# Oscillation of a Class of Fractional Differential Equations with Damping Term

Authors: Huizeng Qin; Bin Zheng
Journal: The Scientific World Journal
 (2013)

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## Abstract

We investigate the oscillation of a class of fractional differential equations with damping term.
Based on a certain variable transformation, the fractional differential equations are converted into another
differential equations of integer order with respect to the new variable. Then, using Riccati transformation,
inequality, and integration average technique, some new oscillatory criteria for the equations are established.
As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results.


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## Body

## 1. Introduction

In the investigations of qualitative properties for differential equations, research of oscillation has gained much attention by many authors in the last few decades (e.g., see [1–16]). In these investigations, we notice that very little attention is paid to oscillation of fractional differential equations.In [17], Jumarie proposed a definition for fractional derivative which is known as the modified Riemann-Liouville derivative in the literature. Since then, many authors have investigated various applications of the modified Riemann-Liouville derivative (e.g., see [18–21]) including various fractional calculus formulae, the fractional variational iteration method, the Bäcklund transformation method, and the fractional subequation method for soling fractional partial differential equations. In this paper, based on the modified Riemann-Liouville derivative, we are concerned with oscillation of a class of fractional differential equations with damping term as follows:
(1)Dtα[a(t)Dtα(r(t)Dtαx(t))]+p(t)Dtα(r(t)Dtαx(t))0000+q(t)x(t)=0,t≥t0>0,0<α<1,
where Dtα(·) denotes the modified Riemann-Liouville derivative with respect to the variable t, the function a∈Cα([t0,∞),R+), r∈C2α([t0,∞),R+), p,q∈C([t0,∞),R+) and Cα denotes continuous derivative of order α.The definition and some important properties for the modified Riemann-Liouville derivative of orderα are listed as follows (see also in [20–24]):
(2)Dtαf(t)={1Γ(1-α)ddt×∫0t‍(t-ξ)-α(f(ξ)-f(0))dξ,0<α<1,(f(n)(t))(α-n),n≤α<n+1,n≥1,(3)Dtαtr=Γ(1+r)Γ(1+r-α)tr-α,(4)Dtα(f(t)g(t))=g(t)Dtαf(t)+f(t)Dtαg(t),(5)Dtαf[g(t)]=fg′[g(t)]Dtαg(t)=Dgαf[g(t)](g′(t))α.As usual, a solutionx(t) of (1) is called oscillatory if it has arbitrarily large zeros; otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.We organize the next as follows. In Section2, using Riccati transformation, inequality, and integration average technique, we establish some new oscillatory criteria for (1), while we present some examples for them in Section 3.


## 2. Oscillatory Criteria for (1)

In the following, we denoteξ=tα/Γ(1+α),  ξi=tiα/Γ(1+α), i=0,1,2,3,4,5, a(t)=a~(ξ), r(t)=r~(ξ), p(t)=p~(ξ), q(t)=q~(ξ), R+=(0,∞), δ~1(ξ,ξi)=∫ξiξ‍(1/a~1/γ(s))ds, δ1(t,ti)=δ~1(ξ,ξi), δ~2(ξ,ξi)=∫ξiξ‍(δ~1(s,ξi)/r~(s))ds, δ2(t,ti)=δ~2(ξ,ξi), A(ξ)=exp(∫ξ0ξ‍(p~(s)/a~(s))ds). Let h1,h2,H∈C([ξ0,∞),R) satisfy
(6)H(ξ,ξ)=0,H(ξ,s)>0,ξ>s≥ξ0.H has continuous partial derivatives ∂H(ξ,s)/∂ξ and ∂H(ξ,s)/∂s on [ξ0,∞) such that
(7)∂H(ξ,s)∂ξ=-h1(ξ,s)H(ξ,s),∂H(ξ,s)∂s=-h2(ξ,s)H(ξ,s),ξ>s≥ξ0.Lemma 1.
Assumex(t) is an eventually positive solution of (1), and
(8)∫ξ0∞‍1A(s)a~(s)ds=∞,(9)∫t0∞‍αtα-1Γ(1+α)r(t)dt=∞,(10)∫ξ0∞‍1r~(ζ)∫ζ∞‍1A(τ)a~(τ)∫τ∞‍A(s)q~(s)dsdτdζ=∞.
Then, there exists a sufficiently large T such that Dtα(r(t)Dtαx(t))>0 on [T,∞) and either Dtαx(t)>0 on [T,∞) or limt→∞x(t)=0.Proof.
Letx(t)=x~(ξ), where ξ=tα/Γ(1+α). Then, by use of (3), we obtain Dtαξ(t)=1, and furthermore, by use of the first equality in (5), we have
(11)Dtαx(t)=Dtαx~(ξ)=x~′(ξ)Dtαξ(t)=x~′(ξ).
Similarly, we have Dtαa(t)=a~′(ξ), Dtαr(t)=r~′(ξ). So, (1) can be transformed into the following form:
(12)[a~(ξ)(r~(ξ)x~′(ξ))′]′+p~(ξ)(r~(ξ)x~′(ξ))′0000+q~(ξ)x~(ξ)=0,ξ≥ξ0>0.
Since x(t) is an eventually positive solution of (1), then x~(ξ) is an eventually positive solution of (12), and there exists ξ1>ξ0 such that x~(ξ)>0 on [ξ1,∞). Furthermore, we have
(13)[A(ξ)a~(ξ)(r~(ξ)x~′(ξ))′]′00=A(ξ)[a~(ξ)(r~(ξ)x~′(ξ))′]′0000+A′(ξ)a~(ξ)(r~(ξ)x~′(ξ))′00=A(ξ)[a~(ξ)(r~(ξ)x~′(ξ))′]′0000+A(ξ)p~(ξ)((r~(ξ)x~′(ξ))′)γ00=-A(ξ)q~(ξ)x~(ξ)<0,ξ≥ξ1.
Then, A(ξ)a~(ξ)(r~(ξ)x~′(ξ))′ is strictly decreasing on [ξ1,∞), and thus (r~(ξ)x~′(ξ))′ is eventually of one sign. We claim (r~(ξ)x~′(ξ))′>0 on [ξ2,∞), where ξ2>ξ1 is sufficiently large. Otherwise, assume that there exists a sufficiently large ξ3>ξ2 such that (r~(ξ)x~′(ξ))′<0 on [ξ3,∞). Then, r~(ξ)x~′(ξ) is strictly decreasing on [ξ3,∞), and we have
(14)r~(ξ)x~′(ξ)-r~(ξ3)x~′(ξ3)=∫ξ3ξ‍A(s)a~(s)(r~(s)x~′(s))′A(s)a~(s)ds≤A(ξ3)a~(ξ3)(r~(ξ3)x~′(ξ3))′×∫ξ3ξ‍1A(s)a~(s)ds.
By (8), we have limξ→∞r~(ξ)x~′(ξ)=-∞. So there exists a sufficiently large ξ4 with ξ4>ξ3 such that x~′(ξ)<0, ξ∈[ξ4,∞). Furthermore,
(15)x~(ξ)-x~(ξ4)=∫ξ4ξ‍x~′(s)ds=∫ξ4ξ‍r~(s)x~′(s)r~(s)ds≤r~(ξ4)x~′(ξ4)∫ξ4ξ‍1r~(s)ds=r~(ξ4)x~′(ξ4)∫t4∞‍αtα-1Γ(1+α)r(t)dt.
By (9), we deduce that limξ→∞x~(ξ)=-∞, which contradicts the fact that x~(ξ) is an eventually positive solution of (9). So, (r~(ξ)x~′(ξ))′>0 on [ξ2,∞), and Dtα(r(t)Dtαx(t))>0 on [t2,∞). Thus, Dtαx(t)=x~′(ξ) is eventually of one sign. Now we assume x~′(ξ)<0, ξ∈[ξ5,∞) for some sufficiently large ξ5>ξ4. Since x~(ξ)>0, furthermore we have limξ→∞x~(ξ)=β≥0. We claim β=0. Otherwise, assume β>0. Then x~(ξ)≥β on [ξ5,∞), and, for ξ∈[ξ5,∞), by (12) we have
(16)[A(ξ)a~(ξ)(r~(ξ)x~′(ξ))′]′≤-A(ξ)q~(ξ)x~(ξ)≤-A(ξ)q~(ξ)β.
Substituting ξ with s in the previous inequality, an integration with respect to s from ξ to ∞ yields
(17)-A(ξ)a~(ξ)(r~(ξ)x~′(ξ))′≤-limξ→∞A(ξ)a~(ξ)(r~(ξ)x~′(ξ))′-β∫ξ∞‍A(s)q~(s)ds<-β∫ξ∞‍A(s)q~(s)ds,
which means
(18)(r~(ξ)x~′(ξ))′>βA(ξ)a~(ξ)∫ξ∞‍A(s)q~(s)ds.
Substituting ξ with τ in (18), an integration for (18) with respect to τ from ξ to ∞ yields
(19)-r~(ξ)x~′(ξ)>-limξ→∞r~(ξ)x~′(ξ)+β∫ξ∞‍1A(τ)a~(τ)∫τ∞‍A(s)q~(s)dsdτ>β∫ξ∞‍1A(τ)a~(τ)∫τ∞‍A(s)q~(s)dsdτ;
that is,
(20)x~′(ξ)<-β1r~(ξ)∫ξ∞‍1A(τ)a~(τ)∫τ∞‍A(s)q~(s)dsdτ.
Substituting ξ with ζ in (20), an integration for (20) with respect to ζ from ξ5 to ξ yields
(21)x~(ξ)-x~(ξ5)0000<-β∫ξ5ξ‍1r~(ζ)∫ζ∞‍1A(τ)a~(τ)∫τ∞‍A(s)q~(s)dsdτdζ.
By (10), one can see limt→∞x~(ξ)=-∞, which causes a contradiction. So, the proof is complete.Lemma 2.
Assume thatx is an eventually positive solution of (1) such that
(22)Dtα(r(t)Dtαx(t))>0,Dtαx(t)>0
on [t1,∞)𝕋, where t1≥t0 is sufficiently large. Then, for t≥t1, we have
(23)Dtαx(t)≥A(ξ)δ1(t,t1)a(t)Dtα(r(t)Dtαx(t))r(t),(24)x(t)≥A(ξ)δ2(t,t1)a(t)Dtα(r(t)Dtαx(t)).Proof.
By (13), we obtain that A(ξ)a~(ξ)(r~(ξ)x~′(ξ))′ is strictly decreasing on [ξ1,∞). So,
(25)r~(ξ)x~′(ξ)≥r~(ξ)x~′(ξ)-r~(ξ1)x~′(ξ1)=∫ξ1ξ‍A(s)a~(s)(r~(s)x~′(s))′A(s)a~(s)ds≥A(ξ)a~(ξ)(r~(ξ)x~′(ξ))′∫ξ1ξ‍1A(s)a~(s)ds=A(ξ)a~(ξ)(r~(ξ)x~′(ξ))′δ~1(ξ,ξ1);
that is
(26)r(t)Dtαx(t)≥A(ξ)δ1(t,t1)a(t)Dtα(r(t)Dtαx(t)),
which admits (23). On the other hand, we have
(27)x~(ξ)≥x~(ξ)-x~(ξ1)≥∫ξ1ξ‍A(s)δ~1(s,ξ1)a~(s)(r~(s)x~′(s))′A(s)r~(s)ds≥A(ξ)a~(ξ)(r~(ξ)x~′(ξ))′∫ξ1ξ‍δ~1(s,ξ1)A(s)r~(s)ds=A(ξ)δ~2(ξ,ξ1)a~(ξ)(r~(ξ)x~′(ξ))′,
which can be rewritten as (24). So the proof is complete.Lemma 3 (see [25, Theorem 41]).
Assume thatA and B are nonnegative real numbers. Then,
(28)λABλ-1-Aλ≤(λ-1)Bλ
for all λ>1.Theorem 4.
Assume that (8)–(10) hold. If there exists ϕ∈Cα([t0,∞),R+) such that for any sufficiently large T≥ξ0, there exist a, b, c with T≤a<c<b satisfying
(29)1H(b,c)∫cb‍H(b,s)A(s)ϕ~(s)q~(s)ds000000+1H(c,a)∫ac‍H(s,a)A(s)ϕ~(s)q~(s)ds0000>1H(b,c)∫cb‍r~(s)ϕ~(s)4δ~1(s,ξ2)Q22(b,s)ds000000+1H(c,a)∫ac‍r~(s)ϕ~(s)4δ~1(s,ξ2)Q12(s,a)ds,
where ϕ~(ξ)=ϕ(t), Q1(s,ξ)=h1(s,ξ)-(ϕ~′(s)/ϕ~(s))H(s,ξ),  Q2(ξ,s)=h2(ξ,s)-(ϕ~′(s)/ϕ~(s))H(ξ,s); then, (1) is oscillatory or satisfies limt→∞x(t)=0.Proof.
Assume that (1) has a nonoscillatory solution x on [t0,∞). Without loss of generality, we may assume that x(t)>0 on [t1,∞), where t1 is sufficiently large. By Lemma 1, we have Dtα(r(t)Dtαx(t))>0,t∈[t2,∞), where t2>t1 is sufficiently large, and either Dtαx(t)>0 on [t2,∞) or limt→∞x(t)=0. Now we assume Dtαx(t)>0 on [t2,∞). Define the generalized Riccati function:
(30)ω(t)=ϕ(t)A(ξ)a(t)Dtα(r(t)Dtαx(t))x(t).
Then, for t∈[t2,∞), we have
(31)Dtαω(t)=Dtαϕ(t)A(ξ)a(t)Dtα(r(t)Dtαx(t))x(t)+ϕ(t)Dtα{A(ξ)a(t)Dtα(r(t)Dtαx(t))x(t)}=ϕ(t)Dtα(A(ξ)a(t)Dtα(r(t)Dtαx(t)))x(t)-ϕ(t)Dtαx(t)A(ξ)a(t)Dtα(r(t)Dtαx(t))x2(t)+Dtαϕ(t)ϕ(t)ω(t)=(ϕ(t)[A(ξ)Dtα(a(t)Dtα(r(t)Dtαx(t)))000000+a(t)Dtα(r(t)Dtαx(t))DtαA(ξ)])×(x(t))-1-ϕ(t)Dtαx(t)A(ξ)a(t)Dtα(r(t)Dtαx(t))x2(t)+Dtαϕ(t)ϕ(t)ω(t)=(ϕ(t)[a(t)Dtα(r(t)Dtαx(t))A′(ξ)DtαξA(ξ)Dtα(a(t)Dtα(r(t)Dtαx(t)))000000+a(t)Dtα(r(t)Dtαx(t))A′(ξ)Dtαξ])×(x(t))-1-ϕ(t)Dtαx(t)A(ξ)a(t)Dtα(r(t)Dtαx(t))x2(t)+Dtαϕ(t)ϕ(t)ω(t).
Using Dtαξ=1 and (23), we obtain
(32)Dtαω(t)≤(ϕ(t)[p~(ξ)a~(ξ)A(ξ)Dtα(a(t)Dtα(r(t)Dtαx(t)))000000+a(t)Dtα(r(t)Dtαx(t))A(ξ)p~(ξ)a~(ξ)])×(x(t))-1-ϕ(t)((r(t)x2(t))-1(A(ξ)δ1(t,t2)a(t)Dtα(r(t)Dtαx(t))00000000×Dtαx(t)A(ξ)a(t)Dtα(r(t)Dtαx(t)))0000000×(r(t)x2(t))-1)+Dtαϕ(t)ϕ(t)ω(t)=(ϕ(t)A(ξ)[Dtα(a(t)Dtα(r(t)Dtαx(t)))00000000000+p(t)Dtα(r(t)Dtαx(t))])×(x(t))-1-δ1(t,t2)ϕ(t)r(t)ω2(t)+Dtαϕ(t)ϕ(t)ω(t)=-A(ξ)q(t)ϕ(t)-δ1(t,t2)ϕ(t)r(t)ω2(t)+Dtαϕ(t)ϕ(t)ω(t).
Let ω(t)=ω~(ξ). Then Dtαw(t)=w~′(ξ), and Dtαϕ(t)=ϕ~′(ξ). So (32) is transformed into the following form:
(33)ω~′(ξ)≤-A(ξ)ϕ~(ξ)q~(ξ)-δ~1(ξ,ξ2)r~(ξ)ϕ~(ξ)ω~2(ξ)+ϕ~′(ξ)ϕ~(ξ)ω~(ξ),ξ≥ξ2.
Choosea, b, c arbitrarily in [ξ2,∞) with b>c>a. Substituting ξ with s, multiplying both sides of (33) by H(ξ,s), and integrating it with respect to s from c to ξ for ξ∈[c,b), we get that
(34)∫cξ‍H(ξ,s)A(s)ϕ~(s)q~(s)ds00≤-∫cξ‍H(ξ,s)w~′(s)ds0000+∫cξ‍H(ξ,s)ϕ~′(s)ϕ~(s)w~(s)ds0000-∫cξ‍H(ξ,s)δ~1(s,ξ2)r~(s)ϕ~(s)ω~2(s)ds00=H(ξ,c)w~(c)0000-∫cξ‍[(δ~1(s,ξ2)H(ξ,s)r~(s)ϕ~(s))1/2w~(s)000000000+12(r~(s)ϕ~(s)δ~1(s,ξ2))1/2Q2(ξ,s)]2ds0000+∫cξ‍r~(s)ϕ~(s)4δ~1(s,ξ2)Q22(ξ,s)ds00≤H(ξ,c)w~(c)+∫cξ‍r~(s)ϕ~(s)4δ~1(s,ξ2)Q22(ξ,s)ds.
Dividing both sides of the inequality (34) by H(ξ,c) and letting ξ→b-, we obtain
(35)1H(b,c)∫cb‍H(b,s)A(s)ϕ~(s)q~(s)ds00≤w~(c)+1H(b,c)∫cb‍r~(s)ϕ~(s)4δ~1(s,ξ2)Q22(b,s)ds.
On the other hand, substitutingξ with s, multiplying both sides of (33) by H(s,ξ), and integrating it with respect to s from ξ to c for ξ∈(a,c], we get that
(36)∫ξc‍H(s,ξ)A(s)ϕ~(s)q~(s)ds00≤-∫ξc‍H(s,ξ)w~′(s)ds+∫ξc‍H(s,ξ)ϕ~′(s)ϕ~(s)w~(s)ds0000-∫ξc‍H(s,ξ)δ~1(s,ξ2)r~(s)ϕ~(s)ω~2(s)ds00=-H(c,ξ)w~(c)0000-∫ξc‍[(δ~1(s,ξ2)H(s,ξ)r~(s)ϕ~(s))1/2w~(s)000000000+12(r~(s)ϕ~(s)δ~1(s,ξ2))1/2Q1(ξ,s)]2ds0000+∫ξc‍r~(s)ϕ~(s)4δ~1(s,ξ2)Q12(s,ξ)ds00≤-H(c,ξ)w~(c)+∫ξc‍r~(s)ϕ~(s)4δ~1(s,ξ2)Q12(s,ξ)ds.
Dividing both sides of the inequality (36) by H(c,ξ) and letting ξ→a+, we obtain
(37)1H(c,a)∫ac‍H(s,a)A(s)ϕ~(s)q~(s)ds0000≤-w~(c)+1H(c,a)∫ac‍r~(s)ϕ~(s)4δ~1(s,ξ2)Q12(s,a)ds.
A combination of (35) and (37) yields
(38)1H(b,c)∫cb‍H(b,s)A(s)ϕ~(s)q~(s)ds000000+1H(c,a)∫ac‍H(s,a)A(s)ϕ~(s)q~(s)ds0000≤1H(b,c)∫cb‍r~(s)ϕ~(s)4δ~1(s,ξ2)Q22(b,s)ds000000+1H(c,a)∫ac‍r~(s)ϕ~(s)4δ~1(s,ξ2)Q12(s,a)ds,
which contradicts (29). So, the proof is complete.Theorem 5.
Under the conditions of Theorem4, if for any sufficiently large l≥ξ0,
(39)limξ→∞sup∫lξ‍[r~(s)ϕ~(s)4δ~1(s,ξ2)H(s,l)A(s)ϕ~(s)q~(s)0-r~(s)ϕ~(s)4δ~1(s,ξ2)Q12(s,l)]ds>0,(40)limξ→∞sup∫lξ‍[r~(s)ϕ~(s)4δ~1(s,ξ2)H(ξ,s)A(s)ϕ~(s)q~(s)0-r~(s)ϕ~(s)4δ~1(s,ξ2)Q22(ξ,s)]ds>0,
then (1) is oscillatory.Proof.
For anyT≥ξ0, let a=T. In (39), we choose l=a. Then, there exists c>a such that
(41)∫ac‍[H(s,a)A(s)ϕ~(s)q~(s)-r~(s)ϕ~(s)4δ~1(s,ξ2)Q12(s,a)]ds>0.
In (40), we choose l=c>a. Then there exists b>c such that
(42)∫cb‍[H(b,s)A(s)ϕ~(s)q~(s)-r~(s)ϕ~(s)4δ~1(s,ξ2)Q22(b,s)]ds>0.
Combining (41) and (42), we obtain (29). The conclusion thus comes from Theorem 4, and the proof is complete.In Theorems4 and 5, if we choose H(ξ,s)=(ξ-s)λ, ξ≥s≥ξ0, where λ>1 is a constant, then we obtain the following two corollaries.Corollary 6.
Under the conditions of Theorem4, if for any sufficiently large T≥ξ0, there exist a, b, c with T≤a<c<b satisfying
(43)1(c-a)λ∫ac‍(s-a)λA(s)ϕ~(s)q~(s)ds0000+1(b-c)λ∫cb‍(b-s)λA(s)ϕ~(s)q~(s)ds00>1(c-a)λ∫ac‍r~(s)ϕ~(s)4δ~1(s,ξ2)(s-a)λ-2(λ+ϕ~′(s)ϕ~(s)(s-a))2ds0000+1(b-c)λ∫cb‍r~(s)ϕ~(s)4δ~1(s,ξ2)(b-s)λ-20000000000000000×(λ-ϕ~′(s)ϕ~(s)(b-s))2ds,
then (1) is oscillatory.Corollary 7.
Under the conditions of Theorem5, if for any sufficiently large l≥ξ0,
(44)limξ→∞sup∫lξ‍[(λ+ϕ~′(s)ϕ~(s)(s-l))2(s-l)λA(s)ϕ~(s)q~(s)0-r~(s)ϕ~(s)4δ~1(s,ξ2)(s-l)λ-20×(λ+ϕ~′(s)ϕ~(s)(s-l))2]ds>0,limξ→∞sup∫lξ‍[(λ-ϕ~′(s)ϕ~(s)(ξ-s))2(ξ-s)λA(s)ϕ~(s)q~(s)0-r~(s)ϕ~(s)4δ~1(s,ξ2)(ξ-s)λ-20×(λ-ϕ~′(s)ϕ~(s)(ξ-s))2]ds>0,
then (1) is oscillatory.Theorem 8.
Assume (8)–(10) hold, and
(45)∫ξ0∞‍[A(s)ϕ~(s)q~(s)-r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)]ds=∞,
where ϕ~ is defined as in Theorem 4. Then every solution of (1) is oscillatory or satisfies limt→∞x(t)=0.Proof.
Assume (1) has a nonoscillatory solution x on [t0,∞). Without loss of generality, we may assume x(t)>0 on [t1,∞), where t1 is sufficiently large. By Lemma 1, we have Dtα(r(t)Dtαx(t))>0, t∈[t2,∞), where t2>t1 is sufficiently large, and either Dtαx(t)>0 on [t2,∞) or limt→∞x(t)=0. Now we assume that Dtαx(t)>0 on [t2,∞). Let ω(t), ω~(ξ) be defined as in Theorem 4. Then we obtain (33), and furthermore,
(46)ω~′(ξ)≤-A(ξ)ϕ~(ξ)q~(ξ)-δ~1(ξ,ξ2)r~(ξ)ϕ~(ξ)ω~2(ξ)+ϕ~′(ξ)ϕ~(ξ)ω~(ξ)≤-A(ξ)ϕ~(ξ)q~(ξ)+r~(ξ)[ϕ~′(ξ)]24δ~1(ξ,ξ2)ϕ~(ξ),ξ≥ξ2.
Substitutingξ with s in (46) and integrating (46) with respect to s from ξ2 to ξ yield
(47)∫ξ2ξ‍[A(s)ϕ~(s)q~(s)-r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)]ds≤ω(ξ2)-ω(ξ)≤ω(ξ2)<∞,
which contradicts (45). So, the proof is complete.Theorem 9.
Assume (8)–(10) hold, and there exists a function G∈C([ξ0,∞),ℝ) such that G(ξ,ξ)=0,  for  ξ≥ξ0, G(ξ,s)>0,  for  ξ>s≥ξ0, and G has a nonpositive continuous partial derivative Gs′(ξ,s). If
(48)limξ→∞sup1G(ξ,ξ0)×{([(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)]γ+1)(γ+1)γ+1[ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1[ϕ~(s)δ~1(s,ξ2)]γ)-1∫ξ0ξ‍G(ξ,s){×r~(s)(γ+1)γ+1[ϕ~(s)δ~1(s,ξ2)]γ)-1)}([(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)]γ+1)(γ+1)γ+1[ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1[ϕ~(s)δ~1(s,ξ2)]γ)-1Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)00000000000+ϕ~(s)δ~1(s,ξ2)φ~1+1/γ(s)r~(s)-[(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)]γ+1(γ+1)γ+1[ϕ~(s)δ~1(s,ξ2)]γr~(s)}ds}=∞,
where ϕ~ is defined as in Theorem 4, then every solution of (1) is oscillatory or satisfies limt→∞x(t)=0.Proof.
Assume (1) has a nonoscillatory solution x on [t0,∞). Without loss of generality, we may assume x(t)>0 on [t1,∞), where t1 is sufficiently large. By Lemma 1, we have Dtα(r(t)Dtαx(t))>0, t∈[t2,∞), where t2>t1 is sufficiently large, and either Dtαx(t)>0 on [t2,∞) or limt→∞x(t)=0. Now we assume Dtαx(t)>0 on [t2,∞). Let ω(t), ω~(ξ) be defined as in Theorem 4. By (46), we have
(49)A(ξ)ϕ~(ξ)q~(ξ)-r~(ξ)[ϕ~′(ξ)]24δ~1(ξ,ξ2)ϕ~(ξ)≤ω~′(ξ),ξ≥ξ2.
Substituting ξ with s in (49), multiplying both sides by G(ξ,s), and then integrating both sides of (49) with respect to s from ξ2 to ξ yield
(50)∫ξ2ξ‍G(ξ,s){A(s)ϕ~(s)q~(s)-r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)}ds0000≤-∫ξ2ξ‍G(ξ,s)ω~′(s)ds0000=G(ξ,ξ2)ω(ξ2)+∫ξ2ξ‍Hs′(ξ,s)ω(s)Δs0000≤G(ξ,ξ2)ω(ξ2)0000≤G(ξ,ξ0)ω(ξ2).
Then,
(51)∫ξ0ξ‍G(ξ,s){A(s)ϕ~(s)q~(s)-r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)}ds00=∫ξ0ξ2‍G(ξ,s){A(s)ϕ~(s)q~(s)-r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)}ds0000+∫ξ2ξ‍G(ξ,s){A(s)ϕ~(s)q~(s)-r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)}ds00≤G(ξ,ξ0)ω~(ξ2)+G(ξ,ξ0)0000×∫ξ0ξ2‍|A(s)ϕ~(s)q~(s)-r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)|ds.
So,
(52)limξ→∞sup1G(ξ,ξ0){∫ξ0ξ‍G(ξ,s){r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)A(s)ϕ~(s)q~(s)000000000000000000000000000-r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)}ds}00≤ω~(ξ2)+∫ξ0ξ2‍|A(s)ϕ~(s)q~(s)-r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)|ds<∞,
which contradicts (48). So the proof is complete.


## 3. Applications of the Results

Example 10.
Consider the following fractional differential equation:(53)Dt1/3[t1/9Dt1/3Dt1/3x(t)]+t-1/3Dt1/3Dt1/3x(t)0000+t-2/3x(t)=0,t≥2.
In (1), if we set t0=2, α=1/3, a(t)=t1/9, r(t)≡1, (t)=t-1/3, q(t)=t-2/3, then we obtain (53). So ξ0=21/3/Γ(4/3), a~(ξ)=a(t)=t1/9=[Γ(4/3)ξ]1/3, r~(ξ)≡1, p~(ξ)=[Γ(4/3)ξ]-1, q~(ξ)=[Γ(4/3)ξ]-2. Furthermore, A(ξ)=exp([Γ(4/3)]-4/3∫ξ0ξ‍s-4/3ds)=exp(3[Γ(4/3)]-4/3[ξ0-1/3-ξ-1/3]), which implies 1≤A(ξ)≤exp(3[Γ(4/3)]-4/3ξ0-1/3). On the other hand, δ~1(ξ,ξ2)=∫ξ2ξ‍(1/A(s)a~(s))ds≥[Γ(4/3)]-1/3exp(-3[Γ(4/3)]-4/3ξ0-1/3)∫ξ2ξ‍s-1/3ds=(3/2)[Γ(4/3)]-1/3exp(-3[Γ(4/3)]-(4/3)ξ0-1/3)(ξ2/3-ξ22/3), which implies limξ→∞δ~1(ξ,ξ2)=∞, and then (8) holds. So, there exists a sufficiently large T>ξ2 such that δ~1(ξ,ξ2)>[Γ(4/3)]2 on [T,∞). In (9),
(54)∫t0∞‍αtα-1Γ(1+α)r(t)dt=∫ξ0∞‍1r~(s)ds=∫ξ0∞‍ds=∞.
In (10),
(55)∫ξ0∞‍1r~(ζ)∫ζ∞‍1A(τ)a~(τ)∫τ∞‍A(s)q~(s)dsdτdζ00≥[Γ(43)]-1/3exp(-3[Γ(43)]-4/3ξ0-1/3)0000×∫ξ0∞‍∫ζ∞‍τ-1/3∫τ∞‍s-1dsdτdζ=∞.
In (48), letting ϕ~(ξ)=ξ, we obtain
(56)∫ξ0∞‍[A(s)ϕ~(s)q~(s)-r~(s)[ϕ~′(s)]24δ~1(s,ξ2)ϕ~(s)]ds00=∫ξ0∞‍{A(s)[Γ(43)]-2-14δ~1(s,ξ2)}1sds00=∫ξ0T‍{A(s)[Γ(43)]-2-14δ~1(s,ξ2)}1sds0000+∫T∞‍{A(s)[Γ(43)]-2-14δ~1(s,ξ2)}1sds00≥∫ξ0T‍{A(s)[Γ(43)]-2-14δ~1(s,ξ2)}1sds0000+∫T∞‍34[Γ(43)]-21sds=∞.
Therefore, (53) is oscillatory by Theorem 8.Example 11.
Consider the following fractional differential equation:(57)Dt1/2[t1/4Dt1/2Dt1/2x(t)]+Γ(3/2)tDt1/2Dt1/2x(t)0000+x(t)=0,t≥2.
In (1), if we set t0=2, α=1/2, a(t)=t1/4, r(t)≡1, p(t)=Γ(3/2)/t, q(t)≡1, then we obtain (57). So ξ0=21/2/Γ(3/2), a~(ξ)=a(t)=t1/4=Γ(3/2)ξ, r~(ξ)≡1, p~(ξ)=ξ-1, q~(ξ)≡1. Furthermore, A(ξ)=exp([Γ(3/2)]-1/2∫ξ0ξ‍s-3/2ds)=exp([Γ(3/2)]-1/2[2ξ0-1/2-2ξ-1/2]), which implies 1≤A(ξ)≤exp(2[Γ(3/2)]-1/2ξ0-1/2). On the other hand, δ~1(ξ,ξ2)=∫ξ2ξ‍(1/A(s)a~(s))ds≥[Γ(3/2)]-1/2exp(-2[Γ(3/2)]-1/2ξ0-1/2)∫ξ2ξ‍(1/s)ds=2[Γ(3/2)]-1/2exp(-2[Γ(3/2)]-1/2ξ0-1/2)(ξ-ξ2), which implies limξ→∞δ~1(ξ,ξ2)=∞. So, there exists a sufficiently large T>ξ2 such that δ~1(ξ,ξ2)>1 on [T,∞).
From the analysis above, one can see the (8) holds. We now test (9) and (10). In (9),
(58)∫t0∞‍αtα-1Γ(1+α)r(t)dt=∫ξ0∞‍1r~(s)ds=∫ξ0∞‍ds=∞.
In (10),
(59)∫ξ0∞‍1r~(ζ)∫ζ∞‍1A(τ)a~(τ)∫τ∞‍A(s)q~(s)dsdτdζ0000≥[Γ(32)]-1/2exp(-2[Γ(32)]-1/2ξ0-1/2)000000×∫ξ0∞‍∫ζ∞‍1τ∫τ∞‍dsdτdζ=∞.
So, (9) and (10) hold. On the other hand, in (44), after putting ϕ~(ξ)≡1, λ=2, for any sufficiently large l, we have
(60)limξ→∞sup∫lξ‍[(λ-ϕ~′(s)ϕ~(s)(ξ-s))2(s-l)λA(s)ϕ~(s)q~(s)00000000000-r~(s)ϕ~(s)4δ~1(s,ξ2)(s-l)λ-2(λ+ϕ~′(s)ϕ~(s)(s-l))2]ds0000≥limξ→∞sup∫lξ‍[(s-l)2s-1-1]ds=∞,limξ→∞sup∫lξ‍[(λ-ϕ~′(s)ϕ~(s)(ξ-s))2(ξ-s)λA(s)ϕ~(s)q~(s)0000000000-r~(s)ϕ~(s)4δ~1(s,ξ2)(ξ-s)λ-2(λ-ϕ~′(s)ϕ~(s)(ξ-s))2]ds0000≥limξ→∞sup∫lξ‍[(ξ-s)2s-1-1]ds=∞.
So (44) holds, and then by Corollary 7 we deduce that (57) is oscillatory.



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*Source: 685621-2013-08-21.xml*