Abstract
In this paper, we find the formula of general solution for a generalized impulsive differential equations of fractional-order q ∈ (2, 3).
1 Introduction
Fractional differential equations play an important part in modeling of many phenomena in various fields of science and engineering, and the subject of fractional differential equations is extensively researched (see [1–19] and the references therein).
On the other hand, impulsive differential equation is a key tool to describe some systems and processes with impulsive effects. There have appeared many papers focused on the subject of impulsive differential equations with Caputo fractional derivative [20–31].
Recently, we have found that there exist general solutions for several kinds of impulsive fractional differential equations in [32–38]. Based on these works, we will further study the general solution of the generalized impulsive differential equations of fractional-order q ∈ (2,3).
where
Next, take
Let
With simplification of system (1), we get
Let J0 = [a, t1] and Ji = (ti, ti + 1] (i = 1,2,…, L). Considering some limiting cases in (1), we have
This means that the solution of (1) satisfies:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
Thus, we present the definition of solution for (1) as follows
A function z(t) : [a, T] → ℝ is said to be the solution of impulsive system (1) if z(a) = xa,
Define a function
By the definition of Caputo fractional derivative, we have
Therefore,
Next, we provide some definitions and conclusions in Section 2, and prove the formula of general solution for (1) in Section 3. Finally, an example is provided to expound the main result in Section 4.
2 Preliminaries
Definition 2.1 ([2])
The fractional integral of order q for function x is defined as
where Γ is the gamma function.
Definition 2.2 ([2])
The Caputo fractional derivative of order q for a function x can be written as
Lemma 2.3 ([39])
If the function h(t, x) is continuous, then the initial value problem
is equivalent to the following nonlinear Volterra integral equation of the second kind,
and its solutions are continuous.
Lemma 2.4 ([32])
Let q ∈ (0, 1) and ξ is a constant. Impulsive system
is equivalent to the fractional integral equation
provided that the integral in (10) exists.
3 Main results
For convenience, let f = f(s, x(s)) and
Let q ∈ (2, 3) and ξ0is a constant. System (4) is equivalent to the fractional integral equation
provided that the integral in (11) exists.
“Necessity”, for system (4), there exist an implicit condition
That is
In fact, we can verify that Eq. (11) satisfies condition (12).
Next, we can obtain
Finally, using Eq. (11) for each t ∈ (tk, tk+1] (where k = 0,1,…, L), we have
So, Eq. (11) satisfies the condition of fractional derivative in (4). Thus, Eq. (11) satisfies all conditions of system (4).
“Sufficiency”, we will prove that the solutions of (4) satisfy Eq. (11) by using mathematical induction. For t ∈ [a, t1], it is certain that the solution of system (4) satisfies Eq. (11) by Lemma 2.3 and
Using (13), we have
Therefore, the approximate solution
Let
we get
Then, by (15), we assume
where σ is an undetermined function with σ(0) = 1. Therefore
Let θ (z) = 1 − σ (z) for z ∈ ℝ in the above equation, then
Using (17), we get
Thus,
Let
Thus,
Thus,
Letting t2 → t1, we have
Using (17) and (22) for (23), we obtain
Therefore θ (z) = ξ0z for ∀z ∈ ℝ (where ξ0 is a constant). Thus
and
Next, suppose
Using (27), we get
Thus,
Let
Then,
Thus,
Therefore, the solutions of system (4) satisfy Eq. (11). So, system (4) is equivalent to Eq. (11). The proof is now completed. □
With similarity to Lemma 3.1, the following two conclusions can be proved.
Let q ∈ (2,3) and ξ1 is a constant. System (5) is equivalent to the fractional integral equation
provided that the integral in (32) exists.
Let q ∈ (2, 3) ξ2and is a constant. System (6) is equivalent to the fractional integral equation
provided that the integral in (33) exists.
Let q ∈ (2, 3) and ξ0is a constant. If a function x is the general solution of system (4) then
Let q ∈ (2, 3) and ξ1is a constant. If a function x is the general solution of system (5) then
Let q ∈ (2, 3) and ξ2is a constant. If a function x is the general solution of system (6) then
Impulses
Let q ∈ (2, 3) and ξb (where b ∈ {0,1,2}) are three constants. If a function x is the general solution of system (1) then
By Definition 2.2, we have
Moreover,
Let q ∈ (2,3) and ξb (where b ∈ {0, 1, 2}) are three constants. System (1) is equivalent to the fractional integral equation
here k ∈{0, 1, 2,…, K}, provided that the integral in (35) exists.
For t ∈ J′0, it is clear that system (1) is equivalent to
For t ∈ J′1 (1 ≤ k ≤ K), by Lemma 3.8, we have
Integrating both sides of the above equation twice, we get
here C, C1 are two constants. Supposing
and
Thus, for t ∈ J′k (here k = 1, 2,…, K), we get
So, the solution of system (1) satisfies Eq. (35).
Next, we can verify that Eq. (35) satisfies all conditions (including conditions (i)-(vii)) in system (1). So, system (1) is equivalent to Eq. (35). The proof is completed. □
For impulsive system (1), we have
On the other hand, using (35), we get
Moreover, we can verify thatEq. (37)is the solution of (36), and indirectly supports our results.
Let q ∈ (2, 3) and ξb (where b ∈ {0,1,2}) are three constants. System (2) is equivalent to the fractional integral equation
provided that the integral in (38) exists.
4 Example
The analytical solution of system (1) is difficult to obtain when f is a nonlinear function in (1). So, let us consider a linear impulsive system.
Next, we give the general solution by
where ξ0, ξ1and ξ2are arbitrary constants.
Next, forEq. (40), we have
and
So, Eq. (40)satisfies Caputo fractional derivative condition in (39).
Secondly, we can verify thatEq. (40)satisfies
Moreover, we can verify thatEq. (40)satisfies the corresponding conditions (i)-(vii) of (39). Therefore, Eq. (40)is the general solution of (39).
Acknowledgement
The authors are deeply grateful to the anonymous referees for their kind comments, correcting errors and improving written language, which have been very useful for improving the quality of this paper. The work described in this paper is financially supported by the National Natural Science Foundation of China (Grant No. 21576033) and the Natural Science Foundation of Jiangxi Province (Grant No. 20151BAB207013) and Jiujiang University Research Foundation (Grant No. 8400183).
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