# Some New Oscillation Criteria for a Class of Nonlinear Fractional Differential Equations with Damping Term

Authors: Bin Zheng; Qinghua Feng
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We are concerned with oscillation of solutions of a class of nonlinear fractional differential equations with damping term. Based on a generalized Riccati function and inequality technique, we establish some new oscillation criteria for it. Some applications are also presented for the established results.


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## Body

## 1. Introduction

Fractional differential equations are generalizations of classical differential equations of integer order, and one can find their applications in many fields of science and engineering. In the last few decades, research for various aspects of fractional differential equations, for example, the existence, uniqueness, and stability of solutions of fractional differential equations, the numerical methods for fractional differential equations, and so on, has been paid much attention by many authors (e.g., we refer the reader to see [1–8] 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 include Chen's work [9], in which some new oscillation criteria are established for the following fractional differential equation:
(1)[r(t)(D-αx(t))η]′-q(t)f(∫t∞‍(v-t)-αy(v)dv)=0,t>0,
where r and q are positive functions and η is a quotient of two odd positive numbers.In this paper, we are concerned with oscillation of solutions of the following nonlinear (2+α)-order fractional differential equation with damping term:
(2)(a(t)[(r(t)D-αx(t))′]γ)′+p(t)[(r(t)D-αx(t))′]γ-q(t)f(∫t∞‍(ξ-t)-αx(ξ)dξ)=0,t∈[t0,∞),
where a∈C1([t0,∞),ℝ+), r∈C2([t0,∞),ℝ+), p,q∈C([t0,∞),ℝ+), f∈C(ℝ,ℝ) satisfying xf(x)>0, f(x)/xγ≥L>0 for x≠0, γ is a quotient of two odd positive integers, α∈(0,1), D-αx(t) denotes the Liouville right-sided fractional derivative of order α of x, and D-αx(t)=-(1/Γ(1-α))(d/dt)∫t∞‍(ξ-t)-αx(ξ)dξ.A nontrivial solution of (2) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (2) is said to be oscillatory in case all its solutions are oscillatory.Motivated by the idea in [10], we will establish some new oscillation criteria for (2) by a generalized Riccati function and inequality technique in Section 2, and we will present some applications for our results in Section 3. Throughout this paper, ℝ denotes the set of real numbers and ℝ+=(0,∞). For more details about the theory of fractional differential equations, we refer the reader to [11–13]. For the sake of convenience, in the rest of this paper, we set X(t)=∫t∞‍(ξ-t)-αx(ξ)dξ, A(t)=∫t0t‍(p(s)/a(s))ds,
(3)θ1(t,a)=∫at‍1[eA(s)a(s)]1/γds,θ2(t,a)=∫at‍θ1(s,a)r(s)ds.


## 2. Main Results

The following lemmas are useful for proving our results.Lemma 1.
Assume thatx is a solution of (2). Then, X′(t)=-Γ(1-α)D-αx(t).Lemma 2.
Assume thatx is an eventually positive solution of (2) and
(4)∫t0∞‍1[eA(s)a(s)]1/γds=∞,(5)∫t0∞‍1r(s)ds=∞,(6)∫t0∞‍1r(ξ)∫ξ∞‍[1eA(τ)a(τ)∫τ∞‍eA(s)q(s)ds]1/γdτdξ=∞.
Then, there exists a sufficiently large T such that
(7)(r(t)D-αx(t))′<0on[T,∞)
and either D-αx(t)<0 on [T,∞) or limt→∞X(t)=0.Proof.
Sincex is an eventually positive solution of (2), there exists t1 such that x(t)>0 on [t1,∞). So X(t)>0 on [t1,∞) and we have
(8)(eA(t)a(t)[(r(t)D-αx(t))′]γ)′=eA(t)(a(t)[(r(t)D-αx(t))′]γ)′+eA(t)p(t)[(r(t)D-αx(t))′]γ=eA(t){(a(t)[(r(t)D-αx(t))′]γ)′+p(t)[(r(t)D-αx(t))′]γ}=eA(t)q(t)f(X(t))≥LeA(t)q(t)Xγ(t)>0.
Then, eA(t)a(t)[(r(t)D-αx(t))′]γ is strictly increasing on [t1,∞), and thus (r(t)D-αx(t))′ is eventually of one sign. We claim that (r(t)D-αx(t))′<0 on [t2,∞), where t2>t1 is sufficiently large. Otherwise, assume there exists a sufficiently large t3>t2 such that (r(t)D-αx(t))′>0 on [t3,∞). Then, for t∈[t3,∞), we have
(9)r(t)D-αx(t)-r(t3)D-αx(t3)=∫t3t‍[eA(s)a(s)]1/γ(r(s)D-αx(s))′[eA(s)a(s)]1/γds≥[eA(t3)a(t3)]1/γ(r(t3)D-αx(t3))′×∫t3t‍1[eA(s)a(s)]1/γds.
By (4), we have
(10)limt→∞r(t)D-αx(t)=∞,
which implies, for some sufficiently large t4>t3, D-αx(t)>0, t∈[t4,∞). By Lemma 1, we have
(11)X(t)-X(t4)=∫t4t‍X′(s)ds=-Γ(1-α)∫t4t‍D-αx(s)ds=-Γ(1-α)∫t4t‍r(s)D-αx(s)r(s)ds≤-Γ(1-α)r(t4)D-αx(t4)∫t4t‍1r(s)ds.
By (5), we obtain limt→∞X(t)=-∞, which contradicts X(t)>0 on [t1,∞). So (r(t)D-αx(t))′<0 on [t2,∞). Thus, D-αx(t) is eventually of one sign. Now, we assume that D-αx(t)>0, t∈[t5,∞), for some sufficiently t5>t4. Then, by Lemma 1, X′(t)<0 for t∈[t5,∞). Since X(t)>0, furthermore we have limt→∞X(t)=β≥0. We claim that β=0. Otherwise, assume that β>0. Then, X(t)≥β on [t5,∞), and, for t∈[t5,∞), by (8) we have
(12)(eA(t)a(t)[(r(t)D-αx(t))′]γ)′≥LeA(t)q(t)Xγ(t)≥LβγeA(t)q(t).
Substituting t with s in (12), an integration for (12) with respect to s from t to ∞ yields
(13)-eA(t)a(t)[(r(t)D-αx(t))′]γ≥-limt→∞eA(t)a(t)[(r(t)D-αx(t))′]γ+Lβγ∫t∞‍eA(s)q(s)ds>Lβγ∫t∞‍eA(s)q(s)ds;
which means
(14)(r(t)D-αx(t))′<-L1/γβ[1eA(t)a(t)∫t∞‍eA(s)q(s)ds]1/γ.
Substituting t with τ in (14), an integration for (14) with respect to τ from t to ∞ yields
(15)-r(t)D-αx(t)<-limt→∞r(t)D-αx(t)-L1/γβ∫t∞‍[1eA(τ)a(τ)∫τ∞‍eA(s)q(s)ds]1/γdτ<-L1/γβ∫t∞‍[1eA(τ)a(τ)∫τ∞‍eA(s)q(s)ds]1/γdτ;
that is,
(16)X′(t)<-L1/γΓ(1-α)β1r(t)×∫t∞‍[1eA(τ)a(τ)∫τ∞‍eA(s)q(s)ds]1/γdτ.
Substituting t with ξ in (16), an integration for (16) with respect to ξ from t5 to t yields
(17)X(t)-X(t5)<-L1/γΓ(1-α)β×∫t5t‍1r(ξ)∫ξ∞‍[1eA(τ)a(τ)∫τ∞‍eA(s)q(s)ds]1/γdτdξ.
By (6), one can see that limt→∞X(t)=-∞, which is a contradiction. So the proof is complete.Lemma 3.
Assume thatx is an eventually positive solution of (2) such that
(18)(r(t)D-αx(t))′<0,D-αx(t)<0on[t1,∞)𝕋,
where t1≥t0 is sufficiently large. Then, one has
(19)X′(t)≥-Γ(1-α)θ1(t,t1)[eA(t)a(t)]1/γ(r(t)D-αx(t))′r(t),X(t)≥-Γ(1-α)θ2(t,t1)[eA(t)a(t)]1/γ(r(t)D-αx(t))′.Proof.
By Lemma2, we obtain that eA(t)a(t)[(r(t)D-αx(t))′]γ is strictly increasing on [t1,∞). So
(20)r(t)D-αx(t)≤r(t)D-αx(t)-r(t1)D-αx(t1)=∫t1t‍[eA(s)a(s)]1/γ[r(s)D-αx(s)]′[eA(s)a(s)]1/γds≤[eA(t)a(t)]1/γ(r(t)D-αx(t))′∫t1t‍1[eA(s)a(s)]1/γds=θ1(t,t1)[eA(t)a(t)]1/γ(r(t)D-αx(t))′.
Using Lemma 1, we obtain that
(21)X′(t)≥-Γ(1-α)θ1(t,t1)[eA(t)a(t)]1/γ(r(t)D-αx(t))′r(t).
Then,
(22)X(t)≥X(t)-X(t1)≥-∫t1t‍Γ(1-α)θ1(s,t1)[eA(s)a(s)]1/γ(r(t)D-αx(s))′r(s)ds≥-Γ(1-α)[eA(t)a(t)]1/γ(r(t)D-αx(t))′∫t1t‍θ1(s,t1)r(s)ds=-Γ(1-α)θ2(t,t1)[eA(t)a(t)]1/γ(r(t)D-αx(t))′.Lemma 4 ([14, Theorem 41]).
Assume thatA and B are nonnegative real numbers. Then,
(23)λABλ-1-Aλ≤(λ-1)Bλ,∀λ>1.Theorem 5.
Assume (4)–(6) hold and there exist two functions ρ∈C1([t0,∞),ℝ+) and η∈C1([t0,∞),[0,∞)) such that
(24)∫T∞‍{((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,T)]γr(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+ρ(s)Γ(1-α)θ1(s,T)η1+(1/γ)(s)r(s)-(+r(s)ρ′(s)]γ+1[(γ+1)η1/γ(s)ρ(s)Γ(1-α)θ1(s,T)+r(s)ρ′(s)]γ+1)×((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,T)]γr(s))-1}ds=∞,
for all sufficiently large T. Then, every solution of (2) is oscillatory or satisfies limt→∞X(t)=0.Proof.
Assume (2) 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 2, we have (r(t)D-αx(t))′<0, t∈[t2,∞), where t2>t1 is sufficiently large, and either D-αx(t)<0 on [t2,∞) or limt→∞X(t)=0. Define the generalized Riccati function as follows:
(25)ω(t)=ρ(t){-eA(t)a(t)[(r(t)D-αx(t))′]γXγ(t)+η(t)}.
Then, for t∈[t2,∞), we have
(26)ω′(t)=-ρ′(t)eA(t)a(t)[(r(t)D-αx(t))′]γXγ(t)+ρ(t){-eA(t)a(t)[(r(t)D-αx(t))′]γXγ(t)}′+ρ′(t)η(t)+ρ(t)η′(t)=-ρ(t){(Xγ(t)(eA(t)a(t)[(r(t)D-αx(t))′]γ)′-γXγ-1(t)X′(t)eA(t)a(t)[(r(t)D-αx(t))′]γ(eA(t)a(t)[(r(t)Dαx(t))′]γ)′)×(X2γ(t))-1(Xγ(t)(eA(t)a(t)[(r(t)D-αx(t))′]γ)′}+ρ′(t)ρ(t)ω(t)+ρ(t)η′(t)=-ρ(t){γXγ-1(t)X′(t)eA(t)a(t)[(r(t)D-αx(t))′]γX2γ(t)(Xγ(t){eA(t)(a(t)[(r(t)D-αx(t))′]γ)′+eA(t)p(t)[(r(t)D-αx(t))′]γ(Xγ(t){eA(t)(a(t)[(r(t)Dαx(t))′]γ)′})×(X2γ(t))-1-γXγ-1(t)X′(t)eA(t)a(t)[(r(t)D-αx(t))′]γX2γ(t)}+ρ′(t)ρ(t)ω(t)+ρ(t)η′(t)=-ρ(t)q(t)eA(t)f(X(t))Xγ(t)+γρ(t)X′(t)a(t)eA(t)[(r(t)D-αx(t))′]γXγ+1(t)+ρ′(t)ρ(t)ω(t)+ρ(t)η′(t).
By Lemma 3 and the definition of f, we get that
(27)ω′(t)≤-Lρ(t)q(t)eA(t)-(γρ(t)Γ(1-α)θ1(t,t2)[eA(t)a(t)]1/γ×(r(t)D-αx(t))′a(t)eA(t)[(r(t)D-αx(t))′]γ[eA(t)a(t)]1/γ)×(r(t)Xγ+1(t))-1+ρ′(t)ρ(t)ω(t)+ρ(t)η′(t)=-Lρ(t)q(t)eA(t)-γρ(t)Γ(1-α)θ1(t,t2)r(t)×[-(ω(t)ρ(t)-η(t))]1+(1/γ)+ρ′(t)ρ(t)ω(t)+ρ(t)η′(t)=-Lρ(t)q(t)eA(t)-γρ(t)Γ(1-α)θ1(t,t2)r(t)×[ω(t)ρ(t)-η(t)]1+(1/γ)+ρ′(t)ρ(t)ω(t)+ρ(t)η′(t).
Using the following inequality (see [15, Equation (2.17)])
(28)(u-v)1+(1/γ)≥u1+(1/γ)+1γv1+(1/γ)-(1+1γ)v1/γu,
we obtain
(29)[ω(t)ρ(t)-η(t)]1+(1/γ)≥ω1+(1/γ)(t)ρ1+(1/γ)(t)+1γη1+(1/γ)(t)-(1+1γ)η1/γ(t)ω(t)ρ(t).
A combination of (27) and (29) yields the following:
(30)ω′(t)≤-Lρ(t)q(t)eA(t)+ρ′(t)ρ(t)ω(t)+ρ(t)η′(t)-γρ(t)Γ(1-α)θ1(t,t2)r(t)×[ω1+(1/γ)(t)ρ1+(1/γ)(t)+1γη1+(1/γ)(t)-(1+1γ)η1/γ(t)ω(t)ρ(t)]=-Lρ(t)q(t)eA(t)+ρ(t)η′(t)-ρ(t)Γ(1-α)θ1(t,t2)η1+(1/γ)(t)r(t)-γρ(t)Γ(1-α)θ1(t,t2)r(t)ω1+(1/γ)(t)ρ1+(1/γ)(t)+(γ+1)η1/γ(t)ρ(t)Γ(1-α)θ1(t,t2)+r(t)ρ′(t)r(t)ρ(t)ω(t).
Setting
(31)λ=1+1γ,Aλ=γρ(t)Γ(1-α)θ1(t,t2)r(t)ω1+(1/γ)(t)ρ1+(1/γ)(t),Bλ-1=γ1/(γ+1)×(γ+1)η1/γ(t)ρ(t)Γ(1-α)θ1(t,t2)+r(t)ρ′(t)(γ+1)[Γ(1-α)ρ(t)θ1(t,t2)]γ/(γ+1)r1/(γ+1)(t).
Using Lemma 4 in (30), we get that
(32)ω′(t)≤-Lρ(t)q(t)eA(t)+ρ(t)η′(t)-ρ(t)Γ(1-α)θ1(t,t2)η1+(1/γ)(t)r(t)+[(γ+1)η1/γ(t)ρ(t)Γ(1-α)θ1(t,t2)+r(t)ρ′(t)]γ+1(γ+1)γ+1[Γ(1-α)ρ(t)θ1(t,t2)]γr(t).
Substituting t with s in (32), an integration for (32) with respect to s from t2 to t yields
(33)∫t2t‍{((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+ρ(s)Γ(1-α)θ1(s,t2)η1+(1/γ)(s)r(s)-(+r(s)ρ′(s)]γ+1[(γ+1)η1/γ(s)ρ(s)Γ(1-α)θ1(s,t2)+r(s)ρ′(s)]γ+1)×((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1}ds≤ω(t2)-ω(t)≤ω(t2),
which contradicts (24). So the proof is complete.Theorem 6.
Assume (4)–(6) hold and, for all sufficiently large T,
(34)∫T∞‍{(4γ[Γ(1-α)]γθ1(s,T)θ2γ-1(s,T)r(s)ρ(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+γρ(s)[Γ(1-α)]γθ1(s,T)θ2γ-1(s,T)η2(s)r(s)-(+r(s)ρ′(s)}2{2γρ(s)η(s)[Γ(1-α)]γθ1(s,T)θ2γ-1(s,T)+r(s)ρ′(s)}2)×(4γ[Γ(1-α)]γθ1(s,T)θ2γ-1(s,T)r(s)ρ(s))-1}ds=∞,
where ρ and η are defined as in Theorem 5. Then, every solution of (2) is oscillatory or satisfies limt→∞X(t)=0.Proof.
Assume (2) 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 2, we have (r(t)D-αx(t))′<0,t∈[t2,∞), where t2>t1 is sufficiently large, and either D-αx(t)<0 on [t2,∞) or limt→∞X(t)=0. Let ω(t) be defined as in Theorem 5. Proceeding as in Theorem 5, we obtain (26). By Lemma 3, we have the following observation:
(35)X′(t)X(t)≥-Γ(1-α)θ1(t,t2)[eA(t)a(t)]1/γ(r(t)D-αx(t))′r(t)X(t)=-Γ(1-α)θ1(t,t2)[eA(t)a(t)]1/γ(r(t)D-αx(t))′r(t)Xγ(t)×Xγ-1(t)≥-Γ(1-α)θ1(t,t2)[eA(t)a(t)]1/γ(r(t)D-αx(t))′r(t)Xγ(t)×{-Γ(1-α)θ2(t,t2)[eA(t)a(t)]1/γ(r(t)D-αx(t))′}γ-1=-[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)r(t)×{eA(t)a(t)[(r(t)D-αx(t))′]γXγ(t)}.
Using (35) in (26) we get that
(36)ω′(t)≤-Lρ(t)q(t)eA(t)-γρ(t)[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)r(t)×{eA(t)a(t)[(r(t)D-αx(t))′]γXγ(t)}2+ρ′(t)ρ(t)ω(t)+ρ(t)η′(t)=-Lρ(t)q(t)eA(t)-γρ(t)[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)r(t)×[ω(t)ρ(t)-η(t)]2+ρ′(t)ρ(t)ω(t)+ρ(t)η′(t)=-Lρ(t)q(t)eA(t)-γρ(t)[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)η2(t)r(t)-γ[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)r(t)ρ(t)ω2(t)+2γρ(t)η(t)[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)+r(t)ρ′(t)r(t)ρ(t)×ω(t)+ρ(t)η′(t)≤-Lρ(t)q(t)eA(t)+ρ(t)η′(t)-γρ(t)[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)η2(t)r(t)+({+r(t)ρ′(t)}22γρ(t)η(t)[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)+r(t)ρ′(t)}2)×(4γ[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)r(t)ρ(t))-1.
Substituting t with s in (36), an integration for (36) with respect to s from t2 to t yields
(37)∫t2t‍{Lρ(s)q(s)eA(s)-ρ(s)η′(s)+γρ(s)[Γ(1-α)]γθ1(s,t2)θ2γ-1(s,t2)η2(s)r(s)-({+r(s)ρ′(s)}22γρ(s)η(s)[Γ(1-α)]γθ1(s,t2)θ2γ-1(s,t2)+r(s)ρ′(s)}2+r(s)ρ′(s)}2)×(4γ[Γ(1-α)]γθ1(s,t2)θ2γ-1(s,t2)r(s)ρ(s))-1}ds≤ω(t2)-ω(t)≤ω(t2),
which contradicts (34). So the proof is complete.Theorem 7.
Define𝔻={(t,s)∣t≥s≥t0}. Assume (4)–(6) hold and there exists a function H∈C1(𝔻,ℝ) such that
(38)H(t,t)=0,fort≥t0,H(t,s)>0,fort>s≥t0,H has a nonpositive continuous partial derivative Hs′(t,s), and
(39)limt→∞sup1H(t,t0)×{∫t0t‍H(t,s)×{r(s)(γ+1)γ+1)-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+ρ(s)Γ(1-α)θ1(s,T)η1+(1/γ)(s)r(s)-(+r(s)ρ′(s)]γ+1[(γ+1)η1/γ(s)ρ(s)Γ(1-α)θ1(s,T)+r(s)ρ′(s)]γ+1)×((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,T)]γ×r(s)(γ+1)γ+1)-1}ds∫t0t‍}=∞,
for all sufficiently large T, where ρ and η are defined as in Theorem 5. Then, every solution of (2) is oscillatory or satisfies limt→∞X(t)=0.Proof.
Assume (2) 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 2, we have either D-αx(t)<0 on [t2,∞), for some sufficiently large t2>t1, or limt→∞X(t)=0. Now we assume D-αx(t)<0. Let ω(t) be defined as in Theorem 5. By (32), we have
(40)Lρ(t)q(t)eA(t)-ρ(t)η′(t)+ρ(t)Γ(1-α)θ1(t,t2)η1+(1/γ)(t)r(t)-[(γ+1)η1/γ(t)ρ(t)Γ(1-α)θ1(t,t2)+r(t)ρ′(t)]γ+1(γ+1)γ+1[Γ(1-α)ρ(t)θ1(t,t2)]γr(t)≤-ω′(t).
Substituting t with s in (40), multiplying both sides by H(t,s), and then integrating with respect to s from t2 to t yield
(41)∫t2t‍H(t,s)×{((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+ρ(s)Γ(1-α)θ1(s,t2)η1+(1/γ)(s)r(s)-([(γ+1)η1/γ(s)ρ(s)Γ(1-α)θ1(s,t2)+r(s)ρ′(s)]γ+1+r(s)ρ′(s)]γ+1)×((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1}ds≤-∫t2t‍H(t,s)ω′(s)ds=H(t,t2)ω(t2)+∫t2t‍Hs′(t,s)ω(s)Δs≤H(t,t2)ω(t2)≤H(t,t0)ω(t2).
Then
(42)∫t0t‍H(t,s)×{((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+ρ(s)Γ(1-α)θ1(s,t2)η1+(1/γ)(s)r(s)-(+r(s)ρ′(s)]γ+1[(γ+1)η1/γ(s)ρ(s)Γ(1-α)θ1(s,t2)+r(s)ρ′(s)]γ+1)×((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1}ds=∫t0t2‍H(t,s)×{((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+ρ(s)Γ(1-α)θ1(s,t2)η1+(1/γ)(s)r(s)-([(γ+1)η1/γ(s)ρ(s)Γ(1-α)θ1(s,t2)+r(s)ρ′(s)]γ+1+r(s)ρ′(s)]γ+1)×((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1}ds+∫t2t‍H(t,s)×{((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+ρ(s)Γ(1-α)θ1(s,t2)η1+(1/γ)(s)r(s)-([(γ+1)η1/γ(s)ρ(s)Γ(1-α)θ1(s,t2)+r(s)ρ′(s)]γ+1+r(s)ρ′(s)]γ+1)×((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1}ds≤H(t,t0)ω(t2)+H(t,t0)×∫t0t2‍|((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+ρ(s)Γ(1-α)θ1(s,t2)η1+(1/γ)(s)r(s)-([(γ+1)η1/γ(s)ρ(s)Γ(1-α)θ1(s,t2)+r(s)ρ′(s)]γ+1+r(s)ρ′(s)]γ+1)×((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1|ds.
So
(43)limt→∞sup1H(t,t0)×{∫t0t‍H(t,s)×{r(s)(γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γ)-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+ρ(s)Γ(1-α)θ1(s,t2)η1+(1/γ)(s)r(s)-([(γ+1)η1/γ(s)ρ(s)Γ(1-α)θ1(s,t2)+r(s)ρ′(s)]γ+1+r(s)ρ′(s)]γ+1)×((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γ×r(s)(γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γ)-1}ds∫t0t‍}≤ω(t2)+∫t0t2‍|((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+ρ(s)Γ(1-α)θ1(s,t2)η1+(1/γ)(s)r(s)-([(γ+1)η1/γ(s)ρ(s)Γ(1-α)θ1(s,t2)+r(s)ρ′(s)]γ+1+r(s)ρ′(s)]γ+1)×((γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,t2)]γr(s))-1|ds<∞,
which contradicts (39). So the proof is complete.Theorem 8.
LetH, ρ, and η be defined as in Theorem 7. If (4)–(6) hold and
(44)limt→∞sup1H(t,t0)×∫t0t‍H(t,s)×{×(s,T)r(s)ρ(s)4γ[Γ(1-α)]γθ1(s,T)θ2γ-1)-1}dsLρ(s)q(s)eA(s)-ρ(s)η′(s)4γ[Γ(1-α)]γθ1(s,T)θ2γ-1+γρ(s)[Γ(1-α)]γθ1(s,T)θ2γ-1(s,T)η2(s)r(s)-({2γρ(s)η(s)[Γ(1-α)]γθ1(s,T)θ2γ-1(s,T)+r(s)ρ′(s)}2×(s,T)r(s)ρ(s)4γ[Γ(1-α)]γθ1(s,T)θ2γ-1)-1+r(s)ρ′(s)}2)×(4γ[Γ(1-α)]γθ1(s,T)θ2γ-1(s,T)r(s)ρ(s))-1}ds=∞,
for all sufficiently large T, then every solution of (2) is oscillatory or satisfies limt→∞X(t)=0.Proof.
Assume (2) 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 2, we have either D-αx(t)<0 on [t2,∞), for some sufficiently large t2>t1, or limt→∞X(t)=0. Now, we assume D-αx(t)<0. Let ω(t) be defined as in Theorem 5. By (36), we have
(45)Lρ(t)q(t)eA(t)-ρ(t)η′(t)+γρ(t)[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)η2(t)r(t)-{2γρ(t)η(t)[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)+r(t)ρ′(t)}24γ[Γ(1-α)]γθ1(t,t2)θ2γ-1(t,t2)r(t)ρ(t)≤-ω′(t).
Substituting t with s in (45), multiplying both sides by H(t,s), and then integrating with respect to s from t2 to t yield
(46)∫t2t‍H(t,s)×{×(4γ[Γ(1-α)]γθ1(s,t2)θ2γ-1(s,t2)r(s)ρ(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+γρ(s)[Γ(1-α)]γθ1(s,t2)θ2γ-1(s,t2)η2(s)r(s)-({2γρ(s)η(s)[Γ(1-α)]γθ1(s,t2)θ2γ-1(s,t2)+r(s)ρ′(t)}2+r(s)ρ′(t)}2)×(4γ[Γ(1-α)]γθ1(s,t2)θ2γ-1(s,t2)r(s)ρ(s))-1}ds≤-∫t2t‍H(t,s)ω′(s)ds=H(t,t2)ω(t2)+∫t2t‍Hs′(t,s)ω(s)Δs≤H(t,t2)ω(t2)≤H(t,t0)ω(t2).
Then, similar to the process of Theorem 7, we get that
(47)limt→∞sup1H(t,t0)×{∫t0t‍H(t,s)×{×(4γ[Γ(1-α)]γθ1(s,t2)θ2γ-1(s,t2)r(s)ρ(s))-1Lρ(s)q(s)eA(s)-ρ(s)η′(s)+γρ(s)[Γ(1-α)]γθ1(s,t2)θ2γ-1(s,t2)η2(s)r(s)-({2γρ(s)η(s)[Γ(1-α)]γθ1(s,t2)θ2γ-1×θ2γ-1(s,t2)+r(s)ρ′(s)}2×θ2γ-1(s,t2)+r(s)ρ′(s)}2)×(4γ[Γ(1-α)]γθ1(s,t2)θ2γ-1×(s,t2)r(s)ρ(s)4γ[Γ(1-α)]γθ1(s,t2)θ2γ-1)-1}ds∫t0t‍H(t,s)}<∞,
which contradicts (44). So the proof is complete.Remark 9.
In Theorems7 and 8, if we take H(t,s) for some special functions such as (t-s)m or ln(t/s), then we can obtain some corollaries, which are omitted here.Remark 10.
The established oscillation criteria for (2) above are new results so far in the literature to the best of our knowledge.


## 3. Applications

In this section, we will present some applications for the established results above.Example 1.
Consider the following:(48)(t5/3[(D-αx(t))′]5/3)′+t-8/3[(D-αx(t))′]5/3-t-8/3[M+e∫t∞‍(ξ-t)-αx(ξ)dξ]×(∫t∞‍(ξ-t)-αx(ξ)dξ)5/3=0,t∈[2,∞),
where M>0 is a constant.
We have in (2) γ=5/3, a(t)=t5/3, p(t)=q(t)=t-8/3, f(x)=x5/3[ex+M], r(t)=1, t0=2. Then, f(x)/xγ≥M=L. Moreover, 1≤eA(t)=e∫2t(p(s)/a(s))‍ds=e∫2t‍s-13/3ds=e-(3/10)[t-10/3-2-10/3]≤e. Then, we have
(49)∫t0∞‍1[eA(s)a(s)]1/γds=∫2∞‍1[eA(s)a(s)]3/5ds≥e-3/5∫2∞‍1sds=∞,∫t0∞‍1r(s)ds=∞.
Furthermore,
(50)∫t0∞‍1r(ξ)∫ξ∞‍[1eA(τ)a(τ)∫τ∞‍eA(s)q(s)ds]1/γdτdξ=∫2∞‍∫ξ∞‍[1eA(τ)τ5/3∫τ∞‍eA(s)s-8/3ds]3/5dτdξ≥e-3/5∫2∞‍∫ξ∞‍[1τ5/3∫τ∞‍s-8/3ds]3/5dτdξ=(5e3)-3/5∫2∞‍[∫ξ∞‍1τ2dτ]dξ=(5e3)-3/5∫2∞‍1ξdξ=∞.
On the other hand, for a sufficiently large T, we have
(51)θ1(t,T)=∫Tt‍1[eA(s)a(s)]1/γds=∫Tt‍1[eA(s)s5/3]3/5ds≥e-3/5∫Tt‍1sds⟶∞.
So we can take T*>T such that θ1(t,T)>1 for t∈[T*,∞). Taking ρ(t)=t5/3 and η(t)=0 in (24), we get that
(52)∫T∞‍{[r(s)ρ′(s)]γ+1(γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,T)]γr(s)}Lρ(s)q(s)-[r(s)ρ′(s)]γ+1(γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,T)]γr(s)}ds=∫T∞‍{M-(58)8/31[Γ(1-α)θ1(s,T)]5/3}1sds=∫TT*‍{M-(58)8/31[Γ(1-α)θ1(s,T)]5/3}1sds+∫T*∞‍{M-(58)8/31[Γ(1-α)θ1(s,T)]5/3}1sds≥∫TT*‍{M-(58)8/31[Γ(1-α)θ1(s,T)]5/3}1sds+∫T*t‍{M-(58)8/31[Γ(1-α)]5/3}1sds⟶∞,
provided that M>(5/8)8/3(1/[Γ(1-α)]5/3). So (4)–(6) and (24) all hold, and by Theorem 5 we deduce that every solution of (48) is oscillatory or satisfies limt→∞X(t)=0 under the condition M>(5/8)8/3(1/[Γ(1-α)]5/3).Example 2.
Consider the following:(53)(t3[(D-αx(t))′]3)′+t-4[(D-αx(t))′]3-Mt-4[∫t∞‍(ξ-t)-αx(ξ)dξ]3=0,t∈[2,∞),
where α∈(0,1) and M>0 is a constant.
We have in (2) γ=3, a(t)=t3, p(t)=q(t)=t-4, f(x)=Mx3, r(t)=1, t0=2. Then, f(x)/xγ≥M=L. Moreover, 1≤eA(t)=e∫2t‍(p(s)/a(s))ds=e∫2t‍s-7ds=e-(1/6)[t-6-2-6]≤e1/384. Then, we have
(54)∫t0∞‍1[eA(s)a(s)]1/γds=∫2∞‍1[eA(s)a(s)]1/3ds≥e-1/1152∫2∞‍1sds=∞,∫t0∞‍1r(s)ds=∞.
Furthermore,
(55)∫t0∞‍1r(ξ)∫ξ∞‍[1eA(τ)a(τ)∫τ∞‍eA(s)q(s)ds]1/γdτdξ=∫2∞‍∫ξ∞‍[1eA(τ)τ3∫τ∞‍eA(s)s-4ds]1/3dτdξ≥e-1/1152∫2∞‍∫ξ∞‍[1τ3∫τ∞‍s-4ds]1/3dτdξ=e-1/115233∫2∞‍[∫ξ∞‍1τ2dτ]dξ=e-1/115233∫2∞‍1ξdξ=∞.
On the other hand, for a sufficiently large T, we have
(56)θ1(t,T)=∫Tt‍1[eA(s)a(s)]1/γds=∫Tt‍1[eA(s)s3]1/3ds≥e-1/1152∫Tt‍1sds⟶∞.
So we can take T*>T such that θ1(t,T)>1 for t∈[T*,∞). Taking ρ(t)=t3, η(t)=0, H(t,s)=t-s in (39), we get that
(57)limt→∞sup1t-t0×{[r(s)ρ′(s)]γ+1(γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,T)]γr(s)∫t0t‍(t-s)×{[r(s)ρ′(s)]γ+1(γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,T)]γr(s)Lρ(s)q(s)-[r(s)ρ′(s)]γ+1(γ+1)γ+1[Γ(1-α)ρ(s)θ1(s,T)]γr(s)}ds}=limt→∞sup1t-2×{{M-(34)41[Γ(1-α)θ1(s,T)]3}∫2t‍(t-s)×{M-(34)41[Γ(1-α)θ1(s,T)]3}1sds}=limt→∞sup1t-2×{{M-(34)41[Γ(1-α)θ1(s,T)]3}∫2T*‍(t-s)×{M-(34)41[Γ(1-α)θ1(s,T)]3}1sds+∫T*t‍(t-s)×{M-(34)41[Γ(1-α)θ1(s,T)]3}1sds}≥limt→∞sup1t-2×{∫2T*‍(t-s){M-(34)41[Γ(1-α)]3}1sds+∫T*t‍(t-s){M-(34)41[Γ(1-α)]3}1sds∫2T*‍{M-(34)41[Γ(1-α)]3}}=∞,
provided that M>(3/4)4(1/[Γ(1-α)]3). So (4)–(6) and (39) all hold, and by Theorem 7 we deduce that every solution of (48) is oscillatory or satisfies limt→∞X(t)=0 under the condition M>(3/4)4(1/[Γ(1-α)]3).



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