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

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 fractionalorder 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.


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][2][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:  0    () +  ( ( − )) = 0, where 0 <  < 1.Some oscillation results are also given for the delay parabolic differential equation as follows: ( For the purpose, known methods are expanded for our results.

Preliminaries
We first give some definitions and make some preliminaries.The fractional-order derivative   () can be defined in different ways, wherein Riemann-Liouville's definition and Caputo's definition are two main expressions as follows.
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.

Main Result
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.
Proof.The equation  0   () + ( − ) = 0 has a solution    , if and only if  is a root of characteristic of equation () =  +  −  = 0. Using the conditions for (11), we have the following.
(2) If  > /   , (12) does not have a real root, and the solution    is oscillatory.So we can obtain  > /   , 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 All the conditions of Theorem 6 are satisfied.Fractional derivative of some special functions is as  1/2 sin  = sin( + /4).Thus, all the solutions of (15) are oscillatory.One of such solutions is () = sin .
We consider the eventually positive solution of the following inequality: Assume the inequality (18) has a solution    : If  > /   , the inequality  +   −  ≤ 0 does not have a real root.That means every solution of the inequality (18) is oscillatory.Thus, if  > /   , every solution of the delay differential equation (17) oscillates.

The Fractional-Order Delay Differential Equation with
Oscillating Coefficients.We consider oscillation of solutions of the fractional-order delay differential equation with oscillating coefficients: We have the sufficient conditions for oscillation of all solutions of (20) as follows.
Then, the solution of (20) is oscillatory.
Proof.Assume that (20) has eventually positive solution.If there exists  0 > 0 such that  >  0 , then () > 0, ( − ) > 0, Similar to the proof process of Theorem 8, every solution of the inequality (24) is oscillatory due to  > /   .Thus, the inequality (24) does not have eventually positive solution.Therefore, if  is sufficiently large,  > /   , and every solution of (20) is oscillatory.The proof of the theorem is complete.
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.

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]: where (, ) ∈ Ω × (, ∞) = .Ω is the bounded region with piecewise smooth in   .The hypotheses are always true as follows.
Consider the boundary conditions as follows: where  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 fractionalorder delay partial differential equation with respect to time variable .
Some sufficient conditions for oscillation of all solutions of (35) with ( 2 ) and ( 3 ) are established in Theorem 15.Before presenting the theorem, the lemma is given as follows.
Lemma 13.For the Dirichlet problem, where  is a constant.
where Φ() > 0, which satisfies Lemma 13.By using Green's formula and boundary condition ( 2 ), we obtain where  is the surface element on Ω.Substituting with (37), we obtain Based on Corollary 10, the proof of the theorem can be immediately completed.Let V() be defined as V() = ∫ Ω (, ),  >  1 , where (, ) denotes a solution of problem (35) with the boundary condition ( 1 ) or ( 3 ).Then, we obtain By using Green's formula and boundary condition ( 1 ), we obtain where  is the surface element on Ω.Let V() = ∫ Ω (, ) ( >  1 ) be substituted into (42).We obtain The detail of the proof process is similar to the proof process of Theorem 15.
Theorem 16.Assume that ( 1 ) and ( 2 ) hold and ( 3 ) is satisfied.Every solution of the delay differential equation with the coefficients (25) oscillates.Then, every solution of (35) with the boundary condition ( 3 ) oscillates.
Proof.Let (, ) ∈ Ω × [ 1 , ∞) on the  be the definite integral in Ω.For the sake of contradiction, assume that (35) with the boundary condition ( 3 ) has a nonoscillatory solution and (35) with the boundary condition ( 1 ) has an eventually positive solution: By using Green's formula and boundary condition ( 3 ), we obtain where  is the surface element on Ω.Let V() = ∫ Ω (, ) ( >  1 ) be substituted into (45).We obtain The detail of the proof process is similar to the proof process of Theorem 15.By using Theorem 9, the proof of the theorem can be completed.

Table 1 :
Values of () with different .