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

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.

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: 𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^𝜂′−𝑞(𝑡)𝑓^∞_𝑡(𝑣−𝑡)^−𝛼𝑦(𝑣)𝑑𝑣=0,𝑡>0, (1) 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: 𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾′+𝑝(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾−𝑞(𝑡)𝑓^∞_𝑡(𝜉−𝑡)^−𝛼𝑥(𝜉)𝑑𝜉=0,𝑡∈𝑡₀,∞, (2) where 𝑎∈𝐶¹([𝑡₀,∞),ℝ₊), 𝑟∈𝐶²([𝑡₀,∞),ℝ₊), 𝑝,𝑞∈𝐶([𝑡₀,∞),ℝ₊), 𝑓∈𝐶(ℝ,ℝ) satisfying 𝑥𝑓(𝑥)>0, 𝑓(𝑥)/𝑥^𝛾≥𝐿>0 for 𝑥≠0, 𝛾 is a quotient of two odd positive integers, 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 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)=∫^𝑡_𝑡0(𝑝(𝑠)/𝑎(𝑠))𝑑𝑠, 𝜃₁(𝑡,𝑎)=^𝑡_𝑎1𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾𝑑𝑠,𝜃₂(𝑡,𝑎)=^𝑡_𝑎𝜃₁(𝑠,𝑎)𝑟(𝑠)𝑑𝑠. (3)

2. Main Results

The following lemmas are useful for proving our results.

Lemma 1. Assume that 𝑥 is a solution of (2). Then, 𝑋^′(𝑡)=−Γ(1−𝛼)𝐷^𝛼₋𝑥(𝑡).

Lemma 2. Assume that 𝑥 is an eventually positive solution of (2) and ^∞_𝑡01𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾𝑑𝑠=∞,_∞𝑡01𝑟(𝑠)𝑑𝑠=∞,_∞𝑡01𝑟(𝜉)^∞_𝜉1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠^1/𝛾𝑑𝜏𝑑𝜉=∞. (4)(5)(6) Then, there exists a sufficiently large 𝑇 such that 𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′<0𝑜𝑛[𝑇,∞) (7) 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)^′𝛾′=𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾′+𝑒^𝐴(𝑡)𝑝(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾=𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾′+𝑝(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾=𝑒^𝐴(𝑡)𝑞(𝑡)𝑓(𝑋(𝑡))≥𝐿𝑒^𝐴(𝑡)𝑞(𝑡)𝑋^𝛾(𝑡)>0. (8) Then, 𝑒^𝐴(𝑡)𝑎(𝑡)[(𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′]^𝛾 is strictly increasing on [𝑡₁,∞), and thus (𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′ is eventually of one sign. We claim that (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)^1/𝛾𝑟(𝑠)𝐷^𝛼₋𝑥(𝑠)^′𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾≥𝑒𝑑𝑠_𝐴(𝑡3)𝑎𝑡₃^1/𝛾𝑟𝑡₃𝐷^𝛼₋𝑥𝑡₃^′×^𝑡_𝑡31𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾𝑑𝑠. (9)
By (4), we have lim_𝑡→∞𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)=∞, (10) which implies, for some sufficiently large 𝑡₄>𝑡₃, 𝐷^𝛼₋𝑥(𝑡)>0, 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. (11) By (5), 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 that 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 that 𝛽=0. Otherwise, assume that 𝛽>0. Then, 𝑋(𝑡)≥𝛽 on [𝑡₅,∞), and, for 𝑡∈[𝑡₅,∞), by (8) we have 𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾′≥𝐿𝑒^𝐴(𝑡)𝑞(𝑡)𝑋^𝛾(𝑡)≥𝐿𝛽^𝛾𝑒^𝐴(𝑡)𝑞(𝑡). (12) Substituting 𝑡 with 𝑠 in (12), an integration for (12) with respect to 𝑠 from 𝑡 to ∞ yields −𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾≥−lim_𝑡→∞𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾+𝐿𝛽^𝛾∞_𝑡𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠>𝐿𝛽^𝛾∞_𝑡𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠; (13) which means 𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′<−𝐿^1/𝛾𝛽1𝑒^𝐴(𝑡)𝑎(𝑡)^∞_𝑡𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠^1/𝛾. (14) Substituting 𝑡 with 𝜏 in (14), an integration for (14) with respect to 𝜏 from 𝑡 to ∞ yields −𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)<−lim_𝑡→∞𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)−𝐿^1/𝛾𝛽^∞_𝑡1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠^1/𝛾𝑑𝜏<−𝐿^1/𝛾𝛽^∞_𝑡1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠^1/𝛾𝑑𝜏; (15) that is, 𝑋^′(𝑡)<−𝐿^1/𝛾Γ()𝛽11−𝛼𝑟(𝑡)×^∞_𝑡1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠^1/𝛾𝑑𝜏. (16) Substituting 𝑡 with 𝜉 in (16), an integration for (16) with respect to 𝜉 from 𝑡₅ to 𝑡 yields 𝑋(𝑡)−𝑋𝑡₅<−𝐿^1/𝛾Γ(1−𝛼)𝛽×^𝑡_𝑡51𝑟(𝜉)^∞_𝜉1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠^1/𝛾𝑑𝜏𝑑𝜉. (17) By (6), one can see that 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,∞_𝕋, (18) where 𝑡₁≥𝑡₀ is sufficiently large. Then, one has 𝑋^′(𝑡)Γ()𝜃≥−1−𝛼₁𝑡,𝑡₁𝑒^𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝑟(𝑡),𝑋(𝑡)≥−Γ(1−𝛼)𝜃₂𝑡,𝑡₁𝑒^𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′. (19)

Proof. By Lemma 2, we obtain that 𝑒^𝐴(𝑡)𝑎(𝑡)[(𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′]^𝛾 is strictly increasing on [𝑡₁,∞). So 𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)≤𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)−𝑟𝑡₁𝐷^𝛼₋𝑥𝑡₁=^𝑡_𝑡1𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾𝑟(𝑠)𝐷^𝛼₋𝑥(𝑠)^′𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾≤𝑒𝑑𝑠_𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝑡_𝑡11𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾𝑑𝑠=𝜃₁𝑡,𝑡₁𝑒^𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′. (20) Using Lemma 1, we obtain that 𝑋^′(𝑡)Γ()𝜃≥−1−𝛼₁𝑡,𝑡₁𝑒^𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝑟(𝑡). (21) Then, 𝑋(𝑡)≥𝑋(𝑡)−𝑋𝑡₁≥−^𝑡_𝑡1Γ(1−𝛼)𝜃₁𝑠,𝑡₁𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑠)^′𝑟(𝑠)(𝑒𝑑𝑠≥−Γ1−𝛼)^𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝑡_𝑡1𝜃₁𝑠,𝑡₁𝑟(𝑠)()𝜃𝑑𝑠=−Γ1−𝛼₂𝑡,𝑡₁𝑒^𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′. (22)

Lemma 4 ([14, Theorem 41]). Assume that 𝐴 and 𝐵 are nonnegative real numbers. Then, 𝜆𝐴𝐵^𝜆−1−𝐴^𝜆≤(𝜆−1)𝐵^𝜆,∀𝜆>1. (23)

Theorem 5. Assume (4)–(6) hold and there exist two functions 𝜌∈𝐶¹([𝑡₀,∞),ℝ₊) and 𝜂∈𝐶¹([𝑡₀,∞),[0,∞)) such that 𝐿𝜌(^∞_𝑇𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝜌(𝑠)Γ(1−𝛼)𝜃₁(𝑠,𝑇)𝜂^1+(1/𝛾)(𝑠)𝑟(𝑠)−𝛾+1𝜂^1/𝛾(𝑠)𝜌(𝑠)Γ(1−𝛼)𝜃₁()(𝑠)𝜌𝑠,𝑇+𝑟^′(𝑠)^𝛾+1×𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁(𝑠,𝑇)^𝛾𝑟(𝑠)⁻¹𝑑𝑠=∞, (24) 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 as follows: 𝜔(𝑡)(𝑡)−𝑒=𝜌^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾𝑋^𝛾(𝑡)+𝜂(𝑡). (25) Then, for 𝑡∈[𝑡₂,∞), we have 𝜔^′(𝑡)=−𝜌^′(𝑡)𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾𝑋^𝛾(𝑡)(𝑡)−𝑒+𝜌^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾𝑋^𝛾(𝑡)^′+𝜌^′(𝑡)𝜂(𝑡)+𝜌(𝑡)𝜂^′(𝑡)=−𝜌(𝑡)𝑋^𝛾(𝑡)𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾′−𝛾𝑋^𝛾−1(𝑡)𝑋^′(𝑡)𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾×𝑋^2𝛾(𝑡)⁻¹+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)+𝜌(𝑡)𝜂^′(𝑡)=−𝜌(𝑡)𝑋^𝛾(𝑡)𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾′+𝑒^𝐴(𝑡)𝑝(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾×𝑋^2𝛾(𝑡)⁻¹−𝛾𝑋^𝛾−1(𝑡)𝑋^′(𝑡)𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾𝑋^2𝛾(𝑡)+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)+𝜌(𝑡)𝜂^′(𝑡)𝜌(𝑡)𝑞(𝑡)𝑒=−^𝐴(𝑡)𝑓(𝑋(𝑡))𝑋^𝛾(𝑡)+𝛾𝜌(𝑡)𝑋^′(𝑡)𝑎(𝑡)𝑒^𝐴(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾𝑋^𝛾+1(𝑡)+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)+𝜌(𝑡)𝜂^′(𝑡). (26) By Lemma 3 and the definition of 𝑓, we get that 𝜔^′(𝑡)≤−𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)−𝛾𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂𝑒^𝐴(𝑡)𝑎(𝑡)^1/𝛾×𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝑎(𝑡)𝑒^𝐴(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾×𝑟(𝑡)𝑋^𝛾+1(𝑡)⁻¹+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)+𝜌(𝑡)𝜂^′(𝑡)=−𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)−𝛾𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂𝑟(𝑡)×−𝜔(𝑡)𝜌(𝑡)−𝜂(𝑡)^1+(1/𝛾)+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)+𝜌(𝑡)𝜂^′(𝑡)=−𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)−𝛾𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂𝑟(𝑡)×𝜔(𝑡)𝜌(𝑡)−𝜂(𝑡)^1+(1/𝛾)+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)+𝜌(𝑡)𝜂^′(𝑡). (27)
Using the following inequality (see [15, Equation (2.17)]) (𝑢−𝑣)^1+(1/𝛾)≥𝑢^1+(1/𝛾)+1𝛾𝑣^1+(1/𝛾)−11+𝛾𝑣^1/𝛾𝑢, (28) we obtain 𝜔(𝑡)𝜌(𝑡)−𝜂(𝑡)^1+(1/𝛾)≥𝜔^1+(1/𝛾)(𝑡)𝜌^1+(1/𝛾)(𝑡)+1𝛾𝜂^1+(1/𝛾)(𝑡)−11+𝛾𝜂^1/𝛾(𝑡)𝜔(𝑡)𝜌(𝑡). (29) A combination of (27) and (29) yields the following: 𝜔^′(𝑡)≤−𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)+𝜌(𝑡)𝜂^′(𝑡)−𝛾𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂𝑟(𝑡)×𝜔^1+(1/𝛾)(𝑡)𝜌^1+(1/𝛾)(𝑡)+1𝛾𝜂^1+(1/𝛾)(𝑡)−11+𝛾𝜂^1/𝛾(𝑡)𝜔(𝑡)𝜌(𝑡)=−𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)+𝜌(𝑡)𝜂^′(𝑡)−𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂𝜂^1+(1/𝛾)(𝑡)𝑟(𝑡)−𝛾𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂𝑟(𝑡)𝜔^1+(1/𝛾)(𝑡)𝜌^1+(1/𝛾)(𝑡)+𝛾+1𝜂^1/𝛾(𝑡)𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂+𝑟(𝑡)𝜌^′(𝑡)𝑟(𝑡)𝜌(𝑡)𝜔(𝑡). (30) Setting 1𝜆=1+𝛾,𝐴^𝜆=𝛾𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂𝑟(𝑡)𝜔^1+(1/𝛾)(𝑡)𝜌^1+(1/𝛾)(𝑡),𝐵^𝜆−1=𝛾^1/(𝛾+1)×𝛾+1𝜂^1/𝛾(𝑡)𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂+𝑟(𝑡)𝜌^′(𝑡)𝛾+1Γ(1−𝛼)𝜌(𝑡)𝜃₁𝑡,𝑡₂^{𝛾/(𝛾+1)}𝑟^1/(𝛾+1)(𝑡). (31) Using Lemma 4 in (30), we get that 𝜔^′(𝑡)≤−𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)+𝜌(𝑡)𝜂^′(𝑡)−𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂𝜂^1+(1/𝛾)(𝑡)𝑟(𝑡)+𝛾+1𝜂^1/𝛾(𝑡)𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂+𝑟(𝑡)𝜌^′(𝑡)^𝛾+1𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑡)𝜃₁𝑡,𝑡₂^𝛾𝑟(𝑡). (32) Substituting 𝑡 with 𝑠 in (32), an integration for (32) with respect to 𝑠 from 𝑡₂ to 𝑡 yields 𝐿𝜌(^𝑡_𝑡2𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂𝜂^1+(1/𝛾)(𝑠)𝑟(𝑠)−𝛾+1𝜂^1/𝛾(𝑠)𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑠)^𝛾+1×𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁𝑠,𝑡₂^𝛾𝑟(𝑠)⁻¹𝑡𝑑𝑠≤𝜔₂−𝜔(𝑡)≤𝜔𝑡₂, (33) which contradicts (24). So the proof is complete.

Theorem 6. Assume (4)–(6) hold and, for all sufficiently large 𝑇, 𝐿𝜌(^∞_𝑇𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝛾𝜌(𝑠)[Γ(1−𝛼)]^𝛾𝜃₁(𝑠,𝑇)𝜃₂^𝛾−1(𝑠,𝑇)𝜂²(𝑠)𝑟(𝑠)−2𝛾𝜌(𝑠)𝜂(𝑠)[Γ(1−𝛼)]^𝛾𝜃₁(𝑠,𝑇)𝜃₂^𝛾−1()(𝑠)𝜌𝑠,𝑇+𝑟^′(𝑠)²×4𝛾[Γ(1−𝛼)]^𝛾𝜃₁(𝑠,𝑇)𝜃₂^𝛾−1(𝑠,𝑇)𝑟(𝑠)𝜌(𝑠)⁻¹𝑑𝑠=∞, (34) where 𝜌 and 𝜂 are defined as in Theorem 5. 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. 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: 𝑋^′(𝑡)𝑋(𝑡)Γ()𝜃≥−1−𝛼₁𝑡,𝑡₂𝑒^𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝑟(𝑡)𝑋(𝑡)Γ()𝜃=−1−𝛼₁𝑡,𝑡₂𝑒^𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝑟(𝑡)𝑋^𝛾(𝑡)×𝑋^𝛾−1(𝑡)Γ()𝜃≥−1−𝛼₁𝑡,𝑡₂𝑒^𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝑟(𝑡)𝑋^𝛾(𝑡)×−Γ(1−𝛼)𝜃₂𝑡,𝑡₂𝑒^𝐴(𝑡)𝑎(𝑡)^1/𝛾𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾−1[Γ(=−1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂𝑟(𝑡)×𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾𝑋^𝛾(𝑡). (35) Using (35) in (26) we get that 𝜔^′(𝑡)≤−𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)−𝛾𝜌(𝑡)[Γ(1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂𝑟(𝑡)×𝑒^𝐴(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝛾𝑋^𝛾(𝑡)²+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)+𝜌(𝑡)𝜂^′(𝑡)=−𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)−𝛾𝜌(𝑡)[Γ(1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂𝑟(𝑡)×𝜔(𝑡)𝜌(𝑡)−𝜂(𝑡)²+𝜌^′(𝑡)𝜌(𝑡)𝜔(𝑡)+𝜌(𝑡)𝜂^′(𝑡)=−𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)−𝛾𝜌(𝑡)[Γ(1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂𝜂²(𝑡)𝑟(𝑡)−𝛾[Γ(1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂𝑟(𝑡)𝜌(𝑡)𝜔²(𝑡)+2𝛾𝜌(𝑡)𝜂(𝑡)[Γ(1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂+𝑟(𝑡)𝜌^′(𝑡)𝑟(𝑡)𝜌(𝑡)×𝜔(𝑡)+𝜌(𝑡)𝜂^′(𝑡)≤−𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)+𝜌(𝑡)𝜂^′(𝑡)−𝛾𝜌(𝑡)[Γ(1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂𝜂²(𝑡)𝑟(𝑡)+2𝛾𝜌(𝑡)𝜂(𝑡)[Γ(1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂+𝑟(𝑡)𝜌^′(𝑡)²×4𝛾[Γ(1−𝛼)]^𝛾𝜃₁(𝑡,𝑡₂)𝜃₂^𝛾−1(𝑡,𝑡₂)𝑟(𝑡)𝜌(𝑡)⁻¹. (36) Substituting 𝑡 with 𝑠 in (36), an integration for (36) with respect to 𝑠 from 𝑡₂ to 𝑡 yields 𝐿𝜌(^𝑡_𝑡2𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝛾𝜌(𝑠)[Γ(1−𝛼)]^𝛾𝜃₁𝑠,𝑡₂𝜃₂^𝛾−1𝑠,𝑡₂𝜂²(𝑠)𝑟(𝑠)−2𝛾𝜌(𝑠)𝜂(𝑠)[Γ(1−𝛼)]^𝛾𝜃₁𝑠,𝑡₂𝜃₂^𝛾−1𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑠)²×4𝛾[Γ(1−𝛼)]^𝛾𝜃₁(𝑠,𝑡₂)𝜃₂^𝛾−1(𝑠,𝑡₂)𝑟(𝑠)𝜌(𝑠)⁻¹𝑡𝑑𝑠≤𝜔₂−𝜔(𝑡)≤𝜔𝑡₂, (37) which contradicts (34). So the proof is complete.

Theorem 7. Define 𝔻={(𝑡,𝑠)∣𝑡≥𝑠≥𝑡₀}. Assume (4)–(6) 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, (38)𝐻 has a nonpositive continuous partial derivative 𝐻^′_𝑠(𝑡,𝑠), and lim_𝑡→∞1sup𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻()×(𝑠)𝑞(𝑠)𝑒𝑡,𝑠𝐿𝜌^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝜌(𝑠)Γ(1−𝛼)𝜃₁(𝑠,𝑇)𝜂^1+(1/𝛾)(𝑠)𝑟(𝑠)−𝛾+1𝜂^1/𝛾(𝑠)𝜌(𝑠)Γ(1−𝛼)𝜃₁()(𝑠)𝜌𝑠,𝑇+𝑟^′(𝑠)^𝛾+1×𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁(𝑠,𝑇)^𝛾×𝑟(𝑠)⁻¹𝑑𝑠=∞, (39) for all sufficiently large 𝑇, where 𝜌 and 𝜂 are defined as in Theorem 5. 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 either 𝐷^𝛼₋𝑥(𝑡)<0 on [𝑡₂,∞), for some sufficiently large 𝑡₂>𝑡₁, or lim_𝑡→∞𝑋(𝑡)=0. Now we assume 𝐷^𝛼₋𝑥(𝑡)<0. Let 𝜔(𝑡) be defined as in Theorem 5. By (32), we have 𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)−𝜌(𝑡)𝜂^′(𝑡)+𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂𝜂^1+(1/𝛾)(𝑡)𝑟(𝑡)−𝛾+1𝜂^1/𝛾(𝑡)𝜌(𝑡)Γ(1−𝛼)𝜃₁𝑡,𝑡₂+𝑟(𝑡)𝜌^′(𝑡)^𝛾+1𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑡)𝜃₁𝑡,𝑡₂^𝛾𝑟(𝑡)≤−𝜔^′(𝑡). (40) 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 ^𝑡_𝑡2𝐻()×(𝑠)𝑞(𝑠)𝑒𝑡,𝑠𝐿𝜌^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂𝜂^1+(1/𝛾)(𝑠)𝑟(𝑠)−𝛾+1𝜂^1/𝛾(𝑠)𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑠)^𝛾+1×𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁𝑠,𝑡₂^𝛾𝑟(𝑠)⁻¹𝑑𝑠≤−^𝑡_𝑡2𝐻(𝑡,𝑠)𝜔^′(𝑠)𝑑𝑠=𝐻𝑡,𝑡₂𝜔𝑡₂+^𝑡_𝑡2𝐻^′_𝑠(𝑡,𝑠)𝜔(𝑠)Δ𝑠≤𝐻𝑡,𝑡₂𝜔𝑡₂≤𝐻𝑡,𝑡₀𝜔𝑡₂. (41) Then ^𝑡_𝑡0𝐻()×(𝑠)𝑞(𝑠)𝑒𝑡,𝑠𝐿𝜌^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂𝜂^1+(1/𝛾)(𝑠)𝑟(𝑠)−𝛾+1𝜂^1/𝛾(𝑠)𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑠)^𝛾+1×𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁𝑠,𝑡₂^𝛾𝑟(𝑠)⁻¹=𝑑𝑠_𝑡2𝑡0𝐻()×(𝑠)𝑞(𝑠)𝑒𝑡,𝑠𝐿𝜌^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂𝜂^1+(1/𝛾)(𝑠)𝑟(𝑠)−𝛾+1𝜂^1/𝛾(𝑠)𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑠)^𝛾+1×𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁𝑠,𝑡₂^𝛾𝑟(𝑠)⁻¹𝑑𝑠+^𝑡_𝑡2𝐻()×(𝑠)𝑞(𝑠)𝑒𝑡,𝑠𝐿𝜌^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂𝜂^1+(1/𝛾)(𝑠)𝑟(𝑠)−𝛾+1𝜂^1/𝛾(𝑠)𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑠)^𝛾+1×𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁𝑠,𝑡₂^𝛾𝑟(𝑠)⁻¹𝑑𝑠≤𝐻𝑡,𝑡₀𝜔𝑡₂+𝐻𝑡,𝑡₀×^𝑡2_𝑡0𝐿𝜌(𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂𝜂^1+(1/𝛾)(𝑠)𝑟(𝑠)−𝛾+1𝜂^1/𝛾(𝑠)𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑠)^𝛾+1×𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁𝑠,𝑡₂^𝛾𝑟(𝑠)⁻¹𝑑𝑠. (42) So lim_𝑡→∞1sup𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻()×(𝑠)𝑞(𝑠)𝑒𝑡,𝑠𝐿𝜌^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂𝜂^1+(1/𝛾)(𝑠)𝑟(𝑠)−𝛾+1𝜂^1/𝛾(𝑠)𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑠)^𝛾+1×𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁𝑠,𝑡₂^𝛾×𝑟(𝑠)⁻¹𝑡𝑑𝑠≤𝜔₂+^𝑡2_𝑡0𝐿𝜌(𝑠)𝑞(𝑠)𝑒^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂𝜂^1+(1/𝛾)(𝑠)𝑟(𝑠)−𝛾+1𝜂^1/𝛾(𝑠)𝜌(𝑠)Γ(1−𝛼)𝜃₁𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑠)^𝛾+1×𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁𝑠,𝑡₂^𝛾𝑟(𝑠)⁻¹𝑑𝑠<∞, (43) which contradicts (39). So the proof is complete.

Theorem 8. Let 𝐻, 𝜌, and 𝜂 be defined as in Theorem 7. If (4)–(6) hold and lim_𝑡→∞1sup𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻()×(𝑠)𝑞(𝑠)𝑒𝑡,𝑠𝐿𝜌^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝛾𝜌(𝑠)[Γ(1−𝛼)]^𝛾𝜃₁(𝑠,𝑇)𝜃₂^𝛾−1(𝑠,𝑇)𝜂²(𝑠)𝑟(𝑠)−2𝛾𝜌(𝑠)𝜂(𝑠)[Γ(1−𝛼)]^𝛾𝜃₁(𝑠,𝑇)𝜃₂^𝛾−1()(𝑠)𝜌𝑠,𝑇+𝑟^′(𝑠)²×4𝛾[Γ(1−𝛼)]^𝛾𝜃₁(𝑠,𝑇)𝜃₂^𝛾−1(𝑠,𝑇)𝑟(𝑠)𝜌(𝑠)⁻¹𝑑𝑠=∞, (44) 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 either 𝐷^𝛼₋𝑥(𝑡)<0 on [𝑡₂,∞), for some sufficiently large 𝑡₂>𝑡₁, or lim_𝑡→∞𝑋(𝑡)=0. Now, we assume 𝐷^𝛼₋𝑥(𝑡)<0. Let 𝜔(𝑡) be defined as in Theorem 5. By (36), we have 𝐿𝜌(𝑡)𝑞(𝑡)𝑒^𝐴(𝑡)−𝜌(𝑡)𝜂^′(𝑡)+𝛾𝜌(𝑡)[Γ(1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂𝜂²(𝑡)𝑟(𝑡)−2𝛾𝜌(𝑡)𝜂(𝑡)[Γ(1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂+𝑟(𝑡)𝜌^′(𝑡)²4𝛾[Γ(1−𝛼)]^𝛾𝜃₁𝑡,𝑡₂𝜃₂^𝛾−1𝑡,𝑡₂𝑟(𝑡)𝜌(𝑡)≤−𝜔^′(𝑡). (45) 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 ^𝑡_𝑡2𝐻()×(𝑠)𝑞(𝑠)𝑒𝑡,𝑠𝐿𝜌^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝛾𝜌(𝑠)[Γ(1−𝛼)]^𝛾𝜃₁𝑠,𝑡₂𝜃₂^𝛾−1𝑠,𝑡₂𝜂²(𝑠)𝑟(𝑠)−2𝛾𝜌(𝑠)𝜂(𝑠)[Γ(1−𝛼)]^𝛾𝜃₁𝑠,𝑡₂𝜃₂^𝛾−1𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑡)²×4𝛾[Γ(1−𝛼)]^𝛾𝜃₁(𝑠,𝑡₂)𝜃₂^𝛾−1(𝑠,𝑡₂)𝑟(𝑠)𝜌(𝑠)⁻¹𝑑𝑠≤−^𝑡_𝑡2𝐻(𝑡,𝑠)𝜔^′(𝑠)𝑑𝑠=𝐻𝑡,𝑡₂𝜔𝑡₂+^𝑡_𝑡2𝐻^′_𝑠(𝑡,𝑠)𝜔(𝑠)Δ𝑠≤𝐻𝑡,𝑡₂𝜔𝑡₂≤𝐻𝑡,𝑡₀𝜔𝑡₂. (46) Then, similar to the process of Theorem 7, we get that lim_𝑡→∞1sup𝐻𝑡,𝑡₀×^𝑡_𝑡0𝐻()×(𝑠)𝑞(𝑠)𝑒𝑡,𝑠𝐿𝜌^𝐴(𝑠)−𝜌(𝑠)𝜂^′(𝑠)+𝛾𝜌(𝑠)[Γ(1−𝛼)]^𝛾𝜃₁𝑠,𝑡₂𝜃₂^𝛾−1𝑠,𝑡₂𝜂²(𝑠)𝑟(𝑠)−2𝛾𝜌(𝑠)𝜂(𝑠)[Γ(1−𝛼)]^𝛾𝜃₁𝑠,𝑡₂×𝜃₂^𝛾−1𝑠,𝑡₂+𝑟(𝑠)𝜌^′(𝑠)²×4𝛾[Γ(1−𝛼)]^𝛾𝜃₁𝑠,𝑡₂𝜃₂^𝛾−1×(𝑠,𝑡₂)𝑟(𝑠)𝜌(𝑠)⁻¹𝑑𝑠<∞, (47) which contradicts (44). So the proof is complete.

Remark 9. In Theorems 7 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: 𝑡^5/3𝐷^𝛼₋𝑥(𝑡)^′5/3′+𝑡^−8/3𝐷^𝛼₋𝑥(𝑡)^′5/3−𝑡^−8/3𝑀+𝑒^{∫∞𝑡(𝜉−𝑡)−𝛼𝑥(𝜉)𝑑𝜉}×^∞_𝑡(𝜉−𝑡)^−𝛼𝑥(𝜉)𝑑𝜉^5/3=0,𝑡∈[2,∞), (48) where 𝑀>0 is a constant.
We have in (2) 𝛾=5/3, 𝑎(𝑡)=𝑡^5/3, 𝑝(𝑡)=𝑞(𝑡)=𝑡^−8/3, 𝑓(𝑥)=𝑥^5/3[𝑒^𝑥+𝑀], 𝑟(𝑡)=1, 𝑡₀=2. Then, 𝑓(𝑥)/𝑥^𝛾≥𝑀=𝐿. Moreover, 1≤𝑒^𝐴(𝑡)=𝑒^{∫𝑡2(𝑝(𝑠)/𝑎(𝑠))𝑑𝑠}=𝑒^{∫𝑡2𝑠−13/3𝑑𝑠}=𝑒^{−(3/10)[𝑡−10/3−2−10/3]}≤𝑒. Then, we have ^∞_𝑡01𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾𝑑𝑠=^∞₂1𝑒^𝐴(𝑠)𝑎(𝑠)^3/5𝑑𝑠≥𝑒^−3/5∞₂1𝑠𝑑𝑠=∞,_∞𝑡01𝑟(𝑠)𝑑𝑠=∞. (49) Furthermore, ^∞_𝑡01𝑟(𝜉)^∞_𝜉1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠^1/𝛾=𝑑𝜏𝑑𝜉_∞2∞𝜉1𝑒^𝐴(𝜏)𝜏^5/3∞_𝜏𝑒^𝐴(𝑠)𝑠^−8/3𝑑𝑠^3/5𝑑𝜏𝑑𝜉≥𝑒^−3/5∞₂^∞_𝜉1𝜏^5/3∞_𝜏𝑠^−8/3𝑑𝑠^3/5=𝑑𝜏𝑑𝜉5𝑒3^−3/5∞₂^∞_𝜉1𝜏²𝑑𝜏𝑑𝜉=5𝑒3^−3/5∞₂1𝜉𝑑𝜉=∞. (50) On the other hand, for a sufficiently large 𝑇, we have 𝜃₁(𝑡,𝑇)=^𝑡_𝑇1𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾=𝑑𝑠_𝑡𝑇1𝑒^𝐴(𝑠)𝑠^5/33/5𝑑𝑠≥𝑒^−3/5𝑡_𝑇1𝑠𝑑𝑠⟶∞. (51) So we can take 𝑇^∗>𝑇 such that 𝜃₁(𝑡,𝑇)>1 for 𝑡∈[𝑇^∗,∞). Taking 𝜌(𝑡)=𝑡^5/3 and 𝜂(𝑡)=0 in (24), we get that ^∞_𝑇(𝑠)𝑞(𝑠)−𝑟(𝑠)𝜌𝐿𝜌_′(𝑠)^𝛾+1𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁(𝑠,𝑇)^𝛾𝑟(𝑠)=𝑑𝑠_∞𝑇5𝑀−8^8/31Γ(1−𝛼)𝜃₁(𝑠,𝑇)^5/31𝑠𝑑𝑠=^𝑇∗_𝑇5𝑀−8^8/31Γ(1−𝛼)𝜃₁(𝑠,𝑇)^5/31𝑠+𝑑𝑠_∞𝑇∗5𝑀−8^8/31Γ(1−𝛼)𝜃₁(𝑠,𝑇)^5/31𝑠≥𝑑𝑠_𝑇∗𝑇5𝑀−8^8/31Γ(1−𝛼)𝜃₁(𝑠,𝑇)^5/31𝑠+𝑑𝑠_𝑡𝑇∗5𝑀−8^8/31[Γ(1−𝛼)]^5/31𝑠𝑑𝑠⟶∞, (52) 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 lim_𝑡→∞𝑋(𝑡)=0 under the condition 𝑀>(5/8)^8/3(1/[Γ(1−𝛼)]^5/3).

Example 2. Consider the following: 𝑡³𝐷^𝛼₋𝑥(𝑡)^′3′+𝑡⁻⁴𝐷^𝛼₋𝑥(𝑡)^′3−𝑀𝑡^−4∞_𝑡(𝜉−𝑡)^−𝛼𝑥(𝜉)𝑑𝜉³=0,𝑡∈[2,∞), (53) 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)/𝑥^𝛾≥𝑀=𝐿. Moreover, 1≤𝑒^𝐴(𝑡)=𝑒^{∫𝑡2(𝑝(𝑠)/𝑎(𝑠))𝑑𝑠}=𝑒^{∫𝑡2𝑠−7𝑑𝑠}=𝑒^{−(1/6)[𝑡−6−2−6]}≤𝑒^1/384. Then, we have ^∞_𝑡01𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾𝑑𝑠=^∞₂1𝑒^𝐴(𝑠)𝑎(𝑠)^1/3𝑑𝑠≥𝑒^−1/1152∞₂1𝑠𝑑𝑠=∞,_∞𝑡01𝑟(𝑠)𝑑𝑠=∞. (54) Furthermore, ^∞_𝑡01𝑟(𝜉)^∞_𝜉1𝑒^𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝑒^𝐴(𝑠)𝑞(𝑠)𝑑𝑠^1/𝛾=𝑑𝜏𝑑𝜉_∞2∞𝜉1𝑒^𝐴(𝜏)𝜏^3∞_𝜏𝑒^𝐴(𝑠)𝑠⁻⁴𝑑𝑠^1/3𝑑𝜏𝑑𝜉≥𝑒^−1/1152∞₂^∞_𝜉1𝜏^3∞_𝜏𝑠⁻⁴𝑑𝑠^1/3=𝑒𝑑𝜏𝑑𝜉_−1/11523√3^∞2∞𝜉1𝜏²𝑒𝑑𝜏𝑑𝜉=^−1/1152₃√3^∞21𝜉𝑑𝜉=∞. (55) On the other hand, for a sufficiently large 𝑇, we have 𝜃₁(𝑡,𝑇)=^𝑡_𝑇1𝑒^𝐴(𝑠)𝑎(𝑠)^1/𝛾=𝑑𝑠_𝑡𝑇1𝑒^𝐴(𝑠)𝑠^31/3𝑑𝑠≥𝑒^{−1/1152𝑡}_𝑇1𝑠𝑑𝑠⟶∞. (56) 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 lim_𝑡→∞1sup𝑡−𝑡₀×^𝑡_𝑡0()×(𝑠)𝑞(𝑠)−𝑟(𝑠)𝜌𝑡−𝑠𝐿𝜌_′(𝑠)^𝛾+1𝛾+1^𝛾+1Γ(1−𝛼)𝜌(𝑠)𝜃₁(𝑠,𝑇)^𝛾𝑟(𝑠)𝑑𝑠=lim_𝑡→∞1sup×𝑡−2_𝑡2()×3𝑡−𝑠𝑀−4⁴1Γ(1−𝛼)𝜃₁(𝑠,𝑇)³1𝑠𝑑𝑠=lim_𝑡→∞1sup×𝑡−2_𝑇∗2()×3𝑡−𝑠𝑀−4⁴1Γ(1−𝛼)𝜃₁(𝑠,𝑇)³1𝑠+𝑑𝑠_𝑡𝑇∗()×3𝑡−𝑠𝑀−4⁴1Γ(1−𝛼)𝜃₁(𝑠,𝑇)³1𝑠𝑑𝑠≥lim_𝑡→∞1sup×𝑡−2_𝑇∗2(3𝑡−𝑠)𝑀−4⁴1[Γ(1−𝛼)]³1𝑠𝑑𝑠+^𝑡_𝑇∗(3𝑡−𝑠)𝑀−4⁴1[Γ(1−𝛼)]³1𝑠𝑑𝑠=∞, (57) 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 lim_𝑡→∞𝑋(𝑡)=0 under the condition 𝑀>(3/4)⁴(1/[Γ(1−𝛼)]³).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the Natural Science Foundation of Shandong Province (in China) (Grant no. ZR2013AQ009) and the Doctoral Initializing Foundation of Shandong University of Technology (in China) (Grant no. 4041-413030).