Oscillation Criteria for Nonlinear Fractional Differential Equations

Abstract

Several oscillation criteria are established for nonlinear fractional differential equations of the form {𝑎(𝑡)[(𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′]^𝜂}^′−𝐹(𝑡,∫^∞_𝑡(𝑣−𝑡)^−𝛼𝑥(𝑣)𝑑𝑣)=0, where 𝐷^𝛼₋𝑥 is the Liouville right-side fractional derivative of order 𝛼∈(0,1) of 𝑥 and 𝜂 is a quotient of two odd positive integers. We also give some examples to illustrate the main results. To the best of our knowledge, the results are initiation for the oscillatory behavior of the equations.

1. Introduction

In this paper, we are dealing with the oscillation problem of nonlinear fractional differential equations of the form 𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂′−𝐹𝑡,^∞_𝑡(𝑣−𝑡)^−𝛼𝑥(𝑣)𝑑𝑣=0,𝑡≥𝑡₀>0, (1) where 𝛼∈(0,1) is a constant, 𝜂 is a ratio of two odd positive integers. 𝐷^𝛼₋𝑥(𝑡) is the Liouville right-side fractional derivative of order 𝛼 of 𝑥(𝑡) defined by 𝐷^𝛼₋𝑥(𝑡)1=−Γ()𝑑1−𝛼×𝑑𝑡_∞𝑡(𝑣−𝑡)^−𝛼𝑥(𝑣)𝑑𝑣,𝑡∈𝑅₊=(0,∞). (2)

Throughout this paper, we will suppose that the following conditions hold:(𝐴₁)𝑎(𝑡)∈𝐶¹([𝑡₀,∞);𝑅₊), 𝑟(𝑡)∈𝐶²([𝑡₀,∞);𝑅₊), ∫^∞_𝑡0(𝑑𝑠/𝑎^1/𝜂(𝑠))=∞, ∫^∞_𝑡0(𝑑𝑠/𝑟(𝑠))=∞;(𝐴₂)𝐹(𝑡,𝐺)∈𝐶([𝑡₀,∞)×𝑅;𝑅), there exists a function 𝑞(𝑡)∈𝐶([𝑡₀,∞);𝑅₊) such that 𝐹(𝑡,𝐺)/𝐺^𝜂≥𝑞(𝑡) for 𝐺≠0 and 𝑥≠0, 𝑡≥𝑡₀.

By a solution of (1), we mean a function 𝑥(𝑡)∈𝐶(𝑅₊;𝑅) such that ∫^∞_𝑡(𝑣−𝑡)^−𝛼𝑥(𝑣)𝑑𝑣∈𝐶¹(𝑅₊;𝑅), 𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)∈𝐶²(𝑅₊;𝑅) and satisfies (1) on [𝑇_𝑥,∞). A nontrivial solution of (1) is called oscillatory if it has arbitrary large zero. Otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all of its solutions are oscillatory.

The theory of fractional derivatives goes back to Leibniz’s note in [1], and this led to the appearance of the theory of derivatives and integrals of arbitrary order. The theory had taken a more or less finished form due primarily to Liouville, Grünwald, Letnikov, and Riemann by the end of nineteenth century. We can see some of the books such as [2, 3] on the subject of fractional derivatives and fractional integrals.

Fractional differential equations are generalizations of classical differential equations of integer order. In the last few decades, many researchers found that fractional derivatives and fractional integrals were applied in widespread fields of science and engineering, especially in mathematical modeling and simulation of systems and processes, instead of simply being applied in pure theoretical fields of mathematics. Nowadays, many articles have investigated some aspects of fractional differential equations, such as the existence, the uniqueness and stability of solutions, the methods for explicit and numerical solutions, and the stability of solutions (we refer the reader to see [4–9] and the references cited in there). In very recent days, the research on oscillation of various fractional differential equations is being a hot topic; see [10–16].

In [10], Grace et al. discussed the oscillation of the following question: 𝐷^𝑞_𝑎𝑥+𝑓₁(𝑡,𝑥)=𝑣(𝑡)+𝑓₂(𝑡,𝑥),lim_𝑡→𝑎+𝐽_𝑎^1−𝑞𝑥(𝑡)=𝑏₁, (3) where 𝐷^𝑞_𝑎 denotes the Riemann-Liouville differential operator of order 𝑞 with 0<𝑞≤1, and the functions 𝑓₁, 𝑓₂, and 𝑣 are continuous functions.

In [11], Chen et al. established several oscillation criteria for (3) with some additional initial conditions and 𝑚−1≤𝑞≤𝑚, 𝑚≥1 is an integer. They improved and extended some results of [10].

In [12], Chen considered the oscillation of the fractional differential equation 𝑟(𝑡)𝐷^𝛼₋𝛾^𝜂(𝑡)^′−𝑞(𝑡)𝑓^∞_𝑡(𝑣−𝑡)^−𝛼𝛾(𝑣)𝑑𝑣=0,𝑡>0 (4) with 0<𝛼≤1.

In [13], Han et al. have established some oscillation criteria for the equation 𝑟(𝑡)𝑔𝐷^𝛼₋𝑦(𝑡)^′−𝑝(𝑡)𝑓^∞_𝑡(𝑠−𝑡)^−𝛼𝑦(𝑠)𝑑𝑠=0,𝑡>0. (5)

In [14], Qi and Cheng studied the following equation: 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 (6) with 0<𝛼<1 and established some new interval oscillation criteria by using a generalized Riccati transformation and inequality technique.

In [15], Feng and Meng paid attention to the oscillation of the fractional differential equation 𝐷^𝛼_𝑡𝑟(𝑡)𝜓(𝑥(𝑡))𝐷^𝛼_𝑡𝑥(𝑡(𝑡)𝑓(𝑥(𝑡(𝑡))+𝑞))=𝑒,𝑡>𝑡₀>0,0<𝛼<1. (7)

In [16], Chen considered the oscillation of the fractional differential equation 𝑟(𝑡)𝐷^𝛼₋𝑦^𝜂′−𝑞(𝑡)𝑓^∞_𝑡(𝑠−𝑡)^−𝛼𝑥(𝑠)𝑑𝑠=0,𝑡>0,0<𝛼<1. (8)

The purpose of this paper is to establish some oscillation criteria for (1) by generalized Riccati function and present some applications for our results.

In order to prove our theories, we use the general weighted functions from the class 𝑋. We say that a function 𝐻=𝐻(𝑡,𝑠) belongs to the function class 𝑋, if 𝐻∈𝐶(𝐷;𝑅₊), where 𝐷={(𝑡,𝑠)∶𝑡₀≤𝑠≤𝑡<∞}, which satisfies 𝐻(𝑡,𝑡)=0, 𝐻(𝑡,𝑠)>0 for 𝑡>𝑠, and has nonpositive continuous partial derivative 𝐻^′_𝑠(𝑡,𝑠) on 𝐷.

2. Main Results

First, we set 𝐺(𝑡)=^∞_𝑡(𝑣−𝑡)^−𝛼𝑥(𝑣)𝑑𝑣, (9) then it follows that 𝐺^′(𝑡)=−Γ(1−𝛼)𝐷^𝛼₋𝑥(𝑡). (10)

We give the following lemmas for our results.

Lemma 1 (see [17]). Let 𝑋 and 𝑌 be nonnegative; then 𝑚𝑋𝑌^𝑚−1−𝑋^𝑚≤(𝑚−1)𝑌^𝑚𝑓𝑜𝑟𝑚>1. (11)

Lemma 2. Assume that 𝑥(𝑡) is an eventually positive solution of (1), and ^∞_𝑡01𝑟(𝜉)^∞_𝜉1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝜂𝑑𝜏𝑑𝜉=∞; (12) then there exists a sufficiently large 𝑇 such that (𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′<0 on [𝑇,∞), and one of the following two conditions hold:(i)𝐷^𝛼₋𝑥(𝑡)<0 on [𝑇,∞),(ii)𝐷^𝛼₋𝑥(𝑡)>0 on [𝑇,∞) and lim_𝑡→∞𝐺(𝑡)=0.

Proof. From the hypothesis, there exists a 𝑡₁ such that 𝑥(𝑡)>0 on [𝑡₁,∞), so that 𝐺(𝑡)>0 on [𝑡₁,∞), and we have 𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂′=𝐹𝑡,^∞_𝑡(𝑣−𝑡)^−𝛼𝑥(𝑣)(𝑡)𝐺𝑑𝑣≥𝑞^𝜂(𝑡)>0,𝑡>𝑡₁. (13) Then 𝑎(𝑡)[(𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′]^𝜂 is strictly increasing on [𝑡₁,∞), and we can conclude that (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, if there exists a 𝑡₃>𝑡₂ such that (𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′|_𝑡3>0, then we can get (𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′>0 on [𝑡₃,∞); from those conditions we get 𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂≥𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂_𝑡3≐𝛿>0,𝑡≥𝑡₃, (14) that is, 𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′≥𝛿(𝑎(𝑡))^1/𝜂>0. (15) Integrating two sides of the previous inequality from 𝑡₃ to 𝑡 leads to 𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)−𝑟𝑡₃𝐷^𝛼₋𝑥𝑡₃≥𝛿^𝑡_𝑡3𝑑𝑠(𝑎(𝑠))^1/𝜂. (16) Then from (𝐴₁) we have lim_𝑡→∞𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)=+∞, which implies that for a certain constant 𝑡₄>𝑡₃, 𝐷^𝛼₋𝑥(𝑡)>0, 𝑡∈[𝑡₄,∞), then 𝐺(𝑡)−𝐺𝑡₄=^𝑡_𝑡4𝐺^′(𝑠)()𝑑𝑠=−Γ1−𝛼^𝑡_𝑡4𝑟(𝑠)𝐷^𝛼₋𝑥(𝑠)𝑟(𝑠)()𝑟𝑡𝑑𝑠≤−Γ1−𝛼₄𝐷^𝛼₋𝑥𝑡₄^𝑡_𝑡4𝑑𝑠𝑟(𝑠); (17) by (𝐴₁) we obtain lim_𝑡→∞𝐺(𝑡)=−∞, which contradicts to 𝐺(𝑡)>0 on [𝑡₁,∞).
So we have (𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′<0 on [𝑡₂,∞), and 𝐷^𝛼₋𝑥(𝑡) is eventually of one sign. There are two possibilities: (i) 𝐷^𝛼₋𝑥(𝑡)<0 on [𝑇,∞), (ii) 𝐷^𝛼₋𝑥(𝑡)>0 on [𝑇,∞), where 𝑇 is sufficiently large.
Assume that 𝐷^𝛼₋𝑥(𝑡)>0, 𝑡∈[𝑡₅,∞) for certain sufficiently large constant 𝑡₅>𝑡₄; then 𝐺^′(𝑡)<0, 𝑡∈[𝑡₅,∞) and we have lim_𝑡→∞𝐺(𝑡)=𝛽≥0. We claim that 𝛽=0. Otherwise, let 𝛽>0; then 𝐺(𝑡)≥𝛽 on [𝑡₅,∞), and by (13), we have 𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂′≥𝑞(𝑡)𝐺^𝜂(𝑡)≥𝑞(𝑡)𝛽^𝜂,𝑡∈𝑡₅,∞. (18)
Integrating two sides of above inequality from 𝑡 to ∞ leads to −𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂≥^∞_𝑡𝑞(𝑠)𝛽^𝜂𝑑𝑠−lim_𝑡→∞𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂≥𝛽^𝜂∞_𝑡𝑞(𝑠)𝑑𝑠, (19) which means 𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′1≤−𝛽𝑎(𝑡)^∞_𝑡𝑞(𝑠)𝑑𝑠^1/𝜂,𝑡∈𝑡₅,∞. (20)
Integrating two sides of (20) from 𝑡 to ∞ yields −𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)≤−lim_𝑡→∞𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)−𝛽^∞_𝑡1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝜂𝑑𝜏≤−𝛽^∞_𝑡1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝜂𝑑𝜏,𝑡∈𝑡₅,∞. (21)
That is, 𝐺^′(𝑡)()1≤−𝛽Γ1−𝛼𝑟(𝑡)^∞_𝑡1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝜂𝑑𝜏. (22) Integrating two sides of (22) from 𝑡₅ to 𝑡, we have 𝐺(𝑡)−𝐺𝑡₅()×≤−𝛽Γ1−𝛼_𝑡𝑡51𝑟(𝜉)×^∞_𝜉1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝜂𝑑𝜏𝑑𝜉. (23) By (12), we have lim_𝑡→∞𝐺(𝑡)=−∞, which contradicts to the fact that 𝐺(𝑡)>0. Then we get that 𝛽=0, which is lim_𝑡→∞𝐺(𝑡)=0. The proof is complete.

Lemma 3. Assume that 𝑥(𝑡) is an eventually positive solution of (1) such that (𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′<0, 𝐷^𝛼₋𝑥(𝑡)<0 on [𝑡₁,∞), where 𝑡₁ is sufficiently large and 𝑡₁>𝑡₀. Then one has 𝐺^′(𝑡)()1≥−Γ1−𝛼𝑟(𝑡)𝐴𝑡₁,𝑡𝑎^1/𝜂(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′, (24) where 𝐴(𝑡₁,𝑡)=∫^𝑡_𝑡1𝑎^−(1/𝜂)(𝑠)𝑑𝑠.

Proof. From (13), we get that 𝑎(𝑡)[(𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′]^𝜂 is strictly increasing on [𝑡₁,∞), so we get 𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)≤𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)−𝑟𝑡₁𝐷^𝛼₋𝑥𝑡₁=^𝑡_𝑡1𝑎^1/𝜂(𝑠)𝑟(𝑠)𝐷^𝛼₋𝑥(𝑠)^′𝑎^1/𝜂(𝑠)𝑑𝑠≤𝑎^1/𝜂(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝑡_𝑡1𝑑𝑠𝑎^1/𝜂(𝑠)=𝐴(𝑡)𝑎^1/𝜂(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′. (25) Then we get 𝐺^′(𝑡)()1≥−Γ1−𝛼𝑟(𝑡)𝐴(𝑡)𝑎^1/𝜂(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′. (26)

Theorem 4. Assume that (𝐴₁), (𝐴₂), and (12) hold; if there exists a function 𝜌∈𝐶¹([𝑡₀,∞);𝑅₊) such that ^∞_𝑇𝜌(𝑠)𝑞(𝑠)−1𝜂+1^𝜂+1𝜌^′₊(𝑠)𝜌(𝑠)^𝜂+1×𝜌(𝑠)𝑟^𝜂(𝑠)Γ(1−𝛼)𝐴𝑡₂,𝑠^𝜂𝑑𝑠=∞ (27) for all sufficiently large constants 𝑡₂ and 𝑇, where 𝜌^′₊(𝑠)=max{𝜌^′(𝑠),0}, 𝐴(𝑡₂,𝑡) is defined in Lemma 3, 𝑡≥𝑇≥𝑡₂≥𝑡₀>0, then every solution of (1) is oscillatory or satisfies lim_𝑡→∞𝐺(𝑡)=0.

Proof. Suppose to the contrary that (1) has a nonoscillatory solution 𝑥(𝑡) on [𝑡₀,∞); without loss of generality, we assume that 𝑥(𝑡)>0 on [𝑡₁,∞), where 𝑡₁ is sufficiently large. By Lemma 2 we have (𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡))^′<0 on [𝑡₂,∞), where 𝑡₂ is sufficiently large and 𝑡₂>𝑡₁, and either 𝐷^𝛼₋𝑥(𝑡)<0 on [𝑡₂,∞) or lim_𝑡→∞𝐺(𝑡)=0. (In the case 𝑥(𝑡) is eventually negative, it can be proved similarly.)
If 𝐷^𝛼₋𝑥(𝑡)<0 on [𝑡₂,∞), then the conclusion of Lemma 3 holds. We define 𝑤(𝑡)(𝑡)𝑎(𝑡𝑟(𝑡)𝐷=−𝜌)^𝛼₋𝑥(𝑡)^′𝜂𝐺^𝜂(𝑡), (28) then 𝑤(𝑡)>0 on [𝑡₂,∞), and from (1), (𝐴₂), and Lemma 3, we have 𝑤^′(𝑡)=𝜌^′(𝑡)𝜌(𝑡)𝑤(𝑡)−𝜌(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂′𝐺^𝜂(𝑡)−𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂𝜂𝐺^𝜂−1(𝑡)𝐺^′(𝑡)×𝐺^2𝜂(𝑡)⁻¹≤𝜌^′₊(𝑡)𝜌(𝑡)𝑤(𝑡)(𝑡)𝐹((𝑡−𝜌𝑡,𝐺))𝐺^𝜂(𝑡)+𝜂𝜌(𝑡)𝑎(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂×()𝐴𝑡−Γ1−𝛼²,𝑡𝑟(𝑡)𝑎^1/𝜂(𝑡)×𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′×𝐺^𝜂+1(𝑡)⁻¹≤𝜌^′₊(𝑡)𝜌(𝑡)𝑤(𝑡)(𝑡)𝑞(𝑡)()𝜌(𝑡)×𝐴𝑡−𝜌−𝜂Γ1−𝛼₂,𝑡𝑟(𝑡)𝑎^{(𝜂+1)/𝜂}(𝑡)𝑟(𝑡)𝐷^𝛼₋𝑥(𝑡)^′𝜂+1𝐺^𝜂+1(𝑡)=𝜌^′₊(𝑡)𝜌(𝑡)𝑤(𝑡)(𝑡)𝑞(𝑡)()𝐴𝑡−𝜌−𝜂Γ1−𝛼²,𝑡𝑟(𝑡)1𝜌^1/𝜂(𝑡)𝑤^{(𝜂+1)/𝜂}(𝑡). (29)
Let 𝑚=1+(1/𝜂), 𝑋=[𝜂Γ(1−𝛼)𝐴(𝑡₂,𝑡)/𝜌^1/𝜂(𝑡)𝑟(𝑡)]^1/𝑚𝑤(𝑡), 𝑌^𝑚−1=(𝜂/(𝜂+1))(𝜌^′₊(𝑡)/𝜌(𝑡))[𝜌^1/𝜂(𝑡)𝑟(𝑡)/𝜂Γ(1−𝛼)𝐴(𝑡)]^{𝜂/(𝜂+1)}; apply Lemma 1 to (29), we get 𝑤^′(𝑡)(𝑡)𝑞(𝑡)+1≤−𝜌𝜂+1^𝜂+1×𝜌^′₊(𝑡)𝜌(𝑡)^𝜂+1𝜌(𝑡)𝑟^𝜂(𝑡)Γ(1−𝛼)𝐴𝑡₂,𝑡^𝜂. (30) Integrating two sides of (30) from 𝑡₂ to 𝑡, we have ^𝑡_𝑡2𝜌(𝑠)𝑞(𝑠)−1𝜂+1^𝜂+1𝜌^′₊(𝑠)𝜌(𝑠)^𝜂+1×𝜌(𝑠)𝑟^𝜂(𝑠)Γ(1−𝛼)𝐴𝑡₂,𝑠^𝜂≤𝑤𝑡₂−𝑤(𝑡)≤𝑤𝑡₂, (31)
which contradicts to (27).
If 𝐷^𝛼₋𝑥(𝑡)>0 on [𝑡₂,∞), then from Lemma 2, we get that lim_𝑡→∞𝐺(𝑡)=0. This completes the proof.

Corollary 5. Assume that (𝐴₁), (𝐴₂), and (12) hold; if there exists a function 𝜌∈𝐶¹([𝑡₀,∞);𝑅₊) such that ^∞_𝑇𝜌(𝑠)𝑞(𝑠)𝑑𝑠=∞,_∞𝑇𝜌^′₊(𝑠)𝜌(𝑠)^𝜂+1𝜌(𝑠)𝑟^𝜂(𝑠)𝐴^𝜂𝑡₂,𝑠𝑑𝑠<∞ (32) for a sufficiently large constant 𝑇, where 𝜌^′₊(𝑠)=max{𝜌^′(𝑠),0}, 𝐴(𝑡₂,𝑡) is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies lim_𝑡→∞𝐺(𝑡)=0.

Theorem 6. Assume that (𝐴₁), (𝐴₂), and (12) hold; if there exist functions 𝜌∈𝐶¹([𝑡₀,∞);𝑅₊) and 𝐻(𝑡,𝑠)∈𝑋 such that limsup_𝑡→∞1𝐻𝑡,𝑡₀×^𝑡_𝑡2𝜌(𝑠𝐻()𝑞(𝑠)−𝑟)𝑡,𝑠_𝜂(𝑠)ℎ₊^𝜂+1(𝑡,𝑠)𝜂+1^𝜂+1Γ(1−𝛼)𝐴𝑡₂,𝑠𝐻(𝑡,𝑠)^𝜂𝑑𝑠=∞, (33) where 𝑡₂ is sufficiently large, 𝑡≥𝑡₂≥𝑡₀,𝐴(𝑡₂,𝑡) is defined in Lemma 3, ℎ(𝑡,𝑠)=𝐻(𝑡,𝑠)(𝜌^′₊(𝑠)/𝜌(𝑠))+𝐻^′_𝑠(𝑡,𝑠),ℎ₊(𝑡,𝑠)=max{0,ℎ(𝑡,𝑠)}, then every solution of (1) is oscillatory or satisfies lim_𝑡→∞𝐺(𝑡)=0.

Proof. Suppose that 𝑥(𝑡) is a nonoscillatory solution of (1); without loss of generality, we let 𝑥(𝑡) be an eventually positive solution of (1). According to the proof of Theorem 4, if 𝐷^𝛼₋𝑥(𝑡)<0 on [𝑡₂,∞), then (29) holds. Multiplying two sides of (29) by 𝐻(𝑡,𝑠) and integrating it from 𝑡₂ to 𝑡, we get ^𝑡_𝑡2𝜌(𝑠)𝑞(𝑠)𝐻(𝑡,𝑠)𝑑𝑠≤−^𝑡_𝑡2𝐻(𝑡,𝑠)𝑤^′(𝑠)𝑑𝑠+^𝑡_𝑡2𝐻()𝜌𝑡,𝑠^′+(𝑠)𝜌(𝑠)𝑤(𝑠)()𝑑𝑠−𝜂Γ1−𝛼^𝑡_𝑡2𝐻()𝐴𝑡𝑡,𝑠²,𝑠𝜌^1/𝜂(𝑠)𝑟(𝑠)𝑤^{(𝜂+1)/𝜂}(𝑠)𝑑𝑠≤𝐻𝑡,𝑡₂𝑤𝑡₂+^𝑡_𝑡2𝐻()𝜌𝑡,𝑠^′+(𝑠)𝜌(𝑠)+𝐻^′_𝑠()𝑤(𝑠)−()𝐴𝑡𝑡,𝑠𝜂Γ1−𝛼₂,𝑠𝐻(𝑡,𝑠)𝜌^1/𝜂(𝑠)𝑟(𝑠)𝑤^{(𝜂+1)/𝜂}(𝑠)𝑑𝑠≤𝐻𝑡,𝑡₂𝑤𝑡₂+^𝑡_𝑡2ℎ₊()𝑤(𝑠)−()𝐴𝑡𝑡,𝑠𝜂Γ1−𝛼₂,𝑠𝐻(𝑡,𝑠)𝜌^1/𝜂(𝑠)𝑟(𝑠)𝑤^{(𝜂+1)/𝜂}(𝑠)𝑑𝑠. (34)
Taking 1𝑚=1+𝜂,𝑋=𝜂Γ(1−𝛼)𝐴(𝑡₂,𝑠)𝐻(𝑡,𝑠)𝜌^1/𝜂(𝑠)𝑟(𝑠)^{𝜂/(𝜂+1)}𝑤(𝑠),𝑌^𝑚−1=𝜂×𝜌(𝑠)𝑟𝜂+1_{𝜂/(𝜂+1)}(𝑠)𝜂Γ(1−𝛼)𝐴(𝑡₂,𝑠)𝐻(𝑡,𝑠)^{𝜂/(𝜂+1)}ℎ₊(𝑡,𝑠), (35) applying Lemma 1 to (34), we obtain that for 𝑡≥𝑡₂, ^𝑡_𝑡2𝜌(𝑠)𝑞(𝑠)𝐻(𝑡,𝑠)𝑑𝑠≤𝐻𝑡,𝑡₂𝑤𝑡₂+1𝜂+1^𝜂+1Γ^𝜂()×1−𝛼_𝑡𝑡2ℎ₊^𝜂+1()𝜌(𝑠)𝑟𝑡,𝑠^𝜂(𝑠)𝐴𝑡₂,𝑠𝐻(𝑡,𝑠)^𝜂𝑑𝑠. (36) Then we have ^𝑡_𝑡2𝜌(𝑠)𝑞(𝑠)𝐻()−𝜌(𝑠)𝑟𝑡,𝑠_𝜂(𝑠)ℎ₊^𝜂+1(𝑡,𝑠)𝜂+1^𝜂+1[Γ(1−𝛼)𝐴(𝑠)𝐻(𝑡,𝑠)]^𝜂𝑑𝑠≤𝐻𝑡,𝑡₂𝑤𝑡₁≤𝐻𝑡,𝑡₀𝑤𝑡₂. (37) From above inequality we have 1𝐻𝑡,𝑡₀×^𝑡_𝑡2𝜌(𝑠)𝑞(𝑠)𝐻()−𝑟𝑡,𝑠_𝜂(𝑠)ℎ₊^𝜂+1(𝑡,𝑠)𝜂+1^𝜂+1Γ(1−𝛼)𝐴𝑡₂,𝑠𝐻(𝑡,𝑠)^𝜂𝑡𝑑𝑠≤𝑤₂. (38) Letting 𝑡→∞ we have limsup_𝑡→∞1𝐻𝑡,𝑡₀×^𝑡_𝑡2𝜌(𝑠)𝑞(𝑠)𝐻()−𝑟𝑡,𝑠_𝜂(𝑠)ℎ₊^𝜂+1(𝑡,𝑠)𝜂+1^𝜂+1Γ(1−𝛼)𝐴𝑡₂,𝑠𝐻(𝑡,𝑠)^𝜂𝑑𝑠<∞, (39) which contradicts to (33). If 𝐷^𝛼₋𝑥(𝑡)>0 on [𝑡₂,∞), from Lemma 2, we have lim_𝑡→∞𝐺(𝑡)=0. So the proof is complete.

Corollary 7. Assume that (𝐴₁), (𝐴₂), and (12) hold; if there exist functions 𝜌∈𝐶¹([𝑡₀,∞);𝑅₊) and 𝐻(𝑡,𝑠)∈𝑋 such that limsup_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡2𝜌(𝑠)𝑞(𝑠)𝐻(𝑡,𝑠)𝑑𝑠=∞,limsup_𝑡→∞1𝐻𝑡,𝑡₀^𝑡_𝑡2𝜌(𝑠)𝑟^𝜂(𝑠)ℎ₊^𝜂+1(𝑡,𝑠)𝐴𝑡₂,𝑠𝐻(𝑡,𝑠)^𝜂𝑑𝑠<∞, (40) where 𝑡₂ is sufficiently large, 𝑡≥𝑡₂≥𝑡₀, 𝐴(𝑡₂,𝑡) is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies lim_𝑡→∞𝐺(𝑡)=0.

With an appropriate choice of the functions 𝐻 and 𝜌, one can derive from Theorem 6 a number of oscillation criteria for (1). For example, let 𝐻(𝑡,𝑠)=ln(𝑡/𝑠), (𝑡,𝑠)∈𝐷, and 𝜌(𝑡)=𝑡^𝜇; then ℎ(𝑡,𝑠)=(1/𝑠)(𝜇ln(𝑡/𝑠)−1), ℎ₊(𝑡,𝑠)=max{ℎ(𝑡,𝑠),0}. Based on the above results we obtain the following corollary.

Corollary 8. Assume that (𝐴₁), (𝐴₂), and (12) hold; if limsup_𝑡→∞1×ln𝑡_𝑡𝑡2𝑠^𝜇𝑞(𝑠)𝑡ln𝑠−𝑟^𝜂(𝑠)ℎ₊(𝑡,𝑠)^𝜂+1𝜂+1^𝜂+1Γ^𝜂(1−𝛼)𝐴(𝑡₂,𝑠)ln(𝑡/𝑠)^𝜂𝑑𝑠=∞, (41) where 𝑡₂ is sufficiently large, 𝑡≥𝑡₂≥𝑡₀,𝐴(𝑡₂,𝑡) is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies lim_𝑡→∞𝐺(𝑡)=0.

Corollary 9. Assume that (𝐴₁), (𝐴₂), and (12) hold; if limsup_𝑡→∞1ln𝑡^𝑡_𝑡2𝑠^𝜇𝑞(𝑠)𝑡ln𝑠𝑑𝑠=∞,limsup_𝑡→∞1ln𝑡^𝑡_𝑡2𝑠^𝜇𝑟^𝜂(𝑠)(1/𝑠)𝜇ln(𝑡/𝑠)−1^𝜂+1𝐴(𝑡₂,𝑠)ln(𝑡/𝑠)^𝜂𝑑𝑠<∞, (42) where 𝑡₂ is sufficiently large, 𝑡≥𝑡₂≥𝑡₀, 𝐴(𝑡₂,𝑡) is defined in Lemma 3, then every solution of (1) is oscillatory or satisfies lim_𝑡→∞𝐺(𝑡)=0.

3. Examples

Example 1. Consider the fractional differential equation 𝑡^𝜂/2𝐷^𝛼₋𝑥(𝑡)^′𝜂′−𝐹𝑡,^∞_𝑡(𝑣−𝑡)^−𝛼𝑥(𝑣)𝑑𝑣=0,𝑡>𝑡₀, (43) where 𝛼∈(0,1) and 𝜂 is a quotient of two odd positive integers.

Here, 𝑎(𝑡)=𝑡^𝜂/2, 𝑟(𝑡)=1, 𝐹(𝑡,𝐺)/𝐺^𝜂≥𝑞(𝑡) for 𝐺≠0 and 𝑥≠0, 𝑡≥𝑡₀, 𝑞(𝑡)≥𝑡^−(𝜂+1).

We set 𝜌(𝑡)=𝑡^𝜇; if 𝑡 is sufficiently large, then 𝐴(𝑡₁,𝑡)=∫^𝑡_𝑡1(𝑑𝑠/𝑠^1/2)=2(√𝑡−𝑡₁)≥√𝑡, and ∫^∞_𝑡0(𝑑𝑠/𝑎^1/𝜂(𝑠))=∫^∞_𝑡0(𝑑𝑠/𝑠^1/2)=∞,∫^∞_𝑡0(𝑑𝑠/𝑟(𝑠))=∞. Consider ^∞_𝑡01𝑟(𝜉)^∞_𝜉1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝜂≥𝑑𝜏𝑑𝜉_{∞𝑡0∞𝜉}1𝜏^𝜂/2∞_𝜏𝑠^−(𝜂+1)𝑑𝑠^1/𝜂𝑑𝜏𝑑𝜉=∞, (44) so (𝐴₁),(𝐴₂), and (12) hold.

Furthermore, for a sufficiently large constant 𝑇 and 𝜂<𝜇<(3/2)𝜂, we have ^∞_𝑇𝜌(𝑠)𝑞(𝑠)≥^∞_𝑇𝑠^𝜇𝑠^−(𝜂+1)𝑑𝑠=∞,_∞𝑇𝜌^′₊(𝑠)𝜌(𝑠)^𝜂+1𝜌(𝑠)𝑟^𝜂(𝑠)𝐴𝑡₁,𝑠^𝜂≤𝑑𝑠_∞𝑇𝜇𝑠^𝜂+1𝑠^𝜇𝑠^𝜂/2𝑑𝑠=𝜇^𝜂+1𝑠^{𝜇−(3/2)𝜂}𝜇−(3/2)𝜂^∞_𝑇<∞. (45)

So the conditions of Corollary 5 hold, and we deduce that every solution of (43) is oscillatory or satisfies lim_𝑡→∞𝐺(𝑡)=0.

Example 2. Consider the fractional differential equation 𝑡^𝜂ln𝑡𝑡𝐷^𝛼₋𝑥(𝑡)^′𝜂′−𝐹𝑡,^∞_𝑡(𝑣−𝑡)^−𝛼𝑥(𝑣)𝑑𝑣=0,𝑡>𝑡₀, (46) where 𝛼∈(0,1) and 𝜂 is a quotient of two odd positive integers.

Here, 𝑎(𝑡)=𝑡^𝜂, 𝑟(𝑡)=ln𝑡/𝑡, 𝐹(𝑡,𝐺)/𝐺^𝜂≥𝑞(𝑡) for 𝐺≠0 and 𝑥≠0, 𝑡≥𝑡₀, 𝑞(𝑡)≥𝑡^−(𝜂+1).

We set 𝜌(𝑡)=1, 𝐻(𝑡,𝑠)=ln(𝑡/𝑠); then ℎ(𝑡,𝑠)=(1/𝑠)(𝜇ln(𝑡/𝑠)−1), and ∫^∞_𝑡0(𝑑s/𝑎^1/𝜂(𝑠))=∫^∞_𝑡0(𝑑𝑠/𝑠)=∞,∫^∞_𝑡0(𝑑𝑠/𝑟(𝑠))=∫^∞_𝑡0(𝑠𝑑𝑠/ln𝑠)=∞.

May as well we assume that 𝑡₂<√𝑡; then we have 𝐴(𝑡₂,𝑡)=ln(𝑡/𝑡₂)>(1/2)ln𝑡. By calculation we have ^∞_𝑡01𝑟(𝜉)^∞_𝜉1𝑎(𝜏)^∞_𝜏𝑞(𝑠)𝑑𝑠^1/𝜂≥𝑑𝜏𝑑𝜉_∞𝑡0𝜉ln𝜉^∞_𝜉1𝜏^𝜂∞_𝜏𝑠^−(𝜂+1)𝑑𝑠^1/𝜂𝑑𝜏𝑑𝜉=∞, (47) so (𝐴₁),(𝐴₂), and (12) hold.

Furthermore, limsup_𝑡→∞1ln𝑡^𝑡_𝑡0𝑠^𝜇𝑡ln𝑠𝑞(𝑠)𝑑𝑠≥limsup_𝑡→∞1ln𝑡^𝑡_𝑡0𝑠^{𝜇−𝜂−1}𝑡ln𝑠𝑑𝑠=∞, (48) and for 𝜂<𝜇<2𝜂, limsup_𝑡→∞1×ln𝑡_𝑡𝑡2𝑠^𝜇𝑟^𝜂(𝑠)(1/𝑠)𝜇ln(𝑡/𝑠)−1^(𝜂+1)𝐴𝑡₂,𝑠ln(𝑡/𝑠)^𝜂𝑑𝑠≤limsup_𝑡→∞1×ln𝑡_𝑡𝑡2𝑠^𝜇ln^𝜂𝑠(1/𝑠)𝜇ln(𝑡/𝑠)−1^(𝜂+1)𝑠^𝜂((ln𝑠/2)ln(𝑡/𝑠))^𝜂𝑑𝑠≤2^(1/𝜂)limsup_𝑡→∞1ln𝑡^𝑡_𝑡0𝑠^{𝜇−2𝜂−1}𝜇ln(𝑡/𝑠)^𝜂+1(ln(𝑡/𝑠))^𝜂𝑑𝑠=2^(1/𝜂)limsup_𝑡→∞𝜇^𝜂+1ln𝑡^𝑡_𝑡0𝑠^{𝜇−2𝜂−1}𝑡ln𝑠𝑑𝑠=2^(1/𝜂)limsup_𝑡→∞𝜇^𝜂+11ln𝑡×𝑡𝜇−2𝜂−ln𝑡₀𝑡₀^𝜇−2𝜂+1𝑡𝜇−2𝜂^𝜇−2𝜂−𝑡₀^𝜇−2𝜂<∞, (49) the conditions of Corollary 9 hold, and we get that every solution of (46) is oscillatory or satisfies lim_𝑡→∞𝐺(𝑡)=0.

Acknowledgment

This research is supported by National Science Foundation of China (11171178 and 11271225). The authors sincerely thank the reviewers for their valuable suggestions and useful comments.