The Periodic Solutions of the Compound Singular Fractional Differential System with Delay

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:

𝐸𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏₁)+𝐶𝑥(𝑡−𝜏₂), (1.1) 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: 𝐻𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏₁)+𝐶𝑥(𝑡+𝜏₂), (1.2) where 𝐷^𝛼 denotes Nishimoto fractional derivative of order 𝛼,  𝛼>0. 𝐻,𝐴,𝐵, and 𝐶 are constant system matrices of appropriate dimensions, and 𝜏₁ and 𝜏₂ are constants with 𝜏₁>0,𝜏₂>0,|𝐻|=0.

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 𝑓(𝑧) is analytic (regular) inside and on 𝐶, here 𝐶∶={𝐶⁻,𝐶⁺},  𝐶⁻ is a contour along the cut joining the points 𝑧 and −∞+𝑖ℑ(𝑧), which starts from the point at −∞, encircles the point 𝑧 once counter-clockwise, and returns to the point at −∞, 𝐶⁺ is a contour along the cut joining the points 𝑧 and +∞+𝑖ℑ(𝑧), which starts from the point at +∞, encircles the point 𝑧 once counter-clockwise, and returns to the point at +∞, 𝑓_𝜐(𝑧)=(𝑓(𝑧))_𝜐∶=Γ(𝜐+1)2𝜋𝑖_𝑐𝑓(𝜍)(𝜍−𝑧)^𝜐+1𝑅𝑑𝜍(𝜐∈𝑍⁻;𝑍⁻𝑓∶={−1,−2,−3,…}),_−𝑛(𝑧)∶=lim_{𝜐→−𝑛}{𝑓_𝜐(𝑧)}(𝑛∈𝑁∶={1,2,3,…}), (2.1) where 𝜍,  −𝜋≤arg(𝜍−𝑧)≤𝜋, for 𝐶⁻, and 0≤arg(𝜍−𝑧)≤2𝜋, for 𝐶⁺.
Then 𝑓_𝜐(𝑧)(𝜐>0) is said to be the fractional derivative of 𝑓(𝑧) of order 𝜐 and 𝑓_𝜐(𝑧)(𝜐<0) is said to be the fractional integral of 𝑓(𝑧) of order −𝜐, provided that |𝑓_𝜐(𝑧)|<∞(𝜐∈𝑅).
Let us recall the following useful properties associated with the definition introduced above [9].

Property 1. For a constant 𝜆, (𝑒^𝜆𝑧)_𝜐=𝜆^𝜐𝑒^𝜆𝑧(𝜆≠0;𝜐∈𝑅;𝑧∈𝐶). (2.2)

Property 2. For a constant 𝜆, (𝑒^−𝜆𝑧)_𝜐=𝑒^{−𝑖𝜋𝜐}𝜆^𝜐𝑒^−𝜆𝑧(𝜆≠0;𝜐∈𝑅;𝑧∈𝐶). (2.3)

Property 3. If the function 𝑓(𝑧) is singlevalued and analytic in some domain Ω⊆𝐶, then (𝑓_𝜇(𝑧))_𝜐=𝑓_𝜇+𝜐(𝑧)=(𝑓_𝜐(𝑧))_𝜇. (2.4)

Property 4. For a constant 𝜆, (𝑧^𝜆)_𝜐=𝑒^{−𝑖𝜋𝜐}Γ(𝜐−𝜆)𝑧Γ(−𝜆)^𝜆−𝜐(𝜐∈𝑅;𝑧∈𝐶;|Γ(𝜐−𝜆)Γ(−𝜆)|<∞). (2.5) In the following section of this paper, we let 𝐷^𝛼 denote the 𝛼 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 det(𝜆^𝛼𝐻−𝐴−𝐵𝑒^−𝜏1𝜆−𝐶𝑒^𝜏2𝜆)=0. (3.1)

Proof. Assume that 𝜂𝑖 is pure imaginary root of (3.1), let 𝑥(𝑡)=𝐾𝑒^𝜂𝑖𝑡(𝐾∈𝑅^𝑛), substituting 𝑥(𝑡) in (1.2), then ((𝜂𝑖)^𝛼𝐻−𝐴−𝐵𝑒^{−𝜏1(𝜂𝑖)}−𝐶𝑒^{𝜏2(𝜂𝑖)})𝐾=0. (3.2) As 𝜂𝑖 is pure imaginary roots of (3.1), note that det((𝜂𝑖)^𝛼𝐻−𝐴−𝐵𝑒^{−𝜏1(𝜂𝑖)}−𝐶𝑒^{𝜏2(𝜂𝑖)})=0. (3.3) So (3.1) exists nonzero solution 𝐾, then, 𝑥(𝑡)=𝐾𝑒^𝜂𝑖𝑡 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 𝑓(𝑡) is a continuous smooth periodic function with period 2l,then it can be expressed as its fourier series form 𝑓(𝑡)=^+∞_𝑘=−∞𝐶_𝑘𝑒^{𝑖𝑘𝜋𝑡/𝑙}, (3.4) where 𝐶_𝑘=(1/2𝑙)∫^𝑙_−𝑙𝑓(𝜉)𝑒^{−𝑖𝑘𝜋𝑡/𝑙}𝑑𝜉.
And as we also know that its fourier series expansion has the following properties: 𝑓^′(𝑡)=^+∞_𝑘=−∞𝐶_𝑘𝑖𝑘𝜋𝑙𝑒^{𝑖𝑘𝜋𝑡/𝑙}, (3.5) we can even get the following relation if 𝑓(𝑡) satisfy some more strictly condition: 𝑓^(𝑘)(𝑡)=^+∞_𝑘=−∞𝐶_𝑘(𝑖𝑘𝜋𝑙)^𝑘𝑒^{𝑖𝑘𝜋𝑡/𝑙}. (3.6) 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 𝑓(𝑡) is continuous and smooth on 𝑅, its 𝛼(𝛼>1) order Nishimoto derivative exists, then we let Ω(𝑡) denote the corresponding function space whose elements have the following property: 𝐷^𝛼𝑓(𝑡)=𝐷^𝛼(^+∞_𝑘=−∞𝐶_𝑘𝑒^{𝑖𝑘𝜋𝑡/𝑙})=^+∞_𝑘=−∞𝐶_𝑘𝐷^𝛼(𝑒^{𝑖𝑘𝜋𝑡/𝑙})=^+∞_𝑘=−∞𝐶_𝑘(𝑖𝑘𝜋𝑡𝑙)^𝛼𝑒^{𝑖𝑘𝜋𝑡/𝑙}, (3.7) it is easy to know from the definition that 𝑒^𝑖𝜆𝑡∈Ω(𝑡)(𝜆∈𝑅), and so Ω(𝑡) is nonempty.

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

Proof. Suppose the period of 𝑥(𝑡) is 2l, 𝑥(𝑡) is continuous and differentiable because of 𝛼>1, then we can denote it in the form of its fourier series: 𝑥(𝑡)=^+∞_𝑘=−∞𝐶_𝑘𝑒^{𝑖𝑘𝜋𝑡/𝑙}. (3.8) Since 𝑥(𝑡)∈Ω(𝑡), we have 𝐷^𝛼𝑥(𝑡)=^+∞_𝑘=−∞𝐶_𝑘(𝑖𝑘𝜋𝑡𝑙)^𝛼𝑒^{𝑖𝑘𝜋𝑡/𝑙}, (3.9) where 𝐶_𝑘=(1/2𝑙)∫^𝑙_−𝑙𝑥(𝜉)𝑒^{−𝑖𝑘𝜋𝑡/𝑙}𝑑𝜉.
We put (3.8) and (3.9) into (1.2), and obtain ^∞_𝑘=−∞[(𝑖𝑘𝜋𝑙)^𝛼𝐻−𝐴−𝐵𝑒^{−𝑖𝑘𝜋𝜏1/𝑙}−𝐶𝑒^{𝑖𝑘𝜋𝜏2/𝑙}]𝐶_𝑘𝑒^{𝑖𝑘𝜋𝑡/𝑙}=0, (3.10) then multiply (1/2𝑙)𝑒^{−𝑖𝑚𝜋𝑡/𝑙}(𝑚=0,±1,±2,…) on both sides of (3.10) and integrate it from −𝑙 to 𝑙, hence ^∞_𝑘=−∞[(𝑖𝑘𝜋𝑙)^𝛼𝐻−𝐴−𝐵𝑒^{−𝑖𝑘𝜋𝜏1/𝑙}−𝐶𝑒^{𝑖𝑘𝜋𝜏2/𝑙}]𝐶_𝑘12𝑙^𝑙_−𝑙𝑒^{𝑖(𝑘−𝑚)𝜋𝑡/𝑙}𝑑𝑡=0. (3.11) It is easy to deduce that 12𝑙^𝑙_−𝑙𝑒^{𝑖(𝑘−𝑚)𝜋𝑡/𝑙}𝑑𝑡=1,𝑘=𝑚,0,𝑘≠𝑚, (3.12) recalling (3.10), it reduces to [(𝑖𝑘𝜋𝑙)^𝛼𝐻−𝐴−𝐵𝑒^{−𝑖𝑘𝜋𝜏1/𝑙}−𝐶𝑒^{𝑖𝑘𝜋𝜏2/𝑙}]𝐶_𝑚=0(𝑚=0,±1,±2,…). (3.13) Thus, if there are no pure imaginary roots in (3.1), then for every 𝑘 we have 𝐶_𝑘=0, according to (3.8), we conclude that 𝑥(𝑡)=constan𝑡 vector which conflicts the suppose that 𝑥(𝑡) 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 𝐻𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−𝜏₁)+𝐶𝑥(𝑡+𝜏₂), (4.1) where 𝛼>0,𝐻=(¹⁰₀₀),𝐴=(^𝑎11𝑎12_𝑎21𝑎22),𝐵=(^𝑏11𝑏12_𝑏21𝑏22),𝐶=(^𝑐11𝑐12_𝑐21𝑐22),𝑥(𝑡)=(^𝑥1_𝑥^(𝑡)_2(𝑡)), and𝑥₁(𝑡),𝑥₂(𝑡) 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 𝑥(𝑡)∈Ω(𝑡), then the conclusion is sufficient and necessary |𝐴|+|𝐵|cos(2𝜏₁𝑦)+|𝐶|cos(2𝜏₂𝑦)+𝐸cos(𝜏₁𝑦)+𝐹cos(𝜏₂𝑦)+𝐺cos((𝜏₂−𝜏₁)𝑦)−𝑏₂₂|𝑦|^𝛼cos(−𝜏₁𝜋𝑦+2𝛼)−𝑐₂₂|𝑦|^𝛼cos(𝜏₂𝜋𝑦+2𝛼)−𝑎₂₂|𝑦|^𝛼𝜋cos(2𝛼)=0−|𝐵|sin(2𝜏₁𝑦)+|𝐶|sin(2𝜏₂𝑦)−𝐸sin(𝜏₁𝑦)+𝐹sin(𝜏₂𝑦)+𝐺sin((𝜏₂−𝜏₁)𝑦)−𝑏₂₂|𝑦|^𝛼sin(−𝜏₁𝜋𝑦+2𝛼)−𝑐₂₂|𝑦|^𝛼sin(𝜏₂𝜋𝑦+2𝛼)−𝑎₂₂|𝑦|^𝛼𝜋sin(2𝛼)=0, (4.2) or |𝐴|+|𝐵|cos(2𝜏₁𝑦)+|𝐶|cos(2𝜏₂𝑦)+𝐸cos(𝜏₁𝑦)+𝐹cos(𝜏₂𝑦)+𝐺cos((𝜏₂−𝜏₁)𝑦)−𝑏₂₂|𝑦|^𝛼cos(−𝜏₁𝜋𝑦−2𝛼)−𝑐₂₂|𝑦|^𝛼cos(𝜏₂𝜋𝑦−2𝛼)−𝑎₂₂|𝑦|^𝛼𝜋cos(−2𝛼)=0−|𝐵|sin(2𝜏₁𝑦)+|𝐶|sin(2𝜏₂𝑦)−𝐸sin(𝜏₁𝑦)+𝐹sin(𝜏₂𝑦)+𝐺sin((𝜏₂−𝜏₁)𝑦)−𝑏₂₂|𝑦|^𝛼sin(−𝜏₁𝜋𝑦−2𝛼)−𝑐₂₂|𝑦|^𝛼sin(𝜏₂𝜋𝑦−2𝛼)−𝑎₂₂|𝑦|^𝛼𝜋sin(−2𝛼)=0, (4.3) where 𝐸=|^𝑎11𝑎12_𝑏21𝑏22|+|^𝑏11𝑏12_𝑎21𝑎22|,  𝐹=|^𝑎11𝑎12_𝑐21𝑐22|+|^𝑐11𝑐12_𝑎21𝑎22|,  𝐺=|^𝑏11𝑏12_𝑐21𝑐22|+|^𝑐11𝑐12_𝑏21𝑏22|.

Proof. First of all we know that (𝑦𝑖)^𝛼=𝑦^𝛼𝑒^{𝜋𝛼𝑖/2},𝑦≥0,|𝑦|^𝛼𝑒^{−𝜋𝛼𝑖/2},𝑦<0, (4.4) according to Theorem 3.1, we have ℎ(𝑦𝑖)=|𝑦𝑖𝐼−𝐴−𝐵𝑒^{−𝜏1𝑦𝑖}−𝐶𝑒^{𝜏2𝑦𝑖}|=(𝑦𝑖)^𝛼−𝑎₁₁−𝑏₁₁𝑒^{−𝜏1𝑦𝑖}−𝑐₁₁𝑒^{𝜏2𝑦𝑖}−𝑎₁₂−𝑏₁₂𝑒^{−𝜏1𝑦𝑖}−𝑐₁₂𝑒^{𝜏2𝑦𝑖}−𝑎₂₁−𝑏₂₁𝑒^{−𝜏1𝑦𝑖}−𝑐₂₁𝑒^{𝜏2𝑦𝑖}−𝑎₂₂−𝑏₂₂𝑒^{−𝜏1𝑦𝑖}−𝑐₂₂𝑒^{𝜏2𝑦𝑖}=|𝐴|+|𝐵|𝑒^{−𝜏1𝑦𝑖}+|𝐶|𝑒^{2𝜏2𝑦𝑖}+|𝐸|𝑒^{−𝜏1𝑦𝑖}+𝐹𝑒^{𝜏2𝑦𝑖}+𝐺𝑒^{(𝜏2−𝜏1)𝑦𝑖}−𝑏₂₂(𝑦𝑖)^𝛼𝑒^{−𝜏1𝑦𝑖}−𝑐₂₂(𝑦𝑖)^𝛼𝑒^{𝜏2𝑦𝑖}−𝑎₂₂(𝑦𝑖)^𝛼. (4.5) then ℜ[ℎ(𝑦𝑖)]=|𝐴|+|𝐵|cos(2𝜏₁𝑦)+|𝐶|cos(2𝜏₂𝑦)+𝐸cos(𝜏₁𝑦)+𝐹cos(𝜏₂𝑦)+𝐺cos((𝜏₂−𝜏₁)𝑦)−𝑏₂₂|𝑦|^𝛼cos(−𝜏₁𝜋𝑦±2𝛼)−𝑐₂₂|𝑦|^𝛼cos(𝜏₂𝜋𝑦±2𝛼)−𝑎₂₂|𝑦|^𝛼𝜋cos(±2𝛼)=0,ℑ[ℎ(𝑦𝑖)]=−|𝐵|sin(2𝜏₁𝑦)+|𝐶|sin(2𝜏₂𝑦)−𝐸sin(𝜏₁𝑦)+𝐹sin(𝜏₂𝑦)+𝐺sin((𝜏₂−𝜏₁)𝑦)−𝑏₂₂|𝑦|^𝛼sin(−𝜏₁𝜋𝑦±2𝛼)−𝑐₂₂|𝑦|^𝛼sin(𝜏₂𝜋𝑦±2𝛼)−𝑎₂₂|𝑦|^𝛼𝜋sin(±2𝛼)=0, (4.6) where ℜ[𝑧] and ℑ[𝑧] denote the real and imaginary parts of 𝑧, respectively. Then using Theorem 3.1, if there exists 𝑦∈𝑅,𝑦≠0,ℎ(𝑦𝑖)=0, we obtain that ℜ[ℎ(𝑦𝑖)]=0,ℑ[ℎ(𝑦𝑖)]=0, (4.7) 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: 𝐷^𝛼𝑥₁(𝑡)=−𝑥₂(𝑡)+𝑥₁(𝑡−1)+𝑥₂(𝑡−1)+𝑥₂(𝑡+1),0=𝑥₁(𝑡)+𝑥₁(𝑡−1)+𝑥₁(𝑡+1), (5.1) where 𝐻=(¹⁰₀₀),𝐴=(⁰⁻¹₁₀),𝐵=(¹¹₁₀),𝐶=(⁰¹₁₀),𝜏₁=1,𝜏₂=1,𝛼>1. so we have |𝐴|=1,|𝐵|=−1,|𝐶|=−1,𝐸=0,𝐹=0,𝐺=−2.
Using the discriminant of Theorem 4.1, we have 2cos(2𝑦)=−1,0=0, (5.2) the solution is𝑦=𝑘𝜋+𝜋/3(𝑘=0,±1,±2,…).
According to Theorem 4.1 , system (5.1) has non-constant periodic solution. We suppose that𝑦=𝜋/3, as ((𝑦𝑖)^𝛼𝐼−𝐴−𝐵𝑒^−𝑦𝑖−𝐶𝑒^𝑦𝑖)𝐾=0 (5.3) exists non-zero solution, it means that 𝜋[(3)^𝛼cos(𝜋𝛼2√)−32𝜋]+[(3)^𝛼sin(𝜋𝛼2√)+32]𝑖0𝑘00𝑘¹₂=00. (5.4) So we have 𝑘₁=0,  𝑘₂ for any real number. Supposed that 𝑘₂=1, then we obtained a non-constant periodic solution of system (5.1): 𝑥(𝑡)=𝑒^{(𝜋/3)𝑖𝑡}01. (5.5)
We can verify that 𝑥(𝑡) 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: 𝐷^1/2𝑥₁(𝑡)=−𝑥₂(𝑡)+𝑥₁(𝑡−1)+𝑥₂(𝑡+1),0=𝑥₁(𝑡)+𝑥₂(𝑡)+𝑥₁(𝑡−1)+𝑥₂(𝑡−1)+𝑥₁(𝑡+1)+𝑥₂(𝑡+1). (5.6) where 𝐻=(¹⁰₀₀),𝐴=(⁰⁻¹₁₁),𝐵=(⁰¹₁₁),𝐶=(⁰¹₁₁),𝜏₁=1,𝜏₂=1,𝛼=1/2.
So we have |𝐴|=1,|𝐵|=−1,|𝐶|=−1,𝐸=0,𝐹=0,𝐺=−2.
Using the discriminant of Theorem 4.1, we obtained (2cos𝑦+1)|𝑦|^1/2𝜋cos4+2cos2𝑦+1=0,(2cos𝑦+1)|𝑦|^1/2𝜋sin4=2sin2𝑦. (5.7) Through the simplication of this equation, we have √2sin(2𝑦+(𝜋/4))=−1/2, (5.8) and the solution is 𝑦=𝑘𝜋+(1/2)arcsin(−1/2√2)−𝜋/8,𝑘=(0,±1,±2,…).
According to Theorem 4.1, there exists non-constant periodic solution in the system.