e-(1/α)∫t0τ‍(p1(s)/r(s))dsZ(α+1)/α(τ)dτ≤Z(t).
Let
(2.16)F(t)=∫Tt‍r-1/α(τ)e-(1/α)∫t0τ‍(p1(s)/r(s))dsZ(α+1)/α(s)ds>0,F′(t)=r-1/α(t)e-(1/α)∫t0t‍(p1(s)/r(s))dsZ(α+1)/α(t),t>T.
We will discuss in the following two cases.
(1) By (2.1) and (2.5), we see that
(2.17)C1(α+1)/αC2-(α+1)/α2r-1/α(t)e-(1/α)∫t0t(‍p1(s)/r(s))ds≤F′(t)F-(α+1)/α(t),t≥T1>T.
Integrating the above inequality from T1 to t≥T1, we have
(2.18)αF-1/α(T1)>αF-1/α(T1)-αF-1/α(t)>C1(α+1)/αC2-(α+1)/α2∫T1t‍r-1/α(s)e-(1/α)∫t0s(‍p1(τ)/r(τ))dτds→∞,t→∞.
But, it is impossible that the above inequality holds.
(2) Observe thatW(t)<0 and by (2.8), we have x′(t)<0, so that x=x(t) is monotonic decreasing function for t≥T, and if limt→∞x(t)=c, then c=0. Otherwise, if c>0, by (2.15) and (2.9), we have
(2.19)Q(t)f(c)≤Z(t)f(c)≤Z(t)f(x(t))=-e∫t0τ‍(p1(s)/r(s))dsr(t)k1(x(t),x′(t))|x′(t)|α,t≥T.
By condition (B), we have
(2.20)-x′(t)≥C2-1/αf1/α(c)Q1/α(t)r-1/α(t)e-(1/α)∫t0t(‍p1(s)/r(s))ds,t≥T.
Integrating the above inequality from T1 to t≥T1  leads to
(2.21)x(T)-c≥x(T)-x(t)≥C2-1/αf1/α(c)∫Tt‍Q1/α(s)r-1/α(s)e-(1/α)∫t0τ‍(p1(s)/r(s))dsds→∞,t→∞.
But, this is impossible. We choose ρ(t)=1, thus (2.15) has the following form:
(2.22)Z(T)+Q(t)+C1C2-1/α∫Tt‍r-1/α(τ)e-(1/α)∫t0τ‍(p1(s)/r(s))dsZ(α+1)/α(τ)dτ≤Z(t),Z(t)=-e∫t0t‍(p1(s)/r(s))dsW(t).
When∫Tt‍r-1/α(τ)e-(1/α)∫t0τ‍(p1(s)/r(s))dsZ(α+1)/α(τ)dτ<Mα+1, by considering Hölder′s inequality and (2.10), we have
(2.23)-lnx(t)x(T)=∫Tt‍-x′(s)x-1(s)ds≤(∫Tt‍r(s)e-∫t0τ‍(p1(s)/r(s))ds|x′(s)|α+1xα+1(s)ds)1/(α+1)(∫Tt‍r-1/α(s)e-(1/α)∫t0τ(‍p1(s)/r(s))dsds)α/(α+1)≤C0-(α+1)/α(∫tNt‍r-1/α(s)e-(1/α)∫t0τ(‍p1(s)/r(s))dsZ(α+1)/α(s)ds)1/(α+1)(R(t)-R(T))α/(α+1)≤MC0-(α+1)/αRα/(α+1)(t),
such that MC0-(α+1)/αRα/(α+1)(∞)≥MC0-(α+1)/αRα/(α+1)(t)≥-ln(x(t)/x(T))→∞,t→∞. However, this is also impossible.
If∫Tt‍r-1/α(τ)e-(1/α)∫t0τ(‍p1(s)/r(s))dsZ(α+1)/α(τ)dτ→∞,t→∞, then by (2.8), we see that limt→∞Q(t)=∞. From (2.15), we obtain
(2.24)Z(t)≥Z(T)+Q(T2)+C1C2-1/αF(t),t≥T2≥T,C1C2-1/αF′(t)(Z(T)+Q(T2)+C1C2-1/αF(t))-(α+1)/α≥C1C2-1/αr-1/α(t)e-(1/α)∫t0t‍(p1(s)/r(s))ds,t≥T2.
Integrating the above inequality from T2 to t≥T2, we have
(2.25)αQ-1/α(T2)>α(Z(T)+Q(T2)+C1C2-1/αF(T2))-1/α≥C1C2-1/α(R(∞)-R(T2)),T2≥T.
Let T2→∞; the above inequality contradicts (2.8); this completes the proof.Theorem 2.3.
Suppose thatα≤β and
(2.26)∫t0∞‍((C1β)βq̃(s)-C2αα|p1(s)|α+1(α+1)α+1βαrα(s))ds=∞,i=1,2.
If there exists 0<ε<((α+1)β/α)C2-1/α,t0≤t1<t2<⋯<tn<⋯→∞, such that
(2.27)∫tnt‍θ(s)ds>0,t>tn,limt→∞(∫tnt‍θ(s)ds)1/α(R(∞)-R(t))(β-α)/λ(α+1)+1=∞,n=1,2,…,
where
(2.28)θ(t)=(C1β)βq̃(t)-1α+1|p1(t)|α+1εαrα(t),R(t)=∫t0t‍r-1/α(s)ds,Cε=(βC2-1/α-εαα+1),δ=(C0C2)(α+1)/α,λ=β(δ-1)α+1-εδα+1,
then every solution of (1.9) is oscillatory.Note
From (2.27), it is easy to obtain the following equation:
(2.29)limt→∞(∫tnt‍θ(sds))1/α(R(∞)-R(t))=∞,limt→∞(∫tnt‍θ(s)ds)1/αlnR(∞)R(t)=∞,n=1,2,….Proof.
Letx=x(t) be a nonoscillatory solution of (1.9). Then, there exists T≥t0 such that x=x(t)≠0,t>T. We may assume that x=x(t)>0,t>T.
Introduce the Riccati transformationW(t)=r(t)U(t)/x(t)β,t≥T. From conditions (A) and (B), we have
(2.30)(W(t)xβ(t)C0r(t))1/α≥|x′(t)|≥(W(t)xβ(t)C2r(t))1/α,|f(x)|≥(C1β)β|x|β,t≥T.
Differentiating W(t), and applying (1.9) and the above inequality, leads to
(2.31)W′(t)=-p(t)k2(x(t),x′(t))x(β-α)/(α+1)(t)W(t)r(t)-q(t)ϕ(x(g1(t)),x′(g2(t)))xβ(t)-βW(t)x′(t)x(t)≤{-p1(t)r(t)x(β-α)/(α+1)(t)W(t)-(C1β)βq̃(t)-βC2-1/αr-1/α(t)x(β-α)/α(t)|W(t)|(α+1)/α,W(t)<0-p2(t)r(t)x(β-α)/(α+1)(t)W(t)-(C1β)βq̃(t)-βC2-1/αr-1/α(t)x(β-α)/α(t)W(α+1)/α(t),W(t)>0.
By (2.1), we see that
(2.32)W′(t)≤{-(C1β)βq̃(t)+C2αα|p1(t)|α+1(α+1)α+1βαrα(t),Z′(t)<0-(C1β)βq̃(t)+C2αα|p2(t)|α+1(α+1)α+1βαrα(t),Z′(t)>0.
The following proof is similar to that in Theorem 2.2, using (2.26), we find that limt→∞W(t)=-∞. Thus there exists T1≥T, such that W(t)<0,x′(t)<0,t≥T1. Because x=x(t) is monotonic decreasing function on [T1,∞); hence limt→∞x(t)=c≥0. By (2.31), we have
(2.33)W′(t)≤|p1(t)|r(t)x(β-α)/(α+1)(t)|W(t)|-(C1β)βq̃(t)-βC2-1/αr-1/α(t)x(β-α)/(α+1)(t)|W(t)|(α+1)/α,W(t)<0,t≥T1.
By using of weighted mean inequality, we can transform the above inequality into(2.34)W′(t)≤-θ(t)-Cεr-1/α(t)|x(β-α)/(α+1)(t)W(t)|(α+1)/α,W(t)<0.
We need to show that c=0. Otherwise, if c>0,  by the above inequality, we have
(2.35)W′(t)≤-θ(t)-Cεc(β-α)/(α+1)r-1/α(t)|W(t)|(α+1)/α,W(t)<0.
By choosing N≥1, such that tN≥T1, integrating the above inequality from TN to t≥TN and by (2.27), we can get
(2.36)W(t)≤W(tN)-∫tNtn‍θ(s)ds-Cεc(β-α)/(α+1)∫tNt‍r1/α(s)|W(s)|(α+1)/αds,t≥tn,n>N,W(t)<0,
or
(2.37)Cεc(β-α)/(α+1)r-1/α(t)|W(t)|(α+1)/α(|W(tN)|+∫tNtn‍θ(s)ds+Cεc(β-α)/(α+1)∫tNt‍r-1/α(s)|W(s)|(α+1)/αds)(α+1)/α≥Cεc(β-α)/(α+1)r-1/α(t),t≥tn,n>N.
Differentiating the above inequality on the interval [tn,∞], we have
(2.38)α(|W(tN)|+∫tNtn‍θ(s)ds+Cεc(β-α)/(α+1)∫tNtn‍r-1/α(s)|W(s)|(α+1)/αds)-1/α≥Cεc(β-α)/(α+1)(R(∞)-R(tn)),t>tn,n>N.
This is a contradiction to (2.29); hence, we have limt→∞x(t)=0. According to the above discussion, we have
(2.39)|W(t)|≥|W(tN)|+∫tNtn‍θ(s)ds+Cε∫tNt‍r-1/α(s)|x(β-α)/(α+1)(t)W(s)|(α+1)/αds,t≥tn,n>N,W(t)<0,(2.40)(|W(tN)|+∫tNtn‍θ(s)ds+Cε∫tNtn‍r-1/α(s)|x(β-α)/(α+1)(t)W(s)|(α+1)/αds)-1/α≥Cε∫tn∞‍r-1/α(s)x(β-α)/(α+1)(s)ds,n>N.
We will discuss in the following two cases.
If there existsM>0, such that ∫tNt‍r-1/α(s)x(β-α)/α(s)|W(s)|(α+1)/αds≤Mα+1,t≥tN, by considering Hölder's inequality and (2.39), we have
(2.41)α+1β-α(x(α-β)/(α+1)(t)-x(α-β)/(α+1)(tN))=∫tNt‍-x′(s)x-(β+1)/(α+1)(s)ds≤(∫tNt‍r(s)|x′(s)|α+1xβ+1(s)ds)1/(α+1)(∫tNt‍r-1/α(s)ds)α/(α+1)≤C0-(α+1)/α(∫tNt‍r-1/α(s)x(β-α)/α(s)|W(s)|(α+1)/αds)1/(α+1)(R(t)-R(tN))α/(α+1)≤MC0-(α+1)/αRα/(α+1)(t).
By choosing n>N, such that 2(α+1)/(β-α)x(t)<x(tN),t≥tn,n>N, from the above inequality, we have x(β-α)/(α+1)(t)≥((α+1)C0(α+1)/α/2(β-α)M)R-α/(α+1)(t),t≥tn,n>N. Inserting it in (2.40), we can get
(2.42)(∫tNtn‍θ(s)ds)-1/α≥(|W(tN)|+∫tNtn‍θ(s)ds+∫tNtn‍C̃ε|W(s)|(α+1)/αR(s)r1/α(s)ds)-1/α≥C̃εαlnR(∞)R(tn),C̃ε=Cε((α+1)C0(α+1)/α2(β-α)M)(α+1)/α.
This is contradiction to (2.29).
If∫tN∞‍(x(β-α)/α(s)|W(s)|(α+1)/α/(ρ(s)r(s))1/α)ds=∞ for t≥tn,n>N, along with (2.39) and (2.30), we have
(2.43)C2r(t)|x′(t)|α|x(t)|β≥|W(t)|≥|W(tN)|+CεC0(α+1)/α∫tNt‍r(s)|x′(s)|α+1|x(s)|β+1ds,t≥tn,n>N,
leading  to
(2.44)CεC0(α+1)/αr(t)|x′(t)|α+1|x(t)|β+1(|W(tN)|+CεC0(α+1)/α∫tNt‍r(s)|x′(s)|α+1|x(s)|β+1ds)-1≥-CεC2C0(α+1)/αx′(t)x(t).
Integrating the above inequality from tn to t>tn and by (2.40), we can get
(2.45)lnC2r(t)|x′(t)|α-W(tN)|x(t)|β≥CεC2C0(α+1)/αlnx(tn)x(t),
or
(2.46)-x′(t)x(β/α-ε/(α+1))δ-β/α(t)≥|W(tN)|1/αC2-1/αx(β/α-ε/(α+1))δ(tn)(r(t))-1/α.
Integrating the above inequality on the interval [t,∞), we have
(2.47)x(t)≥C1/λ(R(∞)-R(t))1/λ,C=λ|W(tN)|1/αC2-1/αx(β/α-ε/(α+1))δ(tn).
IfRρ(∞)=∞, the above inequality cannot be satisfied; hence, Rρ(∞)<∞. Inserting it in (2.40),  we can get
(2.48)(β-αλ(α+1)+1)≥CεC(β-α)/λ(α+1)(∫tNtn‍θ(sds))1/α(R(∞)-R(tn))(β-α)/λ(α+1)+1.
This is contradiction to (2.26).
Hence, we complete the proof of Theorem2.3.


## 3. Some Examples

Example 3.1.
Let us consider the oscillatory behavior of the following differential equation:(3.1)(tλ|x′(t)|α-1x′(t))′+ptλ1|x′(t)|α-1x′(t)+qtλ2|x(t)|α-1x(t)ϕ(x(t),x′(t))=0,t≥t0.
Comparing (3.1) with (1.9), we can find that
(3.2)r(t)=tλ,C2=C0=1,p(t)=ptλ1,p>0,C1=1,C3=C4=1,q(t)=qtλ2,β=α,p1(t)=ptλ1,p2(t)=0,q̃(t)=qnϕtλ2.
Letρ(t)=tμ, (2.4)–(2.7) are transformed into the equations
(3.3)∫t0t‍(qnϕ-μα+1(α+1)α+1sλ-λ2-α-1)ep∫t0s‍τλ1-λdτsλ2+μds→∞,t→∞,(3.4)∫Tt‍sλ2ep∫t0s‍τλ1-λdτds>0,(3.5)R(t)=∫t0t‍s-λ/αe-(p/α)∫t0s‍τλ1-λdτds→∞,t→∞,(3.6)∫Tt‍(s-λ∫Ts‍ςλ2e-p∫ςs‍τλ1-λdτdς)1/αds→∞,t→∞,(3.7)limt→∞(∫Tt‍sλ2ep∫t0s‍τλ1-λdτds)1/α(R(∞)-R(t))>αC21/αqnϕC1.
We will discuss inthe following cases.
(1)λ1+1<λ, choosing μ=-min{λ2+1,0}, provided that λ≤α,λ-λ2<α+1, or qnϕ>μα+1/(α+1)α+1 is satisfied for α≥λ=λ2+α+1, then (3.3)–(3.5) hold, and the solution of (3.1) is oscillatory.
(2)λ=λ1+1,p+λ+α≥0, choosing μ=-min{λ2+1,0}, provided that λ≤α, and λ-λ2<α+1 or qnϕ>|μ-p|α+1/(α+1)α+1,α≥λ=λ2+α+1, then (3.3)–(3.5) hold, the solution of (3.1) is oscillatory.
(3)λ=λ1+1,p+λ+α<0, we can see that R(∞)<∞,  choosing μ=-min{λ2+1,0}, and therefore, (3.6) and (3.7) are transformed into the following equation:
(3.8)∫Tt‍(qnϕλ2+p+1(s-λ+λ2+1-s-λ-pTλ2+p+1))1/αds→∞,t→∞,λ2+p+1>0,limt→∞t(λ2-λ+α+1)/α>(λ+p-α)C21/αqnϕC1(λ2+p+1)1/α,λ2+p+1>0,
provided that λ2+p+1>0 and λ<λ2+α+1, or λ=λ2+α+1,((λ+p-α)C21/α/qnϕC1)(λ2+p+1)1/α<1, then (3.3)-(3.4) and (3.6)-(3.7) hold, the solution of (3.1) is oscillatory.
In particular, we choseα=1,p=0,λ=0,λ2=-2,ϕ(x(t),x′(t))=1=nϕ,μ=1,q>1/4, thus the conditions of the case (1) can be satisfied. This is the sufficient condition for all solutions of x′′(t)+(q/t2)x(t)=0 to be oscillatory.
If we chooseα=P-1,p=2(P-1),λ=0,λ1=-1,λ2=-P,ϕ(x,y)≥((P-1)/P)P+λ̃/ln2|x|>((P-1)/P)P=nϕ,λ̃>0,μ=P-1,q≥1, the conditions of the case (2) can be satisfied. Compared with the conditions: q=1,λ̃>(1/2)((P-1)/P)P+1 in [6], our results are more general.Example 3.2.
Let us consider the oscillatory behavior of the following differential equation:(3.9)x′′(t)-sintx′(t)+1+cost1+sin2tx(t)(1+|x(t)|2),t≥0.
Comparing (3.9) with (1.9), we can see that q̃(t)=(1+cost)/(1+sin2t),α=1,r(t)=1,f(x)=x(1+x2),C1=3/23,C0=C2=C3=C4=1,p1(t)=p2(t)=-sint,β=3. Choosing ε=1, clearly, the conditions (2.26) and (2.27) of Theorem 6 can be satisfied. Therefore, we may conclude that (3.9) is oscillatory. Example 3.2 is Example 2 of [5]. It is easy to verify that Example 1 of  [5] also satisfies with Theorem 6.Example 3.3.
Let us consider the oscillatory behavior of the following differential equation:(3.10)[11+t2x′(t)]′-a+bsinttx′(t)+q(t)|x(t)|β-1x(t),t≥0.
Comparing (3.10) with (1.9), we can see that q̃(t)=q(t),α=1,r(t)=1/(1+t2),f(x)=|x|β-1x,C1=β,C0=C2=C3=C4=1,andp1(t)=p2(t)=(a+bsint)/t. Thus, we have R(t)=t-t0+(1/3)(t3-t03)→∞,t→∞, so the second condition of Theorem 2.3 in (2.27) is satisfied. Therefore, another condition is
(3.11)∫t0∞‍(q(s)-(a+bsins)24β(1+1s2))ds=∞,
and there exists 0<ε<2β,t0≤t1<t2<⋯<tn<⋯→∞ such that
(3.12)∫tnt‍(q(s)-(a+bsins)22ε(1+1s2))ds>0,t>tn,n=1,2,….
If only ∫t0∞‍(q(s)-(2a2+b2)/8β)ds=∞,∫t0∞‍(q(s)-(2a2+b2)/4ε)ds>M,  where M is a sufficiently large constant, then the conditions (3.11) and (3.12) can be satisfied, the solution of (3.10) is oscillatory.
By takingq(t)=sin2t, provided that 2a2+b2<4β,ε=(1/2β+1/(2a2+b2))-1, (3.11) and (3.12) hold. By the way, we also note that for (3.10), the example given in [3],
(3.13)q(t)={3+t24[2t-(6n-4)π+1+t2t]2,(6n-4)π≤t≤(6n-72)π,3+t24[2(6n-3)π-t-1+t2t]2,(6n-72)π≤t≤(6n-3)π,n={1,2,…}
are not continuous at t=(6n-4)π,(6n-3)π,n=1,2,…, and
(3.14)∑n=1[t/6π]∫(6n-4)π(6n-7/2)π‍3+t24[2t-(6n-4)π+1+t2t]2dt=∞,‍∫(6n-7/3)π(6n-3)π‍3+t24[2(6n-2)π-t+1+t2t]2dt=∞.
Though (3.11) and (3.12) hold, but it is the general requirement that for fixed t,∫t0t‍q(s)ds is bounded; hence, this example is not appropriate.



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*Source: 714357-2009-06-15.xml*