# Interval Oscillation Criteria for a Class of Fractional Differential Equations with Damping Term

Authors: Chunxia Qi; Junmo Cheng
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Some new interval oscillation criteria are established based onthe certain Riccati transformation and inequality technique for a class of fractional differential equations with damping term. For illustrating the validity of the established results, we also present some applications for them.


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## Body

## 1. Introduction

Fractional differential equations are generalizations of classical differential equations of integer order and can find their applications in many fields of science and engineering. In the last few decades, research on 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 only include Chen’s work [9], in which some new oscillation criteria are established for the following fractional differential equation:
(1)[r(t)(D_αy(t))η]′-q(t)f(∫t∞‍(v-t)-αy(v)dv)=0,[r(t)(D_αy(t))η]′-q(t)f(∫t∞‍(v-t)-αybbbb)t>0,
where r,q are positive-valued functions and η is the quotient of two odd positive numbers.In this paper, we are concerned with oscillation of solutions of fractional differential equations of the following form:(2)(a(t)[r(t)D_αx(t)]′)′+p(t)[r(t)D_αx(t)]′-q(t)∫t∞‍(ξ-t)-αx(ξ)dξ=0,t∈[t0,∞),
where a∈C1([t0,∞),ℝ+),r∈C2([t0,∞),ℝ+),p,q∈C([t0,∞),ℝ+),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 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.The organization of the rest of this paper is as follows. In Section2, we establish some new interval oscillation criteria for (2) by a generalized Riccati transformation and inequality technique and 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 [10–12].


## 2. Main Results

For the sake of convenience, in the rest of this paper, we setX(t)=∫t∞‍(ξ-t)-αx(ξ)dξ,A(t)=∫t0t‍(p(s)/a(s))ds,δ1(t,a)=∫at‍(1/eA(s)a(s))ds, and  δ2(t,a)=∫at‍(δ1(s,a)/r(s))ds.Lemma 1.
Assumex is a solution of (2). Then X′(t)=-Γ(1-α)D_αx(t).Lemma 2.
Assumex is an eventually positive solution of (2), and
(3)∫t0∞‍1eA(s)a(s)ds=∞,(4)∫t0∞‍1r(s)ds=∞,(5)∫t0∞‍1r(ξ)∫ξ∞‍1eA(τ)a(τ)∫τ∞‍eA(s)q(s)dsdτdξ=∞.
Then there exists a sufficiently large T such that
(6)[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
(7)(eA(t)a(t)[r(t)D_αx(t)]′)′=eA(t)(a(t)[r(t)D_αx(t)]′)′+eA(t)p(t)[r(t)D_αx(t)]′=eA(t){(a(t)[r(t)D_αx(t)]′)′mmmmmm+p(t)[r(t)D_αx(t)]′(a(t)[r(t)D_αx(t)]′)′}=eA(t)q(t)X(t)>0.
Then eA(t)a(t)[r(t)D_αx(t)]′ is strictly increasing on [t1,∞), and thus [r(t)D_αx(t)]′ is eventually of one sign. We claim [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
(8)r(t)D_αx(t)-r(t3)D_αx(t3)=∫t3t‍eA(s)a(s)[r(s)D_αx(s)]′eA(s)a(s)ds≥eA(t3)a(t3)[r(t3)D_αx(t3)]′∫t3t‍1eA(s)a(s)ds.
By (3), we have
(9)limt⟶∞r(t)D_αx(t)=∞,
which implies for some sufficiently large t4>t3,D_αx(t)>0, and  t∈[t4,∞). By Lemma 1, we have
(10)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 (4), 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 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 β=0. Otherwise, assume β>0. Then X(t)≥β on [t5,∞), and, for t∈[t5,∞), by (7) we have
(11)(eA(t)a(t)[r(t)D_αx(t)]′)′≥eA(t)q(t)X(t)≥βeA(t)q(t).
Substituting t with s in (11), an integration for (11) with respect to s from t to ∞ yields
(12)-eA(t)a(t)[r(t)D_αx(t)]′≥-limt⟶∞eA(t)a(t)[r(t)D_αx(t)]′+β∫t∞‍eA(s)q(s)ds>β∫t5t‍eA(s)q(s)ds,
which means
(13)(r(t)D_αx(t))′<-βeA(t)a(t)∫t∞‍eA(s)q(s)ds.
Substituting t with τ in (13), an integration for (13) with respect to τ from t to ∞ yields
(14)-r(t)D_αx(t)<-limt⟶∞r(t)D_αx(t)<-β∫t∞‍1eA(τ)a(τ)∫τ∞‍eA(s)q(s)dsdτ<-β∫t∞‍1eA(τ)a(τ)∫τ∞‍eA(s)q(s)dsdτ.
That is,
(15)X′(t)<-Γ(1-α)βr(t)∫t∞‍1eA(τ)a(τ)∫τ∞‍eA(s)q(s)dsdτ.
Substituting t with ξ in (15), an integration for (15) with respect to ξ from t5 to t yields
(16)X(t)-X(t5)<-Γ(1-α)β∫t5t‍1r(ξ)∫ξ∞‍1eA(τ)a(τ)nnnnnnnnnnnnnnn×∫τ∞‍eA(s)q(s)dsdτdξ.
By (5), one can see 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
(17)[r(t)D_αx(t)]′<0,D_αx(t)<0on[t1,∞),
where t1≥t0 is sufficiently large. Then we have
(18)X′(t)≥-Γ(1-α)δ1(t,t1)eA(t)a(t)(r(t)D_αx(t))′r(t),(19)X(t)≥-Γ(1-α)δ2(t,t1)eA(t)a(t)[r(t)D_αx(t)]′.Proof.
By Lemma2 we have eA(t)a(t)[r(t)D_αx(t)]′  is strictly increasing on [t1,∞). So
(20)r(t)D_αx(t)≤r(t)D_αx(t)-r(t1)D_αx(t)=∫t1t‍eA(s)a(s)[r(s)D_αx(s)]′eA(s)a(s)ds≤eA(t)a(t)[r(t)D_αx(t)]′∫t1t‍1eA(s)a(s)ds=δ1(t,t1)eA(t)a(t)[r(t)D_αx(t)]′.
Using Lemma 1 we obtain that
(21)X′(t)≥-Γ(1-α)δ1(t,t1)eA(t)a(t)[r(t)D_αx(t)]′r(t).
Then
(22)X(t)≥X(t)-X(t1)≥-∫t1t‍Γ(1-α)δ1(s,t1)eA(s)a(s)[r(s)D_αx(s)]′r(s)ds≥-Γ(1-α)eA(t)a(t)[r(t)D_αx(t)]′∫t1t‍δ1(s,t1)r(s)ds=-Γ(1-α)δ2(t,t1)eA(t)a(t)[r(t)D_αx(t)]′.Theorem 4.
Assume (3)–(5) hold, and there exist two functions ϕ∈C1([t0,∞),ℝ+) and φ∈C1([t0,∞),[0,∞)) such that
(23)∫T∞{ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)+ϕ(s)Γ(1-α)δ1(s,T)φ2(s)r(s)-[2φ(s)ϕ(s)Γ(1-α)δ1(s,T)+r(s)ϕ′(s)]2×(4Γ(1-α)ϕ(s)δ1(s,T)r(s))-1ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)}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:
(24)ω(t)=ϕ(t){-eA(t)a(t)[r(t)D_αx(t)]′X(t)+φ(t)}.
Then for t∈[t2,∞), we have
(25)ω′(t)=-ϕ′(t)eA(t)a(t)[r(t)D_αx(t)]′X(t)+ϕ(t){-eA(t)a(t)[r(t)D_αx(t)]′X(t)}′+ϕ′(t)φ(t)+ϕ(t)φ′(t)=-ϕ(t){(X(t)(eA(t)a(t)[r(t)D_αx(t)]′)′-X′(t)eA(t)a(t)[r(t)Dαx(t)]′X(t)(eA(t)a(t)[r(t)D_αx(t)]′)′)×(X2(t))-1(X(t)(eA(t)a(t)[r(t)D_αx(t)]′)′}+ϕ′(t)ϕ(t)ω(t)+ϕ(t)φ′(t)=-ϕ(t){(X(t){eA(t)(a(t)[r(t)D_αx(t)]′)′mmmmmmmmmmm+eA(t)p(t)[r(t)D_αx(t)]′eA(t)(a(t)[r(t)D_αx(t)]′)′}mmmmmmmm-X′(t)eA(t)a(t)[r(t)Dαx(t)]′X(t){eA(t)(a(t)[r(t)D_αx(t)]′)′)mmmmmmmm×(X2(t))-1(X(t){eA(t)(a(t)[r(t)D_αx(t)]′)′}+ϕ′(t)ϕ(t)ω(t)+ϕ(t)φ′(t)=-ϕ(t)q(t)eA(t)+ϕ(t)X′(t)a(t)eA(t)[r(t)D_αx(t)]′X2(t)+ϕ′(t)ϕ(t)ω(t)+ϕ(t)φ′(t).
By Lemma 3 and the definition of f we get that
(26)ω′(t)≤-ϕ(t)q(t)eA(t)-((r(t)D_αx(t))′a(t)eA(t)(r(t)D_αx(t))′ϕ(t)Γ(1-α)δ1(t,t2)eA(t)a(t)×(r(t)D_αx(t))′a(t)eA(t)(r(t)D_αx(t))′)×(r(t)X2(t))-1+ϕ′(t)ϕ(t)ω(t)+ϕ(t)φ′(t)=-ϕ(t)q(t)eA(t)-ϕ(t)Γ(1-α)δ1(t,t2)r(t)×[ω(t)ϕ(t)-φ(t)]2+ϕ′(t)ϕ(t)ω(t)+ϕ(t)φ′(t)=-ϕ(t)q(t)eA(t)+ϕ(t)φ′(t)-ϕ(t)Γ(1-α)δ1(t,t2)φ2(t)r(t)-ϕ(t)Γ(1-α)δ1(t,t2)r(t)ω2(t)ϕ2(t)+2φ(t)ϕ(t)Γ(1-α)δ1(t,t2)+r(t)ϕ′(t)r(t)ϕ(t)ω(t)≤-ϕ(t)q(t)eA(t)+ϕ(t)φ′(t)-ϕ(t)Γ(1-α)δ1(t,t2)φ2(t)r(t)+[2φ(t)ϕ(t)Γ(1-α)δ1(t,t2)+r(t)ϕ′(t)]24Γ(1-α)ϕ(t)δ1(t,t2)r(t).
Substituting t with s in (26), an integration for (26) with respect to s from t2 to t yields
(27)∫t2t{ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)+ϕ(s)Γ(1-α)δ1(s,t2)φ2(s)r(s)-[2φ(s)ϕ(s)Γ(1-α)δ1(s,t2)+r(s)ϕ′(s)]2×(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1}ds≤ω(t2)-ω(t)≤ω(t2)<∞,
which contradicts (23). So the proof is complete.Theorem 5.
Define𝔻={(t,s)∣t≥s≥t0}. Assume (3)–(5) hold, and there exists a function H∈C1(𝔻,ℝ) such that
(28)H(t,t)=0,fort≥t0,H(t,s)>0,fort>s≥t0,
and H has a nonpositive continuous partial derivative Hs′(t,s), and
(29)limt⟶∞sup1H(t,t0)×{∫t0tH(t,s){ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)nnnnnnnnnnnnnnnn+ϕ(s)Γ(1-α)δ1(s,T)φ2(s)r(s)nnnnnnnnnnnnnnnn-[+r(s)ϕ′(s)2φ(s)ϕ(s)Γ(1-α)nnnnnnnnnnnnnnnnnnn×δ1(s,T)+r(s)ϕ′(s)]2nnnnnnnnnnnnnnnn×(4Γ(1-α)ϕ(s)δ1nnnnnnnnnnnnnnnnnnnn×(s,T)r(s)4Γ(1-α)ϕ(s)δ1)-1ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)}ds{∫t0tH(t,s){ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)}=∞,
for all sufficiently large T, where ϕ,φ are defined as in Theorem 4. 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 D_αx(t)<0 on [t2,∞) for some sufficiently large t2>t1. Let ω(t) be defined as in Theorem 4. By (26) we have
(30)ϕ(t)q(t)eA(t)+ϕ(t)φ′(t)-ϕ(t)Γ(1-α)δ1(t,t2)φ2(t)r(t)cii+[2φ(t)ϕ(t)Γ(1-α)δ1(t,t2)+r(t)ϕ′(t)]24Γ(1-α)ϕ(t)δ1(t,t2)r(t)≤-ω′(t).
Substituting t with s in (30), multiplying both sides by H(t,s), and then integrating it with respect to s from t2 to t yield
(31)∫t2t‍H(t,s)×{(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1[2φ(s)ϕ(s)Γ(1-α)δ1(s,t2)+r(s)ϕ′(s)]24Γ(1-α)ϕ(s)δ1(s,t2)r(s)ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)+ϕ(s)Γ(1-α)δ1(s,t2)φ2(s)r(s)-[2φ(s)ϕ(s)Γ(1-α)δ1(s,t2)+r(s)ϕ′(s)]24Γ(1-α)ϕ(s)δ1(s,t2)r(s)}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
(32)∫t0tH(t,s)×{(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)+ϕ(s)Γ(1-α)δ1(s,t2)φ2(s)r(s)-[×δ1(s,t2)+r(s)ϕ′(s)2φ(s)ϕ(s)Γ(1-α)×δ1(s,t2)+r(s)ϕ′(s)]2×(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)}ds=∫t0t2H(t,s)×{(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1[2φ(s)ϕ(s)Γ(1-α)δ1(s,t2)+r(s)ϕ′(s)]24Γ(1-α)ϕ(s)δ1(s,t2)r(s)ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)i+ϕ(s)Γ(1-α)δ1(s,t2)φ2(s)r(s)-[2φ(s)ϕ(s)Γ(1-α)δ1(s,t2)+r(s)ϕ′(s)]24Γ(1-α)ϕ(s)δ1(s,t2)r(s)}ds+∫t2tH(t,s)×{(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1[2φ(s)ϕ(s)Γ(1-α)δ1(s,t2)+r(s)ϕ′(s)]24Γ(1-α)ϕ(s)δ1(s,t2)r(s)ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)i+ϕ(s)Γ(1-α)δ1(s,t2)φ2(s)r(s)i-[2φ(s)ϕ(s)Γ(1-α)δ1(s,t2)+r(s)ϕ′(s)]24Γ(1-α)ϕ(s)δ1(s,t2)r(s)}i×ds≤H(t,t0)ω(t2)+H(t,t0)×∫t0t2|ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)+(ϕ(s)Γ(1-α)δ1(s,t2)φ2(s))×(r(s))-1-[(s)ϕ′(s)2φ(s)ϕ(s)Γ(1-α)δ1×(s,t2)+r(s)ϕ′(s)]2×(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1|ds.
So
(33)limt⟶∞sup1H(t,t0)×{∫t0tH(t,s){(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)nnnnnnnnnnnnn+(ϕ(s)Γ(1-α)δ1(s,t2)φ2(s))nnnnnnnnnnnnn×(r(s))-1-[×δ1(s,t2)+r(s)ϕ′(s)2φ(s)ϕ(s)Γ(1-α)nnnnnnnnnnnnnnnnnnnnnnnnnnn×δ1(s,t2)+r(s)ϕ′(s)]2nnnnnnnnnnnnn×(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1(s)q(s)eA(s)-ϕ(s)φ′(s)}ds∫t0tH(t,s){(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)}≤ω(t2)+∫t0t2|ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)+ϕ(s)Γ(1-α)δ1(s,t2)φ2(s)r(s)-[2φ(s)ϕ(s)Γ(1-α)δ1(s,t2)+r(s)ϕ′(s)]2×(4Γ(1-α)ϕ(s)δ1(s,t2)r(s))-1|ds<∞,
which contradicts (29). So the proof is complete.In Theorems5, 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 as follows.Corollary 6.
Assume (3)–(5) hold, and
(34)limt⟶∞sup1(t-t0)m×{∫t0t(t-s)m{(4Γ(1-α)ϕ(s)δ1(s,T)r(s))-1ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)nnnnnnnnnnnnnnnn+(ϕ(s)Γ(1-α)δ1(s,T)φ2(s))nnnnnnnnnnnnnnnn×(r(s))-1nnnnnnnnnnnnnnnn-[+r(s)ϕ′(s)2φ(s)ϕ(s)Γ(1-α)δ1(s,T)nnnnnnnnnnnnnnnnnnn+r(s)ϕ′(s)]2nnnnnnnnnnnnnnnn×(4Γ(1-α)ϕ(s)δ1nnnnnnnnnnnnnnnnnnn×(s,T)r(s)4Γ(1-α)ϕ(s)δ1)-1}ds∫t0t(t-s)m{(4Γ(1-α)ϕ(s)δ1(s,T)r(s))-1ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)}=∞,
for all sufficiently large T. Then every solution of (2) is oscillatory or satisfies limt→∞X(t)=0.Corollary 7.
Assume (3)–(5) hold, and
(35)limt⟶∞sup1(lnt-lnt0)×{∫t0t(lnt-lns){δ1(s,T)r(s))-1ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)nnnnnnnnnnnnnnnnn+ϕ(s)Γ(1-α)δ1(s,T)φ2(s)r(s)nnnnnnnnnnnnnnnnn-[+r(s)ϕ′(s)2φ(s)ϕ(s)Γ(1-α)δ1(s,T)nnnnnnnnnnnnnnnnnnnnnn+r(s)ϕ′(s)]2nnnnnnnnnnnnnnnnn×(4Γ(1-α)ϕ(s)nnnnnnnnnnnnnnnnnnnn×δ1(s,T)r(s))-1}ds∫t0t(lnt-lns){δ1(s,T)r(s))-1ϕ(s)q(s)eA(s)-ϕ(s)φ′(s)}=∞,
for all sufficiently large T. Then every solution of (2) is oscillatory or satisfies limt→∞X(t)=0.


## 3. Applications

In this section, we will present some applications for the above established results.Example 8.
Consider(36)(t(D_αx(t))′)′+t-2(D_αx(t))′-Mt-2∫t∞‍(ξ-t)-αx(ξ)dξ=0,t∈[2,∞),
where M>0 is a quotient of two odd positive integers.We have in (2) a(t)=t,p(t)=t-2,q(t)=Mt-2,r(t)=1,andt0=2. Then
(37)1≤eA(t)=e∫2t‍(p(s)/a(s))ds=e∫2t‍s-3ds=e-(1/2)[t-2-2-2]≤e1/8.
Moreover, we have
(38)∫t0∞‍1eA(s)a(s)ds=∫2∞‍1eA(s)a(s)ds≥e-1/8∫2∞‍1sds=∞,∫t0∞‍1r(s)ds=∞.
Furthermore,
(39)∫t0∞‍1r(ξ)∫ξ∞‍1eA(τ)a(τ)∫τ∞‍eA(s)q(s)dsdτdξ=M∫2∞‍∫ξ∞‍1eA(τ)τ∫τ∞‍eA(s)s2dsdτdξ≥Me-1/8∫2∞‍∫ξ∞‍1τ∫τ∞‍1s2dsdτdξ=Me-1/8∫2∞‍∫ξ∞‍1τ2dτdξ=Me-1/8∫2∞‍1ξdξ=∞.
On the other hand, for a sufficiently large T, we have
(40)δ1(t,T)=∫Tt‍1eA(s)a(s)ds=∫Tt‍1eA(s)sds≥e-1/8∫Tt‍1sds⟶∞.
So we can take T*>T such that δ1(t,T)>1 for t∈[T*,∞). Taking ϕ(t)=t,φ(t)=0 in (23), we get that
(41)∫T∞‍{ϕ(s)q(s)eA(s)-[r(s)ϕ′(s)]24[Γ(1-α)ϕ(s)δ1(s,T)]r(s)}ds≥∫T∞‍[M-14Γ(1-α)δ1(s,T)]1sds=∫TT*‍[M-14Γ(1-α)δ1(s,T)]1sdsccci+∫T*∞‍[M-14Γ(1-α)δ1(s,T)]1sds≥∫TT*‍[M-14Γ(1-α)δ1(s,T)]1sdsccci+∫T*∞‍[M-14Γ(1-α)]1sds=∞,
provided that M>1/(4Γ(1-α)). So (3)–(5) and (23) all hold, and by Theorem 4 we deduce that every solution of (36) is oscillatory or satisfies limt→∞X(t)=0 under condition M>1/4Γ(1-α).Example 9.
Consider(42)c(t(D_αx(t))′)′+e-t(D_αx(t))′cc-(Mt-5/2+lnt)∫t∞‍(ξ-t)-αx(ξ)dξ=0,t∈[2,∞),
where α∈(0,1), and M>0 is a constant.We have in (2) a(t)=t,p(t)=e-t,q(t)=Mt-5/2+lnt,r(t)=1,andt0=2. Then
(43)1≤eA(t)=e∫2t‍(p(s)/a(s))ds=e∫2t‍(1/ess)ds≤e∫2t‍s-(3/2)ds=e-2[t-1/2-2-1/2]≤e2.
So we have
(44)∫t0∞‍1eA(s)a(s)ds=∫2∞‍1eA(s)sds≥e-2∫2∞‍1sds=∞,∫t0∞‍1r(s)ds=∞.
Furthermore,
(45)∫t0∞‍1r(ξ)∫ξ∞‍1eA(τ)a(τ)∫τ∞‍eA(s)q(s)dsdτdξ=∫2∞‍∫ξ∞‍1eA(τ)τ∫τ∞‍eA(s)(Ms-5/2+lns)dsdτdξ≥Me-2∫2∞‍∫ξ∞‍1τ∫τ∞‍s-5/2dsdτdξ=2Me-23∫2∞‍[∫ξ∞‍1τ2dτ]dξ=2Me-23∫2∞‍1ξdξ=∞.
On the other hand, Taking ϕ(t)=t2,φ(t)=0,andH(t,s)=t-s in (29), we get that
(46)limt⟶∞sup1t-t0{-[r(s)ϕ′(s)]24Γ(1-α)ϕ(s)δ1(s,T)r(s)}ds∫t0t(t-s)×{[r(s)ϕ′(s)]24Γ(1-α)ϕ(s)δ1(s,T)r(s)ϕ(s)q(s)eA(s)-[r(s)ϕ′(s)]24Γ(1-α)ϕ(s)δ1(s,T)r(s)}ds}≥limt⟶∞sup1t-2×{∫2t(t-s){1Γ(1-α)δ1(s,T)(Ms-5/2+lns)s2nnnnnnnnnnnnnnnnnnnn-1Γ(1-α)δ1(s,T)}ds}≥limt⟶∞sup1t-2{∫2t‍(t-s)Ms-1/2ds}=∞.
So (3)–(5) and (29) all hold, and by Corollary 6 with m=1 we deduce that every solution of (42) is oscillatory or satisfies limt→∞X(t)=0.



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*Source: 301085-2013-04-10.xml*