#### Abstract

The paper gives sufficient conditions on the existence of periodic solution for a class of compound singular fractional differential systems with delay, involving Nishimoto fractional derivative. Furthermore, for the particular functions, the necessary conditions on the existence of periodic solution are also derived. Especially, for two-dimensional compound singular fractional differential equation with delay, the criteria of existence of periodic solution are obtained. Finally, two examples are presented to verify the validity of criteria.

#### 1. Introduction

In real life, there are many phenomena with time delay. The mathematical model derived from engineering, physics, mechanics, control theory, chemical reactions, biology, and medicine was made with a significant amount of delay, such as the limited signal transmission speed human reaction time to the outside world. Therefore, the delay is widespread in nature and society, in the introduction of time-delay differential equations there can be a more accurate description and explanation of various phenomena and processes.

Fractional calculus is the promotion of classical calculus. The study found that fractional calculus was very suitable to describe long memory and hereditary properties of various materials and processes [1, 2]. In the recent years, fractional calculus becomes a research hotspot, its field of concern has become wide, such as the numerical method of the equation in [3], the existence and uniqueness of equations in [4], fractional Brownian motion, fractional reaction-diffusion equation and random walk [5, 6], fractional wavelet transform [7], and fractional control [8].

Most of the above-mentioned studies, utilize the Riemann-liouville fractional derivative definition, which due to its nature of its definition is simple and relatively good. But Nishimoto definition of fractional calculus [9, 10], has not received a lot of attention, this may be part of the naturalization due to the complexity of its definition, but compared to Riemann-liouville fractional calculus, it has a better nature, relevant results more concise useful.

The existence of periodic solutions of differential equations is one of the important research directions of biomathematics [11–15], which has a wide range of applications, such as the existence of periodic orbits of celestial movement and its stability.

In [12], the author discussed the following system:

and obtained sufficient and necessary conditions for the existence of periodic solutions for the system. Taking into account the periodic solutions of the fractional time-delay system will be a very important practical significance; we are tried to generalize the corresponding results to the case of fractional order.

For the above reasons we consider the following compound singular fractional differential system with delay: where denotes Nishimoto fractional derivative of order , . , and are constant system matrices of appropriate dimensions, and and are constants with .

#### 2. Definitions and Notations

In this section we introduce the definitions of fractional derivative/integral and related basic properties used in the paper; more information can be obtained from [9, 10].

*Definition 2.1 (see [9]). *If the function is analytic (regular) inside and on , here , is a contour along the cut joining the points and , which starts from the point at , encircles the point once counter-clockwise, and returns to the point at , is a contour along the cut joining the points and , which starts from the point at , encircles the point once counter-clockwise, and returns to the point at ,
where , , for , and , for .

Then is said to be the fractional derivative of of order and is said to be the fractional integral of of order , provided that .

Let us recall the following useful properties associated with the definition introduced above [9].

*Property 1. *For a constant ,

*Property 2. *For a constant ,

*Property 3. *If the function is singlevalued and analytic in some domain , then

*Property 4. *For a constant ,
In the following section of this paper, we let denote the order Nishimoto derivative.

#### 3. The Main Results

In this section, we discuss some problems to the system of the system (1.2).

Theorem 3.1. *The sufficient condition for the existence of the nonconstant periodic solutions of system (1.2) is that the following equation exists pure imaginary roots
*

*Proof. *Assume that is pure imaginary root of (3.1), let , substituting in (1.2), then
As is pure imaginary roots of (3.1), note that
So (3.1) exists nonzero solution , then, is the nonconstant periodic solution of (1.2).

If the system have nonconstant periodic solutions, then we may wonder whether the solution satisfy (3.1), in fact, as you will see it holds when the function satisfy some conditions. We know that if the function is a continuous smooth periodic function with period 2l,then it can be expressed as its fourier series form
where .

And as we also know that its fourier series expansion has the following properties:
we can even get the following relation if satisfy some more strictly condition:
To obtain the similar property of our fractional derivative, what conditions the function should satisfy? We give the following function space.

*Definition 3.2. *If the periodic function is continuous and smooth on , its order Nishimoto derivative exists, then we let denote the corresponding function space whose elements have the following property:
it is easy to know from the definition that , and so is nonempty.

Theorem 3.3. *If is the non-constant periodic solution of (1.2), and further , one can obtain the necessity of Theorem 3.1.*

*Proof. *Suppose the period of is 2l, is continuous and differentiable because of , then we can denote it in the form of its fourier series:
Since , we have
where .

We put (3.8) and (3.9) into (1.2), and obtain
then multiply on both sides of (3.10) and integrate it from to , hence
It is easy to deduce that
recalling (3.10), it reduces to
Thus, if there are no pure imaginary roots in (3.1), then for every we have , according to (3.8), we conclude that vector which conflicts the suppose that is the non-constant periodic solution of (1.2).

#### 4. Two-Dimensional Case

For the case of the two-dimensional compound singular fractional differential system with delay, there is where , and is scalar function.

Using Theorem 3.1, we obtained the following theorem.

Theorem 4.1. *If one of the following equations exists non-zero real root, then system(4.1) exists non-constant periodic solution, further more, if , then the conclusion is sufficient and necessary
**
or
**
where , , .*

*Proof. *First of all we know that
according to Theorem 3.1, we have
then
where and denote the real and imaginary parts of , respectively. Then using Theorem 3.1, if there exists , we obtain that
hence system (4.1) exists non-constant periodic solution. This proved the theorem.

#### 5. Examples

In this section we give some concrete examples to illustrate our conclusions.

*Example 5.1. *We consider the following two-dimensional compound singular fractional differential system with delay:
where . so we have .

Using the discriminant of Theorem 4.1, we have
the solution is.

According to Theorem 4.1 , system (5.1) has non-constant periodic solution. We suppose that, as
exists non-zero solution, it means that
So we have , for any real number. Supposed that , then we obtained a non-constant periodic solution of system (5.1):

We can verify that is a non-constant periodic solution of system (5.1).

*Example 5.2. *Consider the following two-dimension compounded with singular fractional differential equation delay system:
where .

So we have .

Using the discriminant of Theorem 4.1, we obtained
Through the simplication of this equation, we have
and the solution is .

According to Theorem 4.1, there exists non-constant periodic solution in the system.