# Oscillation on a Class of Differential Equations of Fractional Order

Authors: Tongbo Liu; Bin Zheng; Fanwei Meng
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Based on Riccati transformation and certain inequality technique,
some new oscillatory criteria are established for the solutions of a
class of sequential differential equations with fractional order defined in the modified Riemann-Liouville derivative. The
oscillatory criteria established are of new forms compared with the
existing results so far in the literature. For illustrating the
validity of the results established, we present some examples for
them.


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## Body

## 1. Introduction

Recently, research for oscillation of various equations including differential equations, difference equations, and dynamic equations on time scales, has been a hot topic in the literature, and much effort has been done to establish new oscillation criteria for these equations so far (e.g., see [1–21] and the references therein). In these investigations, we notice that very little attention is paid to oscillation of fractional differential equations. Recent results in this direction only include Chen’s work [22, 23] and Zheng’s work [24].In this paper, we are concerned with oscillation of a class of fractional differential equations as follows:(1)Dtα[a(t)(Dtα(r(t)Dtαx(t)))γ]+q(t)f(x(t))=0,t≥t0>0,0<α<1,
where Dtα(·) denotes the modified Riemann-Liouville derivative [25] with respect to the variable t, γ is the quotient of two odd positive numbers, the functions a∈Cα([t0,∞),R+),r∈C2α([t0,∞),R+),andq∈C([t0,∞),R+), Cα denotes continuous derivative of order α, and the function f is continuous satisfying f(x)/xγ≥K for some positive constant K and for all x≠0.The definition and some important properties for Jumarie’s modified Riemann-Liouville derivative of orderα are listed as follows (see also in [26–28]):(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,Dtαtr=Γ(1+r)Γ(1+r-α)tr-α,(3)Dtα(f(t)g(t))=g(t)Dtαf(t)+f(t)Dtαg(t),(4)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 this paper 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), q(t)=q~(ξ), R+=(0,∞), δ~1(ξ,ξi)=∫ξiξ(1/a~1/γ(s))ds,‍δ1(t,ti)=δ~1(ξ,ξi), and δ~2(ξ,ξi)=∫ξiξ(δ~1(s,ξi)/r~(s))ds,‍δ2(t,ti)=δ~2(ξ,ξi).Lemma 1.
Assume thatx(t) is an eventually positive solution of (1), and
(5)∫ξ0∞‍1a~1/γ(s)ds=∞,(6)∫t0∞‍αtα-1Γ(1+α)r(t)dt=∞,(7)∫ξ0∞‍1r~(ζ)∫ζ∞‍[1a~(τ)∫τ∞‍q~(s)ds]1/γdτdζ=∞.
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 (2) we obtain Dtαξ(t)=1, and furthermore by use of the first equality in (4), we have
(8)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:
(9)[a~(ξ)((r~(ξ)x~′(ξ))′)γ]′+q~(ξ)f(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 (9), and there exists ξ1>ξ0 such that x~(ξ)>0 on [ξ1,∞). Furthermore, we have
(10)[a~(ξ)((r~(ξ)x~′(ξ))′)γ]′=-q~(ξ)f(x~(ξ))≤-Kq~(ξ)x~γ(ξ)<0,ξ≥ξ1.
Then a~(ξ)((r~(ξ)x~′(ξ))′)γ 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
(11)r~(ξ)x~′(ξ)-r~(ξ3)x~′(ξ3)=∫ξ3ξ‍a~1/γ(s)(r~(s)x~′(s))′a~1/γ(s)ds≤a~1/γ(ξ3)(r~(ξ3)x~′(ξ3))′∫ξ3ξ‍1a~1/γ(s)ds.
By (5), 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,
(12)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 (6) we deduce that limξ→∞x~(ξ)=-∞, which contradicts the fact that x~(ξ) is an eventually positive solution of (6). 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 that  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 that β=0. Otherwise, assume that β>0. Then x~(ξ)≥β on [ξ5,∞), and for ξ∈[ξ5,∞) by (9) we have
(13)[a~(ξ)((r~(ξ)x~′(ξ))′)γ]′≤-Kq~(ξ)x~γ(ξ)≤-Kq~(ξ)βγ.
Substituting ξ with s in the inequality previously, an integration with respect to s from ξ to ∞ yields
(14)-a~(ξ)((r~(ξ)x~′(ξ))′)γ≤-limξ→∞a~(ξ)((r~(ξ)x~(ξ))′)γ-Kβγ∫ξ∞‍q~(s)ds<-Kβγ∫ξ∞‍q~(s)ds,
which means
(15)(r~(ξ)x~′(ξ))′>K1/γβ[1a~(ξ)∫ξ∞‍q~(s)ds]1/γ.
Substituting ξ with τ in (15), an integration for (15) with respect to τ from ξ to ∞ yields
(16)-r~(ξ)x~′(ξ)>-limξ→∞r~(ξ)x~′(ξ)+K1/γβ∫ξ∞‍[1a~(τ)∫τ∞‍q~(s)ds]1/γdτ>K1/γβ∫ξ∞‍[1a~(τ)∫τ∞‍q~(s)ds]1/γdτ;
that is,
(17)x~′(ξ)<-K1/γβ1r~(ξ)∫ξ∞‍[1a~(τ)∫τ∞‍q~(s)ds]1/γdτ.
Substituting ξ with ζ in (17), an integration for (17) with respect to ζ from ξ5 to ξ yields
(18)x~(ξ)-x~(ξ5)<-K1/γβ∫ξ5ξ‍1r~(ζ)∫ζ∞‍[1a~(τ)∫τ∞‍q~(s)ds]1/γdτdζ.
By (7), 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
(19)Dtα(r(t)Dtαx(t))>0,Dtαx(t)>0
on [t1,∞)𝕋, where t1≥t0 is sufficiently large. Then one has
(20)Dtαx(t)≥δ1(t,t1)a1/γ(t)Dtα(r(t)Dtαx(t))r(t),(21)x(t)≥δ2(t,t1)a1/γ(t)Dtα(r(t)Dtαx(t)).Proof.
By (10), we obtain that a~(ξ)((r~(ξ)x~′(ξ))′)γ is strictly decreasing on [ξ1,∞). So
(22)r~(ξ)x~′(ξ)≥r~(ξ)x~′(ξ)-r~(ξ1)x~′(ξ1)=∫ξ1ξ‍a~1/γ(s)(r~(s)x~′(s))′a~1/γ(s)ds≥a~1/γ(ξ)(r~(ξ)x~′(ξ))′∫ξ1ξ‍1a~1/γ(s)ds=a~1/γ(ξ)(r~(ξ)x~′(ξ))′δ~1(ξ,ξ1);
that is
(23)r(t)Dtαx(t)≥δ1(t,t1)a1/γ(t)Dtα(r(t)Dtαx(t)),
which admits (20). On the other hand, we have
(24)x~(ξ)≥x~(ξ)-x~(ξ1)≥∫ξ1ξ‍δ~1(s,ξ1)a~1/γ(s)(r~(s)x~′(s))′r~(s)ds≥a~1/γ(ξ)(r~(ξ)x~′(ξ))′∫ξ1ξ‍δ~1(s,ξ1)r~(s)ds=δ~2(ξ,ξ1)a~1/γ(ξ)(r~(ξ)x~′(ξ))′,
which can be rewritten as (21). So the proof is complete.Lemma 3 (see [29, Theorem 41]).
Assume thatA and B are nonnegative real numbers. Then
(25)λABλ-1-Aλ≤(λ-1)Bλ
for all λ>1.Theorem 4.
Leth1,h2,H^∈C([ξ0,∞),R) satisfying H^(ξ,ξ)=0,H^(ξ,s)>0,ξ>s≥ξ0, and H has continuous partial derivatives H^ξ′(ξ,s) and H^s′(ξ,s) on [ξ0,∞). Assume that (5)–(7) hold, and for any sufficiently large T≥ξ0, there exist ϕ∈C1([t0,∞),R+) and φ∈C1([t0,∞),[0,∞)) and a,b, and c with T≤a<c<b satisfying(26)1H^(b,c)∫cb‍H^(b,s){ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)}ds+1H^(c,a)∫ac‍H^(s,a){ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)}ds>1H^(b,c)∫cb‍H^(b,s){×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~(s)H^s′(b,s)]γ+1[ϕ~(s)H^s′(b,s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^s′(b,s)]γ+1)×(×[ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~(s)H^s′(b,s)]γ+1[ϕ~(s)H^s′(b,s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)}ds+1H^(c,a)∫ac‍H^(s,a){[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~(s)H^s′(s,a)]γ+1[ϕ~(s)H^s′(b,s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^s′(s,a)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~(s)H^s′(b,s)]γ+1[ϕ~(s)H^s′(b,s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)ϕ~(s)H^s′(b,s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1}ds,
where ϕ~(ξ)=ϕ(t),φ~(ξ)=φ(t). Then every solution of (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 that Dtαx(t)>0 on [t2,∞). Define the generalized Riccati function
(27)ω(t)=ϕ(t){a(t)(Dtα(r(t)Dtαx(t)))γxγ(t)+φ(t)}.
Then for t∈[t2,∞), we have
(28)Dtαω(t)=Dtαϕ(t)a(t)(Dtα(r(t)Dtαx(t)))γxγ(t)+ϕ(t)Dtα{a(t)(Dtα(r(t)Dtαx(t)))γxγ(t)}+Dtαϕ(t)φ(t)+ϕ(t)Dtαφ(t)=ϕ(t){(x2γ(t))-1(xγ(t)Dtα(a(t)(Dtα(r(t)Dtαx(t)))γ)-γxγ-1(t)Dtαx(t)a(t)×(Dtα(r(t)Dtαx(t)))γ(Dtα(r(t)Dtαx(t)))γ)×(x2γ(t))-1}+Dtαϕ(t)ϕ(t)ω(t)+ϕ(t)Dtαφ(t)=-ϕ(t)q(t)f(x(t))xγ(t)-γϕ(t)Dtαx(t)a(t)(Dtα(r(t)Dtαx(t)))γxγ+1(t)+Dtαϕ(t)ϕ(t)ω(t)+ϕ(t)Dtαφ(t).
By Lemma 2 and the definition of f we get that
(29)Dtαω(t)≤-Kϕ(t)q(t)-((r(t)xγ+1(t))-1(γϕ(t)δ1(t,t2)a1/γ(t)Dtα(r(t)Dtαx(t))a(t)×(Dtα(r(t)Dtαx(t)))γ)×(r(t)xγ+1(t))-1)+Dtαϕ(t)ϕ(t)ω(t)+ϕ(t)Dtαφ(t)=-Kϕ(t)q(t)-γϕ(t)δ1(t,t2)r(t)[ω(t)ϕ(t)-φ(t)]1+(1/γ)+Dtαϕ(t)ϕ(t)ω(t)+ϕ(t)Dtαφ(t).
Using the following inequality (see [2, Equation (2.17)])
(30)(u-v)1+(1/γ)≥u1+(1/γ)+1γv1+(1/γ)-(1+1γ)v1/γu,
we obtain
(31)[ω(t)ϕ(t)-φ(t)]1+(1/γ)≥ω1+(1/γ)(t)ϕ1+(1/γ)(t)+1γφ1+(1/γ)(t)-(1+1γ)φ1/γ(t)ω(t)ϕ(t).
A combination of (29) and (31) yields:
(32)Dtαω(t)≤-Kϕ(t)q(t)+Dtαϕ(t)ϕ(t)ω(t)+ϕ(t)Dtαφ(t)-γϕ(t)δ1(t,t2)r(t)[ω1+(1/γ)(t)ϕ1+(1/γ)(t)+1γφ1+(1/γ)(t)-(1+1γ)φ1/γ(t)ω(t)ϕ(t)]=-Kϕ(t)q(t)+ϕ(t)Dtαφ(t)-ϕ(t)δ1(t,t2)φ1+(1/γ)(t)r(t)-γϕ(t)δ1(t,t2)r(t)ω1+(1/γ)(t)ϕ1+(1/γ)(t)+(γ+1)φ1/γ(t)ϕ(t)δ1(t,t2)+r(t)Dtαϕ(t)r(t)ϕ(t)×ω(t),t≥t2.
Let ω(t)=ω~(ξ). Then Dtαw(t)=w~′(ξ),Dtαϕ(t)=ϕ~′(ξ),andDtαφ(t)=φ~′(ξ), and (32) is transformed into the following form:
(33)ω~′(ξ)≤-Kϕ~(ξ)q~(ξ)+ϕ~(ξ)φ~′(ξ)-ϕ~(ξ)δ~1(ξ,ξ2)φ~1+(1/γ)(ξ)r(ξ)-γδ~1(ξ,ξ2)r~(ξ)ϕ~1/γ(ξ)ω~1+(1/γ)(ξ)+(γ+1)φ~1/γ(ξ)ϕ~(ξ)δ~1(ξ,ξ2)+r~(ξ)ϕ~′(ξ)r~(ξ)ϕ~(ξ)ω~(ξ),ξ≥ξ2.
Selecta,b,andc 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){ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)}ds≤-∫cξ‍H^(ξ,s)w~′(s)ds+∫cξ‍H^(ξ,s){-γδ~1(s,ξ2)r~(s)ϕ~1/γ(s)ω~1+(1/γ)(s)+((r~(s)ϕ~(s))-1((γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s))×(r~(s)ϕ~(s))-1)ω~(s)γδ~1(s,ξ2)r~(s)ϕ~1/γ(s)}ds=H^(ξ,c)w~(c)+∫cξ‍H^(ξ,s){-γδ~1(s,ξ2)r~(s)ϕ~1/γ(s)ω~1+(1/γ)(s)+((r~(s)ϕ~(s))-1((γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^s′(ξ,s))×(r~(s)ϕ~(s))-1)ω~(s)γδ~1(s,ξ2)r~(s)ϕ~1/γ(s)}ds.
Setting
(35)λ=1+1γ,Aλ=γδ~1(s,ξ2)r~(s)ϕ~1/γ(s)ω1+(1/γ)(s),Bλ-1=γ1/(γ+1)×(((γ+1)[ϕ~(s)δ~1(s,ξ2)]γ/(γ+1)r~1/(γ+1)(s))-1((γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(ξ,s))×((γ+1)[ϕ~(s)δ~1(s,ξ2)]γ/(γ+1)r~1/(γ+1)(s))-1),
by a combination of Lemma 3 and (33), we get that
(36)∫cs‍H^(s,s){ϕ~(s)δ~1(s,s2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,s2)φ~1+(1/γ)(s)r~(s)}ds≤H^(s,c)w~(c)+∫cs‍H^(s,s)×{((γ+1)γ+1[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~(s)H^′s(ξ,s)]γ+1[ϕ~(s)H^′s(ξ,s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(ξ,s)]γ+1)×((γ+1)γ+1[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1ϕ~(s)H^′s(ξ,s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1}ds.
Dividing both sides of inequality (36) by H^(ξ,c) and letting  ξ→b-, we obtain
(37)1H^(b,c)∫cb‍H^(b,s){ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)}ds≤w~(c)+1H^(b,c)∫cb‍H^(b,s)×{×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1[ϕ~(s)H^′s(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(b,s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γ[ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(γ+1)φ~1/γ(s)ϕ~}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], similar to (36)-(37), we get that
(38)∫ξc‍H^(s,ξ){ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)}ds≤-H^(c,ξ)w~(c)+∫ξc‍H^(s,ξ){[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(s,ξ)]γ+1[ϕ~(s)H^′s(s,ξ)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(s,ξ)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1}ds.
Dividing both sides of inequality (38) by H^(c,ξ) and letting ξ→a+, we obtain(39)1H^(c,a)∫ac‍H^(s,a){ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)}ds≤-w~(c)+1H^(c,a)∫ac‍H^(s,a)×{×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~(s)H^′s(s,a)]γ+1[(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(s,a)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)φ~1/γ(s)ϕ~]γr~(s))-1φ~1/γ(s)ϕ~}ds.
A combination of (37) and (39) yields
(40)1H^(b,c)∫cb‍H^(b,s){ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)}ds+1H^(c,a)∫ac‍H^(s,a){ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)}ds≤1H^(b,c)∫cb‍H^(b,s){×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~(s)H^′s(b,s)]γ+1[ϕ~(s)H^′s(b,s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(b,s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1ϕ~(s)H^′s(b,s)}ds+1H^(c,a)∫ac‍H^(s,a)×{×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~(s)H^′s(s,a)]γ+1[ϕ~(s)H^′s(s,a)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(s,a)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1ϕ~(s)H^′s(s,a)}ds,
which contradicts (26). So the proof is complete.Theorem 5.
Under the conditions of Theorem4, if for any sufficiently large l≥ξ0,
(41)limξ→∞sup∫lξ‍H^(s,l){×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-(×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1([+r~(s)ϕ~(s)H^′s(s,l)]γ+1(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(s,l)(γ+1)φ~1/γ(γ+1)φ~1/γ]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)}ds>0,(42)limξ→∞sup∫lξ‍H^(ξ,s){∫lξ‍Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-([ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~(s)H^′s(ξ,s)]γ+1[ϕ~(s)H^′s(ξ,s)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(ξ,s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)∫lξ‍}ds>0,
then (1) is oscillatory.Proof.
For anyT≥ξ0, let a=T. In (41) we choose l=a. Then there exists c>a such that
(43)∫ac‍H^(s,a){×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-([ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(s,a)]γ+1[ϕ~(s)H^′s(s,a)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(s,a)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)}ds>0.
In (42) we choose l=c>a. Then there exists b>c such that
(44)∫cb‍H^(b,s){×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-(×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(b,s)]γ+1[[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+r~(s)ϕ~(s)H^′s(b,s)(γ+1)φ~1/γ]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)}ds>0.
Combining (43) and (44) we obtain (26). 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
(45)1(c-a)λ∫ac‍(s-a)λ{ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)}ds+1(b-c)λ∫cb‍(b-s)λ×{ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)}ds>1(c-a)λ∫ac‍(s-a)λ×{×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+λ(s-a)λ-1r~(s)ϕ~(s)]γ+1[(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+λ(s-a)λ-1r~(s)ϕ~(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s)[ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1)-1}ds+1(b-c)λ×∫cb‍(b-s)λ{(-λ(b-s)λ-1r~(s)ϕ~(s)]γ+1[(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)-λ(b-s)λ-1r~(s)ϕ~(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1}ds,
then (1) is oscillatory.Corollary 7.
Under the conditions of Theorem5, if for any sufficiently large l≥ξ0,
(46)limξ→∞sup∫lξ‍(s-l)λ{×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-([ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(+λ(s-l)λ-1r~(s)ϕ~(s)]γ+1[(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+λ(s-l)λ-1r~(s)ϕ~(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)}ds>0,limξ→∞sup∫lξ‍(ξ-s)λ{×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-((-λ(ξ-s)λ-1r~(s)ϕ~(s)]γ+1[(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)-λ(ξ-s)λ-1r~(s)ϕ~(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)}ds>0,
then (1) is oscillatory.Theorem 8.
Assume (5)–(7) hold, and there exists two functions ϕ∈C1([t0,∞),ℝ+) and φ∈C1([t0,∞),[0,∞)) such that
(47)∫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)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+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~,φ~ are defined as in Theorem 4. Then every solution of (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 that Dtαx(t)>0 on [t2,∞). Let ω(t),ω~(ξ) be defined as in Theorem 4. Then we obtain (32).
Setting(48)λ=1+1γ,Aλ=γϕ(t)δ1(t,t2)r(t)ω1+(1/γ)(t)ϕ1+(1/γ)(t),Bλ-1=γ1/(γ+1)×(γ+1)φ1/γ(t)ϕ(t)δ1(t,t2)+r(t)Dtαϕ(t)(γ+1)[ϕ(t)δ1(t,t2)]γ/(γ+1)r1/(γ+1)(t).
Using Lemma 3 in (32) we get that
(49)Dtαω(t)≤-Kϕ(t)q(t)+ϕ(t)Dtαφ(t)-ϕ(t)δ1(t,t2)φ1+(1/γ)(t)r(t)+[(γ+1)φ1/γ(t)ϕ(t)δ1(t,t2)+r(t)Dtαϕ(t)]γ+1(γ+1)γ+1[ϕ(t)δ1(t,t2)]γr(t),
which is rewritten in the following form:
(50)ω~′(ξ)≤-Kϕ~(ξ)q~(ξ)+ϕ~(ξ)φ~′(ξ)-ϕ~(ξ)δ~1(ξ,ξ2)φ~1+(1/γ)(ξ)r~(ξ)+[(γ+1)φ~1/γ(ξ)ϕ~(ξ)δ~1(ξ,ξ2)+r~(ξ)ϕ~′(ξ)]γ+1(γ+1)γ+1[ϕ~(ξ)δ~1(ξ,ξ2)]γr~(ξ),ξ≥ξ2.
Substitutingξ with s in (50), an integration for (50) with respect to s from ξ2 to ξ yields
(51)∫ξ2ξ‍{[(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)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+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≤ω(ξ2)-ω(ξ)≤ω(ξ2)<∞,
which contradicts (47). So the proof is complete.Theorem 9.
Assume that (5)–(7) hold, and there exists a function H∈C([ξ0,∞),ℝ) such that H(ξ,ξ)=0,forξ≥ξ0,H(ξ,s)>0,forξ>s≥ξ0, and H has a nonpositive continuous partial derivative Hs′(ξ,s). If
(52)limξ→∞sup1H(ξ,ξ0)×{∫ξ0ξ‍H(ξ,s)×{×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-((+r~(s)ϕ~′(s)]γ+1[(s)ϕ~′(s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)}ds∫ξ0ξ‍}=∞,
where ϕ~,φ~ are defined as in Theorem 4, then every solution of (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 that Dtαx(t)>0 on [t2,∞). Let ω(t),ω~(ξ) be defined as in Theorem 4. By (50) we have
(53)Kϕ~(ξ)q~(ξ)-ϕ~(ξ)φ~′(ξ)+ϕ~(ξ)δ~1(ξ,ξ2)φ~1+(1/γ)(ξ)r~(ξ)-[(γ+1)φ~1/γ(ξ)ϕ~(ξ)δ~1(ξ,ξ2)+r~(ξ)ϕ~′(ξ)]γ+1(γ+1)γ+1[ϕ~(ξ)δ~1(ξ,ξ2)]γr~(ξ)≤ω~′(ξ),ξ≥ξ2.
Substituting ξ with s in (53), multiplying both sides by H(ξ,s), and then integrating with respect to s from ξ2 to ξ yield
(54)∫ξ2ξ‍H(ξ,s){((γ+1)γ+1[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-((+r~(s)ϕ~′(s)]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))-1)}ds≤-∫ξ2ξ‍H(ξ,s)ω~′(s)ds=H(ξ,ξ2)ω(ξ2)+∫ξ2ξ‍Hs′(ξ,s)ω(s)Δs≤H(ξ,ξ2)ω(ξ2)≤H(ξ,ξ0)ω(ξ2).
Then
(55)∫ξ0ξ‍H(ξ,s){×((γ+1)γ+1[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-((+r~(s)ϕ~′(s)]γ+1[(s)ϕ~′(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))-1)}ds=∫ξ0ξ2‍H(ξ,s){×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-((+r~(s)ϕ~′(s)]γ+1[(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)}ds+∫ξ2ξ‍H(ξ,s){[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-((+r~(s)ϕ~′(s)]γ+1[(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)}ds≤H(ξ,ξ0)ω~(ξ2)+H(ξ,ξ0)∫ξ0ξ2‍|[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-((+r~(s)ϕ~′(s)]γ+1[r~(s)ϕ~′(s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)|ds.
So
(56)limξ→∞sup1H(ξ,ξ0)×{∫ξ0ξ‍H(ξ,s)×{[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-((+r~(s)ϕ~′(s)]γ+1[r~(s)ϕ~′(s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)}ds∫ξ0ξ‍}≤ω~(ξ2)+∫ξ0ξ2‍|×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-((+r~(s)ϕ~′(s)]γ+1[r~(s)ϕ~′(s)(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)|ds<∞,
which contradicts (52). So the proof is complete.


## 3. Applications of the Results

Example 10.
Consider the following fractional differential equation:(57)Dt1/2[t(Dt1/2(Dt1/2x(t)))5/3]+t3/2Γ3(3/2)[x5/3(t)+x11/3(t)]=0,t≥2.
In (1), if we set t0=2,α=1/2,γ=5/3,a(t)=t,r(t)≡1,q(t)=t3/2/Γ3(3/2), then we obtain (57). So a~(ξ)=a(t)=t=Γ(3/2)ξ, r~(ξ)≡1, and q~(ξ)=ξ3, and f(x)=x5/3+x11/3, which implies that f(x)/x5/3=1+x2≥1=K. Furthermore, δ~1(ξ,ξ2)=∫ξ2ξ‍(1/a~1/γ(s))ds=[Γ(3/2)]-3/5∫ξ2ξ‍s-3/5ds=(5/2)[Γ(3/2)]-3/5(ξ2/5-ξ22/5), which implies that limξ→∞δ~1(ξ,ξ2)=∞. So there exists a sufficiently large T>ξ2 such that δ~1(ξ,ξ2)>1 on [T,∞). Furthermore, one can easily see that (5)–(7) all hold. On the other hand, in (46), after letting ϕ~(ξ)≡1,φ~(ξ)=0,andλ=2, for any sufficiently large l (we may let l≥T without loss of generality), it holds that
(58)limξ→∞sup∫lξ‍(s-l)λ{×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-((+λ(s-l)λ-1r~(s)ϕ~(s)]γ+1[(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)+λ(s-l)λ-1r~(s)ϕ~(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)}ds≥limξ→∞sup∫lξ‍(s-l)2[s3-(34)8/3(s-l)8/3]ds=∞,limξ→∞sup∫lξ‍(ξ-s)λ{[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+ϕ~(s)δ~1(s,ξ2)φ~1+(1/γ)(s)r~(s)-(-λ(ξ-s)λ-1r~(s)ϕ~(s)]γ+1([(γ+1)φ~1/γ(s)ϕ~(s)δ~1(s,ξ2)+r~(s)ϕ~′(s)-λ(ξ-s)λ-1r~(s)ϕ~(s)]γ+1)×([ϕ~(s)δ~1(s,ξ2)]γr~(s)(γ+1)γ+1×[ϕ~(s)δ~1(s,ξ2)]γr~(s))-1)}ds≥limξ→∞sup∫lξ‍(ξ-s)2[s3-(34)8/3(ξ-s)8/3]ds=∞.
So according to Corollary 7 we deduce that (57) is oscillatory.Example 11.
Consider the following fractional differential equation:(59)Dt1/2[t-1/6(Dt1/2(Dt1/2x(t)))1/3]+Γ1/3(3/2)t1/6x1/3(t)[1+ex2(t)]=0,t≥2.
In (1), if we set t0=2,α=1/2,γ=1/3,a(t)=t-1/6,r(t)≡1,q(t)=Γ1/3(3/2)/t1/6, then we obtain (57). So a~(ξ)=a(t)=t-1/6=[Γ(3/2)]-1/3ξ-1/3, r~(ξ)≡1, and q~(ξ)=ξ-1/3, and f(x)=x1/3[1+ex2], which implies that f(x)/x1/3=1+ex2≥1=K. Furthermore, δ~1(ξ,ξ2)=∫ξ2ξ‍(1/a~1/γ(s))ds=∫ξ2ξ‍Γ(3/2)sds=(1/2)Γ(3/2)(ξ2-ξ22), which implies that limξ→∞δ~1(ξ,ξ2)=∞. So there exists a sufficiently large T>ξ2 such that δ~1(ξ,ξ2)>1 on [T,∞). Furthermore, after some computation one can see that (5)–(7) all hold, and in (47), letting ϕ~(ξ)=ξ,φ~(ξ)=0, we obtain
(60)∫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)Kϕ~(s)q~(s)-ϕ~(s)φ~′(s)+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=∫ξ0T‍{s2/3-1(4/3)4/3[sδ~1(s,ξ2)]1/3}ds+∫T∞‍{s2/3-1(4/3)4/3[sδ~1(s,ξ2)]1/3}ds≥∫ξ0T‍{s2/3-1(4/3)4/3[sδ~1(s,ξ2)]1/3}ds+∫T∞‍{s2/3-1(4/3)4/3s-1/3}ds=∞.
Therefore, (59) is oscillatory by Theorem 8.



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*Source: 830836-2013-09-12.xml*