On the concept of general solution for 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 aDtq denote Caputo fractional derivative of order q in interval [a, t], f : J × ℝ → ℝ and, I_iĪ_j, Î_i : ℝ → ℝ are appropriate functions (here i = 1,2,…, L and j = 1,2,…, M and l = 1,2, …, N, respectively), a = t₀ < t₁ < … < t_L < t_L+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+)=limε→0+x(ti+ε) and x(ti−)=limε→0−x(ti+ε) represent the right and left limits of x(t) at t = t_i, 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 J_i = (t_i, t_i + 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

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

3 Main results

For convenience, let f = f(s, x(s)) and ∑i=10yi=0 in this section.