Comparison theorem of nonlinear fractional differential equations and application

Toán học, Cơ học & Ứng dụng 
 D. X. Hung, N. T. Hong, “Comparison theorem of  equations and application.” 232 
COMPARISON THEOREM OF NONLINEAR FRACTIONAL 
DIFFERENTIAL EQUATIONS AND APPLICATION 
Dao Xuan Hung1,*, Nguyen Thi Hong2 
Abstract: In this paper, we first establish a comparison theorem for the 
nonlinear factional differential equations, then derive a local asymptotical stability 
theorem for the nonlinear differential equation under some suitable conditions. 
Keywords: Caputo derivative; Equilibrium; Asympticalability. 
1. INTRODUCTION 
In recent decades, fractional calculus and fractional differential equations attract 
much attention and increasing interests due to their potential applications in sciences and 
engineering ([1, 6, 9, 11, 12] and many references cited therein). Fractional 
differenital/integral operator is one kind of the pseudo-differential operators [10]. It plays 
an important role in fractional equations. In general three kinds of fractional derivative 
operators are used, including Gr¨unwald-Letnikov fractional derivative operator, Riemann 
-Liouville fractional derivative operator and Caputo fractional derivative operator. They 
are not equivalent but have close relations [6, 7, 12]. In applied sciences and engineering, 
the Caputo derivative is often used. In this paper the involved fractional derivative 
indicates the Caputo derivative, which is defined as 
 = 
in which m = , i.e., m is the first integer not less than α, is the usual mth-order 
derivative with respect to τ . In real applications, the fractional order α is often less than 1, 
here we restrict α ∈ (0, 1) as usual. For the case α > 1, we can often translate the fractional 
systems into systems with the same actional order which lies in (0,1) provided some 
suitable conditions are satisfied [2]. 
The Cauchy problem of the fractional differential equation reads as 
 = f(t, x), x(0) = , (1) 
If we request the vector function f is continuous and satisfies a Lipschitz condition 
with respect to the second argument x on a suitable set G, then the initial value problem 
(1) determines a unique solution on some interval [0, T] [4]. Throughout the paper, we 
always assume f fulfils the above condition on a suitable set which we call G, so equation 
(1) exists one and only one solution defined on [0, T]. 
In [8], Matignon studied stability of zero solution to the following fractional system 
(t) = AX 
with X(0) = = , where X = , α ∈ (0, 1), A ∈ 
. Afterwards, Li, et al., further studied stability of zero solution of the above system 
with multiple time-delays [3]. All this stability belongs to linear stability. In this paper, we 
mainly study nonlinear stability of fractional differential equation. 
The outline of the present paper is arranged as follows. In Section 2, some basic 
notions are introduced and the Comparison Theorem is established. In the last section, we 
derive a symptotical stability theorem of the fractional nonlinear differential equation. 
2. COMPARISON THEOREM 
The autonomous differential equation with Caputo derivative is in the following form, 
 = f(x), (2) 
with initial condition x(0) = . In this paper, we mainly consider stability of equilibrium 
Nghiên cứu khoa học công nghệ 
Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san CBES2, 04 - 2018 233
to equation (2). We assume that the initial problem (2.1) has one and only one solution. 
This solution is denoted by x(t) = φ(t; ). 
In the following, we give some definitions. 
2.1. Definition 2.1. If there exists a constant e such that f(e) = 0, then e is called an 
equilibrium point to equation (2). 
2.2. Definition 2.2. The equilibrium point e to (2.1) is said to be: 1)locally stable if ∀ > 0, 
there exists a δ > 0 such that |x(t) − e| < holds for ∀ ∈ {z : |z − e| < δ} and ∀t ≥ 0; 2) 
local asymptotically stability if x(t) is locally stable and = e. 
The theorem below can be found in [4]. 
2.3. Theorem 2.1. If f is continuous and α ∈ (0, 1), then the Cauchy problem 
 is equivalent to a nonlinear integral equation of the second kind, 
 . 
Next, we establish a comparison theorem for the fractional differential equations. 
Consider the following fractional equations, 
 = f(t, x), (3) 
 = F(t, x), (4) 
2.4. Theorem 2.2. Consider equations (3) and (4) with initial value condition ( , x( )) = 
(0, ). If f, F are continuous then (3) and (4) have one and only one solution x(t), y(t) 
passing through (0, ), respectively. Besides, if the vector functions satisfy f(t, x) F(t, 
x), then x(t) y(t) for t > 0. 
Proof. The solutions of systems (5) and (6) can be expressed in the following form: 
 (5) 
And 
 (6) 
Subtracting Eq. (6) from Eq. (5), one has 
 y(t) – x(t) = (7) 
Since is a nonnegative function, it then 
follows from Eq. (7) that x(t) . The proof is completed. 
Similar to the ordinary differential equation, the solution of equation (1) can be 
extended to +∞. This fact means that the conclusions of Theorem 2.2 hold for (0,+∞). In 
the last section, we discuss the stability of equilibrium to the autonomous nonlinear 
differential equation with Caputo derivative. Since the non-zero equilibrium of differential 
equation with Caputo derivative can be moved to the origin, we only study stability of the 
zero solution of the considered equation. 
3. ASYMPTOTICAL STABILITY ANALYSIS 
In this section, we derive the following stability theorem. 
Theorem 3.1. If the nonlinear equation = f(x) with α ∈ (0, 1) satisfies following 
conditions: (i) f(0) = 0, (ii) the 2th-order derivative f’’(x) of f(x) is continuous and f’(0) < 
0, (iii) if its non-zero solution x(t) = φ(t; ) has a zero point but its derivative value at this 
zero point is non-zero, then the zero solution is locally asymptotically stable. 
Proof. Set f’(0) = λ, s(x)x = f(x) – f’(0)x, then near the zero solution one has 
f(x) = f’(0)x + o(x) = (λ + s(x))x, 
Toán học, Cơ học & Ứng dụng 
 D. X. Hung, N. T. Hong, “Comparison theorem of  equations and application.” 234 
where s(x) = O(x). 
Since f’(0) m > 0 such that −M < 
λ+s(x) 0, or −Mx > f(x) > −mx if x < 0, for all 
x ∈ (−δ, δ) with δ > 0 small enough. Now, we assert that for the initial value | | small 
enough then |x(t)| 0. Because λ < 0, there exits a positive constant 
small enough such that ∀ ∈ {z : |z| 
0. For ≤ 0, this case can be similarly proved so it is omitted here. 
Firstly, following Theorem 2.1 we have that equation (2.1) is equivalent to 
 . 
It is obvious that x(t) is continuous with respect to t, so we can take > 0 small 
enough such that 
 < 
i.e., 
 . 
Therefore, 
 . 
Thus, there exits a small such that the solution x(t) of (2.1) satisfies 
 . 
Now, we shall prove that before x(t) passes through time-axis t, that is, before it 
becomes negative if it does indeed, it is always less than . Infact, if x(t) equals to in 
the first time, say, at time > 0, x( ) = and > x(t) ≥ 0 for all t ∈ (0, ), then 
 . 
From the above formula, we know 
but, clearly 
due to assumption. So the conclusion holds. 
Next, we show that x(t) never transverses the time-axis t, i.e., it is always non-
negative. 
Assume x(t) transverses the time-axis t in the first time , then for > 0 small 
enough, 
]. 
Thus, we get 
 , 
 , 
In which then 
 = 
Nghiên cứu khoa học công nghệ 
Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san CBES2, 04 - 2018 235
 = 
 + . (8) 
For the right hand of (8), 
And 
So 
 + . 
But the left hand of (3.1) is negative, this is a contradiction. 
Therefore, 0 ≤ x(t) 0 and > 0. Based on assumption (iii), x(t) is strictly 
positive for this situation. 
So, we draw a conclusion: for the initial value | | small enough, the non-zero 
solutions |x(t)| 0 if > 0, x(t) < 0 if < 0. 
Now for small δ > 0, small initial value | | > 0, and (t, x) ∈(0,+∞) × {(−δ, δ)\{0}}, one 
has −Mx 0, or, −Mx > f(x) > −mx if < 0. 
Now, consider following three equations: 
 (8) 
 (9) 
and 
 (10) 
These equations satisfy the conditions of Theorem 2.2, therefore 
x(t) 0 and x(t) > y(t) > z(t) if 0, where x(t), y(t) and 
z(t) solve (9), (2) and (10), respectively. 
It is evident that 
 = = 0. 
Thus, 
 . 
This completes the proof. 
4. CONCLUSIONS 
In this paper, we first give Definitions about locally stable, local asymptotically 
stability, then we derive and prove a comparison theorem. At last, we aply the comparison 
theorem to give and prove a local asymptotical stability theorem for the nonlinear 
differential equation under some suitable conditions. These results are helpful to study 
fractional differential equations and establishing fractional models in science and 
engineering. 
REFERENCES 
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TÓM TẮT 
ĐỊNH LÝ SO SÁNH NGHIỆM CỦA PHƯƠNG TRÌNH VI PHÂN CẬP 
 PHÂN SỐ PHI TUYẾN VÀ ỨNG DỤNG 
Trong bài báo này, đầu tiên chúng tôi đưa ra và chứng minh định lý so sánh 
nghiệm cho phương trình vi phân cấp phân số phi tuyến, sau đó, chúng tôi áp dụng 
định lý này để thành lập và chứng minh định lí ổn định tiệm cận của phương trình vi 
phân cấp phân số phi tuyến dưới các điều kiện phù hợp. 
Từ Khóa: Đạo hàm Caputo; Đạo hàm cấp phân số; Ổn định tiệm cận. 
Nhận bài ngày 26 tháng 02 năm 2018 
Hoàn thiện ngày 16 tháng 3 năm 2018 
Chấp nhận đăng ngày 20 tháng 3 năm 2018 
Địa chỉ: 1 Khoa Khoa học cơ bản, Trường Đại học Mỏ- Địa chất; 
 2 Khoa Khoa học cơ bản, Học viện Kỹ thuật Mật mã. 
 * Email: daoxuanhung@humg.edu.vn.