Oscillation on a Class of Differential Equations of Fractional Order

Abstract

Based on Riccati transformation and certain inequality technique, some new oscillatory criteria are established for the solutions of a class of sequential differential equations with fractional order defined in the modified Riemann-Liouville derivative. The oscillatory criteria established are of new forms compared with the existing results so far in the literature. For illustrating the validity of the results established, we present some examples for them.

1. Introduction

Recently, research for oscillation of various equations including differential equations, difference equations, and dynamic equations on time scales, has been a hot topic in the literature, and much effort has been done to establish new oscillation criteria for these equations so far (e.g., see [1–21] 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 [22, 23] and Zheng’s work [24].

In this paper, we are concerned with oscillation of a class of fractional differential equations as follows: 𝐷^𝛼_𝑡𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)^𝛾+𝑞(𝑡)𝑓(𝑥(𝑡))=0,𝑡≥𝑡₀>0,0<𝛼<1, (1) where 𝐷^𝛼_𝑡(⋅) denotes the modified Riemann-Liouville derivative [25] with respect to the variable 𝑡, 𝛾 is the quotient of two odd positive numbers, the functions 𝑎∈𝐶^𝛼([𝑡₀,∞),𝑅₊),𝑟∈𝐶^2𝛼([𝑡₀,∞),𝑅₊),and𝑞∈𝐶([𝑡₀,∞),𝑅₊), 𝐶^𝛼 denotes continuous derivative of order 𝛼, and the function 𝑓 is continuous satisfying 𝑓(𝑥)/𝑥^𝛾≥𝐾 for some positive constant 𝐾 and for all 𝑥≠0.

The definition and some important properties for Jumarie’s modified Riemann-Liouville derivative of order 𝛼 are listed as follows (see also in [26–28]):𝐷^𝛼_𝑡𝑓(𝑡)=1Γ()𝑑1−𝛼𝑑𝑡×∫^𝑡₀(𝑡−𝜉)^−𝛼×𝑓(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)

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 this paper 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), 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), and 𝛿₂(𝜉,𝜉_𝑖)=∫^𝜉_𝜉𝑖(𝛿₁(𝑠,𝜉_𝑖)/𝑟(̃𝑠))𝑑𝑠,𝛿₂(𝑡,𝑡_𝑖)=𝛿₂(𝜉,𝜉_𝑖).

Lemma 1. Assume that 𝑥(𝑡) is an eventually positive solution of (1), and ^∞_𝜉01𝑎^1/𝛾(𝑠)𝑑𝑠=∞,_∞𝑡0𝛼𝑡^𝛼−1Γ(1+𝛼)𝑟(𝑡)𝑑𝑡=∞,_∞𝜉01𝑟̃(𝜁)^∞_𝜁1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝛾𝑑𝜏𝑑𝜁=∞. (5)(6)(7) 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 (2) we obtain 𝐷^𝛼_𝑡𝜉(𝑡)=1, and furthermore by use of the first equality in (4), we have 𝐷^𝛼_𝑡𝑥(𝑡)=𝐷^𝛼_𝑡𝑥(𝜉)=𝑥^′(𝜉)𝐷^𝛼_𝑡𝜉(𝑡)=𝑥^′(𝜉). (8) 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)𝑟̃(𝜉)𝑥^′(𝜉)^′𝛾′+𝑞(𝜉)𝑓(𝑥(𝜉))=0,𝜉≥𝜉₀>0. (9) Since 𝑥(𝑡) is an eventually positive solution of (1), then 𝑥(𝜉) is an eventually positive solution of (9), and there exists 𝜉₁>𝜉₀ such that 𝑥(𝜉)>0 on [𝜉₁,∞). Furthermore, we have 𝑎(𝜉)𝑟̃(𝜉)𝑥^′(𝜉)^′𝛾′𝑞(𝜉)𝑓(𝑥(𝜉𝑞(𝜉)𝑥=−))≤−𝐾^𝛾(𝜉)<0,𝜉≥𝜉₁. (10) Then 𝑎(𝜉)((𝑟(̃𝜉)𝑥^′(𝜉))^′)^𝛾 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 𝑟̃(𝜉)𝑥^′(𝜉)−𝑟̃𝜉₃𝑥^′𝜉₃=^𝜉_𝜉3𝑎^1/𝛾(𝑠)𝑟̃(𝑠)𝑥^′(𝑠)^′𝑎^1/𝛾(𝑠)≤𝑎𝑑𝑠_1/𝛾𝜉₃𝑟̃𝜉₃𝑥^′𝜉₃^′𝜉_𝜉31𝑎^1/𝛾(𝑠)𝑑𝑠. (11) By (5), 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_𝜉4𝑟̃(𝑠)𝑥^′(𝑠)𝑟̃(𝑠)≤𝑟̃𝜉𝑑𝑠₄𝑥^′𝜉₄^𝜉_𝜉41𝑟̃(𝑠)=𝑟̃𝜉𝑑𝑠₄𝑥^′𝜉₄^∞_𝑡4𝛼𝑡^𝛼−1Γ(1+𝛼)𝑟(𝑡)𝑑𝑡. (12) By (6) we deduce that lim_𝜉→∞𝑥(𝜉)=−∞, which contradicts the fact that 𝑥(𝜉) is an eventually positive solution of (6). 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 that 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 that 𝛽=0. Otherwise, assume that 𝛽>0. Then 𝑥(𝜉)≥𝛽 on [𝜉₅,∞), and for 𝜉∈[𝜉₅,∞) by (9) we have 𝑎(𝜉)𝑟̃(𝜉)𝑥^′(𝜉)^′𝛾′≤−𝐾𝑞(𝜉)𝑥^𝛾(𝜉)≤−𝐾𝑞(𝜉)𝛽^𝛾. (13) Substituting 𝜉 with 𝑠 in the inequality previously, an integration with respect to 𝑠 from 𝜉 to ∞ yields −𝑎(𝜉)𝑟̃(𝜉)𝑥^′(𝜉)^′𝛾≤−lim_𝜉→∞𝑎(𝜉)(𝑟̃(𝜉)𝑥(𝜉))^′𝛾−𝐾𝛽^𝛾∞_𝜉𝑞(𝑠)𝑑𝑠<−𝐾𝛽^𝛾∞_𝜉𝑞(𝑠)𝑑𝑠, (14) which means 𝑟̃(𝜉)𝑥^′(𝜉)^′>𝐾^1/𝛾𝛽1𝑎(𝜉)^∞_𝜉𝑞(𝑠)𝑑𝑠^1/𝛾. (15) Substituting 𝜉 with 𝜏 in (15), an integration for (15) with respect to 𝜏 from 𝜉 to ∞ yields −𝑟̃(𝜉)𝑥^′(𝜉)>−lim_𝜉→∞𝑟̃(𝜉)𝑥^′(𝜉)+𝐾^1/𝛾𝛽^∞_𝜉1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝛾𝑑𝜏>𝐾^1/𝛾𝛽^∞_𝜉1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝛾𝑑𝜏; (16) that is, 𝑥^′(𝜉)<−𝐾^1/𝛾𝛽1𝑟̃(𝜉)^∞_𝜉1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝛾𝑑𝜏. (17) Substituting 𝜉 with 𝜁 in (17), an integration for (17) with respect to 𝜁 from 𝜉₅ to 𝜉 yields 𝑥(𝜉)−𝑥𝜉₅<−𝐾^1/𝛾𝛽^𝜉_𝜉51𝑟̃(𝜁)^∞_𝜁1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝛾𝑑𝜏𝑑𝜁. (18) By (7), 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 (19) on [𝑡₁,∞)_𝕋, where 𝑡₁≥𝑡₀ is sufficiently large. Then one has 𝐷^𝛼_𝑡𝑥(𝑡)≥𝛿₁𝑡,𝑡₁𝑎^1/𝛾(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑟(𝑡),𝑥(𝑡)≥𝛿₂𝑡,𝑡₁𝑎^1/𝛾(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡). (20)(21)

Proof. By (10), we obtain that 𝑎(𝜉)((𝑟(̃𝜉)𝑥^′(𝜉))^′)^𝛾 is strictly decreasing on [𝜉₁,∞). So 𝑟̃(𝜉)𝑥^′(𝜉)≥𝑟̃(𝜉)𝑥^′(𝜉)−𝑟̃𝜉₁𝑥^′𝜉₁=^𝜉_𝜉1𝑎^1/𝛾(𝑠)𝑟̃(𝑠)𝑥^′(𝑠)^′𝑎^1/𝛾(𝑠)≥𝑎𝑑𝑠_1/𝛾(𝜉)𝑟̃(𝜉)𝑥^′(𝜉)^′𝜉_𝜉11𝑎^1/𝛾(𝑠)=𝑎𝑑𝑠_1/𝛾(𝜉)𝑟̃(𝜉)𝑥^′(𝜉)^′𝛿₁𝜉,𝜉₁; (22) that is 𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)≥𝛿₁𝑡,𝑡₁𝑎^1/𝛾(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡), (23) which admits (20). On the other hand, we have 𝑥(𝜉)≥𝑥(𝜉)−𝑥𝜉₁≥^𝜉_𝜉1𝛿₁𝑠,𝜉₁𝑎^1/𝛾(𝑠)𝑟̃(𝑠)𝑥^′(𝑠)^′𝑟̃(𝑠)≥𝑎𝑑𝑠_1/𝛾(𝜉)(𝑟̃(𝜉)𝑥^′(𝜉))^′𝜉_𝜉1𝛿₁𝑠,𝜉₁𝑟̃(𝑠)=𝛿𝑑𝑠₂𝜉,𝜉₁𝑎^1/𝛾(𝜉)𝑟̃(𝜉)𝑥^′(𝜉)^′, (24) which can be rewritten as (21). So the proof is complete.

Lemma 3 (see [29, Theorem 41]). Assume that 𝐴 and 𝐵 are nonnegative real numbers. Then 𝜆𝐴𝐵^𝜆−1−𝐴^𝜆≤(𝜆−1)𝐵^𝜆 (25) for all 𝜆>1.

Theorem 4. Let ℎ₁,ℎ₂,𝐻∈𝐶([𝜉₀,∞),𝑅) satisfying 𝐻(𝜉,𝜉)=0,𝐻(𝜉,𝑠)>0,𝜉>𝑠≥𝜉₀, and 𝐻 has continuous partial derivatives 𝐻^′_𝜉(𝜉,𝑠) and 𝐻^′_𝑠(𝜉,𝑠) on [𝜉₀,∞). Assume that (5)–(7) hold, and for any sufficiently large 𝑇≥𝜉₀, there exist 𝜙∈𝐶¹([𝑡₀,∞),𝑅₊) and 𝜑∈𝐶¹([𝑡₀,∞),[0,∞)) and 𝑎,𝑏, and 𝑐 with 𝑇≤𝑎<𝑐<𝑏 satisfying1𝐻(𝑏,𝑐)^𝑏𝑐𝐻(𝑏,𝑠)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)+1𝑑𝑠𝐻(𝑐,𝑎)^𝑐𝑎𝐻(𝑠,𝑎)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)>1𝑑𝑠𝐻(𝑏,𝑐)^𝑏𝑐𝐻(𝑏,𝑠)𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝑏,𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹+1𝑑𝑠𝐻(𝑐,𝑎)^𝑐𝑎𝐻(𝑠,𝑎)𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝑠,𝑎)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠, (26) where 𝜙(𝜉)=𝜙(𝑡),𝜑(𝜉)=𝜑(𝑡). Then every solution of (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 that 𝐷^𝛼_𝑡𝑥(𝑡)>0 on [𝑡₂,∞). Define the generalized Riccati function 𝜔(𝑡)(𝑡𝑎(𝑡)𝐷=𝜙)^𝛼𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)^𝛾𝑥^𝛾(𝑡)+𝜑(𝑡). (27) Then for 𝑡∈[𝑡₂,∞), we have 𝐷^𝛼_𝑡𝜔(𝑡)=𝐷^𝛼_𝑡𝜙(𝑡)𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)^𝛾𝑥^𝛾(𝑡)+𝜙(𝑡)𝐷^𝛼_𝑡𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)^𝛾𝑥^𝛾(𝑡)+𝐷^𝛼_𝑡𝜙(𝑡)𝜑(𝑡)+𝜙(𝑡)𝐷^𝛼_𝑡𝜑(𝑡)=𝜙(𝑡)𝑥^𝛾(𝑡)𝐷^𝛼_𝑡𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)^𝛾−𝛾𝑥^𝛾−1(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑎(𝑡)×𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)^𝛾×𝑥^2𝛾(𝑡)⁻¹+𝐷^𝛼_𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡)+𝜙(𝑡)𝐷^𝛼_𝑡𝜑(𝑡)𝜙(𝑡)𝑞(𝑡)𝑓(𝑥(𝑡=−))𝑥^𝛾(𝑡)−𝛾𝜙(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑎(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)^𝛾𝑥^𝛾+1(𝑡)+𝐷^𝛼_𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡)+𝜙(𝑡)𝐷^𝛼_𝑡𝜑(𝑡). (28) By Lemma 2 and the definition of 𝑓 we get that 𝐷^𝛼_𝑡𝜔(𝑡)(𝑡)𝑞(𝑡)−(𝑡)𝛿≤−𝐾𝜙𝛾𝜙₁𝑡,𝑡₂𝑎^1/𝛾(𝑡)𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)𝑎(𝑡)×𝐷^𝛼_𝑡𝑟(𝑡)𝐷^𝛼_𝑡𝑥(𝑡)^𝛾×𝑟(𝑡)𝑥^𝛾+1(𝑡)⁻¹+𝐷^𝛼_𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡)+𝜙(𝑡)𝐷^𝛼_𝑡𝜑(𝑡)(𝑡)𝑞(𝑡)−(𝑡)𝛿=−𝐾𝜙𝛾𝜙₁𝑡,𝑡₂𝑟(𝑡)𝜔(𝑡)𝜙(𝑡)−𝜑(𝑡)^1+(1/𝛾)+𝐷^𝛼_𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡)+𝜙(𝑡)𝐷^𝛼_𝑡𝜑(𝑡). (29) Using the following inequality (see [2, Equation (2.17)]) (𝑢−𝑣)^1+(1/𝛾)≥𝑢^1+(1/𝛾)+1𝛾𝑣^1+(1/𝛾)−11+𝛾𝑣^1/𝛾𝑢, (30) we obtain 𝜔(𝑡)𝜙(𝑡)−𝜑(𝑡)^1+(1/𝛾)≥𝜔^1+(1/𝛾)(𝑡)𝜙^1+(1/𝛾)(𝑡)+1𝛾𝜑^1+(1/𝛾)(𝑡)−11+𝛾𝜑^1/𝛾(𝑡)𝜔(𝑡)𝜙(𝑡). (31) A combination of (29) and (31) yields: 𝐷^𝛼_𝑡𝜔(𝑡)(𝑡)𝑞(𝑡)+𝐷≤−𝐾𝜙^𝛼𝑡𝜙(𝑡)𝜙(𝑡)𝜔(𝑡)+𝜙(𝑡)𝐷^𝛼_𝑡𝜑(𝑡)−𝛾𝜙(𝑡)𝛿₁𝑡,𝑡₂𝑟(𝑡)𝜔^1+(1/𝛾)(𝑡)𝜙^1+(1/𝛾)(𝑡)+1𝛾𝜑^1+(1/𝛾)(𝑡)−11+𝛾𝜑^1/𝛾(𝑡)𝜔(𝑡)𝜙(𝑡)=−𝐾𝜙(𝑡)𝑞(𝑡)+𝜙(𝑡)𝐷^𝛼_𝑡𝜑(𝑡)−𝜙(𝑡)𝛿₁𝑡,𝑡₂𝜑^1+(1/𝛾)(𝑡)𝑟(𝑡)−𝛾𝜙(𝑡)𝛿₁𝑡,𝑡₂𝑟(𝑡)𝜔^1+(1/𝛾)(𝑡)𝜙^1+(1/𝛾)(𝑡)+𝛾+1𝜑^1/𝛾(𝑡)𝜙(𝑡)𝛿₁𝑡,𝑡₂+𝑟(𝑡)𝐷^𝛼_𝑡𝜙(𝑡)𝑟(𝑡)𝜙(𝑡)×𝜔(𝑡),𝑡≥𝑡₂. (32) Let 𝜔(𝑡)=𝜔(𝜉). Then 𝐷^𝛼_𝑡𝑤(𝑡)=𝑤^′(𝜉),𝐷^𝛼_𝑡𝜙(𝑡)=𝜙^′(𝜉),and𝐷^𝛼_𝑡𝜑(𝑡)=𝜑^′(𝜉), and (32) is transformed into the following form: 𝜔^′(𝜉)≤−𝐾𝜙(𝜉)𝑞(𝜉)+𝜙(𝜉)𝜑^′(𝜉)−𝜙(𝜉)𝛿₁𝜉,𝜉₂𝜑^1+(1/𝛾)(𝜉)𝑟(𝜉)−𝛾𝛿₁𝜉,𝜉₂𝑟̃(𝜉)𝜙^1/𝛾(𝜉)𝜔^1+(1/𝛾)(𝜉)+𝛾+1𝜑^1/𝛾(𝜉)𝜙(𝜉)𝛿₁𝜉,𝜉₂+𝑟̃(𝜉)𝜙^′(𝜉)𝑟̃(𝜉)𝜙(𝜉)𝜔(𝜉),𝜉≥𝜉₂. (33)
Select 𝑎,𝑏,and𝑐 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)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)𝑑𝑠≤−^𝜉_𝑐𝐻(𝜉,𝑠)𝑤^′(𝑠)+𝑑𝑠_𝜉𝑐𝐻()−𝛾𝛿𝜉,𝑠¹𝑠,𝜉₂𝑟̃(𝑠)𝜙^1/𝛾(𝑠)𝜔^1+(1/𝛾)(𝑠)+𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)×𝑟̃(𝑠)𝜙(𝑠)⁻¹𝜔(𝑠)=𝐻()𝑤(𝑐)+𝑑𝑠𝜉,𝑐_𝜉𝑐𝐻()−𝛾𝛿𝜉,𝑠¹𝑠,𝜉₂𝑟̃(𝑠)𝜙^1/𝛾(𝑠)𝜔^1+(1/𝛾)(𝑠)+𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠()×𝑟̃(𝑠)𝜙(𝑠)𝜉,𝑠₋₁𝜔(𝑠)𝑑𝑠. (34) Setting 1𝜆=1+𝛾,𝐴^𝜆=𝛾𝛿₁𝑠,𝜉₂𝑟̃(𝑠)𝜙^1/𝛾(𝑠)𝜔^1+(1/𝛾)(𝑠),𝐵^𝜆−1=𝛾^1/(𝛾+1)×𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠()×𝜙(𝑠)𝛿𝜉,𝑠𝛾+1₁𝑠,𝜉₂^{𝛾/(𝛾+1)}𝑟̃^1/(𝛾+1)(𝑠)⁻¹, (35) by a combination of Lemma 3 and (33), we get that ^𝑠_𝑐𝐻(𝑠,𝑠)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝑠₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)≤𝐻()𝑤(𝑐)+𝑑𝑠𝑠,𝑐_𝑠𝑐𝐻()×𝜑𝑠,𝑠𝛾+1^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝜉,𝑠)^𝛾+1×𝛾+1^𝛾+1𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠. (36) Dividing both sides of inequality (36) by 𝐻(𝜉,𝑐) and letting 𝜉→𝑏⁻, we obtain 1𝐻(𝑏,𝑐)^𝑏𝑐𝐻(𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑𝑏,𝑠)_′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)≤𝑤(𝑐)+1𝑑𝑠𝐻(𝑏,𝑐)^𝑏𝑐𝐻()×𝜑𝑏,𝑠𝛾+1^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝑏,𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠. (37)
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], similar to (36)-(37), we get that ^𝑐_𝜉𝐻(𝑠,𝜉)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)𝐻()𝑤(𝑐)+𝑑𝑠≤−𝑐,𝜉_𝑐𝜉𝐻(𝑠,𝜉)𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝑠,𝜉)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠. (38) Dividing both sides of inequality (38) by 𝐻(𝑐,𝜉) and letting 𝜉→𝑎⁺, we obtain1𝐻(𝑐,𝑎)^𝑐𝑎𝐻(𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑𝑠,𝑎)_′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)𝑤(𝑐)+1𝑑𝑠≤−𝐻(𝑐,𝑎)^𝑐𝑎𝐻()×𝜑𝑠,𝑎𝛾+1^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝑠,𝑎)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠. (39) A combination of (37) and (39) yields 1𝐻(𝑏,𝑐)^𝑏𝑐𝐻(𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑𝑏,𝑠)_′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)+1𝑑𝑠𝐻(𝑐,𝑎)^𝑐𝑎𝐻(𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑𝑠,𝑎)_′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)≤1𝑑𝑠𝐻(𝑏,𝑐)^𝑏𝑐𝐻(𝑏,𝑠)𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝑏,𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹+1𝑑𝑠𝐻(𝑐,𝑎)^𝑐𝑎𝐻()×𝜑𝑠,𝑎𝛾+1^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝑠,𝑎)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠, (40) which contradicts (26). 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(𝑠,𝑙)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝑠,𝑙)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠>0,lim_𝜉→∞sup^𝜉_𝑙𝐻(𝜉,𝑠)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝜉,𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠>0, (41)(42) then (1) is oscillatory.

Proof. For any 𝑇≥𝜉₀, let 𝑎=𝑇. In (41) we choose 𝑙=𝑎. Then there exists 𝑐>𝑎 such that ^𝑐_𝑎𝐻(𝑠,𝑎)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝑠,𝑎)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠>0. (43) In (42) we choose 𝑙=𝑐>𝑎. Then there exists 𝑏>𝑐 such that ^𝑏_𝑐𝐻(𝑏,𝑠)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝑟̃(𝑠)𝜙(𝑠)𝐻^′_𝑠(𝑏,𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠>0. (44) Combining (43) and (44) we obtain (26). 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)^𝜆𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)+1𝑑𝑠(𝑏−𝑐)^𝜆𝑏𝑐(𝑏−𝑠)^𝜆×𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)>1𝑑𝑠(𝑐−𝑎)^𝜆𝑐𝑎(𝑠−𝑎)^𝜆×𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝜆(𝑠−𝑎)^𝜆−1𝑟̃(𝑠)𝜙(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹+1𝑑𝑠(𝑏−𝑐)^𝜆×^𝑏_𝑐(𝑏−𝑠)^𝜆𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)−𝜆(𝑏−𝑠)^𝜆−1𝑟̃(𝑠)𝜙(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠, (45) 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)^𝜆𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝜆(𝑠−𝑙)^𝜆−1𝑟̃(𝑠)𝜙(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠>0,lim_𝜉→∞sup^𝜉_𝑙(𝜉−𝑠)^𝜆𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)−𝜆(𝜉−𝑠)^𝜆−1𝑟̃(𝑠)𝜙(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠>0, (46) then (1) is oscillatory.

Theorem 8. Assume (5)–(7) hold, and there exists two functions 𝜙∈𝐶¹([𝑡₀,∞),ℝ₊) and 𝜑∈𝐶¹([𝑡₀,∞),[0,∞)) such that ^∞_𝜉0𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1𝛾+1^𝛾+1𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)𝑑𝑠=∞, (47) where 𝜙,𝜑 are defined as in Theorem 4. Then every solution of (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 that 𝐷^𝛼_𝑡𝑥(𝑡)>0 on [𝑡₂,∞). Let 𝜔(𝑡),𝜔(𝜉) be defined as in Theorem 4. Then we obtain (32).
Setting 1𝜆=1+𝛾,𝐴^𝜆=𝛾𝜙(𝑡)𝛿₁𝑡,𝑡₂𝑟(𝑡)𝜔^1+(1/𝛾)(𝑡)𝜙^1+(1/𝛾)(𝑡),𝐵^𝜆−1=𝛾^1/(𝛾+1)×𝛾+1𝜑^1/𝛾(𝑡)𝜙(𝑡)𝛿₁𝑡,𝑡₂+𝑟(𝑡)𝐷^𝛼_𝑡𝜙(𝑡)𝛾+1𝜙(𝑡)𝛿₁𝑡,𝑡₂^{𝛾/(𝛾+1)}𝑟^1/(𝛾+1)(𝑡). (48) Using Lemma 3 in (32) we get that 𝐷^𝛼_𝑡𝜔(𝑡)≤−𝐾𝜙(𝑡)𝑞(𝑡)+𝜙(𝑡)𝐷^𝛼_𝑡𝜑(𝑡)−𝜙(𝑡)𝛿₁𝑡,𝑡₂𝜑^1+(1/𝛾)(𝑡)𝑟(𝑡)+𝛾+1𝜑^1/𝛾(𝑡)𝜙(𝑡)𝛿₁𝑡,𝑡₂+𝑟(𝑡)𝐷^𝛼_𝑡𝜙(𝑡)^𝛾+1𝛾+1^𝛾+1𝜙(𝑡)𝛿₁𝑡,𝑡₂^𝛾𝑟(𝑡), (49) which is rewritten in the following form: 𝜔^′(𝜉)≤−𝐾𝜙(𝜉)𝑞(𝜉)+𝜙(𝜉)𝜑^′(𝜉)−𝜙(𝜉)𝛿₁𝜉,𝜉₂𝜑^1+(1/𝛾)(𝜉)𝑟̃(𝜉)+𝛾+1𝜑^1/𝛾(𝜉)𝜙(𝜉)𝛿₁𝜉,𝜉₂+𝑟̃(𝜉)𝜙^′(𝜉)^𝛾+1𝛾+1^𝛾+1𝜙(𝜉)𝛿₁𝜉,𝜉₂^𝛾𝑟̃(𝜉),𝜉≥𝜉₂. (50)
Substituting 𝜉 with 𝑠 in (50), an integration for (50) with respect to 𝑠 from 𝜉₂ to 𝜉 yields ^𝜉_𝜉2𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1𝛾+1^𝛾+1𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)𝜉𝑑𝑠≤𝜔₂−𝜔(𝜉)≤𝜔𝜉₂<∞, (51) which contradicts (47). So the proof is complete.

Theorem 9. Assume that (5)–(7) hold, and there exists a function 𝐻∈𝐶([𝜉₀,∞),ℝ) such that 𝐻(𝜉,𝜉)=0,𝑓𝑜𝑟𝜉≥𝜉₀,𝐻(𝜉,𝑠)>0,𝑓𝑜𝑟𝜉>𝑠≥𝜉₀, and 𝐻 has a nonpositive continuous partial derivative 𝐻^′_𝑠(𝜉,𝑠). If lim_𝜉→∞1sup𝐻𝜉,𝜉₀×^𝜉_𝜉0𝐻()×𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑𝜉,𝑠_′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠=∞, (52) where 𝜙,𝜑 are defined as in Theorem 4, then every solution of (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 that 𝐷^𝛼_𝑡𝑥(𝑡)>0 on [𝑡₂,∞). Let 𝜔(𝑡),𝜔(𝜉) be defined as in Theorem 4. By (50) we have 𝐾𝜙(𝜉)𝑞(𝜉)−𝜙(𝜉)𝜑^′(𝜉)+𝜙(𝜉)𝛿₁𝜉,𝜉₂𝜑^1+(1/𝛾)(𝜉)𝑟̃(𝜉)−𝛾+1𝜑^1/𝛾(𝜉)𝜙(𝜉)𝛿₁𝜉,𝜉₂+𝑟̃(𝜉)𝜙^′(𝜉)^𝛾+1𝛾+1^𝛾+1𝜙(𝜉)𝛿₁𝜉,𝜉₂^𝛾𝑟̃(𝜉)≤𝜔^′(𝜉),𝜉≥𝜉₂. (53) Substituting 𝜉 with 𝑠 in (53), multiplying both sides by 𝐻(𝜉,𝑠), and then integrating with respect to 𝑠 from 𝜉₂ to 𝜉 yield ^𝜉_𝜉2𝐻(𝜉,𝑠)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1×𝛾+1^𝛾+1𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠≤−^𝜉_𝜉2𝐻(𝜉,𝑠)𝜔^′(𝑠)𝑑𝑠=𝐻𝜉,𝜉₂𝜔𝜉₂+^𝜉_𝜉2𝐻^′_𝑠(𝜉,𝑠)𝜔(𝑠)Δ𝑠≤𝐻𝜉,𝜉₂𝜔𝜉₂≤𝐻𝜉,𝜉₀𝜔𝜉₂. (54) Then ^𝜉_𝜉0𝐻(𝜉,𝑠)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1×𝛾+1^𝛾+1𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹=𝑑𝑠_𝜉2𝜉0𝐻(𝜉,𝑠)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹+𝑑𝑠_𝜉𝜉2𝐻(𝜉,𝑠)𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠≤𝐻𝜉,𝜉₀𝜔𝜉₂+𝐻𝜉,𝜉₀^𝜉2_𝜉0𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠. (55) So lim_𝜉→∞1sup𝐻𝜉,𝜉₀×^𝜉_𝜉0𝐻()×𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑𝜉,𝑠_′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹≤𝜔𝜉𝑑𝑠₂+^𝜉2_𝜉0𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠<∞, (56) which contradicts (52). So the proof is complete.

3. Applications of the Results

Example 10. Consider the following fractional differential equation: 𝐷_𝑡^1/2√𝑡𝐷_𝑡^1/2𝐷_𝑡^1/2𝑥(𝑡)^5/3+𝑡^3/2Γ³()𝑥3/2^5/3(𝑡)+𝑥^11/3(𝑡)=0,𝑡≥2. (57)
In (1), if we set 𝑡₀=2,𝛼=1/2,𝛾=5/3,𝑎(𝑡)=√𝑡,𝑟(𝑡)≡1,𝑞(𝑡)=𝑡^3/2/Γ³(3/2), then we obtain (57). So 𝑎(𝜉)=𝑎(𝑡)=√𝑡=Γ(3/2)𝜉, 𝑟(̃𝜉)≡1, and 𝑞(𝜉)=𝜉³, and 𝑓(𝑥)=𝑥^5/3+𝑥^11/3, which implies that 𝑓(𝑥)/𝑥^5/3=1+𝑥²≥1=𝐾. Furthermore, 𝛿₁(𝜉,𝜉₂)=∫^𝜉_𝜉2(1/𝑎^1/𝛾(𝑠))𝑑𝑠=[Γ(3/2)]^−3/5∫^𝜉_𝜉2𝑠^−3/5𝑑𝑠=(5/2)[Γ(3/2)]^−3/5(𝜉^2/5−𝜉₂^2/5), which implies that lim_𝜉→∞𝛿₁(𝜉,𝜉₂)=∞. So there exists a sufficiently large 𝑇>𝜉₂ such that 𝛿₁(𝜉,𝜉₂)>1 on [𝑇,∞). Furthermore, one can easily see that (5)–(7) all hold. On the other hand, in (46), after letting 𝜙(𝜉)≡1,𝜑(𝜉)=0,and𝜆=2, for any sufficiently large 𝑙 (we may let 𝑙≥𝑇 without loss of generality), it holds that lim_𝜉→∞sup^𝜉_𝑙(𝑠−𝑙)^𝜆𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)+𝜆(𝑠−𝑙)^𝜆−1𝑟̃(𝑠)𝜙(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠≥lim_𝜉→∞sup^𝜉_𝑙(𝑠−𝑙)²𝑠³−34^8/3(𝑠−𝑙)^8/3𝑑𝑠=∞,lim_𝜉→∞sup^𝜉_𝑙(𝜉−𝑠)^𝜆𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)−𝜆(𝜉−𝑠)^𝜆−1𝑟̃(𝑠)𝜙(𝑠)^𝛾+1×𝛾+1^𝛾+1×𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)⁻¹𝑑𝑠≥lim_𝜉→∞sup^𝜉_𝑙(𝜉−𝑠)²𝑠³−34^8/3(𝜉−𝑠)^8/3𝑑𝑠=∞. (58) So according to Corollary 7 we deduce that (57) is oscillatory.

Example 11. Consider the following fractional differential equation: 𝐷_𝑡^1/2𝑡^−1/6𝐷_𝑡^1/2𝐷_𝑡^1/2𝑥(𝑡)^1/3+Γ^1/3(3/2)𝑡^1/6𝑥^1/3(𝑡)1+𝑒^𝑥2(𝑡)=0,𝑡≥2. (59)
In (1), if we set 𝑡₀=2,𝛼=1/2,𝛾=1/3,𝑎(𝑡)=𝑡^−1/6,𝑟(𝑡)≡1,𝑞(𝑡)=Γ^1/3(3/2)/𝑡^1/6, then we obtain (57). So 𝑎(𝜉)=𝑎(𝑡)=𝑡^−1/6=[Γ(3/2)]^−1/3𝜉^−1/3, 𝑟(̃𝜉)≡1, and 𝑞(𝜉)=𝜉^−1/3, and 𝑓(𝑥)=𝑥^1/3[1+𝑒^𝑥2], which implies that 𝑓(𝑥)/𝑥^1/3=1+𝑒^𝑥2≥1=𝐾. Furthermore, 𝛿₁(𝜉,𝜉₂)=∫^𝜉_𝜉2(1/𝑎^1/𝛾(𝑠))𝑑𝑠=∫^𝜉_𝜉2Γ(3/2)𝑠𝑑𝑠=(1/2)Γ(3/2)(𝜉²−𝜉²₂), which implies that lim_𝜉→∞𝛿₁(𝜉,𝜉₂)=∞. So there exists a sufficiently large 𝑇>𝜉₂ such that 𝛿₁(𝜉,𝜉₂)>1 on [𝑇,∞). Furthermore, after some computation one can see that (5)–(7) all hold, and in (47), letting 𝜙(𝜉)=𝜉,𝜑(𝜉)=0, we obtain ^∞_𝜉0𝐾𝜙(𝑠)𝑞(𝑠)−𝜙(𝑠)𝜑^′(𝑠)+𝜙(𝑠)𝛿₁𝑠,𝜉₂𝜑^1+(1/𝛾)(𝑠)𝑟̃(𝑠)−𝛾+1𝜑^1/𝛾(𝑠)𝜙(𝑠)𝛿₁𝑠,𝜉₂+𝑟̃(𝑠)𝜙^′(𝑠)^𝛾+1𝛾+1^𝛾+1𝜙(𝑠)𝛿₁𝑠,𝜉₂^𝛾𝑟̃(𝑠)𝑑𝑠=^𝑇_𝜉0𝑠^2/3−1(4/3)^4/3𝑠𝛿₁𝑠,𝜉₂^1/3+𝑑𝑠_∞𝑇𝑠^2/3−1(4/3)^4/3𝑠𝛿₁𝑠,𝜉₂^1/3≥𝑑𝑠_𝑇𝜉0𝑠^2/3−1(4/3)^4/3𝑠𝛿₁𝑠,𝜉₂^1/3+𝑑𝑠_∞𝑇𝑠^2/3−1(4/3)^4/3𝑠^−1/3𝑑𝑠=∞. (60) Therefore, (59) is oscillatory by Theorem 8.

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (11171178) and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province. The authors would like to thank the reviewers very much for their valuable suggestions on the paper.