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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 14, Issue 1


Volume 13 (2015)

On the concept of general solution for impulsive differential equations of fractional-order q ∈ (2,3)

Xianmin Zhang / Tong Shu / Zuohua Liu / Wenbin Ding / Hui Peng / Jun He
Published Online: 2016-07-08 | DOI: https://doi.org/10.1515/math-2016-0042


In this paper, we find the formula of general solution for a generalized impulsive differential equations of fractional-order q ∈ (2, 3).

Keywords: Fractional differential equations; Impulsive fractional differential equations; Impulse; General solution

MSC 2010: 34A08; 34A37

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 [119] 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 [2031].

Recently, we have found that there exist general solutions for several kinds of impulsive fractional differential equations in [3238]. Based on these works, we will further study the general solution of the generalized impulsive differential equations of fractional-order q ∈ (2,3).


where aDtq denote Caputo fractional derivative of order q in interval [a, t], f : J × ℝ → ℝ and, Ii Īj, Îi : ℝ → ℝ are appropriate functions (here i = 1,2,…, L and j = 1,2,…, M and l = 1,2, …, N, respectively), a = t0 < t1 < … < tL < tL+1 = T, a=t¯0<t¯1<t¯M<t¯M+1=T,a=t^0<t^1<<t^N<t^N+1=T. Here x(ti+)=limε0+x(ti+ε) and x(ti)=limε0x(ti+ε) represent the right and left limits of x(t) at t = ti, respectively, (x(t¯j+),x(t¯j) and (x(t^j+),x(t^j) have similar meaning for x′(t) at t=t¯j and x″(t) at t=t^l, respectively).

Next, take a,t1,t2,tL,t¯1,t¯2,,t¯M,t^1,t^2,,t^N,T to a=t0<t1<<tK<tK+1=T such that


Let J0=[a,t1] and Jk=(tk,tk+1,](k=0,1,2,,K). For each [a,tk] (here k = 0,1,2,…, K), assume [a,tk0][a,tk][a,tk0+1] (here k0 ∈ {1,2,…, L}) and [a,tk1][a,tk][a,t¯k1+1] (here k1 ∈ {1,2,…,M}) and [a,t^k2][a,tk][a,t^k2+1] (here k2 ∈ {1,2,…, N}) respectively.

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) limI(x(tj))0foralli{1,2,,L}andI¯j(x(t¯j))0forallj{1,2,,M}andI^l(x(t^j))0foralll{1,2,,N}{thesolutionofsystem(1)}={thesolutionofsystem(3)),

(ii) limI¯j(x(tj))0forallj{1,2,,M}andI^l(x(t^l))0foralll{1,2,,N}{thesolutionofsystem(1)}={thesolutionofsystem(4)),

(iii) limIi(x(ti))0foralli{1,2,,L}andI^l(x(t^l))0foralll{1,2,,N}{thesolutionofsystem(1)}={thesolutionofsystem(5)),

(iv) limIi(x(ti))0foralli{1,2,,L}andI¯l(x(t^j))0forallj{1,2,,M}{thesolutionofsystem(1)}={thesolutionofsystem(6)),

(v) limI^l(x(t^i))0foralll{1,2,,N}{thesolutionofsystem(1)}={thesolutionofsystem(7)),

(vi) limI¯j(x(t¯i))0forallj{1,2,,M}{thesolutionofsystem(1)}={thesolutionofsystem(8)),

(vii) limIi(x(ti))0foralli{1,2,,L}{thesolutionofsystem(1)}={thesolutionofsystem(9)).

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, z(a)=x¯a and z(a)=x^a, the equation condition aDtqz(t)=f(t,z(t)) for each t ∈ (a, T] is verified, impulsive conditions Δz|t=ti=Ii(z(ti))(here i = 1,2,…, L), Δz|t=t¯i=I¯i(z(t¯i)) (here j = 1,2, …,M) and Δz|t=t^l=I^i(z(t^i)) (here l = 1,2,…,N) are satisfied, the restriction of to the interval Jk (here k = 0,1,2,…, K) is continuous, and conditions (i)-(vii) hold.

Define a function


By the definition of Caputo fractional derivative, we have


Therefore, x~(t) can meet the condition of fractional derivative and impulsive conditions in (1). But, x~(t) is only considered as an approximate solution of (1) since it doesn’t satisfy conditions (i)-(vii).

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


The fractional integral of order q for function x is defined as


where Γ is the gamma function.


The Caputo fractional derivative of order q for a function x can be written as



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.


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 i=10yi=0 in this section.

Let q ∈ (2, 3) and ξ0 is a constant. System (4) is equivalent to the fractional integral equation


provided that the integral in (11) exists.

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) ξ2 and is a constant. System (6) is equivalent to the fractional integral equation


provided that the integral in (33) exists.

Let q ∈ (2, 3) and ξ0 is a constant. If a function x is the general solution of system (4) then


Let q ∈ (2, 3) and ξ1 is a constant. If a function x is the general solution of system (5) then


Let q ∈ (2, 3) and ξ2 is a constant. If a function x is the general solution of system (6) then


Impulses Δxt=ti,Δxt=t¯j and Δxt=t^l have similar effect on x″(t) of (3) by Corollaries 3.4-3.6. Thus, we will consider Δxt=ti and Δxt=t¯j as some special impulses Δxt=t^l for system (3).

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


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 impulsive system (1), we have


On the other hand, using (35), we get


Moreover, we can verify that Eq. (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, ξ1 and ξ2 are arbitrary constants.

Next, for Eq. (40), we have




So, Eq. (40) satisfies Caputo fractional derivative condition in (39).

Secondly, we can verify that Eq. (40) satisfies


Moreover, we can verify that Eq. (40) satisfies the corresponding conditions (i)-(vii) of (39). Therefore, Eq. (40) is the general solution of (39).


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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About the article

E-mail: ,

Received: 2015-10-23

Accepted: 2016-06-08

Published Online: 2016-07-08

Published in Print: 2016-01-01

Citation Information: Open Mathematics, Volume 14, Issue 1, Pages 452–473, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0042.

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© 2016 Zhang et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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