Analysis of a System for Linear Fractional Differential Equations

Abstract

The main purpose of this paper is to obtain the unique solution of the constant coefficient homogeneous linear fractional differential equations 𝐷^𝑞𝑡₀𝑋(𝑡)=𝑃𝑋(𝑡),𝑋(𝑎)=𝐵 and the constant coefficient nonhomogeneous linear fractional differential equations 𝐷^𝑞𝑡₀𝑋(𝑡)=𝑃𝑋(𝑡)+𝐷,𝑋(𝑎)=𝐵 if 𝑃 is a diagonal matrix and 𝑋(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇] and prove the existence and uniqueness of these two kinds of equations for any 𝑃∈𝐿(𝑅^𝑚) and 𝑋(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇]. Then we give two examples to demonstrate the main results.

1. Introduction

System of fractional differential equations has gained a lot of interest because of the challenges it offers compared to the study of system of ordinary differential equations. Numerous applications of this system in different areas of physics, engineering, and biological sciences have been presented in [1–3]. The differential equations involving the Riemman-Liouville differential operators of fractional order 0<𝑞<1 appear to be more important in modeling several physical phenomena and therefore seem to deserve an independent study of their theory parallel to the well-known theory of ordinary differential equations. The existence and uniqueness of solution for fractional differential equations with any 𝑋(𝑡)∈𝐶[𝑡₀,𝑇]×𝐶[𝑡₀,𝑇]×⋯×𝐶[𝑡₀,𝑇] have been studied in many papers, see [4–28]. In [4] Daftradar-Gejji and Babakhani have studied the existence and uniqueness of 𝐷^𝑞0𝑋(𝑡)−𝑋₀=𝑃𝑋(𝑡),(1.1) where 𝐷^𝑞0 denotes the standard Riemman-Liouville fractional derivative, 0<𝑞<1,𝑋(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑚(𝑡))^𝑇, 𝑋(0)=𝑋₀=(𝑥₁₀,𝑥₂₀,…,𝑥_𝑚0)^𝑇, 𝑃∈𝐿(𝑅^𝑚) which is an m dimensional linear space. They have obtained that the system (1.1) has a unique solution defined on [0,𝑇] if 𝑃∈𝐿(𝑅^𝑚) and 𝑋(𝑡)∈𝐶[𝑡₀,𝑇]×𝐶[𝑡₀,𝑇]×⋯×𝐶[𝑡₀,𝑇]. In [17] Belmekki et al. have studied the existence of periodic solution for some linear fractional differential equation in 𝐶_1−𝑞[0,1]. In [21] Ahmad and Nieto have studied the Riemann-Liouwille fractional differential equations with fractional boundary conditions. In comparison with the earlier results of this type we get more general assumptions. We assume 𝑋(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇] instead of 𝑋(𝑡)∈𝐶[𝑡₀,𝑇]×𝐶[𝑡₀,𝑇]×⋯×𝐶[𝑡₀,𝑇] and consider the following system of fractional differential equations: 𝐷𝐷^𝑞𝑡₀𝑋(𝑡)=𝑃𝑋(𝑡),𝑋(𝑎)=𝐵,_𝑞𝑡0𝑋(𝑡)=𝑃𝑋(𝑡)+𝐷,𝑋(𝑎)=𝐵,(1.2) where 𝐷^𝑞0 denotes the standard Riemman-Liouville fractional derivative, 0<𝑞<1, 𝑃∈𝐿(𝑅^𝑚), 𝑋(𝑡)∈𝐶_1−𝑞𝑡₀,𝑇×𝐶_1−𝑞𝑡₀,𝑇×⋯×𝐶_1−𝑞𝑡₀,𝑇,(1.3)(𝑎,𝐵)∈(𝑡₀,𝑇]×𝑅^𝑚 and 𝐷 is a constant vector. We completely generalize the results in [4] and obtain the new results if 𝑋(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇]. Furthermore, we also obtain some results of the unique solution of the homogeneous and nonhomogeneous initial value problems with the classical Mittag-Leffler special function [5] which is similar to the ordinary differential equations. Now we introduce the first Mittag-Leffler function 𝑒_𝑞(𝑡−𝑡₀) defined by 𝑒_𝑞𝑡−𝑡₀=^+∞_𝑘=1𝑡−𝑡₀^𝑘𝑞−1Γ(𝑘𝑞).(1.4) The function 𝑒_𝑞(𝑡−𝑡₀) belongs to 𝐶_1−𝑞[𝑡₀,𝑇]. Indeed, taking the norm in 𝐶_1−𝑞[𝑡₀,𝑇], we have ‖‖𝑒_𝑞𝑡−𝑡₀‖‖_1−𝑞≤^+∞_𝑘=1𝑇−𝑡₀^{(𝑘−1)𝑞}Γ(𝑘𝑞)<+∞.(1.5) The formula remains valid for 𝑞→1⁻. In this case, 𝑒₁(𝑡−𝑡₀)=exp(𝑡−𝑡₀). Then we introduce the second Mittag-Leffler function 𝐸_𝑞(𝑡−𝑡₀) defined by 𝐸_𝑞𝑡−𝑡₀=^+∞_𝑘=1𝑡−𝑡₀^𝑘𝑞Γ(𝑘𝑞+1).(1.6) The formula remains also valid for 𝑞→1⁻. In this case, 𝐸₁(𝑡−𝑡₀)=exp(𝑡−𝑡₀)−1.

The paper is organized as follows. In Section 2 we recall the definitions of fractional integral and derivative and related basic properties and preliminary results used in the text. In Section 3 we obtain the unique solution of the constant coefficient homogeneous and nonhomogeneous linear fractional differential equations for 𝑃 being the diagonal matrix. In Section 4 we prove the existence and uniqueness of these two kinds of equations for any 𝑃∈𝐿(𝑅^𝑚). In Section 5 we give some specific examples to illustrate the results.

2. Definitions and Preliminary Results

Let us denote by 𝐶[𝑡₀,𝑇] the space of all continuous real functions defined on [𝑡₀,𝑇], which turns out to be a Banach space with the norm ‖𝑥‖=max_{𝑡∈[𝑡0,𝑇]}||𝑥(𝑡)||.(2.1)

We define similarly another Banach space 𝐶_1−𝑞[𝑡₀,𝑇], in which function 𝑥(𝑡) is continuous on (𝑡₀,𝑇] and (𝑡−𝑡₀)^1−𝑞𝑥(𝑡) is continuous on [𝑡₀,𝑇] with the norm: ‖𝑥‖_1−𝑞=max_{𝑡∈[𝑡0,𝑇]}𝑡−𝑡₀^1−𝑞||𝑥(𝑡)||.(2.2)

𝐿[𝑡₀,𝑇] is the space of real functions defined on [𝑡₀,𝑇] which are Lebesgue integrable on [𝑡₀,𝑇].

Obviously 𝐶_1−𝑞[𝑡₀,𝑇]⊂𝐿(𝑡₀,𝑇).

The definitions and results of the fractional calculus reported below are not exhaustive but rather oriented to the subject of this paper. For the proofs, which are omitted, we refer the reader to [6] or other texts on basic fractional calculus.

Definition 2.1 (see [6]). The fractional primitive of order 𝑞>0 of function 𝑥(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇] is given by 𝐼^𝑞𝑡₀𝑥(𝑡)=1Γ(𝑞)^𝑡𝑡₀(𝑡−𝑠)^𝑞−1𝑥(𝑠)𝑑𝑠.(2.3)
From [17] we know 𝐼^𝑞𝑡₀𝑥(𝑡) exists for all 𝑞>0, when 𝑥∈𝐶_1−𝑞[𝑡₀,𝑇]; consider also that when 𝑥∈𝐶[𝑡₀,𝑇] then 𝐼^𝑞𝑡₀𝑥(𝑡)∈𝐶[𝑡₀,𝑇] and moreover 𝐼^𝑞𝑡₀𝑥𝑡₀=0.(2.4)

Definition 2.2 (see [6]). The fractional derivative of order 0<𝑞<1 of a function 𝑥(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇] is given by 𝐷^𝑞𝑡₀𝑥(𝑡)=1Γ(1−𝑞)𝑑𝑑𝑥^𝑡𝑡₀(𝑡−𝑠)^−𝑞𝑥(𝑠)𝑑𝑠.(2.5)
We have 𝐷^𝑞𝑡₀𝐼^𝑞𝑡₀𝑥(𝑡)=𝑥(𝑡) for all 𝑥(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇].

Lemma 2.3 (see [6]). Let 0<𝑞<1. If one assumes 𝑥(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇], then the fractional differential equation 𝐷^𝑞𝑡₀𝑥(𝑡)=0(2.6) has 𝑥(𝑡)=𝑐(𝑡−𝑡₀)^𝑞−1,𝑐∈𝑅, as solutions.

From this lemma we can obtain the following law of composition.

Lemma 2.4 (see [6]). Assume that 𝑥(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇] with a fractional derivative of order 0<𝑞<1 that belongs to 𝐶_1−𝑞[𝑡₀,𝑇]. Then 𝐼^𝑞𝑡₀𝐷^𝑞𝑡₀𝑥(𝑡)=𝑥(𝑡)+𝑐𝑡−𝑡₀^𝑞−1,(2.7) for some 𝑐∈𝑅. When the function 𝑥 is in 𝐶[𝑡₀,𝑇], then 𝑐=0.

Lemma 2.5 (see [6]). Let 𝑈 be a nonempty closed subset of a Banach space 𝐸, and let 𝛼_𝑛≥0 for every 𝑛 and such ∑^∞𝑛=0𝛼_𝑛 converges. Moreover, let the mapping 𝐴∶𝑈→𝑈 satisfy the inequality ‖𝐴^𝑛𝑢−𝐴^𝑛𝑣‖≤𝛼_𝑛‖𝑢−𝑣‖,(2.8) for every 𝑛∈𝑁 and any 𝑢,𝑣∈𝑈. Then, 𝐴 has a uniquely defined fixed point 𝑢^∗. Furthermore, for any 𝑢₀∈𝑈, the sequence (𝐴^𝑛𝑢₀)^∞𝑛=1 converges to this fixed point 𝑢^∗.

Lemma 2.6 (see [12]). Let 𝑃∈𝐿(𝑅^𝑚) and have real eigenvalues 𝜆₁,𝜆₂,…,𝜆_𝑟. Then there exists a basis of 𝑅^𝑚 in which the matrix representation of 𝑃 assumes Jordan form, that is, the matrix of 𝑃 is made of diagonal blocks of the form diag(𝐽₁,𝐽₂,…,𝐽_𝑟), where each 𝐽_𝑖 consists of diagonal blocks of the form ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜆1_𝑖𝜆0_𝑖⋯00⋮00⋮10⋱⋯⋯⋯⋮001𝜆⋮00_𝑖⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(2.9)

Lemma 2.7 (see [12]). Let 𝑃∈𝐿(𝑅^𝑚) and have complex eigenvalues 𝜇_𝑗=𝛼_𝑗+𝑖𝛽_𝑗,𝑗=1,2,…,𝑟, with multiplicity. Then there exists a basis of 𝑅^𝑚, where 𝑃 has matrix form diag(𝐽₁,𝐽₂,…,𝐽_𝑟), where each 𝐽_𝑖 consists of diagonal blocks of the type ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐼𝐷0⋯00₂0𝐼𝐷₂⋯00⋮0⋮0⋱⋯⋯𝐼⋮0₂⋮0𝐷⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝐷=⎛⎜⎜⎝𝛼^𝑖−𝛽_𝑖𝛽_𝑖𝛼_𝑖⎞⎟⎟⎠,𝐼₂=⎛⎜⎜⎝1001⎞⎟⎟⎠.(2.10)

Lemma 2.8 (see [12]). Let 𝑃∈𝐿(𝑅^𝑚). Then 𝑅^𝑚 has a basis giving 𝑃 a matrix representation composed of diagonal blocks of type 𝐽_𝑖 and/or matrices 𝐽_𝑖, where 𝐽_𝑖 and 𝐽_𝑖 are as defined in the preceding lemmas.

Now, we will introduce Lemma 2.9 to prove the following Theorem 4.4 in Section 4.

Lemma 2.9. Let 0<𝑞<1. Assume that 𝑥(𝑡) and 𝑓(𝑡) belong to 𝐶_1−𝑞[𝑡₀,𝑇]. Then For the initial value problem 𝐷^𝑞𝑡₀𝑥(𝑡)=𝜆𝑥(𝑡)+𝑓(𝑡),𝑥(𝑎)=𝑏(2.11) has a unique solution 𝑥(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇] provided 𝑡₀<𝑎<𝑎₀, where 𝑎₀ is a suitable constant depending on 𝑡₀, 𝑞, and 𝜆.

Proof. The initial value problem (2.11) will be solved in two steps.  (1) Local existence. Our problem is equivalent to the problem of determination of fixed points of the following operator: 𝐴𝑥(𝑡)=𝑐𝑡−𝑡₀^𝑞−1+1Γ(𝑞)^𝑡𝑡₀(𝑡−𝑠)^𝑞−1(𝜆𝑥(𝑠)+𝑓(𝑠))𝑑𝑠,(2.12) with 𝑐=𝑏−1Γ(𝑞)^𝑎𝑡₀(𝑎−𝑠)^𝑞−1(𝜆𝑥(𝑠)+𝑓(𝑠))𝑑𝑠𝑎−𝑡₀^1−𝑞.(2.13) It is immediate to verify that 𝐴∶𝐶_1−𝑞[𝑡₀,𝑇]→𝐶_1−𝑞[𝑡₀,𝑇] is also well defined. Indeed, |||𝑡−𝑡₀≤||𝜆||||||^1−𝑞1𝐴𝑥(𝑡)|||Γ(𝑞)^𝑡𝑡₀+||||1(𝑡−𝑠)^𝑞−1𝑠−𝑡₀^𝑞−1𝑠−𝑡₀^1−𝑞𝑥(𝑠)𝑑𝑠||||Γ(𝑞)^𝑡𝑡₀≤|𝑐|+||𝜆||‖𝑥(‖𝑡−_1−𝑞𝑠)^𝑞−1𝑠−𝑡₀^𝑞−1𝑠−𝑡₀^1−𝑞𝑓(𝑠)𝑑𝑠|||||||𝐼_𝑞𝑡−𝑡₀^𝑞−1|||+‖𝑓‖_1−𝑞|||𝐼_𝑞𝑡−𝑡₀^𝑞−1≤|𝑐|+||𝜆||‖𝑥‖_1−𝑞|||Γ(𝑞)Γ(2𝑞)𝑡−𝑡⁰^𝑞+‖𝑓‖_1−𝑞Γ(𝑞)Γ(2𝑞)𝑡−𝑡⁰^𝑞,(2.14) for 𝑥(𝑡) and 𝑓(𝑡) belong to ∈𝐶_1−𝑞[𝑡₀,𝑇]. Then we can also prove 𝐴 is a contraction operator. Indeed, 𝑡−𝑡₀^1−𝑞||𝐴𝑥(𝑡)−𝐴𝑦(𝑡)||≤||𝜆||ΓΓ((2𝑞𝑞))𝑎−𝑡⁰^𝑞‖𝑥−𝑦‖_1−𝑞+||𝜆||ΓΓ((2𝑞𝑞))𝑡−𝑡⁰^𝑞‖𝑥−𝑦‖_1−𝑞≤||𝜆||Γ(𝑞)Γ(2𝑞)𝑎−𝑡⁰^𝑞‖𝑥−𝑦‖_1−𝑞+||𝜆||Γ(𝑞)Γ(2𝑞)𝑇−𝑡⁰^𝑞‖𝑥−𝑦‖_1−𝑞,(2.15) for all 𝑥(𝑡),𝑦(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]. Let us assume ||𝜆||Γ(𝑞)Γ(2𝑞)𝑎−𝑡⁰^𝑞<12,(2.16) that is, 𝑎<𝑎₀=Γ(2𝑞)2||𝜆||Γ(𝑞)^1/𝑞+𝑡₀.(2.17) Taking 𝑇−𝑎>0 sufficiently small, we also have ||𝜆||Γ(𝑞)Γ(2𝑞)𝑇−𝑡⁰^𝑞<12,(2.18) and then ‖𝐴𝑥(𝑡)−𝐴𝑦(𝑡)‖_1−𝑞≤𝐿‖𝑥−𝑦‖_1−𝑞,(2.19) with 𝐿<1. Therefore 𝐴 is a contraction operator. This shows that initial problem (2.11) has a unique solution.  (2) Continuation of solution. Since we know the value of 𝑥(𝑡) on (𝑡₀,𝑎], then we can compute 𝑐_∗=𝑏−1Γ(𝑞)^𝑎𝑡₀(𝑎−𝑠)^𝑞−1(𝜆𝑥(𝑠)+𝑓(𝑠))𝑑𝑠𝑎−𝑡₀^1−𝑞.(2.20) We can solve the integral problem 𝑦(𝑡)=𝑐_∗𝑡−𝑡₀^𝑞−1+1Γ(𝑞)^𝑡𝑡₀(𝑡−𝑠)^𝑞−1(𝜆𝑦(𝑠)+𝑓(𝑠))𝑑𝑠,(2.21) obtaining a unique solution 𝑦(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇] for all 𝑇>𝑡₀. Now 𝑥(𝑡) and 𝑦(𝑡) agree on (𝑡₀,𝑎]. Thus the solution admits 𝑦(𝑡) as its continuation. Hence the proof of Lemma 2.9 is complete.

3. Initial Value Problem: Continuous Solutions on (𝑡₀,𝑇]

We open this section with some basic examples, concerning the case when the solutions in 𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇] are submitted to an initial condition.

Theorem 3.1. Let 0<𝑞<1. For all (𝑎,𝐵)∈(𝑡₀,𝑇]×𝑅^𝑚 the initial value problem 𝐷^𝑞𝑡₀𝑋(𝑡)=0,𝑋(𝑎)=𝐵(3.1) admits 𝑋(𝑡)=𝐵𝑎−𝑡₀^1−𝑞𝑡−𝑡₀^𝑞−1,(3.2) as unique solution in 𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇].

Proof. According to Lemma 2.4, the initial value problem (3.1) is equivalent to the following equations: 𝑋(𝑡)=𝐶𝑡−𝑡₀^𝑞−1,𝐶=𝐵𝑎−𝑡₀^1−𝑞.(3.3) Hence the proof of Theorem 3.1 is complete.

Theorem 3.2. Let 0<𝑞<1. Assume 𝐹(𝑡)=𝑓₁(𝑡),𝑓₂(𝑡),…,𝑓_𝑚(𝑡)^𝑇∈𝐶_1−𝑞𝑡₀,𝑇×𝐶_1−𝑞𝑡₀,𝑇×⋯×𝐶_1−𝑞𝑡₀,𝑇.(3.4) Then for all (𝑎,𝐵)∈(𝑡₀,𝑇]×𝑅^𝑚 the initial value problem 𝐷^𝑞𝑡₀𝑋(𝑡)=𝐹(𝑡),𝑋(𝑎)=𝐵(3.5) has a unique solution in 𝐶_1−𝑞𝑡₀,𝑇×𝐶_1−𝑞𝑡₀,𝑇×⋯×𝐶_1−𝑞𝑡₀,𝑇,(3.6) given by 𝑋(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑚(𝑡)^𝑇,(3.7) with 𝑥_𝑖(𝑡)=𝑏_𝑖−1Γ(𝑞)^𝑎𝑡₀(𝑎−𝑠)^𝑞−1𝑓_𝑖(𝑠)𝑑𝑠𝑎−𝑡₀^1−𝑞𝑡−𝑡₀^𝑞−1+1Γ(𝑞)^𝑡𝑡₀(𝑖=(1𝑡,−2,𝑠…),^𝑞−1𝑚𝑓_𝑖)(.𝑠)𝑑𝑠,(3.8)

Proof. According to Lemma 2.4, the initial value problem (3.5) is equivalent to the following equations: 𝑋(𝑡)=𝐶𝑡−𝑡₀^𝑞−1+1Γ(𝑞)^𝑡𝑡₀𝐶=𝐵−1(𝑡−𝑠)^𝑞−1𝐹(𝑠)𝑑𝑠,Γ(𝑞)^𝑎𝑡₀(𝑎−𝑠)^𝑞−1𝐹(𝑠)𝑑𝑠𝑎−𝑡₀^1−𝑞.(3.9) Hence the proof of Theorem 3.2 is complete.

The result remains true even if 𝑞→1⁻. In this case, (3.5) is reduced to the ordinary differential equations 𝑋^(𝑡)=𝐹(𝑡),𝑋(𝑎)=𝐵,(3.10) which have a unique solution in 𝐶𝑡₀,𝑇×𝐶𝑡₀,𝑇×⋯×𝐶𝑡₀,𝑇,(3.11) given by 𝑋(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑚(𝑡)^𝑇,(3.12) with 𝑥_𝑖(𝑡)=𝑏_𝑖−^𝑎𝑡₀𝑓_𝑖(𝑠)𝑑𝑠+^𝑡𝑡₀𝑓_𝑖(𝑠)𝑑𝑠,(𝑖=1,2,…,𝑚).(3.13)

Theorem 3.3. Let 0<𝑞<1. For all (𝑎,𝐵)∈(𝑡₀,𝑇]×𝑅^𝑚 the initial value problem 𝐷^𝑞𝑡₀𝑋(𝑡)=𝑃𝑋(𝑡),𝑋(𝑎)=𝐵,(3.14) where 𝑃=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜆¹0𝜆0₂⋯00⋮00⋮00⋱⋯⋯⋯⋮000𝜆⋮00_𝑚⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠(3.15) has a unique solution in 𝐶_1−𝑞𝑡₀,𝑇×𝐶_1−𝑞𝑡₀,𝑇×⋯×𝐶_1−𝑞𝑡₀,𝑇,(3.16) given by 𝑋(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑚(𝑡)^𝑇,(3.17) with 𝑥_𝑖(𝑡)=𝑏_𝑖𝑒^𝑞−1𝜆^𝑖1/𝑞𝑎−𝑡₀𝑒_𝑞𝜆^𝑖1/𝑞𝑡−𝑡₀,(𝑖=1,2,…,𝑚).(3.18)

Proof. We can write (3.27) in the following form: 𝐷𝐷^𝑞𝑡₀𝑥₁(𝑡)=𝜆₁𝑥₁(𝑡),_𝑞𝑡0𝐷𝑥₂(𝑡)=⋮𝜆₂𝑥₂(𝑡),_𝑞𝑡0𝑥𝑥_𝑚(𝑡)=𝜆_𝑚𝑥_𝑚(𝑡),₁(𝑎)=𝑏₁,…,𝑥_𝑚(𝑎)=𝑏_𝑚.(3.19) According to Lemma 2.4,𝐷^𝑞𝑡₀𝑥_𝑖(𝑡)=𝜆_𝑖𝑥_𝑖(𝑡)(3.20) is equivalent to the following equations: 𝑥_𝑖(𝑡)=𝑐_𝑖𝑡−𝑡₀^𝑞−1+𝐼^𝑞𝑡₀𝜆_𝑖𝑥_𝑖(𝑡),(3.21) for some 𝑐_𝑖∈𝑅. From (3.21) we obtain, by iteration, 𝑥_𝑖(𝑡)=𝑐_𝑖Γ(𝑞)𝑡−𝑡⁰^𝑞−1Γ(𝑞)+𝜆^𝑖𝑡−𝑡₀^2𝑞−1Γ(2𝑞)+⋯+𝜆^𝑖𝑛−1𝑡−𝑡₀^𝑛𝑞−1Γ(𝑛𝑞)+𝜆^𝑛𝑖𝐼^𝑡𝑛𝑞₀𝑥_𝑖(𝑡).(3.22) Letting 𝑛→+∞, ‖𝜆^𝑛𝑖𝐼^𝑡𝑛𝑞₀𝑥_𝑖(𝑡)‖_1−𝑞→0 if 𝑥_𝑖(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]. Indeed, ‖‖𝜆^𝑛𝑖𝐼^𝑡𝑛𝑞₀𝑥_𝑖(𝑡)‖‖_1−𝑞≤||𝜆^𝑛𝑖||‖‖𝑥_𝑖(𝑡)‖‖_1−𝑞Γ(𝑞)Γ((𝑛+1)𝑞)𝑡−𝑡⁰^𝑛𝑞.(3.23)
On the other hand, 𝑒_𝑞𝑡−𝑡₀=^+∞_𝑘=1𝑡−𝑡₀^𝑘𝑞−1Γ(𝑘𝑞),(3.24) then we can obtain 𝑥_𝑖(𝑡)=𝑐_𝑖Γ(𝑞)𝜆^{𝑖(1/𝑞−1)}𝑒_𝑞𝜆^𝑖1/𝑞𝑡−𝑡₀.(3.25) Since 𝑥_𝑖(𝑎)=𝑏_𝑖, 𝑥_𝑖(𝑡)=𝑏_𝑖𝑒^𝑞−1𝜆^𝑖1/𝑞𝑎−𝑡₀𝑒_𝑞𝜆^𝑖1/𝑞𝑡−𝑡₀,(𝑖=1,2,…,𝑚).(3.26) Hence the proof of Theorem 3.3 is complete.
The result remains valid even if 𝑞→1⁻. In this case, 𝑋^(𝑡)=𝑃𝑋(𝑡),𝑋(𝑎)=𝐵(3.27) has a unique solution in 𝐶𝑡₀,𝑇×𝐶𝑡₀,𝑇×⋯×𝐶𝑡₀,𝑇,(3.28) given by 𝑋(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑚(𝑡)^𝑇,(3.29) with 𝑥_𝑖(𝑡)=𝑏_𝑖exp−𝜆_𝑖𝑎−𝑡₀exp𝜆_𝑖𝑡−𝑡₀,(𝑖=1,2,…,𝑚).(3.30)

Theorem 3.4. Let 0<𝑞<1. For all (𝑎,𝐵)∈(𝑡₀,𝑇]×𝑅^𝑚 the initial value problem 𝐷^𝑞𝑡₀𝑋(𝑡)=𝑃𝑋(𝑡)+𝐷,𝑋(𝑎)=𝐵,(3.31) where 𝑃=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜆¹0𝜆0₂⋯00⋮00⋮00⋱⋯⋯⋯⋮000𝜆⋮00_𝑚⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.32) and 𝐷=(𝑑₁,𝑑₂,…,𝑑_𝑚) has a unique solution in 𝐶_1−𝑞𝑡₀,𝑇×𝐶_1−𝑞𝑡₀,𝑇×⋯×𝐶_1−𝑞𝑡₀,𝑇,(3.33) given by 𝑋(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑚(𝑡)^𝑇,(3.34) with 𝑥_𝑖(𝑡)=𝑏_𝑖−𝑑_𝑖𝜆^𝑖−1𝐸_𝑞𝜆^𝑖1/𝑞𝑎−𝑡₀𝑒^𝑞−1𝜆^𝑖1/𝑞𝑎−𝑡₀𝑒_𝑞𝜆^𝑖1/𝑞𝑡−𝑡₀+𝑑_𝑖𝜆(^𝑖−1𝑖𝐸=_𝑞1𝜆,2,^𝑖1/𝑞…𝑡,−𝑚𝑡)₀.(3.35)

Proof. We can write (3.44) in the following form:𝐷𝐷^𝑞𝑡_0𝑞𝑡0𝑥₁(𝑡)=𝜆₁𝑥₁(𝑡)+𝑑₁,𝐷_𝑞𝑡0𝑥₂(𝑡)=⋮𝜆₂𝑥₂(𝑡)+𝑑₂,𝑥_𝑚(𝑡)=𝜆_𝑚𝑥_𝑚(𝑡)+𝑑_𝑚𝑥₁(𝑎)=𝑏₁,…,𝑥_𝑚(𝑎)=𝑏_𝑚.(3.36)
According to Lemma 2.4, the equation 𝐷^𝑞𝑡₀𝑥_𝑖(𝑡)=𝜆_𝑖𝑥_𝑖(𝑡)+𝑑_𝑖(3.37) is equivalent to the following equations: 𝑥_𝑖(𝑡)=𝑐_𝑖𝑡−𝑡₀^𝑞−1+𝐼^𝑞𝑡₀𝜆_𝑖𝑥_𝑖(𝑡)+𝐼^𝑞𝑡₀𝑑_𝑖,(3.38) for some 𝑐_𝑖∈𝑅. From (3.38) we obtain, by iteration, 𝑥_𝑖𝐼^𝑞𝑡₀𝜆_𝑖𝑥_𝑖(𝑡)=𝐼^𝑞𝑡₀𝑐^𝑖𝜆_𝑖𝑡−𝑡₀^𝑞−1+𝐼^𝑡2𝑞₀𝜆^2𝑖𝑥_𝑖(𝑡)+𝐼^𝑡2𝑞₀𝜆_𝑖𝑑_𝑖,(𝑡)=𝑐_𝑖𝑡−𝐼𝑡_0𝑞𝑡0^𝑞−1+𝐼^𝑞𝑡₀𝜆^𝑖𝑐_𝑖𝑡−𝑡₀^𝑞−1+𝐼^𝑡2𝑞₀𝜆^2𝑖𝑥_𝑖(𝑡)+𝐼^𝑞𝑡₀𝑑_𝑖+𝐼^𝑡2𝑞₀𝜆_𝑖𝑑_𝑖,𝑥_𝑖𝑥_𝑖(𝑡)=𝐼^𝑞𝑡₀𝑐^𝑖𝑡−𝑡₀^𝑞−1+𝐼^𝑡2𝑞₀𝜆_𝑖𝑥_𝑖(𝑡)+𝐼^𝑡2𝑞₀𝑑_𝑖,(𝑡)=𝑐_𝑖Γ(𝑞)𝑡−𝑡⁰^𝑞−1Γ(𝑞)+𝜆^𝑖𝑡−𝑡₀^2𝑞−1Γ(2𝑞)+⋯+𝜆^𝑖𝑛−1𝑡−𝑡₀^𝑛𝑞−1+𝑑_𝑖Γ(𝑛𝑞)𝑡−𝑡⁰^𝑞Γ(𝑞+1)+𝜆^𝑖𝑡−𝑡₀^2𝑞Γ(2𝑞+1)+⋯+𝜆^𝑖𝑛−1𝑡−𝑡₀^𝑛𝑞Γ(𝑛𝑞+1)+𝜆^𝑛𝑖𝐼^𝑡𝑛𝑞₀𝑥_𝑖(𝑡).(3.39)
Letting 𝑛→+∞, ‖𝜆^𝑛𝑖𝐼^𝑡𝑛𝑞₀𝑥_𝑖(𝑡)‖_1−𝑞→0 if 𝑥_𝑖(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]. Indeed, ‖‖𝜆^𝑛𝑖𝐼^𝑡𝑛𝑞₀𝑥_𝑖(𝑡)‖‖_1−𝑞=max_{𝑡𝑡∈0,𝑇}𝑡−𝑡₀^1−𝑞||||𝜆_𝑛𝑖1Γ(𝑞)^𝑡𝑡₀≤||𝜆(𝑡−𝑠)^𝑛𝑞−1𝑥_𝑖(𝑠)𝑑𝑠_𝑛𝑖||||||‖‖𝑥_𝑖(𝑡)‖‖_1−𝑞Γ(𝑞)Γ((𝑛+1)𝑞)𝑡−𝑡⁰^𝑛𝑞.(3.40)
On the other hand, 𝑒_𝑞𝑡−𝑡₀=^+∞_𝑘=1𝑡−𝑡₀^𝑘𝑞−1𝐸Γ(𝑘𝑞)_𝑞,𝑡−𝑡₀=^+∞_𝑘=1𝑡−𝑡₀^𝑘𝑞Γ(𝑘𝑞+1).(3.41)
Then we can obtain 𝑥_𝑖(𝑡)=𝑐_𝑖Γ(𝑞)𝜆^{𝑖(1/𝑞−1)}𝑒_𝑞𝜆^𝑖1/𝑞𝑡−𝑡₀+𝑑_𝑖𝜆^𝑖−1𝐸_𝑞𝜆^𝑖1/𝑞𝑡−𝑡₀.(3.42)
We know that 𝑑_𝑖𝜆^𝑖−1𝐸_𝑞(𝜆^𝑖1/𝑞(𝑡−𝑡₀)) is satisfied for the fractional nonhomogeneous linear differential equation 𝐷^𝑞𝑡₀𝑥_𝑖(𝑡)=𝜆₁𝑥_𝑖(𝑡)+𝑑_𝑖. So we can also deduce that the general solution of the fractional nonhomogeneous linear differential equation is equal to the general solution of the corresponding homogeneous linear differential equation plus the special solution of the nonhomogeneous linear differential equation. If 𝑋(𝑎)=𝐵, 𝑥_𝑖(𝑎)=𝑏_𝑖, then 𝑥_𝑖(𝑡)=𝑏_𝑖−𝑑_𝑖𝜆^𝑖−1𝐸_𝑞𝜆^𝑖1/𝑞𝑎−𝑡₀𝑒^𝑞−1𝜆^𝑖1/𝑞𝑎−𝑡₀𝑒_𝑞𝜆^𝑖1/𝑞𝑡−𝑡₀+𝑑_𝑖𝜆(^𝑖−1𝑖𝐸=_𝑞1𝜆,2,^𝑖1/𝑞…𝑡,−𝑚𝑡)₀.(3.43) Hence the proof of Theorem 3.4 is complete.

The result remains valid even if 𝑞→1⁻. In this case, 𝑋^(𝑡)=𝑃𝑋(𝑡)+𝐷,𝑋(𝑎)=𝐵,(3.44) where 𝑃=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜆¹0𝜆0₂⋯00⋮00⋮00⋱⋯⋯⋯⋮000𝜆⋮00_𝑚⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.45) and 𝐷=(𝑑₁,𝑑₂,…,𝑑_𝑚) has a unique solution in 𝐶𝑡₀,𝑇×𝐶𝑡₀,𝑇×⋯×𝐶𝑡₀,𝑇,(3.46) given by 𝑋(𝑡)=𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑚(𝑡)^𝑇,(3.47) with 𝑥_𝑖(𝑡)==𝑏𝑏_𝑖−𝑑_𝑖𝜆^𝑖−1𝐸₁𝜆_𝑖𝑎−𝑡₀𝑒¹⁻¹𝜆_𝑖𝑎−𝑡₀𝑒₁𝜆_𝑖𝑡−𝑡₀+𝑑_𝑖𝜆^𝑖−1𝐸₁𝜆_𝑖𝑡−𝑡₀_𝑖+𝑑_𝑖𝜆^𝑖−1exp−𝜆𝑎−𝑡₀exp𝜆_𝑖𝑡−𝑡₀−𝑑𝜆⁻¹,(𝑖=1,2,…,𝑚).(3.48)

4. Existence and Uniqueness of the Solution

In Section 3 we have obtained the unique solution of the constant coefficient homogeneous and nonhomogeneous linear fractional differential equations for 𝑃 being the diagonal matrix. In the present section we will prove the existence and uniqueness of these two kinds of equations for any 𝑃∈𝐿(𝑅^𝑚).

Theorem 4.1. Let 0<𝑞<1 and 𝑃∈𝐿(𝑅^𝑚). If the matrix 𝑃 has distinct real eigenvalues, then for all (𝑎,𝐵)∈(𝑡₀,𝑇]×𝑅^𝑚 the initial value problem 𝐷^𝑞𝑡₀𝑋(𝑡)=𝑃𝑋(𝑡),𝑋(𝑎)=𝐵(4.1) has the unique solution 𝑋(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇].

Proof. Since the matrix 𝑃 has distinct real eigenvalues, there exists an invertible matrix 𝑄 such that 𝑄⁻¹𝑃𝑄=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜆¹0𝜆0₂⋯00⋮00⋮00⋱⋯⋯⋯⋮000𝜆⋮00_𝑚⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(4.2) where 𝜆₁,𝜆₂,…,𝜆_𝑚 are the eigenvalues of the matrix 𝑃. If we define 𝑌(𝑡)=𝑄⁻¹𝑋(𝑡), 𝐷^𝑞𝑡₀𝑌(𝑡)=𝐷^𝑞𝑡₀𝑄⁻¹𝑋(𝑡)=𝑄⁻¹𝐷^𝑞𝑡₀𝑋(𝑡)=𝑄⁻¹𝑃𝑋(𝑡)=𝑅𝑌(𝑡),(4.3) with 𝑅=diag(𝜆₁,𝜆₂,…,𝜆_𝑚). From the above Theorem 3.3 we know the initial value problem 𝐷^𝑞𝑡₀𝑌(𝑡)=𝑅𝑌(𝑡),𝑌(𝑎)=𝑄⁻¹𝐵(4.4) has a unique solution 𝑌(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇] defined on [𝑡₀,𝑇]. Then 𝑋(𝑡)=𝑄𝑌(𝑡) uniquely solves the equations (4.1), where 𝑡∈[𝑡₀,𝑇]. Hence the proof of Theorem 4.1 is complete.

Theorem 4.2. Let 0<𝑞<1. For all (𝑎,𝐵)∈(𝑡₀,𝑇]×𝑅² the initial value problem 𝐷^𝑞𝑡₀𝑋(𝑡)=𝑃𝑋(𝑡),𝑋(𝑎)=𝐵,(4.5) where 𝑃=⎛⎜⎜⎝𝛼𝛽−𝛼𝛽⎞⎟⎟⎠(4.6) has the unique solution 𝑋(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇] defined on [𝑡₀,𝑇].

Proof. Let us define 𝑍(𝑡)=𝑥₁(𝑡)+𝑖𝑥₂(𝑡),𝜇=𝛼+𝑖𝛽.(4.7) We can find that (4.5) is equivalent to the following equation 𝐷^𝑞𝑡₀𝑍(𝑡)=𝜇𝑍(𝑡),𝑍(𝑎)=𝑥₁(𝑎)+𝑥₂(𝑎)=𝑏₁+𝑖𝑏₂.(4.8) Obviously, 𝑍(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇] if 𝑥₁(𝑡) and 𝑥₂(𝑡) belong to 𝐶_1−𝑞[𝑡₀,𝑇]. From the above Theorem 3.3 in Section 3, we know the complex Equation (4.7) has a unique solution defined on [𝑡₀,𝑇]. Hence the proof of Theorem 4.2 is complete.

Theorem 4.3. Let 0<𝑞<1 and 𝑃∈𝑅². If 𝑃 has eigenvalues 𝛼±𝑖𝛽, for all (𝑎,𝐵)∈(𝑡₀,𝑇]×𝑅² the initial value problem 𝐷^𝑞𝑡₀𝑋(𝑡)=𝑃𝑋(𝑡),𝑋(𝑎)=𝐵(4.9) has a unique solution 𝑋(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇].

Proof. Since 𝑃 has eigenvalues 𝛼±𝑖𝛽, there exists an invertible matrix 𝑄 such that 𝑃=𝑄𝑆𝑄⁻¹ where 𝑆=⎛⎜⎜⎝𝛼𝛽−𝛼𝛽⎞⎟⎟⎠.(4.10) Define 𝑌(𝑡)=𝑄⁻¹𝑋(𝑡),(4.11) then 𝐷^𝑞𝑡₀𝑌(𝑡)=𝐷^𝑞𝑡₀𝑄⁻¹𝑋(𝑡)=𝑄⁻¹𝐷^𝑞𝑡₀𝑋(𝑡)=𝑄⁻¹𝑃𝑋(𝑡)=𝑆𝑌(𝑡).(4.12) From the above Theorem 4.2, we know the initial value problem 𝐷^𝑞𝑡₀𝑌(𝑡)=𝑆𝑌(𝑡),𝑌(𝑎)=𝑄⁻¹𝐵(4.13) has a unique solution defined on [𝑡₀,𝑇]. Hence the proof of result is complete.

Theorem 4.4. Let 0<𝑞<1 and 𝑃∈𝑅^𝑚 be an elementary Jordan matrix: ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝⋮𝜆100⋮𝜆100⋱⋯⋯⋯⋯⋮0010⋮𝜆000⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.(4.14) The initial value problem 𝐷^𝑞𝑡₀𝑋(𝑡)=𝑃𝑋(𝑡),𝑋(𝑎)=𝐵(4.15) has a unique solution 𝑋(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇] provided (𝑎,𝐵)∈(𝑡₀,𝑎₀]×𝑅^𝑚, where 𝑎₀ is a suitable constant depending on 𝑡₀, 𝑞, and 𝜆.

Proof. From the (4.15), we can write the equations in the following form: 𝐷𝐷^𝑞𝑡₀𝑥₁(𝑡)=𝜆𝑥₁(𝑡),_𝑞𝑡0𝐷𝑥₂(𝑡)=𝑥⋮₁(𝑡)+𝜆𝑥₂(𝑡),_𝑞𝑡0𝑥𝑥_𝑚(𝑡)=𝑥_𝑚−1(𝑡)+𝜆𝑥_𝑚(𝑡),₁(𝑎)=𝑏₁,…,𝑥_𝑚(𝑎)=𝑏_𝑚.(4.16)
Consider the first equation 𝐷^𝑞𝑡₀𝑥₁(𝑡)=𝜆𝑥₁(𝑡),𝑥₁(𝑎)=𝑏₁.(4.17)
We can obtain the solution of this equation 𝑥₁(𝑡)=𝑏₁𝑒^𝑞−1𝜆^1/𝑞𝑎−𝑡₀𝑒_𝑞𝜆^1/𝑞𝑡−𝑡₀.(4.18)
Consider the second equation 𝐷^𝑞𝑡₀𝑥₂(𝑡)=𝑥₁(𝑡)+𝜆𝑥₂(𝑡),𝑥₂(𝑎)=𝑏₂,(4.19) where now 𝑥₁(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇] is a known function. Since 𝑥₁(𝑡),𝑥₂(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇], according to Lemma 2.9, (4.19) has a unique solution in 𝐶_1−𝑞[𝑡₀,𝑇]. Now 𝑥₁(𝑡) and 𝑥₂(𝑡) are known functions which will be substituted in 𝐷^𝑞𝑡₀𝑥₃(𝑡)=𝑥₂(𝑡)+𝜆𝑥₃(𝑡),𝑥₃(𝑎)=𝑏₃(4.20) and so on. Thus the system of equations given in (4.15) has unique solution in 𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇].

Theorem 4.5. Let 0<𝑞<1 and 𝑃∈𝐿(𝑅^𝑚). The initial value problem 𝐷^𝑞𝑡₀𝑋(𝑡)=𝑃𝑋(𝑡),𝑋(𝑎)=𝐵(4.21) has the unique solution 𝑋(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇] provided (𝑎,𝐵)∈(𝑡₀,𝑎₀]×𝑅^𝑚, where 𝑎₀ is a suitable constant depending on 𝑡₀, 𝑞, and 𝜆.

Proof. In view of Lemma 2.8, there exists an invertible matrix 𝑄 such that 𝑄⁻¹𝑃𝑄 is composed of diagonal blocks of the type 𝐽_𝑖 and 𝐽_𝑖, as defined in the preceding Lemmas 2.7 and 2.8. Let 𝐵=𝑄⁻¹𝑃𝑄 and 𝑌(𝑡)=𝑄⁻¹𝑋(𝑡). Consider the initial value problem: 𝐷^𝑞𝑡₀𝑌(𝑡)=𝐷^𝑞𝑡₀𝑄⁻¹𝑌𝑋(𝑎(𝑡))==𝑄𝑄⁻¹₋₁𝐷^𝑞𝑡₀𝑋(𝑡)=𝑄⁻¹𝑃𝑋(𝑡)=𝐵𝑌(𝑡),𝑋(𝑎)=𝑄⁻¹𝐵.(4.22) Then in view of Theorems 4.1–4.5, (4.22) has a unique solution: 𝑌(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇]. Therefore (4.21) has a unique solution 𝑄⁻¹𝑌(𝑡)∈𝐶_1−𝑞[𝑡₀,𝑇]×𝐶_1−𝑞[𝑡₀,𝑇]×⋯×𝐶_1−𝑞[𝑡₀,𝑇].

Remark 4.6. All the above results are valid for 𝑞→1⁻. Moreover, we can also discuss the case if 𝑎=𝑡₀, in this case, we cannot consider the usual initial condition 𝑥(𝑡₀)=𝑏, but lim_𝑡→𝑡0(𝑡−𝑡₀)^1−𝑞𝑥(𝑡)=𝑏. We can also obtain some similar results by the same method, So we did not give the detailed process and conclusion in this paper.

5. Illustrative Examples

In this section, we give some specific examples to illustrate the above results.

Example 5.1. Consider the following system, where 0<𝑞<1, 𝑡∈[𝑡₀,𝑇], (𝑎,𝐵)∈(𝑡₀,𝑇]×𝑅³,𝐵=(𝑏₁,𝑏₂,𝑏₃),𝐷𝐷^𝑞𝑡₀𝑥₁(𝑡)=3𝑥₁(𝑡)−𝑥₂(𝑡)+𝑥₃(𝑡),_𝑞𝑡0𝐷𝑥₂(𝑡)=−𝑥₁(𝑡)+5𝑥₂(𝑡)−𝑥₃(𝑡),_𝑞𝑡0𝑥𝑥₃(𝑡)=𝑥₁(𝑡)−𝑥₂(𝑡)+3𝑥₃(𝑡),₁(𝑎)=𝑏₁,𝑥₂(𝑎)=𝑏₂,𝑥₃(𝑎)=𝑏₃.(5.1) Here 𝑃=⎛⎜⎜⎜⎜⎝−311−−511−131⎞⎟⎟⎟⎟⎠,(5.2) having the eigenvalues 2, 3, and 6. Choose the eigenvectors  𝑔₁=(1,0,−1)^𝑇,𝑔₂=(1,1,1)^𝑇, and 𝑔₃=(1,−2,1)^𝑇. Then ⎛⎜⎜⎜⎜⎝200030006⎞⎟⎟⎟⎟⎠=𝑄₋₁⎛⎜⎜⎜⎜⎝−311−−511−131⎞⎟⎟⎟⎟⎠𝑄,(5.3) where 𝑄⁻¹=⎛⎜⎜⎜⎜⎜⎝1210−123113136−3116⎞⎟⎟⎟⎟⎟⎠.(5.4)
Define the 𝑌=𝑄⁻¹𝑋. Then the system of equation in 𝑌 is decoupled, namely, 𝐷𝐷^𝑞𝑡₀𝑦₁(𝑡)=2𝑦₁(𝑡),_𝑞𝑡0𝐷𝑦₂(𝑡)=3𝑦₂(𝑡),_𝑞𝑡0𝑦𝑦₃(𝑡)=6𝑦₃(𝑡),₁(𝑎)=12𝑏¹−12𝑏³𝑦₂,(𝑎)=13𝑏¹+13𝑏²+13𝑏³𝑦₃,(𝑎)=16𝑏¹−13𝑏²+16𝑏³.(5.5)
In view of (3.30), we can obtain 𝑦₁(𝑡)=12𝑏¹−1𝑦2𝑏³𝑒^𝑞−12^1/𝑞𝑎−𝑡₀𝑒_𝑞2^1/𝑞𝑡−𝑡₀₂,(𝑡)=13𝑏¹+13𝑏²+1𝑦3𝑏³𝑒^𝑞−13^1/𝑞𝑎−𝑡₀𝑒_𝑞3^1/𝑞𝑡−𝑡₀₃,(𝑡)=16𝑏¹−13𝑏²+16𝑏³𝑒^𝑞−16^1/𝑞𝑎−𝑡₀𝑒_𝑞6^1/𝑞𝑡−𝑡₀.(5.6)
Hence 𝑥₁(𝑥𝑡)=𝑦₁(𝑡)+𝑦₂(𝑡)+𝑦₃(𝑡),₂𝑥(𝑡)=𝑦₂(𝑡)−2𝑦₃(𝑡),₃(𝑡)=−𝑦₁(𝑡)+𝑦₂(𝑡)+𝑦₃(𝑡).(5.7)

Example 5.2. Consider the following system, where 0<𝑞<1, 𝑡∈[𝑡₀,𝑇], (𝑎,𝐵)∈(𝑡₀,𝑇]×𝑅²,𝐵=(𝑏₁,𝑏₂), 𝐷𝐷^𝑞𝑡₀𝑥₁(𝑡)=−2𝑥₁(𝑡)−𝑥₂(𝑡),_𝑞𝑡0𝑥𝑥₂(𝑡)=13𝑥₁(𝑡)+4𝑥₂(𝑡),₁(𝑎)=𝑏₁,𝑥₂(𝑎)=𝑏₂.(5.8) Here 𝑃=⎜⎝⎛⎜−132−41⎞⎟⎟⎠(5.9) having the eigenvalues 1±2𝑖. Choose the eigenvectors   𝑔₁=(1,−3−2𝑖)^𝑇, and 𝑔₂=(1,−3+2𝑖)^𝑇, Then ⎛⎜⎜⎝1+02𝑖1−02𝑖⎞⎟⎟⎠=𝑄₋₁⎛⎜⎜⎝−132−41⎞⎟⎟⎠𝑄,(5.10) where 𝑄𝑄=⎛⎜⎜⎝−31−2𝑖−31+2𝑖₋₁⎞⎟⎟⎠,=⎛⎜⎜⎝22+−433𝑖𝑖4𝑖4−4𝑖⎞⎟⎟⎠.(5.11) Define the 𝑌=𝑄⁻¹𝑋. Then the system of equation in 𝑌 is decoupled, namely, 𝐷^𝑞𝑡₀𝑦₁(𝑡)=(1+2𝑖)𝑦₁(𝑡),𝑦₁(𝑎)=12𝑏¹+34𝑏¹+1𝐷_𝑞𝑡04𝑏²𝑖𝑦₂(𝑡)=(1−2𝑖)𝑦₂(𝑡),𝑦₂(𝑎)=12𝑏¹−34𝑏¹+14𝑏²𝑖.(5.12) In view of (3.30), we can obtain 𝑦₁(𝑡)=12𝑏¹+34𝑏¹+1𝑦4𝑏²𝑖𝑒^𝑞−1(1+2𝑖)^1/𝑞𝑎−𝑡₀𝑒_𝑞(1+2𝑖)^1/𝑞𝑡−𝑡₀₂(𝑡)=12𝑏¹−34𝑏¹+14𝑏²𝑖𝑒^𝑞−1(1−2𝑖)^1/𝑞𝑎−𝑡₀𝑒_𝑞(1−2𝑖)^1/𝑞𝑡−𝑡₀.(5.13) Hence 𝑥𝑥₁(𝑡)=𝑦₁(𝑡)+𝑦₂(𝑡)₂(𝑡)=−3𝑦₁(𝑡)+𝑦₂(𝑡)−2𝑖𝑦₁(𝑡)−𝑦₂(𝑡).(5.14)

Acknowledgments

The authors are highly grateful for the referee's careful reading and comments on this paper. The present paper was supported by the NNSF of China Grants no. 11271087 and no. 61263006.