The Oscillation of a Class of the Fractional-Order Delay Differential Equations

Abstract

Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this, α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.

1. Introduction

For the past three centuries, fractional calculus has been dealt with by mathematicians. In the recent decades, it is widely used in the fields of engineering, science, and economy. Recently, fractional differential systems have gained scholar’s attention [1–3]. Many researchers demonstrated applications of fractional calculus in the frequency dependent damping behavior of many viscoelastic materials, dynamics of interfaces between nanoparticles and substrates, the nonlinear oscillation of earthquakes, bioengineering, continuum and statistical mechanics, signal processing, filter design, circuit theory, and robotics. In some practical systems, such as economic systems, biological systems, and space-light industry systems, due to the transmission of the signal or the mechanical transmission, the research on fractional differential systems with delay is desired.

In [2], the authors described the application of the background and significance of the differential delay equations oscillation and fractional differential equations. The delay differential equation is different from ordinary differential equations. Moreover, compared with the periodic solution, oscillatory solution has more extensive application prospect and theory value in terms of the general solution of equation. In this paper, the necessary or sufficient conditions for the oscillation of several fractional-order delay differential equations are analyzed and discussed. Some oscillation results are proposed for the fractional-order linear delay differential equation as follows: ^𝑅₀𝐷^𝛼𝑥(𝑡)+𝑓(𝑥(𝑡−𝑇))=0, (1) where 0<𝛼<1. Some oscillation results are also given for the delay parabolic differential equation as follows: ^𝑅₀𝐷^𝛼𝑢(𝑥,𝑡)=𝑎(𝑡)Δ𝑢(𝑥,𝑡)+^𝑚_𝑘=1𝑏_𝑘(𝑡)Δ𝑢𝑥,𝑡−𝜏_𝑘−^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑓𝑢𝑥,𝑡−𝜎_𝑖. (2)

For the purpose, known methods are expanded for our results.

2. Preliminaries

We first give some definitions and make some preliminaries. The fractional-order derivative 𝐷^𝛼𝑥(𝑡) can be defined in different ways, wherein Riemann-Liouville’s definition and Caputo’s definition are two main expressions as follows.

Definition 1. Riemann-Liouville’s fractional order derivative [3, 4] is defined by _RL𝐷^𝛼𝑥(𝑡)=1Γ()𝑑1−𝛼𝑑𝑡^𝑡₀(𝑡−𝑠)^−𝛼𝑥(𝑠)(),𝑑𝑠,0<𝛼<1 (3) where Γ(𝛼)=∫^∞₀𝑒^−𝑡𝑡^𝛼−1𝑑𝑡 is the Gamma function.
Caputo’s fractional order derivative is defined by _𝐶𝐷^𝛼𝑥(𝑡)=1Γ(1−𝛼)^𝑡₀(𝑡−𝑠)^−𝛼𝑥^′(𝑠)𝑑𝑠,(0<𝛼<1). (4)

Definition 2. 𝛼-exponential function is as follows [2]: 𝑒_𝛼^𝜆𝑡=𝑡^∞𝛼−1_𝑘=0𝜆𝑡^𝛼𝑘Γ)𝛼),(),((𝑘+1𝑡>0 (5) where 𝑐𝑒_𝛼^𝜆𝑡 satisfies the differential equation _𝑜^RL𝐷^𝛼𝑥(𝑡)=𝜆𝑥(𝑡), (𝑡>0).
The derivative _𝑜^RL𝐷^𝛼𝑥(𝑡) has the property of _𝑜^RL𝐷^𝑛𝛼𝑥(𝑡)=_𝑜^RL𝐷^{(𝑛−1)𝛼}_𝑜^RL𝐷^𝛼𝑥(𝑡). (6)
Mittag-Lefflerfunction [3, 4] can be defined by 𝐸_𝛼,𝛽(𝑧)=^∞_𝑛=0𝑧^𝑛Γ,𝐸𝛼𝑛+𝛽_𝛼(𝑧)=^+∞_𝑛=0𝑧^𝑛Γ)𝛼).((𝑛+1 (7)
The first order delay differential equation including fractional-order derivative [2] is expressed by 𝑥^′(𝑡)+𝑝^𝑅₀𝐷^𝛼𝑥(𝑡)+𝑞𝑥(𝑡−𝑇)=0, (8) where 𝑝,𝑞,𝑇∈𝑅, 0<𝛼<1. Since 𝛼 is a positive rational number, let 𝛼=𝑘/𝑛=𝑘𝛽 satisfy 0<𝛼<1. The expression for (1) with constant coefficients is given by ^𝑅₀𝐷^𝑛𝛽+𝑝^𝑅₀𝐷^𝑘𝛽+𝑞𝑒^−𝜆𝑇_𝑘𝛽𝑒^𝜆𝑡_𝑘𝛽=𝑓(𝜆)𝑒^𝜆𝑡_𝑘𝛽, (9) where 𝑓(𝜆)=𝜆^𝑛+𝑝𝜆+𝑞𝑒^−𝜆𝑇_𝑘𝛽. (10)
Equation (10) is the characteristic polynomial of (8). The contribution in [2] is considered.

Lemma 3. The following statements are equivalent. (1)Every solution of (8) oscillates.(2)The characteristic equation (10) has no real roots.

In this paper, the main results are derived from the application of the basic theory in [2]. The oscillation or nonoscillation of solutions for several fractional-order delay differential equations can be obtained based on these main results.

3. Main Result

3.1. The Fractional-Order Delay Differential Equation with Constant Coefficients

The general form of the fractional-order linear delay differential equation is ^𝑅₀𝐷^𝛼𝑥(𝑡)+𝑝𝑥(𝑡−𝑇)=0. (11)

Since [^𝑅₀𝐷^𝛼+𝑝𝑒_𝛼^−𝜆𝑇]𝑒_𝛼^𝜆𝑡=[𝜆+𝑝𝑒_𝛼^−𝜆𝑇]𝑒_𝛼^𝜆𝑡=𝑓(𝜆)𝑒_𝛼^𝜆𝑡, the characteristic polynomial of (11) is as follows: 𝑓(𝜆)=𝜆+𝑝𝑒_𝛼^−𝜆𝑇=0. (12)

The detailed derivation can be found in Appendices A, B, and C. Using Lemma 3, we have the necessary and sufficient conditions for oscillation of all solutions of (11) as follows.

Corollary 4. The following statements are equivalent.(1)Every solution of (11) oscillates.(2)The characteristic equation (12) has no real roots.

In order to use Corollary 4 more effectively, Theorem 6 is proposed. Before presenting the theorem, the function 𝑔(𝑡)=𝑡/𝑒^𝑡_𝛼 (0<𝑡<+∞) is given (see Appendices A, B, and C for details), and the maximum of 𝑔(𝑡) is as follows: (𝑡)=𝜔𝑀=max𝑔𝑒^𝜔_𝛼,(1≤𝜔≤2). (13)

Lemma 5. (1)If 0<𝑝𝑇<𝜔/𝑒^𝜔_𝛼 or 𝑝<0, (12) has a real root.(2)If 𝑝𝑇>𝜔/𝑒^𝜔_𝛼, (12) has not a real root.

Proof. (1) If 𝑝<0, it is obvious that (12) has a real root.
As 𝑝>0, from the characteristic equation (12), 𝑝𝑒_𝛼^−𝜆𝑇=−𝜆,𝑝𝑇=−𝜆𝑇𝑒_𝛼^−𝜆𝑇<1𝑒¹_𝛼. (14)
So 𝑝<0 or 0<𝑝𝑇<𝜔/𝑒^𝜔_𝛼, and (12) has a real root.
(2) In the same way, we get 𝑝𝑇>𝜔/𝑒^𝜔_𝛼, and (12) has not a real root.

Theorem 6. (1)If 0<𝑝𝑇<𝜔/𝑒^𝜔_𝛼 or 𝑝<0, the solution of (11) is nonoscillatory.(2)If 𝑝𝑇>𝜔/𝑒^𝜔_𝛼, the solution of (11) is oscillatory.

Proof. The equation ^𝑅₀𝐷^𝛼𝑥(𝑡)+𝑝𝑥(𝑡−𝑇)=0 has a solution 𝑐𝑒_𝛼^𝜆𝑡, if and only if 𝜆 is a root of characteristic of equation 𝑓(𝜆)=𝜆+𝑝𝑒_𝛼^−𝜆𝑇=0. Using the conditions for (11), we have the following.(1)If 0<𝑝𝑇<𝜔/𝑒^𝜔_𝛼 or 𝑝<0, (12) has a real root, and the solution of (11) 𝑐𝑒_𝛼^𝜆𝑡 is nonoscillatory.(2)If 𝑝𝑇>𝜔/𝑒^𝜔_𝛼, (12) does not have a real root, and the solution 𝑐𝑒_𝛼^𝜆𝑡 is oscillatory. So we can obtain 𝑝𝑇>𝜔/𝑒^𝜔_𝛼, and the solutions of (11) are oscillatory. Therefore, the proof of the theorem can be completed.

Example 7. We consider delay differential equation with fractional-order derivatives ^𝑅₀𝐷^1/2𝑥(𝑡)+𝑥𝑡−3𝜋4=0,𝑝𝑇=3𝜋4>𝜔𝑒^𝜔_1/2≈0.179641784⋯. (15)(16)

All the conditions of Theorem 6 are satisfied. Fractional derivative of some special functions is as 𝐷^1/2sin𝑡=sin(𝑡+𝜋/4). Thus, all the solutions of (15) are oscillatory. One of such solutions is 𝑥(𝑡)=sin𝑡.

We consider the following fractional-order linear delay differential equation: ^𝑅₀𝐷^𝛼𝑥(𝑡)+𝑓(𝑥(𝑡−𝑇))=0 (17)

(𝐻₁): 𝑢>0, 𝑓(𝑢)≥𝐶𝑢.

Theorem 8. Assume that (𝐻) holds, and 𝐶𝑇>𝜔/𝑒^𝜔_𝛼 are satisfied. Every solution of the delay differential equation (17) oscillates.

Proof. We may assume that (17) has eventually positive solution. Then, there exists 𝑡₀, and if 𝑡>𝑡₀, 0=^𝑅₀𝐷^𝛼𝑥(𝑡)+𝑓(𝑥(𝑡−𝑇))≥^𝑅₀𝐷^𝛼𝑥(𝑡)+𝐶𝑥(𝑡−𝑇), 𝑥(𝑡)>0, and 𝑥(𝑡−𝑇)>0.
We consider the eventually positive solution of the following inequality: ^𝑅₀𝐷^𝛼𝑥(𝑡)+𝐶𝑥(𝑡−𝑇)≤0. (18)
Assume the inequality (18) has a solution 𝑒_𝛼^𝜆𝑡: ^𝑅₀𝐷^𝛼+𝐶𝑒_𝛼^−𝜆𝑇𝑒_𝛼^𝜆𝑡=𝜆+𝐶𝑒_𝛼^−𝜆𝑇𝑒_𝛼^𝜆𝑡≤0. (19)
If 𝐶𝑇>𝜔/𝑒^𝜔_𝛼, the inequality 𝜆+𝐶𝑒_𝛼^−𝜆𝑇≤0 does not have a real root. That means every solution of the inequality (18) is oscillatory. Thus, if 𝐶𝑇>𝜔/𝑒^𝜔_𝛼, every solution of the delay differential equation (17) oscillates.

3.2. The Fractional-Order Delay Differential Equation with Oscillating Coefficients

We consider oscillation of solutions of the fractional-order delay differential equation with oscillating coefficients: ^𝑅₀𝐷^𝛼𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−𝑇)=0. (20)

Let 𝑝=lim_𝑡→∞inf(1/𝑇)∫_𝑡^𝑡+𝑇𝑝(𝑡)𝑑𝑡>0. If 𝑡 is sufficiently large, ^𝑅₀𝐷^𝛼+𝑝(𝑡)𝑒_𝛼^−𝜆𝑇𝑒_𝛼^𝜆𝑡=𝜆+𝑝(𝑡)𝑒_𝛼^−𝜆𝑇𝑒_𝛼^𝜆𝑡≥𝜆+lim_𝑡→∞1inf𝑇_𝑡^𝑡+𝑇𝑝(𝑡)𝑑𝑡𝑒_𝛼^−𝜆𝑇𝑒_𝛼^𝜆𝑡=𝜆+𝑝𝑒_𝛼^−𝜆𝑇𝑒_𝛼^𝜆𝑡,𝜆+𝑝𝑒_𝛼^−𝜆𝑇𝑒_𝛼^𝜆𝑡≤0. (21)

We have the sufficient conditions for oscillation of all solutions of (20) as follows.

Theorem 9. If 𝑡 is sufficiently large, lim_𝑡→∞inf∫_𝑡^𝑡+𝑇𝑝(𝑡)𝑑𝑡>𝜔/𝑒^𝜔_𝛼. Then, the solution of (20) is oscillatory.

Proof. Assume that (20) has eventually positive solution. If there exists 𝑡₀>0 such that 𝑡>𝑡₀, then 𝑥(𝑡)>0, 𝑥(𝑡−𝑇)>0, ^𝑅₀𝐷^𝛼𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝑇)≥^𝑅₀𝐷^𝛼𝑥(𝑡)+𝑝𝑥(𝑡−𝑇_𝑅0),𝐷^𝛼+𝑝𝑒_𝛼^−𝜆𝑇𝑒_𝛼^𝜆𝑡=𝜆+𝑝𝑒_𝛼^−𝜆𝑇𝑒_𝛼^𝜆𝑡=𝑓(𝜆)𝑒_𝛼^𝜆𝑡=0,_𝑅0𝐷^𝛼𝑥(𝑡)+𝑝𝑥(𝑡−𝑇)≤0. (22)(23)(24)
Similar to the proof process of Theorem 8, every solution of the inequality (24) is oscillatory due to 𝑝𝑇>𝜔/𝑒^𝜔_𝛼. Thus, the inequality (24) does not have eventually positive solution. Therefore, if 𝑡 is sufficiently large, 𝑝𝑇>𝜔/𝑒^𝜔_𝛼, and every solution of (20) is oscillatory. The proof of the theorem is complete.

We consider the following fractional-order linear delay differential equations: ^𝑅₀𝐷^𝛼𝑉(𝑡)+^𝑚_𝑗=1𝑝_𝑗𝑉𝑡−𝑇_𝑗−^𝑚_𝑗=1𝑞_𝑗𝑉𝑡−𝜎_𝑗=0,_𝑅0𝐷^𝛼𝑉(𝑡)+𝑎𝜆₀𝑉(𝑡)+^𝑚_𝑗=1𝑝_𝑗𝑉𝑡−𝑇_𝑗−^𝑚_𝑗=1𝑞_𝑗𝑉𝑡−𝜎_𝑗=0. (25)(26)

Their characteristic equations are expressed by (27) and (28), respectively: 𝜆+^𝑚_𝑘=1𝑝_𝑘𝑒^{−𝜆𝑇𝑘}_𝛼−𝑞_𝑘𝑒^{−𝜆𝜎𝑘}_𝛼=0,𝜆+𝑎𝜆₀+^𝑚_𝑘=1𝑝_𝑘𝑒^{−𝜆𝑇𝑘}_𝛼−𝑞_𝑘𝑒^{−𝜆𝜎𝑘}_𝛼=0, (27)(28) where 𝑎, 𝜆₀, 𝑝_𝑘, 𝑞_𝑘, 𝑇_𝑘, 𝜎_𝑘 are nonnegative constants.

Based on Lemma 3, the statements are equivalent for oscillation of all solutions of (25) and (26) as follows.

Corollary 10. The following statements are equivalent.(1)Every solution of (25) and (26) oscillates.(2)The characteristic equations of (27) and (28) have no real roots.

4. The Fractional-Order Delay Partial Differential Equation

To research the oscillation properties of solutions for delay partial differential equations, we consider the delay parabolic differential equations [5]: ^𝑅₀𝐷^𝛼_𝑡𝑢(𝑥,𝑡)=𝑎(𝑡)Δ𝑢(𝑥,𝑡)+^𝑠_𝑘=1𝑏_𝑘(𝑡)Δ𝑢𝑥,𝑡−𝜎_𝑘−𝑞𝑢(𝑥,𝑡−𝑇), (29) where (𝑥,𝑡)∈Ω×(𝑡,∞)=𝐺. 𝜕Ω is the bounded region with piecewise smooth in 𝑅^𝑛.

The hypotheses are always true as follows. (𝐶₁)𝑎,𝑏_𝑘∈𝐶([0,∞);[0,∞)), and 𝑞 is a nonnegative constant. (𝐶₂)𝑇,𝜎_𝑘 are nonnegative constants.

Consider the boundary conditions as follows: 𝜕𝑢(𝑥,𝑡)()𝑡𝜕𝑁=0,on𝑥,𝑡∈𝜕Ω×₀,∞, 𝐵₁ where 𝑁 is the unit exterior normal vector in 𝜕Ω.

If there are no special instructions in this section, we only discuss the oscillation of solutions of the fractional-order delay partial differential equation with respect to time variable 𝑡.

Theorem 11. Assume that (𝐶₁)and (𝐶₂)hold, and 𝐵₁ is satisfied.(1)As 0<𝑞𝑇<𝜔/𝑒^𝜔_𝛼 or 𝑞<0, the solution of (29) and 𝐵₁ is nonoscillatory.(2)As 𝑞𝑇>𝜔/𝑒^𝜔_𝛼, the solution of (29) and 𝐵₁ is oscillatory.

Proof. Let (𝑥,𝑡)∈Ω×[𝑡₁,∞). Equation (29) on both sides of the integral of 𝑥 in Ω is expressed by 𝐷^𝛼_𝑡Ω𝑢(𝑥,𝑡)𝑑𝑥=𝑎(𝑡)_Ω()+Δ𝑢𝑥,𝑡𝑑𝑥_𝑠𝑘=1𝑎_𝑘(𝑡)_ΩΔ𝑢𝑥,𝑡−𝜎_𝑘𝑑𝑥−𝑞_Ω𝑢(𝑥,𝑡−𝑇)𝑑𝑥. (30)
By using Green’s formula and boundary conditions 𝐵₁, Δ𝑢(_Ω𝑥,𝑡)𝑑𝑥=_𝜕Ω𝜕𝑢(𝑥,𝑡)𝜕𝑁𝑑𝑆=0,_ΩΔ𝑢𝑥,𝑡−𝜌_𝑘𝑑𝑥=_𝜕Ω𝜕𝑢𝑥,𝑡−𝜌_𝑘𝜕𝑁𝑑𝑆=0. (31)
Let 𝑣(𝑡)=∫_Ω𝑢(𝑥,𝑡)𝑑𝑥. Substituting (30) with (31), ^𝑅₀𝐷^𝛼𝑣(𝑡)+𝑞𝑣(𝑡−𝑇)=0. (32)
Based on Corollary 10, (1) and (2) in Theorem 14 are proved.

Example 12. Consider the parabolic equation: ^𝑅₀𝐷_𝑡^2/3𝑢()=1𝑥,𝑡4𝜕²𝑢(𝑥,𝑡)𝜕𝑥²−√3𝑢𝑥,𝑡−5𝜋6. (33)
Because of √3×(5𝜋/6)>𝜔/𝑒^𝜔_2/3≈0.240874076⋯, all the conditions in Theorem 11 (1) are satisfied. By using Theorem 11, every solution of (33), with 𝜕𝑢𝜕𝑥_𝑥=0=𝜕𝑢𝜕𝑥_𝑥=𝜋=0,𝑡≥0, (34) oscillates in (0,𝜋)×𝑅⁺. One of such solutions is 𝑢(𝑥,𝑡)=sin𝑡cos2𝑥.

For the more general equation of (29) with coefficients, ^𝑅₀𝐷^𝛼𝑢(𝑥,𝑡)=𝑎(𝑡)Δ𝑢(𝑥,𝑡)−^𝑚_𝑗=1𝑝_𝑗𝑢𝑥,𝑡−𝑇_𝑗+^𝑚_𝑗=1𝑞_𝑗𝑢𝑥,𝑡−𝜎_𝑗. (35)

There are the boundary conditions as follows: 𝑢(𝑥,𝑡)=0,with(𝑥,𝑡)∈𝜕Ω×𝑡₀,∞,𝜕𝑢(𝑥,𝑡)𝜕𝑁+𝜅𝑢=0,with(𝑥,𝑡)∈𝜕Ω×𝑡₀,∞, 𝐵₂𝐵₃ where 𝑁 is the unit exterior normal vector in 𝜕Ω.

Assume that (𝐶₃)   𝑝_𝑗, 𝑞_𝑗, 𝑇_𝑗, 𝜎_𝑗 are nonnegative constants.

Some sufficient conditions for oscillation of all solutions of (35) with 𝐵₂ and 𝐵₃ are established in Theorem 15. Before presenting the theorem, the lemma is given as follows.

Lemma 13. For the Dirichlet problem, 𝑢|Δ𝑢+𝜆𝑢=0,𝑥∈Ω_𝜕Ω=0, (36) where 𝜆 is a constant. The smallest eigenvalue 𝜆₀ of problem (36) is positive, and the corresponding eigenfunction Φ(𝑥) is also positive on 𝑥∈Ω [5]. With each solution 𝑢(𝑥,𝑡) of problem (35) with the boundary condition 𝐵₂, the function 𝑉(𝑡) is defined as 𝑉(𝑡)=_Ω𝑢(𝑥,𝑡)Φ(𝑥)𝑑𝑥,𝑡>𝑡₁. (37)

Theorem 14. Assume that (𝐶₁) and (𝐶₃)hold, and 𝐵₂ is satisfied. Every solution of the delay differential equation (26) with several coefficients oscillates. Then, every solution of (35) oscillates.

Proof. Let (𝑥,𝑡)∈Ω×[𝑡₁,∞). If we multiply both sides of (35) by Φ(𝑥) and integrate 𝑥 in Ω, we obtain 𝐷^𝛼_𝑡Ω𝑢()Φ(𝑥)(𝑡)𝑥,𝑡𝑑𝑥=𝑎_Ω()Φ(𝑥)−Δ𝑢𝑥,𝑡𝑑𝑥_𝑚𝑗=1𝑝_𝑗Ω𝑢𝑥,𝑡−𝑇_𝑗Φ(𝑥)+𝑑𝑥_𝑚𝑗=1𝑞_𝑗𝑢𝑥,𝑡−𝜎_𝑗Φ(𝑥)𝑑𝑥, (38) where Φ(𝑥)>0, which satisfies Lemma 13.
By using Green’s formula and boundary condition 𝐵₂, we obtain _Ω()Φ(𝑥)=Δ𝑢𝑥,𝑡𝑑𝑥_𝜕ΩΦ(𝑥)𝜕𝑢(𝑥)𝜕𝑁−𝑢𝜕Φ+𝜕𝑁𝑑𝑆_Ω𝑢(𝑥,𝑡)ΔΦ(𝑥)𝑑𝑥=−𝜆_0Ω𝑢(𝑥,𝑡)Φ(𝑥)𝑑𝑥, (39) where 𝑑𝑆 is the surface element on 𝜕Ω. Substituting with (37), we obtain ^𝑅₀𝐷^𝛼𝑉(𝑡)+𝑎𝜆₀𝑉(𝑡)+^𝑚_𝑗=1𝑝_𝑗𝑉𝑡−𝑇_𝑗−^𝑚_𝑗=1𝑞_𝑗𝑉𝑡−𝜎_𝑗=0. (40)
Based on Corollary 10, the proof of the theorem can be immediately completed. Let 𝑣(𝑡) be defined as 𝑣(𝑡)=∫_Ω𝑢(𝑥,𝑡)𝑑𝑥, 𝑡>𝑡₁, where 𝑢(𝑥,𝑡) denotes a solution of problem (35) with the boundary condition 𝐵₁ or 𝐵₃. Then, we obtain ^𝑅₀𝐷^𝛼𝑣(𝑡)+^𝑚_𝑗=1𝑝_𝑗𝑣𝑡−𝑇_𝑗−^𝑚_𝑗=1𝑞_𝑗𝑣𝑡−𝜎_𝑗=0. (41)

Theorem 15. Assume that (𝐶₁)(𝐶₃) hold and 𝐵₁ is satisfied. Every solution of the delay differential equation with several coefficients (25) oscillates. Then, every solution of (35) with the boundary condition 𝐵₁ oscillates.

Proof. Let (𝑥,𝑡)∈Ω×[𝑡₁,∞) on the 𝑥 be the definite integral in Ω. Assume that (35) with the boundary condition 𝐵₁ has a nonoscillatory solution and (35) has an eventually positive solution: 𝐷^𝛼_𝑡Ω𝑢()(𝑡)𝑥,𝑡𝑑𝑥=𝑎_ΩΔ𝑢(𝑥,𝑡)𝑑𝑥−^𝑚_𝑗=1𝑝_𝑗Ω𝑢𝑥,𝑡−𝑇_𝑗+𝑑𝑥_𝑚𝑗=1𝑞_𝑗𝑢𝑥,𝑡−𝜎_𝑗𝑑𝑥. (42)
By using Green’s formula and boundary condition 𝐵₁, we obtain Δ𝑢(_Ω𝑥,𝑡)𝑑𝑥=_𝜕Ω𝜕𝑢(𝑥,𝑡)𝜕𝑁𝑑𝑆=0, (43) where 𝑑𝑆 is the surface element on 𝜕Ω.
Let 𝑣(𝑡)=∫_Ω𝑢(𝑥,𝑡)𝑑𝑥 (𝑡>𝑡₁) be substituted into (42). We obtain ^𝑅₀𝐷^𝛼𝑣(𝑡)+^𝑚_𝑗=1𝑝_𝑗𝑣𝑡−𝑇_𝑗−^𝑚_𝑗=1𝑞_𝑗𝑣𝑡−𝜎_𝑗=0. (44)
The detail of the proof process is similar to the proof process of Theorem 15.

Theorem 16. Assume that (𝐶₁)and (𝐶₂)hold and 𝐵₃ is satisfied. Every solution of the delay differential equation with the coefficients (25) oscillates. Then, every solution of (35) with the boundary condition 𝐵₃ oscillates.

Proof. Let (𝑥,𝑡)∈Ω×[𝑡₁,∞) on the 𝑥 be the definite integral in Ω. For the sake of contradiction, assume that (35) with the boundary condition 𝐵₃ has a nonoscillatory solution and (35) with the boundary condition 𝐵₁ has an eventually positive solution: 𝐷^𝛼_𝑡Ω𝑢(𝑥,𝑡)𝑑𝑥=𝑎_ΩΔ𝑢(𝑥,𝑡)𝑑𝑥−^𝑚_𝑗=1𝑝_𝑗Ω𝑢𝑥,𝑡−𝑇_𝑗+𝑑𝑥_𝑚𝑗=1𝑞_𝑗𝑢𝑥,𝑡−𝜎_𝑗𝑑𝑥. (45)
By using Green’s formula and boundary condition 𝐵₃, we obtain Δ𝑢(_Ω𝑥,𝑡)𝑑𝑥=_𝜕Ω𝜕𝑢(𝑥,𝑡)𝜕𝑁𝑑𝑆=−𝜅_𝜕Ω𝑢(𝑥,𝑡)𝑑𝑆≤0, (46) where 𝑑𝑆 is the surface element on 𝜕Ω. Let 𝑣(𝑡)=∫_Ω𝑢(𝑥,𝑡)𝑑𝑥 (𝑡>𝑡₁) be substituted into (45). We obtain ^𝑅₀𝐷^𝛼𝑣(𝑡)+^𝑚_𝑗=1𝑝_𝑗𝑣𝑡−𝑇_𝑗−^𝑚_𝑗=1𝑞_𝑗𝑣𝑡−𝜎_𝑗≤0. (47)
The detail of the proof process is similar to the proof process of Theorem 15.

The delay parabolic differential equations are expressed by ^𝑅₀𝐷^𝛼𝑢(𝑥,𝑡)=𝑎(𝑡)Δ𝑢(𝑥,𝑡)+^𝑚_𝑘=1𝑏_𝑘(𝑡)Δ𝑢𝑥,𝑡−𝑇_𝑘−^𝑛_𝑖=1𝑞_𝑖(𝑡)𝑓_𝑖𝑢𝑥,𝑡−𝜎_𝑖. (48)

Consider the boundary conditions, as shown in 𝐵₁ where 𝑁 is the unit exterior normal vector in 𝜕Ω. (𝐶₄)𝑇_𝑘,𝜎_𝑖 are nonnegative constants. (𝐶₅)𝑎(𝑡),𝑏_𝑘(𝑡),𝑞_𝑖(𝑡)∈𝐶([0,∞);[0,∞)), 𝑘=1,2,…,𝑚; 𝑖=1,2,…,𝑛; where 𝑞_𝑖=inf𝑞_𝑖(𝑡) is a nonnegative constant. (𝐻₂)𝑓_𝑖(𝑢)∈𝐶(𝑅,𝑅), and 𝑢>0,𝑓_𝑖(𝑢)≥𝑐_𝑖𝑢, 𝑖=1,2,…,𝑛.

Theorem 17. Assume that (𝐶₄), (𝐶₅), and (𝐻₂) hold and 𝐵₁ is satisfied. If ∑^𝑛_𝑖=1𝑐_𝑖𝑞_𝑖𝜎_𝑖>𝜔/𝑒^𝜔_𝛼, the solution of (48) with the boundary condition 𝐵₁ is oscillatory.

Proof. Let (𝑥,𝑡)∈Ω×[𝑡₁,∞). Assume that (48) with the boundary condition 𝐵₂ has no oscillation solutions, and (48) has eventually positive solution. If there exists 𝑡₂>𝑡₁ such that 𝑡>𝑡₂, then 𝑢(𝑥,𝑡)>0,  𝑢(𝑥,𝑡−𝑇_𝑘)>0,  𝑢(𝑥,𝑡−𝜎_𝑗)>0. If (𝐻₂) is satisfied, (48) can be transformed into the inequality as follows: ^𝑅₀𝐷^𝛼𝑢(𝑥,𝑡)≤𝑎(𝑡)Δ𝑢(𝑥,𝑡)+^𝑚_𝑘=1𝑏_𝑘(𝑡)Δ𝑢𝑥,𝑡−𝜏_𝑘−^𝑛_𝑖=1𝑞_𝑖𝐶_𝑖𝑢𝑥,𝑡−𝜎_𝑖. (49)
The inequality (49) on both sides of the integral of 𝑥 in Ω is expressed by 𝐷^𝛼_𝑡Ω𝑢(𝑥,𝑡)𝑑𝑥≤𝑎(𝑡)_Ω()+Δ𝑢𝑥,𝑡𝑑𝑥_𝑚𝑗=1𝑏_𝑘(𝑡)_Ω𝑢𝑥,𝑡−𝑇_𝑘−𝑑𝑥_𝑛𝑗=1Ω𝑞_𝑗𝐶_𝑗𝑢𝑥,𝑡−𝜎_𝑗𝑑𝑥. (50)
By using Green’s formula and boundary conditions 𝐵₁, we obtain Δ𝑢(_Ω𝑥,𝑡)𝑑𝑥=_𝜕Ω𝜕𝑢(𝑥,𝑡)𝜕𝑁𝑑𝑆=0,_ΩΔ𝑢𝑥,𝑡−𝜌_𝑘𝑑𝑥=_𝜕Ω𝜕𝑢𝑥,𝑡−𝜌_𝑘𝜕𝑁𝑑𝑆=0. (51)
Substituting (50) with (51) and 𝑣(𝑡)=∫_Ω𝑢(𝑥,𝑡)𝑑𝑥, we obtain 𝐷^𝛼𝑣(𝑡)+^𝑛_𝑗=1𝑞_𝑗𝐶_𝑗𝑣𝑡−𝑇_𝑗≤0. (52)
By using Theorem 9, the proof of the theorem can be completed.

Appendices

A. Maximum of 𝑔(𝑡)=𝑡/𝑒^𝑡_𝛼

Figure 1 illustrates the curves of the function 𝑔(𝑡)=𝑡/𝑒^𝑡_𝛼(𝑡>0) with different 𝛼. From bottom to top, 𝛼=1/4, 1/3, 1/2, 2/3, 3/4, and 1, respectively. From Figure 1, it can be observed that there exists 𝑀 such that =max𝑔(𝑡)=𝜔/𝑒^𝜔_𝛼  (𝜔≥1). In detail, the values of 𝑔(𝑡) with different 𝛼 are reported in Table 1. 𝑚 is the maximum for a fixed 𝛼 in Table 1.

Table 1 

Values of 𝑔(𝑡) with different 𝛼.

Figure 1 

Curves of function 𝑔(𝑡)=𝑡/𝑒𝑡𝛼 (𝑡>0) with different 𝛼.

B. Supplementary Comment of 𝑔(𝑡)

The curves of function 𝑔₂(𝑡)=∑^∞_𝑖=0(𝑡^𝛼(𝑖+1)/Γ(𝛼(𝑖+1))) and 𝑔₃(𝑡)=𝑡² (𝑡>0) are showed in Figure 2. The function 𝑔₂(𝑡) is a supplementary comment of the maximum value of 𝑔(𝑡). From top to bottom, 𝛼=1/4, 1/3, 1/2, 2/3, 3/4, and 4/5, respectively.

Figure 2 

Curves of function 𝑔2(𝑡) and 𝑔3(𝑡).

C. The Characteristic Equation

The Laplace transforms for two fractional derivatives with 0<𝛼<1 [6] are expressed as follows: 𝕃_𝐶𝐷^𝛼𝑓(𝑡)=𝑠^𝛼𝐹(𝑠)(0),𝕃−𝑓_RL𝐷^𝛼𝑓(𝑡)=𝑠^𝛼𝐹(𝑠)−𝑓₀,𝑓₀=lim_𝑥→0𝑡^𝛼−1𝑓(𝑡). (C.1)

The fractional Laplace transform [2] is 𝕃_𝛼{𝑓(𝑡)}=𝐹_𝛼(𝑠)=∫₀^+∞𝑓(𝑡)𝑒_𝛼^−𝑠𝑡𝑑𝑡. There exist 𝑀 and 𝜇 such that |𝑥(𝑡)|≤𝑀𝑒_𝛼^𝜇𝑡. By taking the Laplace transforms on both sides of (11), 𝑠𝑋(𝑠)−𝑥(0)+𝑝𝑒_𝛼^−𝑠𝑇𝑋(𝑠)+𝑝ℎ(𝑠𝑇)=0,Re𝑠>𝜎₀, (C.2) where ℎ(𝑠𝑇)=𝑒_𝛼^−𝑠𝑇∫^∞₀𝑥(𝑡)𝑒_𝛼^−𝑠𝑡𝑑𝑡. 𝜎₀ denotes the abscissa of convergence of 𝑋(𝑠); that is, 𝜎₀=inf{𝜎∈𝑅∣𝑥(𝜎)exists}.

In the most general form 𝐹(𝑠)𝑋(𝑠)=𝑄(𝑠), we can obtain 𝐹(𝑠)=𝑠+𝑝𝑒_𝛼^−𝑠𝑇 and 𝑄(𝑠)=𝑥(0)−𝑝ℎ(𝑠𝑇). 𝐹(𝑠) and 𝑄(𝑠) are both entire functions. Assume 𝐹(𝑠)≠0 for all real 𝑠. 𝑋(𝑠) can be expressed as 𝑋(𝑠)=𝑄(𝑠)/𝐹(𝑠), Re𝑠>𝑠₀>0. Therefore, 𝜆+𝑝𝑒_𝛼^−𝜆𝑇=0 is the characteristic polynomial of (11).

Conflict of Interests

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

Acknowledgment

The authors thank Mr. Weisheng Xu for useful discussions on the proof process in the paper.