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

Abstract

Some new interval oscillation criteria are established based on the 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.

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: 𝑟(𝑡)𝐷^𝛼_{_}𝑦(𝑡)^𝜂′−𝑞(𝑡)𝑓^∞_𝑡(𝑣−𝑡)^−𝛼𝑦(𝑣)𝑑𝑣=0,𝑡>0, (1) 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: 𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′′+𝑝(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′−𝑞(𝑡)^∞_𝑡(𝜉−𝑡)^−𝛼𝑥(𝜉)𝑑𝜉=0,𝑡∈𝑡₀,∞, (2) where 𝑎∈𝐶¹([𝑡₀,∞),ℝ₊),𝑟∈𝐶²([𝑡₀,∞),ℝ₊),𝑝,𝑞∈𝐶([𝑡₀,∞),ℝ₊),𝛼∈(0,1),𝐷^𝛼_{_}𝑥(𝑡) denotes the Liouville right-sided fractional derivative of order 𝛼 of 𝑥, and 𝐷^𝛼_{_}𝑥(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 Section 2, 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 set 𝑋(𝑡)=∫^∞_𝑡(𝜉−𝑡)^−𝛼𝑥(𝜉)𝑑𝜉,𝐴(𝑡)=∫^𝑡_𝑡0(𝑝(𝑠)/𝑎(𝑠))𝑑𝑠,𝛿₁(𝑡,𝑎)=∫^𝑡_𝑎(1/𝑒^𝐴(𝑠)𝑎(𝑠))𝑑𝑠, and  𝛿₂(𝑡,𝑎)=∫^𝑡_𝑎(𝛿₁(𝑠,𝑎)/𝑟(𝑠))𝑑𝑠.

Lemma 1. Assume 𝑥 is a solution of (2). Then 𝑋^′(𝑡)=−Γ(1−𝛼)𝐷^𝛼_{_}𝑥(𝑡).

Lemma 2. Assume 𝑥 is an eventually positive solution of (2), and ^∞_𝑡01𝑒^𝐴(𝑠)𝑎(𝑠)𝑑𝑠=∞,_∞𝑡01𝑟(𝑠)𝑑𝑠=∞,_∞𝑡01𝑟(𝜉)^∞_𝜉1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏𝑑𝜉=∞. (3)(4)(5) Then there exists a sufficiently large 𝑇 such that 𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′<0𝑜𝑛[𝑇,∞), (6) and either 𝐷^𝛼_{_}𝑥(𝑡)<0 on [𝑇,∞) or lim_𝑡→∞𝑋(𝑡)=0.

Proof. Since 𝑥 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 𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′′=𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′′+𝑒^𝐴(𝑡)𝑝(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′=𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′′+𝑝(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′=𝑒^𝐴(𝑡)𝑞(𝑡)𝑋(𝑡)>0. (7) Then 𝑒^𝐴(𝑡)𝑎(𝑡)[𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)]^′ is strictly increasing on [𝑡₁,∞), and thus [𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)]^′ is eventually of one sign. We claim [𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)]^′<0 on [𝑡₂,∞), where 𝑡₂>𝑡₁ is sufficiently large. Otherwise, assume there exists a sufficiently large 𝑡₃>𝑡₂ such that [𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)]^′>0 on [𝑡₃,∞). Then for 𝑡∈[𝑡₃,∞), we have 𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)−𝑟𝑡₃𝐷^𝛼_{_}𝑥𝑡₃=^𝑡_𝑡3𝑒^𝐴(𝑠)𝑎(𝑠)𝑟(𝑠)𝐷^𝛼_{_}𝑥(𝑠)^′𝑒^𝐴(𝑠)𝑎(𝑠)𝑑𝑠≥𝑒^𝐴(𝑡3)𝑎𝑡₃𝑟𝑡₃𝐷^𝛼_{_}𝑥𝑡₃^′𝑡_𝑡31𝑒^𝐴(𝑠)𝑎(𝑠)𝑑𝑠. (8) By (3), we have lim_𝑡⟶∞𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)=∞, (9) which implies for some sufficiently large 𝑡₄>𝑡₃,𝐷^𝛼_{_}𝑥(𝑡)>0, and  𝑡∈[𝑡₄,∞). By Lemma 1, we have 𝑋(𝑡)−𝑋𝑡₄=^𝑡_𝑡4𝑋^′(𝑠)()𝑑𝑠=−Γ1−𝛼^𝑡_𝑡4𝐷^𝛼_{_}𝑥(𝑠)()𝑑𝑠=−Γ1−𝛼^𝑡_𝑡4𝑟(𝑠)𝐷^𝛼_{_}𝑥(𝑠)𝑟(𝑠)()𝑟𝑡𝑑𝑠≤−Γ1−𝛼₄𝐷^𝛼_{_}𝑥𝑡₄^𝑡_𝑡41𝑟(𝑠)𝑑𝑠. (10) By (4), we obtain lim_𝑡→∞𝑋(𝑡)=−∞, which contradicts 𝑋(𝑡)>0 on [𝑡₁,∞). So [𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)]^′<0 on [𝑡₂,∞). Thus 𝐷^𝛼_{_}𝑥(𝑡) is eventually of one sign. Now we assume 𝐷^𝛼_{_}𝑥(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 lim_𝑡→∞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 𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′′≥𝑒^𝐴(𝑡)𝑞(𝑡)𝑋(𝑡)≥𝛽𝑒^𝐴(𝑡)𝑞(𝑡). (11) Substituting 𝑡 with 𝑠 in (11), an integration for (11) with respect to 𝑠 from 𝑡 to ∞ yields −𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′≥−lim_𝑡⟶∞𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′+𝛽^∞_𝑡𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠>𝛽^𝑡_𝑡5𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠, (12) which means 𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′𝛽<−𝑒^𝐴(𝑡)𝑎(𝑡)^∞_𝑡𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠. (13) Substituting 𝑡 with 𝜏 in (13), an integration for (13) with respect to 𝜏 from 𝑡 to ∞ yields −𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)<−lim_𝑡⟶∞𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)−𝛽^∞_𝑡1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏<−𝛽^∞_𝑡1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏. (14) That is, 𝑋^′(𝑡)Γ()𝛽<−1−𝛼𝑟(𝑡)^∞_𝑡1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏. (15) Substituting 𝑡 with 𝜉 in (15), an integration for (15) with respect to 𝜉 from 𝑡₅ to 𝑡 yields 𝑋(𝑡)−𝑋𝑡₅<−Γ(1−𝛼)𝛽^𝑡_𝑡51𝑟(𝜉)^∞_𝜉1𝑒^𝐴(𝜏)𝑎(𝜏)×^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏𝑑𝜉. (16) By (5), one can see lim_𝑡→∞𝑋(𝑡)=−∞, which is a contradiction. So the proof is complete.

Lemma 3. Assume that 𝑥 is an eventually positive solution of (2) such that 𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′<0,𝐷^𝛼_{_}𝑥(𝑡)<0𝑜𝑛𝑡₁,∞, (17) where 𝑡₁≥𝑡₀ is sufficiently large. Then we have 𝑋^′(𝑡)Γ()𝛿≥−1−𝛼₁𝑡,𝑡₁𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′𝑟(𝑡),𝑋(𝑡)≥−Γ(1−𝛼)𝛿₂𝑡,𝑡₁𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′. (18)(19)

Proof. By Lemma 2 we have 𝑒^𝐴(𝑡)𝑎(𝑡)[𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)]^′  is strictly increasing on [𝑡₁,∞). So 𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)≤𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)−𝑟𝑡₁𝐷^𝛼_{_}𝑥(𝑡)=^𝑡_𝑡1𝑒^𝐴(𝑠)𝑎(𝑠)𝑟(𝑠)𝐷^𝛼_{_}𝑥(𝑠)^′𝑒^𝐴(𝑠)𝑎(𝑠)𝑑𝑠≤𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′𝑡_𝑡11𝑒^𝐴(𝑠)𝑎(𝑠)𝑑𝑠=𝛿₁𝑡,𝑡₁𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′. (20) Using Lemma 1 we obtain that 𝑋^′(𝑡)Γ()𝛿≥−1−𝛼₁𝑡,𝑡₁𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′𝑟(𝑡). (21) Then 𝑋(𝑡)≥𝑋(𝑡)−𝑋𝑡₁≥−^𝑡_𝑡1Γ(1−𝛼)𝛿₁𝑠,𝑡₁𝑒^𝐴(𝑠)𝑎(𝑠)𝑟(𝑠)𝐷^𝛼_{_}𝑥(𝑠)^′𝑟(𝑠)()𝑒𝑑𝑠≥−Γ1−𝛼^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′𝑡_𝑡1𝛿₁𝑠,𝑡₁𝑟(𝑠)()𝛿𝑑𝑠=−Γ1−𝛼₂𝑡,𝑡₁𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′. (22)

Theorem 4. Assume (3)–(5) hold, and there exist two functions 𝜙∈𝐶¹([𝑡₀,∞),ℝ₊) and 𝜑∈𝐶¹([𝑡₀,∞),[0,∞)) such that ^∞_𝑇𝜙(𝑠)𝑞(𝑠)𝑒^𝐴(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)⁻¹𝑑𝑠=∞, (23) for all sufficiently large 𝑇. Then every solution of (2) is oscillatory or satisfies lim_𝑡→∞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^𝛼_{_}𝑥(𝑡))^′<0,𝑡∈[𝑡₂,∞), where 𝑡₂>𝑡₁ is sufficiently large, and either 𝐷^𝛼_{_}𝑥(𝑡)<0 on [𝑡₂,∞) or lim_𝑡→∞𝑋(𝑡)=0. Define the generalized Riccati function: 𝜔(𝑡)(𝑡)−𝑒=𝜙^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′𝑋(𝑡)+𝜑(𝑡). (24) Then for 𝑡∈[𝑡₂,∞), we have 𝜔^′(𝑡)=−𝜙^′(𝑡)𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′𝑋(𝑡)(𝑡)−𝑒+𝜙^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′𝑋(𝑡)^′+𝜙^′(𝑡)𝜑(𝑡)+𝜙(𝑡)𝜑^′(𝑡)=−𝜙(𝑡)𝑋(𝑡)𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′′−𝑋^′(𝑡)𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼𝑥(𝑡)^′×𝑋²(𝑡)⁻¹+𝜙^′(𝑡)𝜙(𝑡)𝜔(𝑡)+𝜙(𝑡)𝜑^′(𝑡)=−𝜙(𝑡)𝑋(𝑡)𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′′+𝑒^𝐴(𝑡)𝑝(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′−𝑋^′(𝑡)𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼𝑥(𝑡)^′×𝑋²(𝑡)⁻¹+𝜙^′(𝑡)𝜙(𝑡)𝜔(𝑡)+𝜙(𝑡)𝜑^′(𝑡)=−𝜙(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)+𝜙(𝑡)𝑋^′(𝑡)𝑎(𝑡)𝑒^𝐴(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′𝑋²(𝑡)+𝜙^′(𝑡)𝜙(𝑡)𝜔(𝑡)+𝜙(𝑡)𝜑^′(𝑡). (25) By Lemma 3 and the definition of 𝑓 we get that 𝜔^′(𝑡)≤−𝜙(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)−𝜙(𝑡)Γ(1−𝛼)𝛿₁𝑡,𝑡₂𝑒^𝐴(𝑡)𝑎(𝑡)×𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′𝑎(𝑡)𝑒^𝐴(𝑡)𝑟(𝑡)𝐷^𝛼_{_}𝑥(𝑡)^′×𝑟(𝑡)𝑋²(𝑡)⁻¹+𝜙^′(𝑡)𝜙(𝑡)𝜔(𝑡)+𝜙(𝑡)𝜑^′(𝑡)=−𝜙(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)−𝜙(𝑡)Γ(1−𝛼)𝛿₁𝑡,𝑡₂𝑟(𝑡)×𝜔(𝑡)𝜙(𝑡)−𝜑(𝑡)²+𝜙^′(𝑡)𝜙(𝑡)𝜔(𝑡)+𝜙(𝑡)𝜑^′(𝑡)=−𝜙(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)+𝜙(𝑡)𝜑^′(𝑡)−𝜙(𝑡)Γ(1−𝛼)𝛿₁𝑡,𝑡₂𝜑²(𝑡)𝑟(𝑡)−𝜙(𝑡)Γ(1−𝛼)𝛿₁𝑡,𝑡₂𝑟(𝑡)𝜔²(𝑡)𝜙²(𝑡)+2𝜑(𝑡)𝜙(𝑡)Γ(1−𝛼)𝛿₁𝑡,𝑡₂+𝑟(𝑡)𝜙^′(𝑡)𝑟(𝑡)𝜙(𝑡)𝜔(𝑡)≤−𝜙(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)+𝜙(𝑡)𝜑^′(𝑡)−𝜙(𝑡)Γ(1−𝛼)𝛿₁𝑡,𝑡₂𝜑²(𝑡)𝑟(𝑡)+2𝜑(𝑡)𝜙(𝑡)Γ(1−𝛼)𝛿₁𝑡,𝑡₂+𝑟(𝑡)𝜙^′(𝑡)²4Γ(1−𝛼)𝜙(𝑡)𝛿₁𝑡,𝑡₂𝑟(𝑡). (26) Substituting 𝑡 with 𝑠 in (26), an integration for (26) with respect to 𝑠 from 𝑡₂ to 𝑡 yields ^𝑡_𝑡2𝜙(𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂𝜑²(𝑠)𝑟(𝑠)−2𝜑(𝑠)𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂+𝑟(𝑠)𝜙^′(𝑠)²×4Γ(1−𝛼)𝜙(𝑠)𝛿₁𝑠,𝑡₂𝑟(𝑠)⁻¹𝑡𝑑𝑠≤𝜔₂−𝜔(𝑡)≤𝜔𝑡₂<∞, (27) which contradicts (23). So the proof is complete.

Theorem 5. Define 𝔻={(𝑡,𝑠)∣𝑡≥𝑠≥𝑡₀}. Assume (3)–(5) hold, and there exists a function 𝐻∈𝐶¹(𝔻,ℝ) such that 𝐻(𝑡,𝑡)=0,𝑓𝑜𝑟𝑡≥𝑡₀,𝐻(𝑡,𝑠)>0,𝑓𝑜𝑟𝑡>𝑠≥𝑡₀, (28) and 𝐻 has a nonpositive continuous partial derivative 𝐻^′_𝑠(𝑡,𝑠), and lim_𝑡⟶∞1sup𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻(𝑡,𝑠)𝜙(𝑠)𝑞(𝑠)𝑒^𝐴(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)⁻¹𝑑𝑠=∞, (29) for all sufficiently large 𝑇, where 𝜙,𝜑 are defined as in Theorem 4. Then every solution of (2) is oscillatory or satisfies lim_𝑡→∞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^𝛼_{_}𝑥(𝑡)<0 on [𝑡₂,∞) for some sufficiently large 𝑡₂>𝑡₁. Let 𝜔(𝑡) be defined as in Theorem 4. By (26) we have 𝜙(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)+𝜙(𝑡)𝜑^′(𝑡)−𝜙(𝑡)Γ(1−𝛼)𝛿₁𝑡,𝑡₂𝜑²(𝑡)𝑟(𝑡)+2𝜑(𝑡)𝜙(𝑡)Γ(1−𝛼)𝛿₁𝑡,𝑡₂+𝑟(𝑡)𝜙^′(𝑡)²4Γ(1−𝛼)𝜙(𝑡)𝛿₁𝑡,𝑡₂𝑟(𝑡)≤−𝜔^′(𝑡). (30) 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 ^𝑡_𝑡2𝐻()×𝜙(𝑠)𝑞(𝑠)𝑒𝑡,𝑠_𝐴(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂𝜑²(𝑠)𝑟(𝑠)−2𝜑(𝑠)𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂+𝑟(𝑠)𝜙^′(𝑠)²4Γ(1−𝛼)𝜙(𝑠)𝛿₁𝑠,𝑡₂𝑟(𝑠)𝑑𝑠≤−^𝑡_𝑡2𝐻(𝑡,𝑠)𝜔^′(𝑠)𝑑𝑠=𝐻𝑡,𝑡₂𝜔𝑡₂+^𝑡_𝑡2𝐻^′_𝑠(𝑡,𝑠)𝜔(𝑠)Δ𝑠≤𝐻𝑡,𝑡₂𝜔𝑡₂≤𝐻𝑡,𝑡₀𝜔𝑡₂. (31) Then ^𝑡_𝑡0𝐻()×𝜙(𝑠)𝑞(𝑠)𝑒𝑡,𝑠_𝐴(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂𝜑²(𝑠)𝑟(𝑠)−2𝜑(𝑠)𝜙(𝑠)Γ(1−𝛼)×𝛿₁𝑠,𝑡₂+𝑟(𝑠)𝜙^′(𝑠)²×4Γ(1−𝛼)𝜙(𝑠)𝛿₁𝑠,𝑡₂𝑟(𝑠)⁻¹=𝑑𝑠_𝑡2𝑡0𝐻()×𝜙(𝑠)𝑞(𝑠)𝑒𝑡,𝑠_𝐴(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂𝜑²(𝑠)𝑟(𝑠)−2𝜑(𝑠)𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂+𝑟(𝑠)𝜙^′(𝑠)²4Γ(1−𝛼)𝜙(𝑠)𝛿₁𝑠,𝑡₂𝑟(𝑠)+𝑑𝑠_𝑡𝑡2𝐻()×𝜙(𝑠)𝑞(𝑠)𝑒𝑡,𝑠_𝐴(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂𝜑²(𝑠)𝑟(𝑠)−2𝜑(𝑠)𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂+𝑟(𝑠)𝜙^′(𝑠)²4Γ(1−𝛼)𝜙(𝑠)𝛿₁𝑠,𝑡₂𝑟(𝑠)×𝑑𝑠≤𝐻𝑡,𝑡₀𝜔𝑡₂+𝐻𝑡,𝑡₀×^𝑡2_𝑡0𝜙(𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂𝜑²(𝑠)×(𝑟(𝑠))⁻¹−2𝜑(𝑠)𝜙(𝑠)Γ(1−𝛼)𝛿₁×𝑠,𝑡₂+𝑟(𝑠)𝜙^′(𝑠)²×(4Γ(1−𝛼)𝜙(𝑠)𝛿₁𝑠,𝑡₂𝑟(𝑠)⁻¹𝑑𝑠. (32) So lim_𝑡⟶∞1sup𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻(𝑡,𝑠)𝜙(𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂𝜑²(𝑠)×(𝑟(𝑠))⁻¹−2𝜑(𝑠)𝜙(𝑠)Γ(1−𝛼)×𝛿₁𝑠,𝑡₂+𝑟(𝑠)𝜙^′(𝑠)²×4Γ(1−𝛼)𝜙(𝑠)𝛿₁𝑠,𝑡₂𝑟(𝑠)⁻¹𝑡𝑑𝑠≤𝜔₂+^𝑡2_𝑡0𝜙(𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂𝜑²(𝑠)𝑟(𝑠)−2𝜑(𝑠)𝜙(𝑠)Γ(1−𝛼)𝛿₁𝑠,𝑡₂+𝑟(𝑠)𝜙^′(𝑠)²×4Γ(1−𝛼)𝜙(𝑠)𝛿₁𝑠,𝑡₂𝑟(𝑠)⁻¹𝑑𝑠<∞, (33) which contradicts (29). So the proof is complete.

In Theorems 5, 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 lim_𝑡⟶∞1sup𝑡−𝑡₀^𝑚×^𝑡_𝑡0(𝑡−𝑠)^𝑚𝜙(𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)Γ(1−𝛼)𝛿₁(𝑠,𝑇)𝜑²(𝑠×(𝑟(𝑠)))⁻¹−2𝜑(𝑠)𝜙(𝑠)Γ(1−𝛼)𝛿₁()(𝑠)𝜙𝑠,𝑇+𝑟^′(𝑠)²×4Γ(1−𝛼)𝜙(𝑠)𝛿₁×(𝑠,𝑇)𝑟(𝑠)⁻¹𝑑𝑠=∞, (34) for all sufficiently large 𝑇. Then every solution of (2) is oscillatory or satisfies lim_𝑡→∞𝑋(𝑡)=0.

Corollary 7. Assume (3)–(5) hold, and lim_𝑡⟶∞1supln𝑡−ln𝑡₀×^𝑡_𝑡0(ln𝑡−ln𝑠)𝜙(𝑠)𝑞(𝑠)𝑒^𝐴(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)𝜙𝑠,𝑇+𝑟^′(𝑠)²×4Γ(1−𝛼)𝜙(𝑠)×𝛿₁(𝑠,𝑇)𝑟(𝑠)⁻¹𝑑𝑠=∞, (35) for all sufficiently large 𝑇. Then every solution of (2) is oscillatory or satisfies lim_𝑡→∞𝑋(𝑡)=0.

3. Applications

In this section, we will present some applications for the above established results.

Example 8. Consider 𝑡𝐷^𝛼_{_}𝑥(𝑡)^′′+𝑡⁻²𝐷^𝛼_{_}𝑥(𝑡)^′−𝑀𝑡^−2∞_𝑡(𝜉−𝑡)^−𝛼𝑥(𝜉)𝑑𝜉=0,𝑡∈[2,∞), (36) 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 1≤𝑒^𝐴(𝑡)=𝑒^{∫𝑡2(𝑝(𝑠)/𝑎(𝑠))𝑑𝑠}=𝑒^{∫𝑡2𝑠−3𝑑𝑠}=𝑒^{−(1/2)[𝑡−2−2−2]}≤𝑒^1/8. (37) Moreover, we have ^∞_𝑡01𝑒^𝐴(𝑠)𝑎(𝑠)𝑑𝑠=^∞₂1𝑒^𝐴(𝑠)𝑎(𝑠)𝑑𝑠≥𝑒^−1/8∞₂1𝑠𝑑𝑠=∞,_∞𝑡01𝑟(𝑠)𝑑𝑠=∞. (38) Furthermore, ^∞_𝑡01𝑟(𝜉)^∞_𝜉1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏𝑑𝜉=𝑀^∞₂^∞_𝜉1𝑒^𝐴(𝜏)𝜏^∞_𝜏𝑒^𝐴(𝑠)𝑠²𝑑𝑠𝑑𝜏𝑑𝜉≥𝑀𝑒^−1/8∞₂^∞_𝜉1𝜏^∞_𝜏1𝑠²𝑑𝑠𝑑𝜏𝑑𝜉=𝑀𝑒^−1/8∞₂^∞_𝜉1𝜏²𝑑𝜏𝑑𝜉=𝑀𝑒^−1/8∞₂1𝜉𝑑𝜉=∞. (39) On the other hand, for a sufficiently large 𝑇, we have 𝛿₁(𝑡,𝑇)=^𝑡_𝑇1𝑒^𝐴(𝑠)𝑎(𝑠)=𝑑𝑠_𝑡𝑇1𝑒^𝐴(𝑠)𝑠𝑑𝑠≥𝑒^−1/8𝑡_𝑇1𝑠𝑑𝑠⟶∞. (40) 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 ^∞_𝑇𝜙(𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝑟(𝑠)𝜙^′(𝑠)²4Γ(1−𝛼)𝜙(𝑠)𝛿₁(𝑟(𝑠)≥𝑠,𝑇)𝑑𝑠_∞𝑇1𝑀−4Γ(1−𝛼)𝛿₁()1𝑠,𝑇𝑠=𝑑𝑠_𝑇∗𝑇1𝑀−4Γ(1−𝛼)𝛿₁()1𝑠,𝑇𝑠+𝑑𝑠_∞𝑇∗1𝑀−4Γ(1−𝛼)𝛿₁()1𝑠,𝑇𝑠≥𝑑𝑠_𝑇∗𝑇1𝑀−4Γ(1−𝛼)𝛿₁()1𝑠,𝑇𝑠+𝑑𝑠_∞𝑇∗1𝑀−()14Γ1−𝛼𝑠𝑑𝑠=∞, (41) 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 lim_𝑡→∞𝑋(𝑡)=0 under condition 𝑀>1/4Γ(1−𝛼).

Example 9. Consider √𝑡𝐷^𝛼_{_}𝑥(𝑡)^′′+𝑒^−𝑡𝐷^𝛼_{_}𝑥(𝑡)^′−𝑀𝑡^−5/2+ln𝑡^∞_𝑡(𝜉−𝑡)^−𝛼𝑥(𝜉)𝑑𝜉=0,𝑡∈[2,∞), (42) 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 1≤𝑒^𝐴(𝑡)=𝑒^{∫𝑡2(𝑝(𝑠)/𝑎(𝑠))𝑑𝑠}=𝑒^{∫𝑡2(1/𝑒𝑠√𝑠)𝑑𝑠}≤𝑒^{∫𝑡2𝑠−(3/2)𝑑𝑠}=𝑒^{−2[𝑡−1/2−2−1/2]}≤𝑒^√2. (43) So we have ^∞_𝑡01𝑒^𝐴(𝑠)𝑎(𝑠)𝑑𝑠=^∞₂1𝑒^𝐴(𝑠)√𝑠𝑑𝑠≥𝑒^−√2∞₂1√𝑠𝑑𝑠=∞,_∞𝑡01𝑟(𝑠)𝑑𝑠=∞. (44) Furthermore, ^∞_𝑡01𝑟(𝜉)^∞_𝜉1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)=𝑑𝑠𝑑𝜏𝑑𝜉_∞2∞𝜉1𝑒^𝐴(𝜏)√𝜏^∞_𝜏𝑒^𝐴(𝑠)𝑀𝑠^−5/2+ln𝑠𝑑𝑠𝑑𝜏𝑑𝜉≥𝑀𝑒^−√2∞₂^∞_𝜉1√𝜏^∞_𝜏𝑠^−5/2=𝑑𝑠𝑑𝜏𝑑𝜉2𝑀𝑒^−√23^∞₂^∞_𝜉1𝜏²=𝑑𝜏𝑑𝜉2𝑀𝑒^−√23^∞₂1𝜉𝑑𝜉=∞. (45) On the other hand, Taking 𝜙(𝑡)=𝑡²,𝜑(𝑡)=0,and𝐻(𝑡,𝑠)=𝑡−𝑠 in (29), we get that lim_𝑡⟶∞1sup𝑡−𝑡₀^𝑡_𝑡0()×𝜙(𝑠)𝑞(𝑠)𝑒𝑡−𝑠_𝐴(𝑠)−𝑟(𝑠)𝜙^′(𝑠)²4Γ(1−𝛼)𝜙(𝑠)𝛿₁(𝑠,𝑇)𝑟(𝑠)𝑑𝑠≥lim_𝑡⟶∞1sup×𝑡−2_𝑡2(𝑡−𝑠)𝑀𝑠^−5/2+ln𝑠𝑠²−1Γ(1−𝛼)𝛿₁(𝑠,𝑇)𝑑𝑠≥lim_𝑡⟶∞1sup𝑡−2^𝑡₂(𝑡−𝑠)𝑀𝑠^−1/2𝑑𝑠=∞. (46) 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 lim_𝑡→∞𝑋(𝑡)=0.

Acknowledgments

This work is partially supported by Planning Fund project of the Ministry of Education of China (10YJA630019). The authors would thank the reviewers very much for their valuable suggestions on this paper.