Oscillation of a Class of Fractional Differential Equations with Damping Term

Abstract

We investigate the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into another differential equations of integer order with respect to the new variable. Then, using Riccati transformation, inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results.

1. Introduction

In the investigations of qualitative properties for differential equations, research of oscillation has gained much attention by many authors in the last few decades (e.g., see [1–16]). In these investigations, we notice that very little attention is paid to oscillation of fractional differential equations.

In [17], Jumarie proposed a definition for fractional derivative which is known as the modified Riemann-Liouville derivative in the literature. Since then, many authors have investigated various applications of the modified Riemann-Liouville derivative (e.g., see [18–21]) including various fractional calculus formulae, the fractional variational iteration method, the Bäcklund transformation method, and the fractional subequation method for soling fractional partial differential equations. In this paper, based on the modified Riemann-Liouville derivative, we are concerned with oscillation of a class of fractional differential equations with damping term as follows: 𝐷^𝛼_𝑡𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)+𝑝(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡(𝑡)𝑥(𝑡))+𝑞=0,𝑡≥𝑡₀>0,0<𝛼<1, (1) where 𝐷^𝛼_𝑡(⋅) denotes the modified Riemann-Liouville derivative with respect to the variable 𝑡, the function 𝑎∈𝐶^𝛼([𝑡₀,∞),𝑅₊), 𝑟∈𝐶^2𝛼([𝑡₀,∞),𝑅₊), 𝑝,𝑞∈𝐶([𝑡₀,∞),𝑅₊) and 𝐶^𝛼 denotes continuous derivative of order 𝛼.

The definition and some important properties for the modified Riemann-Liouville derivative of order 𝛼 are listed as follows (see also in [20–24]): 𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑1−𝛼𝑑𝑡×∫^𝑡₀(𝑡−𝜉)^−𝛼𝑓(𝜉)(0)𝑓−𝑓𝑑𝜉,0<𝛼<1,_(𝑛)(𝑡)^{(𝛼−𝑛)}𝐷,𝑛≤𝛼<𝑛+1,𝑛≥1,_𝛼𝑡𝑡^𝑟=Γ(1+𝑟)Γ()𝑡1+𝑟−𝛼^𝑟−𝛼,𝐷^𝛼_𝑡𝑓(𝑡)𝑔(𝑡)=𝑔(𝑡)𝐷^𝛼_𝑡𝑓(𝑡)+𝑓(𝑡)𝐷^𝛼_𝑡𝑔(𝑡),𝐷^𝛼_𝑡𝑓𝑔(𝑡)=𝑓^′_𝑔𝑔(𝑡)𝐷^𝛼_𝑡𝑔(𝑡)=𝐷^𝛼_𝑔𝑓𝑔(𝑡)𝑔^′(𝑡)^𝛼. (2)(3)(4)(5)

As usual, a solution 𝑥(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 the next as follows. In Section 2, 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 denote 𝜉=𝑡^𝛼/Γ(1+𝛼), 𝜉_𝑖=𝑡^𝛼_𝑖/Γ(1+𝛼), 𝑖=0,1,2,3,4,5, 𝑎(𝑡)=𝑎(𝜉), 𝑟(𝑡)=𝑟(̃𝜉), 𝑝(𝑡)=𝑝(𝜉), 𝑞(𝑡)=𝑞(𝜉), 𝑅₊=(0,∞), 𝛿₁(𝜉,𝜉_𝑖)=∫^𝜉_𝜉𝑖(1/𝑎^1/𝛾(𝑠))𝑑𝑠, 𝛿₁(𝑡,𝑡_𝑖)=𝛿₁(𝜉,𝜉_𝑖), 𝛿₂(𝜉,𝜉_𝑖)=∫^𝜉_𝜉𝑖(𝛿₁(𝑠,𝜉_𝑖)/𝑟(̃𝑠))𝑑𝑠, 𝛿₂(𝑡,𝑡_𝑖)=𝛿₂(𝜉,𝜉_𝑖), 𝐴(𝜉)=exp(∫^𝜉_𝜉0(𝑝(𝑠)/𝑎(𝑠))𝑑𝑠). Let ℎ₁,ℎ₂,𝐻∈𝐶([𝜉₀,∞),𝑅) satisfy 𝐻(𝜉,𝜉)=0,𝐻(𝜉,𝑠)>0,𝜉>𝑠≥𝜉₀. (6)𝐻 has continuous partial derivatives 𝜕𝐻(𝜉,𝑠)/𝜕𝜉 and 𝜕𝐻(𝜉,𝑠)/𝜕𝑠 on [𝜉₀,∞) such that 𝜕𝐻(𝜉,𝑠)𝜕𝜉=−ℎ₁(𝜉,𝑠)𝐻(),()𝜉,𝑠𝜕𝐻𝜉,𝑠𝜕𝑠=−ℎ₂(𝜉,𝑠)𝐻(𝜉,𝑠),𝜉>𝑠≥𝜉₀. (7)

Lemma 1. Assume 𝑥(𝑡) is an eventually positive solution of (1), and ^∞_𝜉01𝐴(𝑠)𝑎(𝑠)𝑑𝑠=∞,_∞𝑡0𝛼𝑡^𝛼−1Γ(1+𝛼)𝑟(𝑡)𝑑𝑡=∞,_∞𝜉01𝑟(̃𝜁)^∞_𝜁1𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏𝑑𝜁=∞. (8)(9)(10) Then, there exists a sufficiently large 𝑇 such that 𝐷^𝛼_𝑡(𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡))>0 on [𝑇,∞) and either 𝐷^𝛼_𝑡𝑥(𝑡)>0 on [𝑇,∞) or lim_𝑡→∞𝑥(𝑡)=0.

Proof. Let 𝑥(𝑡)=𝑥(𝜉), where 𝜉=𝑡^𝛼/Γ(1+𝛼). Then, by use of (3), we obtain 𝐷^𝛼_𝑡𝜉(𝑡)=1, and furthermore, by use of the first equality in (5), we have 𝐷^𝛼_𝑡𝑥(𝑡)=𝐷^𝛼_𝑡𝑥(𝜉)=𝑥^′(𝜉)𝐷^𝛼_𝑡𝜉(𝑡)=𝑥^′(𝜉). (11) Similarly, we have 𝐷^𝛼_𝑡𝑎(𝑡)=𝑎^′(𝜉), 𝐷^𝛼_𝑡𝑟(𝑡)=𝑟^′̃(𝜉). So, (1) can be transformed into the following form: 𝑎(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′′+𝑝(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′+𝑞(𝜉)𝑥(𝜉)=0,𝜉≥𝜉₀>0. (12) Since 𝑥(𝑡) is an eventually positive solution of (1), then 𝑥(𝜉) is an eventually positive solution of (12), and there exists 𝜉₁>𝜉₀ such that 𝑥(𝜉)>0 on [𝜉₁,∞). Furthermore, we have 𝐴(𝜉)𝑎(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′′=𝐴(𝜉)𝑎(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′′+𝐴^′(𝜉)𝑎(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′=𝐴(𝜉)𝑎(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′′+𝐴(𝜉)𝑝(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′𝛾=−𝐴(𝜉)𝑞(𝜉)𝑥(𝜉)<0,𝜉≥𝜉₁. (13) Then, 𝐴(𝜉)𝑎(𝜉)(𝑟(̃𝜉)𝑥^′(𝜉))^′ 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 𝑟(̃𝜉)𝑥^′(𝜉)−𝑟̃𝜉₃𝑥^′𝜉₃=^𝜉_𝜉3𝐴(𝑠)𝑎(𝑠)𝑟(̃𝑠)𝑥^′(𝑠)^′𝐴(𝑠)𝑎(𝑠)𝜉𝑑𝑠≤𝐴₃𝑎𝜉₃𝑟̃𝜉₃𝑥^′𝜉₃^′×^𝜉_𝜉31𝐴(𝑠)𝑎(𝑠)𝑑𝑠. (14) By (8), we have lim_𝜉→∞𝑟(̃𝜉)𝑥^′(𝜉)=−∞. So there exists a sufficiently large 𝜉₄ with 𝜉₄>𝜉₃ such that 𝑥^′(𝜉)<0, 𝜉∈[𝜉₄,∞). Furthermore, 𝑥(𝜉)−𝑥𝜉₄=^𝜉_𝜉4𝑥^′(𝑠)𝑑𝑠=^𝜉_𝜉4𝑟(̃𝑠)𝑥^′(𝑠)𝑟(̃𝑠)≤𝑟̃𝜉𝑑𝑠₄𝑥^′𝜉₄^𝜉_𝜉41𝑟(̃𝑠)=𝑟̃𝜉𝑑𝑠₄𝑥^′𝜉₄^∞_𝑡4𝛼𝑡^𝛼−1Γ(1+𝛼)𝑟(𝑡)𝑑𝑡. (15) By (9), we deduce that lim_𝜉→∞𝑥(𝜉)=−∞, which contradicts the fact that 𝑥(𝜉) is an eventually positive solution of (9). So, (𝑟(̃𝜉)𝑥^′(𝜉))^′>0 on [𝜉₂,∞), and 𝐷^𝛼_𝑡(𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡))>0 on [𝑡₂,∞). Thus, 𝐷^𝛼_𝑡𝑥(𝑡)=𝑥^′(𝜉) is eventually of one sign. Now we assume 𝑥^′(𝜉)<0, 𝜉∈[𝜉₅,∞) for some sufficiently large 𝜉₅>𝜉₄. Since 𝑥(𝜉)>0, furthermore we have lim_𝜉→∞𝑥(𝜉)=𝛽≥0. We claim 𝛽=0. Otherwise, assume 𝛽>0. Then 𝑥(𝜉)≥𝛽 on [𝜉₅,∞), and, for 𝜉∈[𝜉₅,∞), by (12) we have 𝐴(𝜉)𝑎(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′′(𝜉)𝑞(𝜉)𝑥(𝜉)(𝜉)𝑞(𝜉)≤−𝐴≤−𝐴𝛽. (16) Substituting 𝜉 with 𝑠 in the previous inequality, an integration with respect to 𝑠 from 𝜉 to ∞ yields −𝐴(𝜉)𝑎(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′≤−lim_𝜉→∞𝐴(𝜉)𝑎(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′−𝛽^∞_𝜉𝐴(𝑠)𝑞(𝑠)𝑑𝑠<−𝛽^∞_𝜉𝐴(𝑠)𝑞(𝑠)𝑑𝑠, (17) which means 𝑟(̃𝜉)𝑥^′(𝜉)^′>𝛽𝐴(𝜉)𝑎(𝜉)^∞_𝜉𝐴(𝑠)𝑞(𝑠)𝑑𝑠. (18) Substituting 𝜉 with 𝜏 in (18), an integration for (18) with respect to 𝜏 from 𝜉 to ∞ yields −𝑟(̃𝜉)𝑥^′(𝜉)>−lim_𝜉→∞𝑟(̃𝜉)𝑥^′(𝜉)+𝛽^∞_𝜉1𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏>𝛽^∞_𝜉1𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏; (19) that is, 𝑥^′(𝜉)1<−𝛽𝑟(̃𝜉)^∞_𝜉1𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏. (20) Substituting 𝜉 with 𝜁 in (20), an integration for (20) with respect to 𝜁 from 𝜉₅ to 𝜉 yields 𝑥(𝜉)−𝑥𝜉₅<−𝛽^𝜉_𝜉51𝑟(̃𝜁)^∞_𝜁1𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝐴(𝑠)𝑞(𝑠)𝑑𝑠𝑑𝜏𝑑𝜁. (21) By (10), one can see lim_𝑡→∞𝑥(𝜉)=−∞, which causes a contradiction. So, the proof is complete.

Lemma 2. Assume that 𝑥 is an eventually positive solution of (1) such that 𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)>0,𝐷^𝛼_𝑡𝑥(𝑡)>0 (22) on [𝑡₁,∞)_𝕋, where 𝑡₁≥𝑡₀ is sufficiently large. Then, for 𝑡≥𝑡₁, we have 𝐷^𝛼_𝑡𝑥(𝑡)≥𝐴(𝜉)𝛿₁𝑡,𝑡₁𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑟(𝑡),𝑥(𝑡)≥𝐴(𝜉)𝛿₂𝑡,𝑡₁𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡). (23)(24)

Proof. By (13), we obtain that 𝐴(𝜉)𝑎(𝜉)(𝑟(̃𝜉)𝑥^′(𝜉))^′ is strictly decreasing on [𝜉₁,∞). So, 𝑟(̃𝜉)𝑥^′(𝜉)≥𝑟(̃𝜉)𝑥^′(𝜉)−𝑟̃𝜉₁𝑥^′𝜉₁=^𝜉_𝜉1𝐴(𝑠)𝑎(𝑠)𝑟(̃𝑠)𝑥^′(𝑠)^′𝐴(𝑠)𝑎(𝑠)(𝜉)𝑎(𝜉)𝑟(̃𝜉)𝑥𝑑𝑠≥𝐴^′(𝜉)^′𝜉_𝜉11𝐴(𝑠)𝑎(𝑠)(𝜉)𝑎(𝜉)𝑟(̃𝜉)𝑥𝑑𝑠=𝐴^′(𝜉)^′𝛿₁𝜉,𝜉₁; (25) that is 𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)≥𝐴(𝜉)𝛿₁𝑡,𝑡₁𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡), (26) which admits (23). On the other hand, we have 𝑥(𝜉)≥𝑥(𝜉)−𝑥𝜉₁≥^𝜉_𝜉1𝐴(𝑠)𝛿₁𝑠,𝜉₁𝑎(𝑠)𝑟(̃𝑠)𝑥^′(𝑠)^′𝐴(𝑠)𝑟(̃𝑠)(𝜉)𝑎(𝜉𝑟(̃𝜉)𝑥𝑑𝑠≥𝐴)^′(𝜉)^′𝜉_𝜉1𝛿₁𝑠,𝜉₁𝐴(𝑠)𝑟(̃𝑠)(𝜉)𝛿𝑑𝑠=𝐴₂𝜉,𝜉₁𝑎(𝜉)𝑟(̃𝜉)𝑥^′(𝜉)^′, (27) which can be rewritten as (24). So the proof is complete.

Lemma 3 (see [25, Theorem 41]). Assume that 𝐴 and 𝐵 are nonnegative real numbers. Then, 𝜆𝐴𝐵^𝜆−1−𝐴^𝜆≤(𝜆−1)𝐵^𝜆 (28) for all 𝜆>1.

Theorem 4. Assume that (8)–(10) hold. If there exists 𝜙∈𝐶^𝛼([𝑡₀,∞),𝑅₊) such that for any sufficiently large 𝑇≥𝜉₀, there exist 𝑎, 𝑏, 𝑐 with 𝑇≤𝑎<𝑐<𝑏 satisfying 1𝐻(𝑏,𝑐)^𝑏_𝑐𝐻()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)+1𝑏,𝑠𝑑𝑠𝐻(𝑐,𝑎)^𝑐_𝑎𝐻()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)>1𝑠,𝑎𝑑𝑠𝐻(𝑏,𝑐)^𝑏_𝑐𝑟(̃𝑠)𝜙(𝑠)4𝛿₁s,𝜉₂𝑄²₂()+1𝑏,𝑠𝑑𝑠𝐻(𝑐,𝑎)^𝑐_𝑎𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂𝑄²₁(𝑠,𝑎)𝑑𝑠, (29) where 𝜙(𝜉)=𝜙(𝑡), 𝑄₁(𝑠,𝜉)=ℎ₁(𝑠,𝜉)−(𝜙^′(𝑠)/𝜙(𝑠))𝐻(𝑠,𝜉),  𝑄₂(𝜉,𝑠)=ℎ₂(𝜉,𝑠)−(𝜙^′(𝑠)/𝜙(𝑠))𝐻(𝜉,𝑠); then, (1) is oscillatory or satisfies lim_𝑡→∞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 𝐷^𝛼_𝑡(𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡))>0,𝑡∈[𝑡₂,∞), where 𝑡₂>𝑡₁ is sufficiently large, and either 𝐷^𝛼_𝑡𝑥(𝑡)>0 on [𝑡₂,∞) or lim_𝑡→∞𝑥(𝑡)=0. Now we assume 𝐷^𝛼_𝑡𝑥(𝑡)>0 on [𝑡₂,∞). Define the generalized Riccati function: 𝜔(𝑡)(𝑡)𝐴(𝜉)𝑎(𝑡)𝐷=𝜙^𝛼𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑥(𝑡). (30) Then, for 𝑡∈[𝑡₂,∞), we have 𝐷^𝛼_𝑡𝜔(𝑡)=𝐷^𝛼_𝑡𝜙(𝑡)𝐴(𝜉)𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑥(𝑡)+𝜙(𝑡)𝐷^𝛼_𝑡𝐴(𝜉)𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑥(𝑡)=𝜙(𝑡)𝐷^𝛼_𝑡𝐴(𝜉)𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑥(𝑡)(𝑡)𝐷−𝜙^𝛼𝑡𝑥(𝑡)𝐴(𝜉)𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑥²(𝑡)+𝐷^𝛼_𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡)=𝜙(𝑡)𝐴(𝜉)𝐷^𝛼_𝑡𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡(𝑡)𝐷)+𝑎^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝐷^𝛼_𝑡𝐴(𝜉×(𝑥(𝑡)))⁻¹(𝑡)𝐷−𝜙^𝛼𝑡𝑥(𝑡)𝐴(𝜉)𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑥²(𝑡)+𝐷^𝛼_𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡)=𝜙(𝑡)𝐴(𝜉)𝐷^𝛼_𝑡𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)+𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝐴^′(𝜉)𝐷^𝛼_𝑡𝜉×(𝑥(𝑡))⁻¹(𝑡)𝐷−𝜙^𝛼𝑡𝑥(𝑡)𝐴(𝜉)𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑥²(𝑡)+𝐷^𝛼_𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡). (31) Using 𝐷^𝛼_𝑡𝜉=1 and (23), we obtain 𝐷^𝛼_𝑡𝜔(𝑡)≤𝜙(𝑡)𝐴(𝜉)𝐷^𝛼_𝑡𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)+𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝐴(𝜉)𝑝(𝜉)𝑎(𝜉)×(𝑥(𝑡))⁻¹−𝜙(𝑡)𝐴(𝜉)𝛿₁𝑡,𝑡₂𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)×𝐷^𝛼_𝑡𝑥(𝑡)𝐴(𝜉)𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)×𝑟(𝑡)𝑥²(𝑡)⁻¹+𝐷^𝛼_𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡)=𝜙(𝑡)𝐴(𝜉)𝐷^𝛼_𝑡𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡(𝑡)𝐷)+𝑝^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)×(𝑥(𝑡))⁻¹−𝛿₁𝑡,𝑡₂𝜙(𝑡)𝑟(𝑡)𝜔²(𝑡)+𝐷^𝛼_𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡)(𝜉)𝑞(𝑡)𝜙(𝑡)−𝛿=−𝐴¹𝑡,𝑡₂𝜙(𝑡)𝑟(𝑡)𝜔²(𝑡)+𝐷^𝛼_𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡). (32) Let 𝜔(𝑡)=𝜔(𝜉). Then 𝐷^𝛼_𝑡𝑤(𝑡)=𝑤^′(𝜉), and 𝐷^𝛼_𝑡𝜙(𝑡)=𝜙^′(𝜉). So (32) is transformed into the following form: 𝜔^′(𝜉)(𝜉)𝜙(𝜉)𝑞(𝜉)−𝛿≤−𝐴¹𝜉,𝜉₂𝑟(̃𝜉)𝜙(𝜉)𝜔²(𝜉)+𝜙^′(𝜉)𝜙(𝜉)𝜔(𝜉),𝜉≥𝜉₂. (33)
Choose 𝑎, 𝑏, 𝑐 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 ^𝜉_𝑐𝐻(𝜉,𝑠)𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)𝑑𝑠≤−^𝜉_𝑐𝐻(𝜉,𝑠)𝑤^′(𝑠)𝑑𝑠+^𝜉_𝑐𝐻()𝜙𝜉,𝑠^′(𝑠)𝜙(𝑠)𝑤(𝑠)−𝑑𝑠_𝜉𝑐𝐻()𝛿𝜉,𝑠¹𝑠,𝜉₂𝑟(̃𝑠)𝜙(𝑠)𝜔²(𝑠)()𝑤(𝑐)−𝑑𝑠=𝐻𝜉,𝑐_𝜉𝑐𝛿₁𝑠,𝜉₂𝐻(𝜉,𝑠)𝑟(̃𝑠)𝜙(𝑠)^1/2𝑤(𝑠)+12𝑟(̃𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂^1/2𝑄₂(𝜉,𝑠)²𝑑𝑠+^𝜉_𝑐𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂𝑄²₂()()𝑤(𝑐)+𝜉,𝑠𝑑𝑠≤𝐻𝜉,𝑐^𝜉_𝑐𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂𝑄²₂(𝜉,𝑠)𝑑𝑠. (34) Dividing both sides of the inequality (34) by 𝐻(𝜉,𝑐) and letting 𝜉→𝑏⁻, we obtain 1𝐻(𝑏,𝑐)^𝑏_𝑐𝐻()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)≤𝑤(𝑐)+1𝑏,𝑠𝑑𝑠𝐻(𝑏,𝑐)^𝑏_𝑐𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂𝑄²₂(𝑏,𝑠)𝑑𝑠. (35)
On the other hand, substituting 𝜉 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], we get that ^𝑐_𝜉𝐻(𝑠,𝜉)𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)𝑑𝑠≤−^𝑐_𝜉𝐻(𝑠,𝜉)𝑤^′(𝑠)𝑑𝑠+^𝑐_𝜉𝐻()𝜙𝑠,𝜉^′(𝑠)𝜙(𝑠)𝑤(𝑠)−𝑑𝑠_𝑐𝜉𝐻()𝛿𝑠,𝜉¹𝑠,𝜉₂𝑟(̃𝑠)𝜙(𝑠)𝜔²(𝑠)()𝑤(𝑐)−𝑑𝑠=−𝐻𝑐,𝜉_𝑐𝜉𝛿₁𝑠,𝜉₂𝐻(𝑠,𝜉)𝑟(̃𝑠)𝜙(s)^1/2𝑤(𝑠)+12𝑟(̃𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂^1/2𝑄₁(𝜉,𝑠)²+𝑑𝑠_𝑐𝜉𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂𝑄²₁()()𝑤(𝑐)+𝑠,𝜉𝑑𝑠≤−𝐻𝑐,𝜉^𝑐_𝜉𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂𝑄²₁(𝑠,𝜉)𝑑𝑠. (36) Dividing both sides of the inequality (36) by 𝐻(𝑐,𝜉) and letting 𝜉→𝑎⁺, we obtain 1𝐻(𝑐,𝑎)^𝑐_𝑎𝐻()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)𝑤(𝑐)+1𝑠,𝑎𝑑𝑠≤−𝐻(𝑐,𝑎)^𝑐_𝑎𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂𝑄²₁(𝑠,𝑎)𝑑𝑠. (37) A combination of (35) and (37) yields 1𝐻(𝑏,𝑐)^𝑏_𝑐𝐻()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)+1𝑏,𝑠𝑑𝑠𝐻(𝑐,𝑎)^𝑐_𝑎𝐻()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)≤1𝑠,𝑎𝑑𝑠𝐻(𝑏,𝑐)^𝑏_𝑐𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂𝑄²₂()+1𝑏,𝑠𝑑𝑠𝐻(𝑐,𝑎)^𝑐_𝑎𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂𝑄²₁(𝑠,𝑎)𝑑𝑠, (38) which contradicts (29). So, the proof is complete.

Theorem 5. Under the conditions of Theorem 4, if for any sufficiently large 𝑙≥𝜉₀, lim_𝜉→∞sup^𝜉_𝑙𝐻()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙(𝑠)𝑠,𝑙4𝛿₁𝑠,𝜉₂𝑄²₁(𝑠,𝑙)𝑑𝑠>0,lim_𝜉→∞sup^𝜉_𝑙𝐻()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙(𝑠)𝜉,𝑠4𝛿₁𝑠,𝜉₂𝑄²₂(𝜉,𝑠)𝑑𝑠>0, (39)(40) then (1) is oscillatory.

Proof. For any 𝑇≥𝜉₀, let 𝑎=𝑇. In (39), we choose 𝑙=𝑎. Then, there exists 𝑐>𝑎 such that ^𝑐_𝑎𝐻()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙(𝑠)𝑠,𝑎4𝛿₁𝑠,𝜉₂𝑄²₁(𝑠,𝑎)𝑑𝑠>0. (41) In (40), we choose 𝑙=𝑐>𝑎. Then there exists 𝑏>𝑐 such that ^𝑏_𝑐𝐻()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙(𝑠)𝑏,𝑠4𝛿₁𝑠,𝜉₂𝑄²₂(𝑏,𝑠)𝑑𝑠>0. (42) Combining (41) and (42), we obtain (29). The conclusion thus comes from Theorem 4, and the proof is complete.

In Theorems 4 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 Theorem 4, if for any sufficiently large 𝑇≥𝜉₀, there exist 𝑎, 𝑏, 𝑐 with 𝑇≤𝑎<𝑐<𝑏 satisfying 1(𝑐−𝑎)^𝜆𝑐𝑎(𝑠−𝑎)^𝜆𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)+1𝑑𝑠(𝑏−𝑐)^𝜆𝑏𝑐(𝑏−𝑠)^𝜆𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)>1𝑑𝑠(𝑐−𝑎)^𝜆𝑐𝑎𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂(𝑠−𝑎)^𝜆−2𝜙𝜆+^′(𝑠)𝜙(𝑠)(𝑠−𝑎)²+1𝑑𝑠(𝑏−𝑐)^𝜆𝑏𝑐𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂(𝑏−𝑠)^𝜆−2×𝜙𝜆−^′(𝑠)𝜙(𝑠)(𝑏−𝑠)²𝑑𝑠, (43) then (1) is oscillatory.

Corollary 7. Under the conditions of Theorem 5, if for any sufficiently large 𝑙≥𝜉₀, lim_𝜉→∞sup^𝜉_𝑙(𝑠−𝑙)^𝜆𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂(𝑠−𝑙)^𝜆−2×𝜙𝜆+^′(𝑠)𝜙(𝑠)(𝑠−𝑙)²𝑑𝑠>0,lim_𝜉→∞sup^𝜉_𝑙(𝜉−𝑠)^𝜆𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂(𝜉−𝑠)^𝜆−2×𝜙𝜆−^′(𝑠)𝜙(𝑠)(𝜉−𝑠)²𝑑𝑠>0, (44) then (1) is oscillatory.

Theorem 8. Assume (8)–(10) hold, and ^∞_𝜉0𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙^′(𝑠)²4𝛿₁𝑠,𝜉₂𝜙(𝑠)𝑑𝑠=∞, (45) where 𝜙 is defined as in Theorem 4. Then every solution of (1) is oscillatory or satisfies lim_𝑡→∞𝑥(𝑡)=0.

Proof. Assume (1) 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 1, we have 𝐷^𝛼_𝑡(𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡))>0, 𝑡∈[𝑡₂,∞), where 𝑡₂>𝑡₁ is sufficiently large, and either 𝐷^𝛼_𝑡𝑥(𝑡)>0 on [𝑡₂,∞) or lim_𝑡→∞𝑥(𝑡)=0. Now we assume that 𝐷^𝛼_𝑡𝑥(𝑡)>0 on [𝑡₂,∞). Let 𝜔(𝑡), 𝜔(𝜉) be defined as in Theorem 4. Then we obtain (33), and furthermore, 𝜔^′(𝜉)(𝜉)𝜙(𝜉)𝑞(𝜉)−𝛿≤−𝐴¹𝜉,𝜉₂𝑟(̃𝜉)𝜙(𝜉)𝜔²(𝜉)+𝜙^′(𝜉)𝜙(𝜉)𝜔(𝜉)(𝜉)𝜙(𝜉)𝑞(𝜉)+𝑟(̃𝜉𝜙≤−𝐴)^′(𝜉)²4𝛿₁𝜉,𝜉₂𝜙(𝜉),𝜉≥𝜉₂. (46)
Substituting 𝜉 with 𝑠 in (46) and integrating (46) with respect to 𝑠 from 𝜉₂ to 𝜉 yield ^𝜉_𝜉2𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙^′(𝑠)²4𝛿₁𝑠,𝜉₂𝜙(𝑠)𝑑𝑠≤𝜔𝜉₂(𝜉)𝜉−𝜔≤𝜔₂<∞, (47) which contradicts (45). So, the proof is complete.

Theorem 9. Assume (8)–(10) hold, and there exists a function 𝐺∈𝐶([𝜉₀,∞),ℝ) such that 𝐺(𝜉,𝜉)=0, for 𝜉≥𝜉₀, 𝐺(𝜉,𝑠)>0, for 𝜉>𝑠≥𝜉₀, and 𝐺 has a nonpositive continuous partial derivative 𝐺^′_𝑠(𝜉,𝑠). If lim_𝜉→∞1sup𝐺𝜉,𝜉₀×^𝜉_𝜉0𝐺(𝜉,𝑠)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+1/𝛾(s)𝑟(̃𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟(̃𝑠)𝜙^′(𝑠)^𝛾+1𝛾+1^𝛾+1𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟(̃𝑠)𝑑𝑠=∞, (48) where 𝜙 is defined as in Theorem 4, then every solution of (1) is oscillatory or satisfies lim_𝑡→∞𝑥(𝑡)=0.

Proof. Assume (1) 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 1, we have 𝐷^𝛼_𝑡(𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡))>0, 𝑡∈[𝑡₂,∞), where 𝑡₂>𝑡₁ is sufficiently large, and either 𝐷^𝛼_𝑡𝑥(𝑡)>0 on [𝑡₂,∞) or lim_𝑡→∞𝑥(𝑡)=0. Now we assume 𝐷^𝛼_𝑡𝑥(𝑡)>0 on [𝑡₂,∞). Let 𝜔(𝑡), 𝜔(𝜉) be defined as in Theorem 4. By (46), we have 𝐴(𝜉)𝜙(𝜉)𝑞(𝜉)−𝑟(̃𝜉)𝜙^′(𝜉)²4𝛿₁𝜉,𝜉₂𝜙(𝜉)≤𝜔^′(𝜉),𝜉≥𝜉₂. (49) Substituting 𝜉 with 𝑠 in (49), multiplying both sides by 𝐺(𝜉,𝑠), and then integrating both sides of (49) with respect to 𝑠 from 𝜉₂ to 𝜉 yield ^𝜉_𝜉2𝐺()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠𝜙𝜉,𝑠)^′(𝑠)²4𝛿₁𝑠,𝜉₂𝜙(𝑠)𝑑𝑠≤−^𝜉_𝜉2𝐺(𝜉,𝑠)𝜔^′(𝑠)𝑑𝑠=𝐺𝜉,𝜉₂𝜔𝜉₂+^𝜉_𝜉2𝐻^′_𝑠(𝜉,𝑠)𝜔(𝑠)Δ𝑠≤𝐺𝜉,𝜉₂𝜔𝜉₂≤𝐺𝜉,𝜉₀𝜔𝜉₂. (50) Then, ^𝜉_𝜉0𝐺()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠𝜙𝜉,𝑠)^′(𝑠)²4𝛿₁𝑠,𝜉₂𝜙(𝑠)=𝑑𝑠_𝜉2𝜉0𝐺()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠𝜙𝜉,𝑠)^′(𝑠)²4𝛿₁𝑠,𝜉₂𝜙(𝑠)+𝑑𝑠_𝜉𝜉2𝐺()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠𝜙𝜉,𝑠)^′(𝑠)²4𝛿₁𝑠,𝜉₂𝜙(𝑠)𝑑𝑠≤𝐺𝜉,𝜉₀𝜔𝜉₂+𝐺𝜉,𝜉₀×^𝜉2_𝜉0𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙^′(𝑠)²4𝛿₁𝑠,𝜉₂𝜙(𝑠)𝑑𝑠. (51) So, lim_𝜉→∞1sup𝐺𝜉,𝜉₀^𝜉_𝜉0𝐺()𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠𝜙𝜉,𝑠)^′(𝑠)²4𝛿₁𝑠,𝜉₂𝜙(𝑠)≤𝜔𝜉𝑑𝑠₂+^𝜉2_𝜉0𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙^′(𝑠)²4𝛿₁𝑠,𝜉₂𝜙(𝑠)𝑑𝑠<∞, (52) which contradicts (48). So the proof is complete.

3. Applications of the Results

Example 10. Consider the following fractional differential equation: 𝐷_𝑡^1/3𝑡^1/9𝐷_𝑡^1/3𝐷_𝑡^1/3𝑥(𝑡)+𝑡^−1/3𝐷_𝑡^1/3𝐷_𝑡^1/3𝑥(𝑡)+𝑡^−2/3𝑥(𝑡)=0,𝑡≥2. (53)
In (1), if we set 𝑡₀=2, 𝛼=1/3, 𝑎(𝑡)=𝑡^1/9, 𝑟(𝑡)≡1, (𝑡)=𝑡^−1/3, 𝑞(𝑡)=𝑡^−2/3, then we obtain (53). So 𝜉₀=2^1/3/Γ(4/3), 𝑎(𝜉)=𝑎(𝑡)=𝑡^1/9=[Γ(4/3)𝜉]^1/3, 𝑟(̃𝜉)≡1, 𝑝(𝜉)=[Γ(4/3)𝜉]⁻¹, 𝑞(𝜉)=[Γ(4/3)𝜉]⁻². Furthermore, 𝐴(𝜉)=exp([Γ(4/3)]^−4/3∫^𝜉_𝜉0𝑠^−4/3𝑑𝑠)=exp(3[Γ(4/3)]^−4/3[𝜉₀^−1/3−𝜉^−1/3]), which implies 1≤𝐴(𝜉)≤exp(3[Γ(4/3)]^−4/3𝜉₀^−1/3). On the other hand, 𝛿₁(𝜉,𝜉₂)=∫^𝜉_𝜉2(1/𝐴(𝑠)𝑎(𝑠))𝑑𝑠≥[Γ(4/3)]^−1/3exp(−3[Γ(4/3)]^−4/3𝜉₀^−1/3)∫^𝜉_𝜉2𝑠^−1/3𝑑𝑠=(3/2)[Γ(4/3)]^−1/3exp(−3[Γ(4/3)]^−(4/3)𝜉₀^−1/3)(𝜉^2/3−𝜉₂^2/3), which implies lim_𝜉→∞𝛿₁(𝜉,𝜉₂)=∞, and then (8) holds. So, there exists a sufficiently large 𝑇>𝜉₂ such that 𝛿₁(𝜉,𝜉₂)>[Γ(4/3)]² on [𝑇,∞). In (9), ^∞_𝑡0𝛼𝑡^𝛼−1Γ(1+𝛼)𝑟(𝑡)𝑑𝑡=^∞_𝜉01𝑟(̃𝑠)𝑑𝑠=^∞_𝜉0𝑑𝑠=∞. (54) In (10), ^∞_𝜉01𝑟(̃𝜁)^∞_𝜁1𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝐴(𝑠)𝑞(𝑠)≥Γ4𝑑𝑠𝑑𝜏𝑑𝜁3^−1/3Γ4exp−33^−4/3𝜉₀^−1/3×^∞_𝜉0^∞_𝜁𝜏^−1/3∞_𝜏𝑠⁻¹𝑑𝑠𝑑𝜏𝑑𝜁=∞. (55) In (48), letting 𝜙(𝜉)=𝜉, we obtain ^∞_𝜉0𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙^′(𝑠)²4𝛿₁𝑠,𝜉₂𝜙(𝑠)=𝑑𝑠_∞𝜉0𝐴(𝑠Γ4)3⁻²−14𝛿₁𝑠,𝜉₂1𝑠=𝑑𝑠_𝑇𝜉0𝐴(𝑠Γ4)3⁻²−14𝛿₁𝑠,𝜉₂1𝑠+𝑑𝑠_∞𝑇𝐴(𝑠Γ4)3⁻²−14𝛿₁𝑠,𝜉₂1𝑠𝑑𝑠≥^𝑇_𝜉0𝐴(𝑠Γ4)3⁻²−14𝛿₁𝑠,𝜉₂1𝑠+𝑑𝑠_∞𝑇34Γ43⁻²1𝑠𝑑𝑠=∞. (56) Therefore, (53) is oscillatory by Theorem 8.

Example 11. Consider the following fractional differential equation: 𝐷_𝑡^1/2𝑡^1/4𝐷_𝑡^1/2𝐷_𝑡^1/2𝑥(𝑡+Γ())3/2√𝑡𝐷_𝑡^1/2𝐷_𝑡^1/2𝑥(𝑡)+𝑥(𝑡)=0,𝑡≥2. (57)
In (1), if we set 𝑡₀=2, 𝛼=1/2, 𝑎(𝑡)=𝑡^1/4, 𝑟(𝑡)≡1, 𝑝(𝑡)=Γ(3/2)/√𝑡, 𝑞(𝑡)≡1, then we obtain (57). So 𝜉₀=2^1/2/Γ(3/2), 𝑎(𝜉)=𝑎(𝑡)=𝑡^1/4=Γ(3/2)𝜉, 𝑟(̃𝜉)≡1, 𝑝(𝜉)=𝜉⁻¹, 𝑞(𝜉)≡1. Furthermore, 𝐴(𝜉)=exp([Γ(3/2)]^−1/2∫^𝜉_𝜉0𝑠^−3/2𝑑𝑠)=exp([Γ(3/2)]^−1/2[2𝜉₀^−1/2−2𝜉^−1/2]), which implies 1≤𝐴(𝜉)≤exp(2[Γ(3/2)]^−1/2𝜉₀^−1/2). On the other hand, 𝛿₁(𝜉,𝜉₂)=∫^𝜉_𝜉2(1/𝐴(𝑠)𝑎(𝑠))𝑑𝑠≥[Γ(3/2)]^−1/2exp(−2[Γ(3/2)]^−1/2𝜉₀^−1/2)∫^𝜉_𝜉2(1/√𝑠)𝑑𝑠=2[Γ(3/2)]^−1/2exp(−2[Γ(3/2)]^−1/2𝜉₀^−1/2)(𝜉−𝜉₂), which implies lim_𝜉→∞𝛿₁(𝜉,𝜉₂)=∞. So, there exists a sufficiently large 𝑇>𝜉₂ such that 𝛿₁(𝜉,𝜉₂)>1 on [𝑇,∞).
From the analysis above, one can see the (8) holds. We now test (9) and (10). In (9), ^∞_𝑡0𝛼𝑡^𝛼−1Γ(1+𝛼)𝑟(𝑡)𝑑𝑡=^∞_𝜉01𝑟(̃𝑠)𝑑𝑠=^∞_𝜉0𝑑𝑠=∞. (58) In (10), ^∞_𝜉01𝑟(̃𝜁)^∞_𝜁1𝐴(𝜏)𝑎(𝜏)^∞_𝜏𝐴(𝑠)𝑞(𝑠)≥Γ3𝑑𝑠𝑑𝜏𝑑𝜁2^−1/2Γ3exp−22^−1/2𝜉₀^−1/2×^∞_𝜉0^∞_𝜁1√𝜏^∞_𝜏𝑑𝑠𝑑𝜏𝑑𝜁=∞. (59) So, (9) and (10) hold. On the other hand, in (44), after putting 𝜙(𝜉)≡1, 𝜆=2, for any sufficiently large 𝑙, we have lim_𝜉→∞sup^𝜉_𝑙(𝑠−𝑙)^𝜆𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂(𝑠−𝑙)^𝜆−2𝜙𝜆+^′(𝑠)𝜙(𝑠)(𝑠−𝑙)²𝑑𝑠≥lim_𝜉→∞sup^𝜉_𝑙(𝑠−𝑙)²𝑠⁻¹−1𝑑𝑠=∞,lim_𝜉→∞sup^𝜉_𝑙(𝜉−𝑠)^𝜆𝐴(𝑠)𝜙(𝑠)𝑞(𝑠)−𝑟(̃𝑠)𝜙(𝑠)4𝛿₁𝑠,𝜉₂(𝜉−𝑠)^𝜆−2𝜙𝜆−^′(𝑠)𝜙(𝑠)(𝜉−𝑠)²𝑑𝑠≥lim_𝜉→∞sup^𝜉_𝑙(𝜉−𝑠)²𝑠⁻¹−1𝑑𝑠=∞. (60) So (44) holds, and then by Corollary 7 we deduce that (57) is oscillatory.