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# Open Mathematics

### formerly Central European Journal of Mathematics

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

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# 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

## Abstract

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

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).

$aDtqx(t)=f(t,x(t)),q∈(2,3),t∈J=[a,T],t≠ti(i=1,2,…,L)andt≠t¯j(j=1,2,…,M)andt≠t^l(l=1,2,…,N),Δxt=ti=x(ti+)−x(ti−)=Iix(ti−),i=1,2,…,L,Δx′t=t¯j=x′(t¯j+)−x′(t¯j−)=I¯jx(t¯j−),j=1,2,…,M,Δx″t=t^l=x″(t^l+)−x″(t^l−)=I^lx(t^l−),l=1,2,…,N,x(a)=xa,x′(a)=x¯a,x″(a)=x^a.$(1)

where ${}_{a}{D}_{t}^{q}$ 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, $a={\overline{t}}_{0}<{\overline{t}}_{1}<\cdots {\overline{t}}_{M}<{\overline{t}}_{M+1}=T,a={\stackrel{^}{t}}_{0}<{\stackrel{^}{t}}_{1}<\cdots <{\stackrel{^}{t}}_{N}<{\stackrel{^}{t}}_{N+1}=T$. Here $x\left({t}_{i}^{+}\right)=\underset{\epsilon \to 0}{lim}+x\left({t}_{i}+\epsilon \right)$ and $x\left({t}_{i}^{-}\right)=\underset{\epsilon \to 0}{lim}-x\left({t}_{i}+\epsilon \right)$ represent the right and left limits of x(t) at t = ti, respectively, $\left({x}^{\prime }\left({\overline{t}}_{j}^{+}\right),\phantom{\rule{thinmathspace}{0ex}}{x}^{\prime }\left({\overline{t}}_{j}^{-}\right)$ and $\left({x}^{″}\left({\stackrel{^}{t}}_{j}^{+}\right),\phantom{\rule{thinmathspace}{0ex}}{x}^{″}\left({\stackrel{^}{t}}_{j}^{-}\right)$ have similar meaning for x′(t) at $t={\overline{t}}_{j}$ and x″(t) at $t={\stackrel{^}{t}}_{l}$, respectively).

Next, take $a,{t}_{1},{t}_{2}\cdots ,{t}_{L},{\overline{t}}_{1},{\overline{t}}_{2},\cdots ,{\overline{t}}_{M},{\stackrel{^}{t}}_{1},{\stackrel{^}{t}}_{2},\cdots ,{\stackrel{^}{t}}_{N},T$ to $a={t}_{0}^{\prime }<{t}_{1}^{\prime }<\cdots <{t}_{K}^{\prime }<{t}_{K+1}^{\prime }=T$ such that

$set{t1,t2,⋯,tL,t¯1,t¯2,⋯,t¯M,t^1,t^2,⋯,t^N}=set{t1′,t2′,⋯tK′}$

Let ${J}_{0}^{\prime }=\left[a,{t}_{1}^{\prime }\right]$ and ${J}_{k}^{\prime }=\left({t}_{k}^{\prime },{t}_{k+1}^{\prime },\right]\phantom{\rule{thinmathspace}{0ex}}\left(k=0,1,2,\cdots ,K\right)$. For each $\left[a,{t}_{k}^{\prime }\right]$ (here k = 0,1,2,…, K), assume $\left[a,{t}_{{k}_{0}}\right]\subseteq \left[a,{t}_{k}^{\prime }\right]\subset \left[a,{t}_{{k}_{0}+1}\right]$ (here k0 ∈ {1,2,…, L}) and $\left[a,{t}_{{k}_{1}}^{\prime }\right]\subseteq \left[a,{t}_{k}^{\prime }\right]\subset \left[a,{\overline{t}}_{{k}_{1}+1}\right]$ (here k1 ∈ {1,2,…,M}) and $\left[a,{\stackrel{^}{t}}_{{k}_{2}}\right]\subseteq \left[a,{t}_{k}^{\prime }\right]\subset \left[a,{\stackrel{^}{t}}_{{k}_{2}+1}\right]$ (here k2 ∈ {1,2,…, N}) respectively.

With simplification of system (1), we get

$aDtqx(t)=f(t,x(t)),q∈(2,3),t∈J=[a,T],t≠ti(k=1,2,…,L),Δxt=ti=Iix(ti−),i=1,2,…,L,Δx′t=ti=I¯ix(ti−),i=1,2,…,L,Δx″t=ti=I^ix(ti−),i=1,2,…,L,x(a)=xa,x′(a)=x¯a,x″(a)=x^a.$(2)

Let J0 = [a, t1] and Ji = (ti, ti + 1] (i = 1,2,…, L). Considering some limiting cases in (1), we have

$limIi(x(ti−))→0foralli∈{1,2,…,L}andI¯j(x(t¯j−))→0forallj∈{1,2,…,M}andI^l(x(t^l−))→0foralll∈{1,2,…,N}{impulsesystem(1)}$

$→aDtqx(t)=f(t,x(t)),q∈(2,3),t∈J=[a,T],x(a)=xa,x′(a)=x¯a,x″(a)=x^a.$(3)

$limI¯j(x(t¯j−))→0forallj∈{1,2,…,M}andI^l(x(t^l−))→0foralll∈{1,2,…,N}{impulsesystem(1)}$

$→aDtqx(t)=f(t,x(t)),q∈(2,3),t∈J=[a,T],t≠ti(i=1,2,…,L),Δxt=ti=Iix(ti−),i=1,2,…,L,x(a)=xa,x′(a)=x¯a,x″(a)=x^a.$(4)

$limIi(x(ti−))→0foralli∈{1,2,…,L}andI^l(x(t^l−))→0foralll∈{1,2,…,N}{impulsesystem(1)}$

$→aDtqx(t)=f(t,x(t)),q∈(2,3),t∈J=[a,T],t≠t¯j(j=1,2,…,M),Δx′t=t¯j=I¯jx(t¯j−),j=1,2,…,M,x(a)=xa,x′(a)=x¯a,x″(a)=x^a.$(5)

$limIi(x(ti−))→0foralli∈{1,2,…,L}andI¯j(x(t¯j−))→0forallj∈{1,2,…,M}{impulsesystem(1)}$

$→aDtqx(t)=f(t,x(t)),q∈(2,3),t∈J=[a,T],t≠t^l(l=1,2,…,N),Δx″t=t^l=I^lx(t^l−),l=1,2,…,N,x(a)=xa,x′(a)=x¯a,x″(a)=x^a.$(6)

$limI^l(x(t^l−))→0foralll∈{1,2,…,N}{impulsesystem(1)}$

$→aDtqx(t)=f(t,x(t)),q∈(2,3),t∈J=[a,T],t≠ti(i=1,2,…,L)andt≠t¯j(j=1,2,…,M),Δxt=ti=Iix(ti−),i=1,2,…,L,Δx′t=t¯j=I¯jx(t¯j−),j=1,2,…,M,x(a)=xa,x′(a)=x¯a,x″(a)=x^a.$(7)

$limI¯j(x(t¯j−))→0forallj∈{1,2,…,M}{impulsesystem(1)}$

$→aDtqx(t)=f(t,x(t)),q∈(2,3),t∈J=[a,T],t≠ti(i=1,2,…,L)andt≠t^l(l=1,2,…,N),Δxt=ti=Iix(ti−),i=1,2,…,L,Δx″t=t^l=I^lx(t^l−),l=1,2,…,N,x(a)=xa,x′(a)=x¯a,x″(a)=x^a.$(8)

$limIi(x(ti−))→0foralli∈{1,2,…,L}{impulsesystem(1)}$

$→aDtqx(t)=f(t,x(t)),q∈(2,3),t∈J=[a,T],t≠t¯j(j=1,2,…,M)andt≠t^l(l=1,2,…,N),Δx′t=t¯j=I¯jx(t¯j−),j=1,2,…,M,Δx″t=t^l=I^lx(t^l−),l=1,2,…,N,x(a)=xa,x′(a)=x¯a,x″(a)=x^a.$(9)

This means that the solution of (1) satisfies:

(i) $\underset{\begin{array}{l}I\left(x\left({t}_{j}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,L\right\}\\ and\phantom{\rule{thinmathspace}{0ex}}{\overline{I}}_{j}\left(x\left({\overline{t}}_{j}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}j\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,M\right\}\\ and\phantom{\rule{thinmathspace}{0ex}}{\stackrel{^}{I}}_{l}\left(x\left({\stackrel{^}{t}}_{j}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}l\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,N\right\}\end{array}}{lim}\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(1\right)\right\}\phantom{\rule{thinmathspace}{0ex}}=\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(3\right)\right),$

(ii) $\underset{\begin{array}{l}{\overline{I}}_{j}\left(x\left({t}_{j}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}j\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,M\right\}\\ and\phantom{\rule{thinmathspace}{0ex}}{\stackrel{^}{I}}_{l}\left(x\left({\stackrel{^}{t}}_{l}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}l\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,N\right\}\end{array}}{lim}\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(1\right)\right\}\phantom{\rule{thinmathspace}{0ex}}=\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(4\right)\right),$

(iii) $\underset{\begin{array}{l}{I}_{i}\left(x\left({t}_{i}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,L\right\}\\ and\phantom{\rule{thinmathspace}{0ex}}{\stackrel{^}{I}}_{l}\left(x\left({\stackrel{^}{t}}_{l}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}l\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,N\right\}\end{array}}{lim}\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(1\right)\right\}\phantom{\rule{thinmathspace}{0ex}}=\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(5\right)\right),$

(iv) $\underset{\begin{array}{l}{I}_{i}\left(x\left({t}_{i}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,L\right\}\\ and\phantom{\rule{thinmathspace}{0ex}}{\overline{I}}_{l}\left(x\left({\stackrel{^}{t}}_{j}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}j\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,M\right\}\end{array}}{lim}\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(1\right)\right\}\phantom{\rule{thinmathspace}{0ex}}=\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(6\right)\right),$

(v) $\underset{\begin{array}{l}{\stackrel{^}{I}}_{l}\left(x\left({\stackrel{^}{t}}_{i}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}l\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,N\right\}\end{array}}{lim}\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(1\right)\right\}\phantom{\rule{thinmathspace}{0ex}}=\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(7\right)\right),$

(vi) $\underset{\begin{array}{l}{\overline{I}}_{j}\left(x\left({\overline{t}}_{i}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}j\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,M\right\}\end{array}}{lim}\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(1\right)\right\}\phantom{\rule{thinmathspace}{0ex}}=\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(8\right)\right),$

(vii) $\underset{\begin{array}{l}{I}_{i}\left(x\left({t}_{i}^{-}\right)\right)\to 0\phantom{\rule{1em}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}all\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}i\phantom{\rule{thinmathspace}{0ex}}\in \left\{1,2,\cdots ,L\right\}\end{array}}{lim}\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(1\right)\right\}\phantom{\rule{thinmathspace}{0ex}}=\left\{\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\mathrm{o}\mathrm{f}\phantom{\rule{thinmathspace}{0ex}}\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m}\left(9\right)\right).$

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, ${z}^{\prime }\left(a\right)={\overline{x}}_{a}$ and ${z}^{″}\left(a\right)={\stackrel{^}{x}}_{a}$, the equation condition ${}_{a}{D}_{t}^{q}z\left(t\right)=f\left(t,z\left(t\right)\right)$ for each t ∈ (a, T] is verified, impulsive conditions $\mathrm{\Delta }z{|}_{t={t}_{i}}={I}_{i}\left(z\left({t}_{i}^{-}\right)\right)$(here i = 1,2,…, L), $\mathrm{\Delta }{z}^{\prime }{|}_{t={\overline{t}}_{i}}={\overline{I}}_{i}\left(z\left({\overline{t}}_{i}^{-}\right)\right)$ (here j = 1,2, …,M) and $\mathrm{\Delta }{z}^{″}{|}_{t={\stackrel{^}{t}}_{l}}={\stackrel{^}{I}}_{i}\left(z\left({\stackrel{^}{t}}_{i}^{-}\right)\right)$ (here l = 1,2,…,N) are satisfied, the restriction of to the interval ${J}_{k}^{\prime }$ (here k = 0,1,2,…, K) is continuous, and conditions (i)-(vii) hold.

Define a function

$x~(t)=x(tk+)+x′(tk+)(t−tk)+x″(tk+)2!(t−tk)2+1Γ(q)∫tkt(t−s)q−1f(s,x(s))dsfort∈(tk,tk+1].$

By the definition of Caputo fractional derivative, we have

$[aDtqx~(t)]t∈(tk,tk+1]=aDtqx(tk+)+x′(tk+)(t−tk)+x″(tk+)2!(t−tk)2+1Γ(q)∫tkt(t−s)q−1f(s,x(s))dst∈(tk,tk+1]=aDtq1Γ(q)∫tkt(t−s)q−1f(s,x(s))dst∈(tk,tk+1]=tkDtq1Γ(q)∫tkt(t−s)q−1f(s,x(s))dst∈(tk,tk+1]=f(t,x(t))|t∈(tk,tk+1].$

Therefore, $\stackrel{~}{x}\left(t\right)$ can meet the condition of fractional derivative and impulsive conditions in (1). But, $\stackrel{~}{x}\left(t\right)$ 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

#### ([2])

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

$aItqx(t)=1Γ(q)∫atx(s)(t−s)1−qds,t>a,q>0,$

where Γ is the gamma function.

#### ([2])

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

$aDtqx(t)=1Γ(n−q)∫atx(n)(s)(t−s)q+1−nds=aItn−qx(n)(t),t>a,0≤n−1

#### ([39])

If the function h(t, x) is continuous, then the initial value problem

$aDtqx(t)=h(t,x(t)),q∈(n,n+1],n∈R+∪{0},x(k)(a)=xak,k=0,1,2,…,n.$

is equivalent to the following nonlinear Volterra integral equation of the second kind,

$x(t)=∑k=0nxakk!(t−a)k+1Γ(q)∫at(t−s)q−1h(s,x(s))ds,$

and its solutions are continuous.

#### ([32])

Let q ∈ (0, 1) and ξ is a constant. Impulsive system

$0Dtqu(t)=h(t,u(t)),t∈J=[0,T],t≠tk,k=1,2,…,m,Δut=tk=Iku(tk−),t=tk,k=1,2,…,m,u(0)=u0.$

is equivalent to the fractional integral equation

$u(t)=u0+1Γ(q)∫0t(t−s)q−1h(s,(u(s))ds,fort∈[0,t1],u0+∑k=1nIk(u(tk−))+1Γ(q)∫0t(t−s)q−1h(s,u(s))ds+ξΓ(q)∑k=1nIk(u(tk−))∫0tk(tk−s)q−1h(s,u(s))ds+∫tkt(t−s)q−1h(s,u(s))ds−∫0t(t−s)q−1h(s,u(s))dsfort∈(tn,tn+1],1≤n≤m,$(10)

provided that the integral in (10) exists.

## 3 Main results

For convenience, let f = f(s, x(s)) and ${\sum }_{i=1}^{0}{y}_{i}=0$ in this section.

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

$x(t)=xa+x¯a(t−a)+x^a2!(t−a)2+∑i=1kIix(ti−)+1Γ(q)∫at(t−s)q−1fds+∑i=1kξ0Ii(x(ti−))1Γ(q)∫ati(ti−s)q−1fds+∫tit(t−s)q−1fds−∫at(t−s)q−1fds+(t−ti)Γ(q−1)∫ati(ti−s)q−2fds+(t−ti)22!Γ(q−2)∫ati(ti−s)q−3fdsfort∈(tk,tk+1],k=0,1,2,…,L.$(11)

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

$x(t)=xa+x¯a(t−a)+x^a2!(t−a)2+∑j=1kI¯jx(t¯j−)(t−t¯j)+1Γ(q)∫at(t−s)q−1fds+∑j=1kξ1I¯j(x(t¯j−))1Γ(q)∫at¯j(t¯j−s)q−1fds+∫t¯jt(t−s)q−1fds−∫at(t−s)q−1fds+(t−t¯j)Γ(q−1)∫at¯j(t¯j−s)q−2fds+(t−t¯j)22!Γ(q−2)∫at¯j(t¯j−s)q−3fdsfort∈(t¯k,t¯k+1],k=0,1,2,...,M.$(32)

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

$x(t)=xa+x¯a(t−a)+x^a2!(t−a)2+12!∑l=1kI^lx(t^l−)(t−t^l)2+1Γ(q)∫at(t−s)q−1fds+∑l=1kξ2I^l(x(t^l−))1Γ(q)∫at^l(t^l−s)q−1fds+∫t^lt(t−s)q−1fds−∫at(t−s)q−1fds+(t−t^l)Γ(q−1)∫at^l(t^l−s)q−2fds+(t−t^l)22!Γ(q−2)∫at^l(t^l−s)q−3fdsfort∈(t^k,t^k+1],k=0,1,2,...,N.$(33)

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

$x″(t)=x^a+1Γ(q−2)∫at(t−s)q−3f(s,x(s))ds+ξ0∑i=1kIi(x(ti−))Γ(q−2)∫ati(ti−s)q−3fds+∫tit(t−s)q−3fds−∫at(t−s)q−3fdsfort∈(tk,tk+1],k=0,1,2,...,L.$

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

$x″(t)=x^a+1Γ(q−2)∫at(t−s)q−3fds+ξ1∑j=1kI¯j(x(t¯j−))Γ(q−2)∫at¯j(t¯j−s)q−3fds+∫t¯jt(t−s)q−3fds−∫at(t−s)q−3fdsfort∈(t¯k,t¯k+1],k=0,1,2,...,M.$

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

$x″(t)=x^a+∑l=1kI^lx(t^l−)+1Γ(q−2)∫at(t−s)q−3fds+ξ2∑l=1kI^l(x(t^l−))Γ(q−2)∫at^l(t^l−s)q−3fds+∫t^lt(t−s)q−3fds−∫at(t−s)q−3fdsfort∈(t^k,t^k+1],k=0,1,2,...,N.$

Impulses ${\mathrm{\Delta }x|}_{t={t}_{i}},\phantom{\rule{thickmathspace}{0ex}}{\mathrm{\Delta }{x}^{\prime }|}_{t={\overline{t}}_{j}}$ and ${\mathrm{\Delta }{x}^{″}|}_{t={\stackrel{^}{t}}_{l}}$ have similar effect on x″(t) of (3) by Corollaries 3.4-3.6. Thus, we will consider ${\mathrm{\Delta }x|}_{t={t}_{i}}$ and ${\mathrm{\Delta }{x}^{\prime }|}_{t={\overline{t}}_{j}}$ as some special impulses ${\mathrm{\Delta }{x}^{″}|}_{t={\stackrel{^}{t}}_{l}}$ 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

$x″(t)=x^a+∑l=1k2I^lx(t^l−)+1Γ(q−2)∫at(t−s)q−3fds+ξ0∑i=1k0Ii(x(ti−))Γ(q−2)∫ati(ti−s)q−3fds+∫tit(t−s)q−3fds−∫at(t−s)q−3fds+ξ1∑j=1k1I¯j(x(t¯j−))Γ(q−2)∫at¯j(t¯j−s)q−3fds+∫t¯jt(t−s)q−3fds−∫at(t−s)q−3fds+ξ2∑l=1k2I^l(x(t^l−))Γ(q−2)∫at^l(t^l−s)q−3fds+∫t^lt(t−s)q−3fds−∫at(t−s)q−3fdsfort∈Jk′,k=0,1,2,...,K.$(34)

Let q ∈ (2,3) and ξb (where b ∈ {0, 1, 2}) are three constants. System (1) is equivalent to the fractional integral equation

$x(t)=xa+x¯a(t−a)+x^a2!(t−a)2+∑i=1k0Iix(ti−)+∑j=1k1I¯jx(t¯j−)(t−t¯j)+12!∑l=1k2I^lx(t^l−)(t−t^l)2+1Γ(q)∫at(t−s)q−1fds+∑i=1k0ξ0Ii(x(ti−))1Γ(q)∫ati(ti−s)q−1fds+∫tit(t−s)q−1fds−∫at(t−s)q−1fds+(t−ti)Γ(q−1)∫ati(ti−s)q−2fds+(t−ti)22!Γ(q−2)∫ati(ti−s)q−3fds+∑j=1k1ξ1I¯j(x(t¯j−))1Γ(q)∫at¯j(t¯j−s)q−1fds+∫t¯jt(t−s)q−1fds−∫at(t−s)q−1fds+(t−t¯j)Γ(q−1)∫at¯j(t¯j−s)q−2fds+(t−t¯j)22!Γ(q−2)∫at¯j(t¯j−s)q−3fds+∑l=1k2ξ2I^l(x(t^l−))1Γ(q)∫at^l(t^l−s)q−1fds+∫t^lt(t−s)q−1fds−∫at(t−s)q−1fds+(t−t^l)Γ(q−1)∫at^l(t^l−s)q−2fds+(t−t^l)22!Γ(q−2)∫at^l(t^l−s)q−3fdsfort∈Jk′,$(35)

here k ∈{0, 1, 2,…, K}, provided that the integral in (35) exists.

For impulsive system (1), we have

$→x(3)(t)=f(t,x(t)),t∈J=[a,T],t≠ti(i=1,2,…,L)andt≠t¯j(j=1,2,…,M)andt≠t^l(l=1,2,…,N),Δxt=ti=Iix(tj−)i=1,2,…,L,Δx′t=t¯j=I¯ix(t¯j−)j=1,2,…,M,Δx″t=t^l=I^lx(t^l−)l=1,2,…,N,x(a)=xa,x′(a)=x¯a,x″(a)=x^a.$(36)

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

$limq→3−⁡x(t)=xa+x¯a(t−a)+x^a2!(t−a)2+12∫at(t−s)2fds,fort∈J0′,xa+x¯a(t−a)+x^a2!(t−a)2+∑i=1k0Iix(ti−)+∑j=1k1I¯jx(t¯j−)(t−t¯j)+12!∑l=1k2I^lx(t^l−)(t−t^l)2+12∫at(t−s)2fds,fort∈Jk′,k=1,2,…K.$(37)

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

$x(t)=xa+x¯a(t−a)+x^a2!(t−a)2+∑i=1kIix(ti−)+I¯ix(t¯i−)(t−t¯i)+12!I^ix(ti−)(t−ti)2+1Γ(q)∫at(t−s)q−1fds+∑i=1kξ0Iix(ti−)+ξ1I¯ix(ti−)+ξ2I^ix(ti−)×1Γ(q)∫ati(ti−s)q−1fds+∫tit(t−s)q−1fds−∫at(t−s)q−1fds+(t−ti)Γ(q−1)∫ati(ti−s)q−2fds+(t−ti)22!Γ(q−2)∫ati(ti−s)q−3fdsfort∈Jk,k=0,1,2,…,K.$(38)

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.

$0Dt9/4x(t)=t,t∈[0,2]∖{1},x(1+)=x(1−)+I(x(1−)),x′(1+)=x′(1−)+I¯(x(1−)),x″(1+)=x″(1−)+I^(x(1−)),x(0)=x0,x′(0)=x¯0,x″(0)=x^0.$(39)

Next, we give the general solution by

$x(t)=x0+x¯0t+x^02!t2+1Γ(94)169×13t134fort∈[0,1],x0+x¯0t+x^02!t2+I(x(1−))+I¯(x(1−))(t−1)+12!I^(x(1−))(t−1)2+1Γ(94)169×13t134t≥0+ξ0I(x(1−))+ξ1I¯(x(1−))+ξ2I^(x(1−))1Γ(94)169×13+1Γ(94)99×13(t−1)94(4t+9)t≥1−1Γ(94)169×13t134t≥0+(t−1)Γ(94+1)t≥1+(t−1)22!Γ(94)t≥1fort∈(1,2].$(40)

where ξ0, ξ1 and ξ2 are arbitrary constants.

Next, for Eq. (40), we have

$0Dt94x(t)=0Dt94x0+x¯0t+x^02!t2+1Γ(94)169×13t134=tfort∈[0,1],$

and

$0Dt94x(t)=0Dt94x0+x¯0t+x^02!t2+I(x(1−))+I¯(x(1−))(t−1)+12!I^(x(1−))(t−1)2+1Γ(94)169×13t134t≥0+ξ0I(x(1−))+ξ1I¯(x(1−))+ξ2I^(x(1−))1Γ(94)169×13+1Γ(94)49×13(t−1)94(4t+9)t≥1−1Γ(94)169×13t134t≥0+(t−1)Γ(94+1)t≥1+(t−1)22!Γ(94)t≥1t∈(1,2]=0Dt941Γ(94)169×13t134t≥0+ξ0I(x(1−))+ξ1I¯(x(1−))+ξ2I^(x(1−))Γ(94)49×13(t−1)94(4t+9)t≥1−4t134t≥0t∈(1,2]=tt∈[0,2]t∈(1,2]+ξ0I(x(1−))+ξ1I¯(x(1−))+ξ2I^(x(1−))[tt∈(1,2]−tt∈[0,2]]t∈(1,2]=t,t∈(1,2].$

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

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

$x(1+)=x(1−)+I(x(1−)),x′(1+)=x′(1−)+I¯(x(1−))andx″(1+)=x″(1−)+I^(x(1−)).$

Moreover, we can verify that Eq. (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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Accepted: 2016-06-08

Published Online: 2016-07-08

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455,

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