Solvability of a boundary value problem for singular multi-term fractional differential system with impulse effects

R E S E A R C H

Open Access

Solvability of a boundary value problem

for singular multi-term fractional differential

system with impulse effects

Xiaohui Yang

_{and Yuji Liu}

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Guangdong University of Finance and Economics, Guangzhou, 510320, P.R. China

Full list of author information is available at the end of the article

Abstract

In this article, we first of all convert the boundary value problems for impulsive fractional differential equations to integral equations. Second, we construct a weighted function space and prove the completely continuous property of a nonlinear operator. Finally, we establish existence results for solutions of a boundary value problem for a nonlinear impulsive fractional differential system. Our analysis relies on the well-known Schauder fixed point theorem. An example is given to illustrate main results.

MSC: 92D25; 34A37; 34K15

Keywords: impulsive multi-term fractional differential system; boundary value problem; Schauder’s fixed point theorem

1 Introduction

Fractional differential equations have many applications in modeling of physical and chemical processes [, ]. In its turn, mathematical aspects of fractional differential equa-tions and methods of their soluequa-tions were discussed by many authors; see the text books [–]. Existence of solutions or positive solutions of boundary value problems (BVPs for short) of fractional differential equations with or without impulse effects have been stud-ied by many authors; see [–] and [–].

In recent years, some authors have studied the solvability or existence of positive solu-tions of BVPs of fractional differential systems [–]. In order to show motivasolu-tions of this paper, we address some of them.

In [], Su investigated the existence of positive solutions (continuous on [, ]) of the following boundary value problem of nonlinear multi-term fractional differential system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα

+u+f(t,v(t),D

p

+v(t)) = ,  <t< ,

Dβ_+v+g(t,u(t),Dq_+u(t)) = ,  <t< , u() = , u() = ,

v() = , v() = ,

()

whereα,β∈(, ),D+is the Riemann-Liouville fractional derivative,  <p<β– ,  <q<

α– ,γ ηα–_{< , andγ η}β–_{< ,f,g: [, ]×R}+_{×R→Rare continuous functions.}

In [], authors studied the existence of multiple positive solutions (continuous on [, ]) of the following boundary value problem ofN-dimension nonlinear fractional differential system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα

+u+f(t,u(t),Dμ+u (t)) = ,  <t< ,

. . . ,

DαN–

+ uN–+fN–(t,uN(t),D

μN

+uN(t)) = ,  <t< , DαN

+uN+fN(t,u(t),D_μ+Nu(t)) = ,  <t< ,

u() =· · ·=uN() = ,

u() =· · ·=uN() = ,

()

whereαi∈(, ),D+is the Riemann-Liouville fractional derivative,  <μi–<αi–  with μ=μN,fi: [, ]×R+×R→R(i= , , . . . ,N) are continuous functions.

In [], the authors investigated the existence of positive solutions (continuous on [, ]) of the following boundary value problem of nonlinear multi-term fractional differential system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα

+u+f(t,v(t),D

p

+v(t)) = ,  <t< ,

Dβ_+v+g(t,u(t),Dq_+u(t)) = ,  <t< , u() = , u() =γu(η),

v() = , v() =γv(η),

()

whereα,β∈(, ),D+is the Riemann-Liouville fractional derivative,  <p≤β– ,  <

q≤α– ,γ ηα–_{< , andγ η}β–_{< ,f,g: [, ]×R}+_{×R→Rare continuous functions.}

In [], the authors studied the existence of solutions of the following four-point coupled boundary value problem for nonlinear fractional differential equation:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα

+u=f(t,u(t),Dα–+ u(t),v(t),D

β–

+ v(t)),  <t< , Dβ_+v=g(t,u(t),Dα–

+ u(t),v(t),Dβ_+–v(t)),  <t< , I–α

+ u() = , u() =av(ξ), I_–+βv() = , v() =bu(η),

()

where  <α,β< ,D∗_+, andI_∗+ are the standard Riemann-Liouville differentiation and integration, f,g: [, ]×R_{→Rare continuous functions, a,b∈R, ξ,η∈(, ) with}

abξβ–_ηα–_{= .}

In [], the following four-point boundary value problem of multi-term fractional dif-ferential system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα

+u=f(t,v(t),Dm+v(t)),  <t< , Dβ_+v=g(t,u(t),Dn_+u(t)),  <t< , u() =γu(ξ), u() =δu(η), v() =γv(ξ), v() =δv(η),

()

was studied, where  <α,β< ,  <m≤β– ,  <n≤α– ,γ > ,δ> ,  <ξ<η< ,D∗_+ is the standard Riemann-Liouville differentiation, f,g: [, ]×R_{→Rare continuous}

In [], the authors studied a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations involving Caputo type fractional deriva-tives. Sufficient conditions for the existence and uniqueness of positive solutions were es-tablished by using Banach’s fixed point theorem and Krasnoselskii’s fixed point theorem.

In [], the authors studied the following m-point boundary value problem for a cou-pled system of impulsive fractional differential equations at resonance:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα

+u=f(t,v(t),D

p

+v(t)),  <t< , Dβ_+v=g(t,u(t),Dq_+u(t)),  <t< ,

u(ti) =Ai(v(ti),Dp_+v(ti)), Dq_+u(ti) =Bi(v(ti),Dp_+v(ti)), i= , , . . . ,k, v(ti) =Ci(u(ti),Dq+u(ti)), Dp+v(ti) =Di(u(ti),Dq+u(ti)), i= , , . . . ,k, Dα–

+ u() = m

i=aiu(ξi), u() =

m

i=biη–i αu(ηi), Dβ_+–v() =

m

i=civ(ζi), v() =

m i=diθ

–β

i v(ηi),

whereα,β∈(, ),α–q≥,β–p≥,  <ξ<· · ·<ξm< ,  <η<· · ·<ηm< ,  <ζ< · · ·<ζm< , and  <θ<· · ·<θm< ,f,g: [, ]×R→Rsatisfy Carathéodory conditions, Ai,Bi,Ci,Di:R×R→R,w(ti) =w(t_i+) –w(t–_i),Dr_+w(ti) =Dr_+w(t_i+) –Dr_+w(t–_i) with w∈ {u,v}andr∈ {p,q},w(t+

i) andw(ti–) denote the right and left limits ofw(t) att=ti, respectively, and the fractional derivative is understood in the Riemann-Liouville sense. k,m,ai,bi,ci,di(i= , , . . . ,m) are fixed constants satisfyingm_i=ai=m_i=bi=m_i=ci= m

i=di=  and

m i=biηi=

m

i=diθi. This system happens to be at resonance in the sense

that the associated linear homogeneous coupled system

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

Dα

+u= , D

β

+v= ,  <t< ,

Dα– + u() =

m

i=aiDα–+ u(ξi), u() = m

i=biη–i αu(ηi), Dβ_+–v() =

m i=ciD

β–

+ v(ζi), v() = m

i=diθ –β

i v(ηi)

has (u(t),v(t)) = (htα–+htα–,htβ–+htβ–) (hi∈R,i= , , , ) as a nontrivial

solu-tion.

We find in the papers mentioned thatf in fractional differential equations is supposed to be continuous, the solutions obtained are also continuous on [, ]. So it is interest-ing to study the solvability of boundary value problems of sinterest-ingular fractional differential equations and to obtain discontinuous solutions of this kind of problems.

In this paper, we study the existence of solutions of the following boundary value prob-lems (BVP for short) for the multi-term impulsive fractional differential system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα

+u(t) =p(t)f(t,v(t),Dθ_+v(t),D_β+–v(t)), t∈(ti,ti+],i∈Nm,

D_β+v(t) =q(t)g(t,u(t),Dδ_+u(t),D_α+–u(t)), t∈(ti,ti+],i∈Nm,

Dα–

+ u() = , u() = , Dβ_+–v() = , v() = ,

lim_t→t+ i(t–ti)

–α_{u(t) =I}

(ti,v(ti),Dδ_+v(ti),Dβ_+–v(ti)), i∈Nm_ ,

limt→t+_i Dα+–u(t) –Dα_–+ u(ti) =J(ti,v(ti),Dδ+v(ti),Dβ_+–v(ti)), i∈Nm_,

lim_t→t+ i(t–ti)

–β_{v(t) =I}

(ti,u(ti),Dδ_+u(ti),Dα_+–u(ti)), i∈Nm_,

limt→t+_i Dβ+–v(t) –Dβ_+–v(ti) =J(ti,u(ti),Dδ+u(ti),Dα_+–u(ti)), i∈Nm_,

where

(a) α,β∈(, ), <δ<α– , <θ<β– ,Db

a+is the Riemann-Liouville fractional derivative of orderb> with the start pointa∈R, see Definition .,

(b) mis a positive integer, =t<t<t<· · ·<tm<tm+= ,Nm ={, , , . . . ,m}, and

Nm

 ={, , . . . ,m},

(c) p,q∈C_{(, )there existski> –,li≤with +ki+li> ,α+l}

–δ> ,

β+l–θ> such that|p(t)| ≤tk( –t)land|q(t)| ≤tk( –t)l for all

t∈(, ),

(d) f : (, )×R→Ris a(θ,β;p)-Carathéodory function,g: (, )×R→Ra (δ,α;q)-Carathéodory function, see Definition .,I,J:{ti} ×R→Rare

(θ,β)-Carathéodory functions,I,J:{ti} ×R→R(δ,α)-Carathéodory functions,

see Definition ..

A pair of functionsu,v: (, ]→Ris called a solution of () if

v|(ti,ti+],D

θ

+v|(ti,ti+],D

β–

+ v|(ti,ti+]∈C

_(ti,ti

+], i∈Nm,

u|(ti,ti+],D

δ

+u|(ti,ti+],D

α–

+ u|(ti,ti+]∈C

_(ti,ti

+], i∈Nm,

lim

t→t+_i(t–ti)

–β

v(t),lim

t→t+_i(t–ti)

+θ–β

Dθ_+v(t),lim t→t_i+D

β–

+ v(t) are finite, i∈Nm_,

lim

t→t+_i(t–ti)

–α

u(t),lim

t→t+_i(t–ti)

+δ–α

Dδ_+u(t),lim t→t_i+D

α–

+ u(t) are finite, i∈Nm_,

Dα

+uandD

β

+vare anα-well integrable function and aβ-well integrable function, respec-tively, and all equations in () are satisfied. Here the definitions of anα-well integrable function and aβ-well integrable function may be found in Definition ..

The purpose of this paper is to obtain the results on the existence of solutions of BVP () by using the Schauder fixed point theorem [] under some suitable assumptions. The solutions obtained may be discontinuous on [, ].

The main features of our paper are as follows. First of all, compared with the well-known papers [–], we construct a new Banach space and establish the existence results of solutions of (). Second, boundary conditions are different from the known ones and im-pulse effects are imposed. Third, boundary conditions and imim-pulse effects in () imply that solutions obtained in this paper are continuous on (ti,ti+] (i∈Nm) but they may be

unbounded on (, ). Finally, bothpf : (t,x,y)→p(t)f(t,x,y) andqg: (t,x,y)→q(t)g(t,x,y) may be singular att= ,  while in the well-known papers mentioned nonlinearities are supposed to be continuous.

The remainder of the paper is organized as follows: In Section , we present some preliminary results. In Section , the existence results for solutions of BVP () are established. Finally, in Section , we present an example to illustrate the main theo-rems.

2 Preliminary results

Definition .[] The Riemann-Liouville fractional integral of orderα>  of a function h: (a, +∞)→Ris given by

Iα

a+h(t) =  (α)

t

a

(t–s)α–_{h(s)ds, ()}

provided that the right-hand side exists.

Definition .[] The Riemann-Liouville fractional derivative of orderα>  of a function h: (a, +∞)→Ris given by

Dα

a+h(t) =  (n–α)

dn dtn

t

a

h(s)

(t–s)α–n+ds, ()

wheren– ≤α<n, provided that the right-hand side exists.

Lemma .(Schauder fixed point theorem) Letbe a closed convex subset of the Banach space X.Suppose T:→and T is compact(i.e.,bounded sets inare mapped into relatively compact sets).Then T has a fixed point in.

Remark . Leta<b. From (.) in [], we haveDγ_a+I_aγ+h(t) =h(t),t∈[a,b],γ >  iff satisfies some suitable assumptions. For example we know (_ath(s)ds) =h(t) for all con-tinuous functionh: [a,b]→R. From (.), in [], for a functionh: [a,b]→R, ifDα

a+h(t) (Dδ

a+h(t),Dα_a+–h(t)) is integrable, then we haveA,B,C,D∈Rsuch that Iα

a+Dαa+h(t) =h(t) +A(t–a)α–+B(t–a)α–, t∈(a,b], I_aδ+Dδ_a+h(t) =h(t) +C(t–a)δ–, t∈(a,b],

I_aα+–Dα_a–+ h(t) =h(t) +D(t–a)α–, t∈(a,b].

These results are generalizations ofh(s)ds=h(t) +Cand [h(t)dt] =h(t) ifhis an ab-solutely continuous function. Ifhis a piecewise continuous function, what are the results? We give the following lemma.

Lemma . Letα∈(, ).Suppose that h: (, ]→Rsatisfies h|(ti,ti+],D

α–

+ h∈C(ti,ti+]

(i∈Nm_),limt→t+_i(t–ti)–αh(t),andlimt→t_i+Dα+–h(t)exist for all i∈Nm_.We can prove that there exist constants ci,j∈R(i∈Nm,j= , )such that

I_α+Dα_+h(t) =h(t) + i

j=

c,j(t–tj)α–+c,j(t–tj)α–

, t∈(ti,ti+],i∈Nm, ()

and

Dα_+I_α+h(t) =h(t), t∈(, ]. ()

that|h(t)| ≤Atα–_{for allt∈(t}

,t]. We have fort∈(t,t]

_t(t–v)α–_h(v)dv≤

t



(t–v)α–_h(v)dv

≤A t



(t–v)α–_vα–_{dv by}v

t =w

=Atα– 



( –w)α–wα–dw.

Then_t(t–v)α–_{h(v)dvis convergent for allt∈(t}

,t]. So fort∈(,t], we have

Dα_+I_α+h(t) =  ( –α)(α)

t



(t–s)–α s



(s–v)α–h(v)dv ds

= 

( –α)(α) t



t

s

(t–s)–α(s–v)α–dsh(v)dv

= 

( –α)(α) t



(t–v) 



( –w)–αwα–dwh(v)dv

= t



(t–v)h(v)dv

=h(t).

ThenDα

+Iα_+h(t) =h(t),t∈(t,t]. Fort∈(ti+,ti+] (i≥), similar to above discussion, use

thatlim_t→t+ j(t–tj)

–α_{h(t) exists, we know that}tj+ tj (t–v)

α–_h(v)dvandt ti+(t–v)

α–_h(v)dv

are convergent. So_t(t–v)α–_{h(v)dvis convergent. Then similarly to the above discussion}

Dα

+Iα+h(t) =h(t). The proof of () is complete.

Step. We prove (). LetDα

+h(t) =H(t). Sincelimt→+t–αh(t) =A_exists, for any> , we haveA–<t–αh(t) <A+for sufficiently smallt∈(,t]. Then for sufficiently small

t∈(,t], we have

(A–)B( –α,α– )≤(A–)

t



(t–s)–α_sα–_ds<

t



(t–s)–α_h(s)ds

< (A+)

t



(t–s)–αsα–ds= (A+)B( –α,α– ).

Then

(A–)B( –α,α– )≤ lim

t→+ t



(t–s)–αh(s)ds

≤ lim

t→+ t



(t–s)–αh(s)ds≤(A+)B( –α,α– ).

Let →, we get limt→+t

(t–s)

–α_h(s)ds=A

B( –α,α – ). Furthermore, since

limt→+Dα–

+ h(t) = B exists, limt→+(t

(t–s)–

α_{h(s)ds) =( –α)B exists by}

Defini-tion .. So

I_α+H(t) =I_α+Dα_+h(t)

= 

( –α)(α) t



(t–s)α–

s



(s–u)–α_h(u)du

=  ( –α)(α)

t



(t–s)α–d s



(s–u)–αh(u)du

= 

( –α)(α)

(t–s)α–

s



(s–u)–α_h(u)du

t



+ (α– ) t



(t–s)α– s



(s–u)–αh(u)du

ds

= 

( –α)(α)

–tα– lim

s→+ s



(s–u)–αh(u)du

+ t



(t–s)α–

s



(s–u)–α_h(u)du

ds

= 

( –α)(α)

–tα– _lim

s→+ s



(s–u)–α_h(u)du

+ t



(t–s)α–d s



(s–u)–αh(u)du

= 

( –α)(α)

–tα– lim

s→+ s



(s–u)–αh(u)du

–

tα– lim

s→+ s



(s–u)–αh(u)du

+ (α– ) t



(t–s)α– s



(s–u)–αh(u)du ds

= 

( –α)(α)

–tα– lim

s→+ s



(s–u)–αh(u)du

– (α– )tα– lim

s→+ s



(s–u)–αh(u)du

+ (α– ) t



t

s

(t–s)α–(s–u)–αdsh(u)du

by s–u t–u=w

= 

( –α)(α)

–tα–_{( –α)B– (α– )t}α–_A

B( –α,α– )

+ (α– ) t







( –w)α–_w–α_dwh(u)du

=:ctα–+ctα–+h(t).

So

Iα

+Dα+h(t) =Iα+H(t) =h(t) +c_tα–+c_tα–, t∈(,t_]. ()

It follows from () that () holds for i= . Now we suppose that () holds for i= , , , . . . ,n<m. We will prove that () holds fori=n+ . Then by the method of mathe-matical induction, we know () holds for alli∈Nm

.

In fact, suppose that

I_α+Dα_+h(t) =h(t) + n

j=

c,j(t–tj)α–+c,j(t–tj)α–

Use the assumption, we get fort∈(tn+,tn+]

H(t) =Dα_+I_α+H(t) =Dα_+I_α+Dα_+h(t)

= 

( –α) t



(t–s)–α_Iα

+Dα+h(s)ds

= 

( –α) _n

j=

tj+

tj

(t–s)–αI_α+Dα_+h(s)ds+ t

tn+

(t–s)–αI_α+Dα_+h(s)ds

= 

( –α) _n

j=

tj+

tj

(t–s)–α

h(s) + j

ν=

c,ν(s–tν)α–+c,ν(s–tν)α–

ds

+ t

tn+

(t–s)–α

h(s) + n

ν=

c,ν(s–tν)α–+c,ν(s–tj)α–

+(s)

ds

=Dα

t+_n+(t) +D

α +h(t) +

 ( –α)

_n

j=

j

ν=

c,ν

tj+

tj

(t–s)–α_(s–t ν)α–ds

+ n

j=

j

ν=

c,ν

tj+

tj

(t–s)–α(s–tν)α–ds

+ n

ν=

c,ν

t

tn+

(t–s)–α_(s–t

ν)α–ds+

n

ν=

c,ν

t

tn+

(t–s)–α_(s–t

j)α–ds

=Dα_t+

n+(t) +D

α +h(t) +

 ( –α)

_n

j=

j

ν=

c,ν(t–tν)

tj+–tν t–tν tj–tν

t–tν

( –w)–αwα–dw

+ n

j=

j

ν=

c,ν

tj+–tν t–tν tj–tν t–tν

( –w)–αwα–dw

+ n

ν=

c,ν(t–tν)



tn+–tν t–tν

( –w)–αwα–dw

+ n

ν=

c,ν



tn+–tν t–tν

( –w)–α_wα–_dw

=Dα_t+

n+(t) +D

α +h(t) +

 ( –α)

_n

ν=

c,ν(t–tν)

n

j=ν

tj+–tν t–tν tj–tν

t–tν

( –w)–αwα–dw

+ n

ν=

c,ν

n

j=ν

tj+–tν t–tν tj–tν t–tν

( –w)–αwα–dw

+ n

ν=

c,ν(t–tν)



tn+–tν t–tν

( –w)–αwα–dw

+ n

ν=

c,ν



tn+–tν t–tν

( –w)–αwα–dw

=Dα

t+_n+(t) +H(t) +  ( –α)

_n

ν=

c,ν(t–tν)





+ n

ν=

c,ν





( –w)–αwα–dw

=Dα_t+

n+(t) +H(t).

It follows thatDα_t+

n+(t) =  on (tn+,tn+]. Similarly to the proof of (), there exist con-stantsc,n+,c,n+∈Rsuch that(t) =c,n+(t–tn+)α–+c,n+(t–tn+)α–. Substituting

(t) into (), we know that () holds fori=n+ . This completes the proof of ().

Remark . Let δ ∈ (, ). Suppose that h: (, ] →R satisfies h|(ti,ti+] ∈ C

_(ti,ti +]

(i∈Nm_) andlimt→t+_i(t–ti)–δh(t) exists for alli∈Nm. We can prove that there exist

con-stantsci∈R(i∈Nm_) such that

Iδ

+Dδ+h(t) =h(t) +

i

j=

cj(t–tj)δ–, t∈(ti,ti+],i∈Nm,

Dδ

+I_δ+h(t) =h(t), t∈(, ].

Proof The proof is similar to that of Lemma . and is omitted.

Definition . Letb> ,c,d∈Rwithc<dbe fixed. A functionh: (c,d)→Ris called ab-well integrable function on (c,d) if both_t(t–s)b–_|h(s)|dsandt

(t–s)b–h(s)dsare

convergent for allt∈[c,d].

Definition . Letb∈(, ) anda∈(,b– ) andμ: (, )→R.h: (, )×R_→Ris

called a (a,b;μ)-Carathéodory function if (i) t→μ(t)h(t,_(t–tx

i)–b, x

(t–ti)+a–b,x)is ab-well integrable function on(, ]for every (x,x,x)∈R,

(ii) (x,x,x)→h(t,_(t–tx

i)–b, x

(t–ti)+a–b,x)is continuous onR

_{for eacht∈(ti,ti} +)

(i∈Nm_),

(iii) for eachr> , there existsMr> such that|xi| ≤r(i= , , ) imply that

h

t, x (t–ti)–b,

x

(t–ti)+a–b,x

≤Mr, t∈(ti,ti+),i∈Nm.

Definition . Letb ∈(, ) and a∈(,b– ). I:{ti :i∈Nm_} ×R _{→Ris a (a,}

b)-Carathéodory function if (i) (x,x,x)→I(ti,_(t x

i–ti–)–b, x

(ti–ti–)+a–b,x)is continuous onR

_{for eachi∈N}m

 ,

(ii) for eachr> , there existsMI,r> such that|xi| ≤r(i= , , ) imply that

I

ti, x (ti–ti–)–b

, x

(ti–ti–)+a–b

,x

≤MI,r, i∈Nm.

Choose

Xδ,α=

u: (, ]→R,u,Dδu,Dα–u∈C(ti,ti+],i∈Nm,

the following limits exist: lim

t→t+_i(t–ti)

–α

u(t),i∈Nm_,

lim

t→t+_i(t–ti)

+δ–α

Dδ_+u(t),i∈Nm_,lim

t→t+_iD

α–

+ u(t),i∈Nm_

Foru∈Xδ,α, define

u=:uδ,α=max

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

sup

t∈(ti,ti+]

(t–ti)–αu(t)

sup

t∈(ti,ti+]

(t–ti)+δ–αDδ_+u(t)

sup

t∈(ti,ti+]

Dα_–+ u(t)

:i∈Nm_ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ .

Lemma . Xδ,αis a Banach space with the norm defined.

Proof In fact, it is easy to see thatXis a normed linear space with the norm · . Let{xu} be a Cauchy sequence inXδ,α. Thenxu–xv →,u,v→+∞. It follows that

sup

t∈(ti,ti+]

(t–ti)–αxu(t) –xv(t)→, v,u→+∞,i∈Nm,

sup

t∈(ti,ti+]

(t–ti)+δ–αDδ+x_u(t) –D_δ+x_v(t)→, v,u→+∞,i∈Nm_,

sup

t∈(ti,ti+]

Dα_–+ xu(t) –Dα_–+ xv(t)→, v,u→+∞,i∈Nm_.

Denotexu,i=xu|(ti,ti+]. Since

lim

t→t+_i(t–ti)

–α_x

u(t),lim t→t_i+(t–ti)

+δ–α_Dδ

+x(t),lim t→t+_iD

α– + x_u(t)

exist, we know thatt→(t–ti)–α_xu

,i(t) are continuous on [ti,ti+]. Thust→(t–ti)–αxu,i(t)

are Cauchy sequences inC[ti,ti+]. So (t–ti)–αxu,i(t) uniformly converges to somex,iin C[ti,ti+] asu→+∞. It follows that

sup

t∈[ti,ti+]

_(t–ti)–α

xu,i(t) –x,i(t)→, u→+∞,i∈Nm.

That is,

sup

t∈[ti,ti+]

(t–ti)–αxu,i(t) – (t–ti)α–x,i(t)→, u→+∞,i∈Nm.

Letx(t) = (t–ti)α–x,i(t) fort∈(ti,ti+] (i∈Nm). It is easy to see thatx∈C(ti,ti+]

(i∈Nm

) and the limitlimt→t_i+(t–ti)–αx(t) exists for alli∈Nm.

Similarly there existy,i,z,i: (, ]→Rsuch that

sup

t∈[ti,ti+]

(t–ti)+δ–αDδ_+xu,i(t) –y,i(t)→, u→+∞,i∈Nm,

sup

t∈[ti,ti+]

Dα_–+ xu,i(t) –z,i(t)→, u→+∞,i∈Nm.

Let y(t) = (t–ti)α–δ–y,i(t) and z(t) =z,i(t) for t∈(ti,ti+] (i∈Nm). Then y,z ∈

C_(t

i,ti+] (i∈Nm) and the limitslimt→t+_i(t–ti)+δ–αy(t) andlimt→t+_i z(t) exist for all

Furthermore, using Lemma ., fort∈(ti,ti+] there existscuj∈Rsuch that

xu(t) +

i

j=

cuj(t–tj)δ––I_δ+y(t)

=Iδ

+Dδ+xu(t) –Iδ+y_(t)

≤

t



(t–s)δ–

(δ) D

δ

+xu(s) –y(s)ds

= i–

j=

tj+

tj

(t–s)δ–

(δ) (s–tj)

α–δ–_(s–tj)+δ–α

Dδ_+xu(s) – (s–tj)+δ–αy(s)ds

+ t

ti

(t–s)δ–

(δ) (s–ti)

α–δ–_(s–ti)+δ–α

Dδ_+xu(s) – (s–ti)+δ–α

y(s)ds

= i–

j=

tj+

tj

(t–s)δ–

(δ) (s–tj)

α–δ–_(s–tj)+δ–α

Dδ_+xu(s) –y,j(s)ds

+ t

ti

(t–s)δ–

(δ) (s–ti)

α–δ–_(s–t

i)+δ–αDδ+xu(s) –y,i(s)ds

≤

_i–

j=

tj+

tj

(t–s)δ–

(δ) (s–tj)

α–δ–_ds+

t

ti

(t–s)δ–

(δ) (s–ti)

α–δ–_ds

×max

sup

t∈(ti,ti+]

(t–ti)+δ–αD_δ+xu(t) –y,i(t):i∈N

= _i–

j=

(t–tj)α– tj+–tj

t–_tj



( –w)δ–

(δ) w

α–δ–_{dw+ (t–t}

i)α– 



( –w)δ–

(δ) w

α–δ–_dw

×max

sup

t∈(ti,ti+]

(t–ti)+δ–αD_δ+xu(t) –y,i(t):i∈N

≤B(δ,α–δ– )

(δ) i

j=

(t–tj)α–max

sup

t∈(ti,ti+]

(t–ti)+δ–αDδ+x_u(t) –y_,i(t):i∈N_

→ asu→+∞.

Thenlimu→+∞[xu(t) + i

j=cuj(t–tj)δ–] =Iδ+y(t). So (t–ti)α–x,i(t) +

i

j=cj(t–tj)δ–=

Iδ

+y(t) (i∈Nm). It follows thatx(t) +

i

j=cj(t–tj)δ–=Iδ+y(t). Thusy(t) =Dδ+x(t)

fort∈(ti,ti+].

We have similarly fort∈(ti,ti+]

xu(t) +

i

j=

cuj(t–tj)α––Iα+–z_(t)

=I_α+–D_α+–xu(t) –I_α+–z(t)

≤ t 

(t–s)α–

(α– )D

α–

+ xu(s) –z(s)ds

= i–

j=

tj+

tj

(t–s)α–

(α– )D

α–

+ t

ti

(t–s)α–

(α– )D

α–

+ xu,i(s) –z,i(s)ds

≤

t



(t–s)α–

(α– ) dsmax

sup

t∈(ti,ti+]

Dα_–+ xu,i(t) –z,i(t):i∈N

= t

α–

(α)max

sup

t∈(ti,ti+] Dα–

+ xu,i(t) –z,i(t):i∈N

→ asu→+∞.

Thenlimu→+∞[xu(t)+ i

j=cuj(t–tj)α–] =Iα+–z(t). So (t–ti)α–x,i(t)+

i

j=cj(t–tj)α–=

Iα–

+ z(t) (i∈Nm). It follows thatx(t) +ij=cj(t–tj)α–=I_α+–z(t). Thusz(t) =Dα_–+ x(t)

fort∈(ti,ti+].

From the above discussion, we know thatxu→xasu→+∞inXδ,α. It follows that

Xδ,αis a Banach space. The proof is complete.

DenoteE=Xδ,α×Xθ,β. Define(x,y)max{xδ,α,yθ,β}. ThenEis a Banach space. For

y∈Xθ,βandx∈Xδ,α, denote

Fy(t) =p(t)f

t,y(t),Dθ +y(t),D

β–

+ y(t)

,

Gx(t) =q(t)gt,x(t),Dδ_+x(t),Dα_+–x(t)

,

Iy(ti) =I

ti,y(ti),Dθ_+y(ti),Dβ_+–y(ti)

,

Jy(ti) =J

ti,y(ti),Dθ_+y(ti),Dβ_+–y(ti)

,

Ix(ti) =I

ti,x(ti),Dδ_+x(ti),Dα_–+ x(ti)

,

Jx(ti) =J

ti,x(ti),Dδ_+x(ti),Dα_+–x(ti)

.

Lemma . Suppose that(a)-(d)hold and(x,y)∈E.Then(u,v)∈E is a solution of ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dα

+u(t) =p(t)f(t,y(t),Dθ_+y(t),D_β+–y(t)), t∈(ti,ti+],i∈Nm,

Dβ_+v(t) =q(t)g(t,x(t),Dδ_+x(t),D_α–+ x(t)), t∈(ti,ti+],i∈Nm,

Dα–

+ u() = , u() = , Dβ_+–v() = , v() = ,

limt→t+_i(t–ti)–αu(t) =I(ti,y(ti),Dδ+y(ti),D

β–

+ y(ti)), i∈Nm,

lim_t→t+ i D

α–

+ u(t) –Dα_–+ u(ti) =J(ti,y(ti),Dδ+y(ti),Dβ_+–y(ti)), i∈Nm_,

lim_t→t+ i(t–ti)

–β_{v(t) =I}

(ti,x(ti),Dδ+x(ti),Dα+–x(ti)), i∈Nm,

limt→t+ i D

β–

+ v(t) –Dβ_+–v(ti) =J(ti,x(ti),Dδ+x(ti),Dα_+–x(ti)), i∈Nm_,

()

if and only if

u(t) = t



(t–s)α–

(α) Fy(s)ds

– 



( –s)α–

(α) Fy(s)ds+ m

j=

( –tj)α–

(α) Jy(tj) + m

j=

( –tj)α–Iy(tj)

tα–

+ i

j=

Iy(tj)(t–tj)α–+

i

j=

Jy(tj)

(α)(t–tj)

α–_{, t∈(ti,ti}

and

v(t) = t



(t–s)β–

(β) Gx(s)ds

– 



( –s)β–

(β) Gx(s)ds+ m

j=

Ix(tj)( –tj)β–+

m

j=

Jx(tj)

(β)( –tj)

β–

tβ–

+ i

j=

Ix(tj)(t–tj)β–+ i

j=

Jx(tj)

(β)(t–tj)

β–_{, t∈(ti,ti}

+],i∈Nm. ()

Proof Fromx∈Xandy∈Y, we see that there exists a constantr>  such thatx=r< +∞. Then there exist constantsMr,f,Mr,g,M,r,I,M,r,I,M,r,J,M,r,J≥ such that

Fy(t)=p(t)ft,y(t),Dθ_+y(t),Dβ_+–y(t)

=p(t)f

t,(t–ti)

–β_y(t)

(t–ti)–β ,

(t–ti)+θ–β_Dθ +y(t) (t–ti)+θ–β ,D

β–

+ x(t)

≤Mr,ftk( –t)l, t∈(ti,ti+),i∈Nm, ()

and similarly

Gx(t)≤Mr,gtk( –t)l, t∈(ti,ti+),i∈Nm,

I

ti,y(ti),Dθ+y(ti),D

β–

+ y(ti)≤M,r,I, i∈Nm,

I

ti,x(ti),Dδ_+x(ti),Dα_–+ x(ti)≤M,r,I, i∈Nm, ()

J

ti,y(ti),Dθ_+y(ti),Dβ_+–y(ti)≤M,r,J, i∈Nm,

J

ti,x(ti),Dδ+x(t_i),D_α+–x(t_i)≤M_,r,J, i∈Nm_.

Hence

t



(t–s)α–

(α) Fy(s)ds ≤

t



(t–s)α–

(α) Fy(s)ds

≤ t 

(t–s)α–

(α) s

k_{( –s)}l_Mr

,fds

≤Mr,f t



(t–s)α–

(α) s

k_(t–s)l_ds

=Mr,ftα+l+k 



( –w)α+l–

(α) w k_dw

=Mr,ftα+l+k

B(α+l,k+ )

(α) <∞, t∈(, ].

This meansFyis anα-integral function. Similarly we get

t



Fy(s)ds

Fromu∈Xand () in Lemma ., we know that there exist constantsAi,Bi(i∈Nm_) such that

u(t) = t



(t–s)α–

(α) Fy(s)ds+ i

j=

Aj(t–tj)α–+Bj(t–tj)α–,

t∈(ti,ti+],i∈Nm. ()

So

Dδ +u(t) =

t



(t–s)α–δ–

(α–δ) Fy(s)ds+ (α) (α–δ)

i

j=

Aj(t–tj)α–δ–

+ (α– ) (α–δ– )

i

j=

Bj(t–tj)α–δ–, t∈(ti,ti+],i∈Nm, ()

and

Dα– + u(t) =

t



Fy(s)ds+(α) i

j=

Aj, t∈(ti,ti+],i∈Nm. ()

FromDα–

+ u() =  andu() =  imply thatA=  and





( –s)α–

(α) Fy(s)ds+ m

j=

Aj( –tj)α–+Bj( –tj)α–

= . ()

Now we get by using the impulse conditions

Bi=I

ti,y(ti),Dθ_+y(ti),Dβ_+–y(ti)

, i∈Nm_, (α)Ai=J

ti,y(ti),Dθ_+y(ti),Dβ_–+ y(ti)

, i∈Nm_. Together with (), we obtain

B= –





( –s)α–

(α) Fy(s)ds– m

j=

( –tj)α–

(α) Jy(tj) + ( –tj)

α–_I y(tj)

.

SubstituteAi,Biinto (), we get (). Similarly we get ().

Now we suppose thatusatisfies () andvsatisfies (). We will prove thatu∈Xand v∈Y,u,vis a solution of BVP ().

It is easy to see thatu∈X,v∈Y. Furthermore, by direct computation, we get

Dα_+u(t) =Fy(t), t∈(ti,ti+],i∈Nm,

Dα_+–u() = , u() = ,

Dβ_+v(t) =Gx(t), t∈(ti,ti+],i∈Nm,

Dβ_+–v() = , v() = ,

lim

t→t+_i(t–ti)

–α

lim

t→t+ i

D_α+–u(t) –D_α+–u(ti) =Jy(ti), i∈Nm,

lim

t→t+_i(t–ti)

–β_{v(t) =I}

y(ti), i∈Nm,

lim

t→t+i

D_β+–v(t) –D_β+–v(ti) =Jy(ti), i∈Nm .

Then (u,v) is a solution of BVP (). The proof is completed.

For (x,y)∈E, defineT(x,y) byT(x,y)(t) = ((Ty)(t), (Tx)(t)) with

(Ty)(t) =

t



(t–s)α–

(α) Fy(s)ds

– 



( –s)α–

(α) Fy(s)ds+ m

j=

( –tj)α– (α) Jy(tj) +

m

j=

( –tj)α–Iy(tj)

tα–

+ i

j=

Iy(tj)(t–tj)α–+

i

j=

Jy(tj)

(α)(t–tj)

α–_{, t∈(ti,ti}

+],i∈Nm, ()

and

(Tx)(t) =

t



(t–s)β–

(β) Gx(s)ds

– 



( –s)β–

(β) Gx(s)ds+ m

j=

Ix(tj)( –tj)β–+

m

j=

Jx(tj)

(β)( –tj)

β–

tβ–

+ i

j=

Ix(tj)(t–tj)β–+ i

j=

Jx(tj)

(β)(t–tj)

β–_{, t∈(t}

i,ti+],i∈Nm. ()

Lemma . Suppose that(a)-(d)hold.Then T:E→E is well defined and is completely continuous, (x,y)is a solution of BVP()if and only if(x,y) =T(x,y).

Proof By Lemma ., we know thatTy∈XandTx∈Y. ThenT:E→Eis well defined.

It is easy to show from Lemma . that (x,y) is a solution of BVP () if and only if (x,y) = T(x,y).

Now, we prove that T is completely continuous. It suffices to prove that T:Y→X

andT:X→Yare completely continuous. We divide the proof of completely continuous

property ofTinto four steps. Similarly we can prove the completely continuous property

ofT.

Step. Prove thatTis continuous.

Letyn∈Y(n= , , , . . .) withyn→yasn→+∞. We will prove thatTyn→Tyas

n→+∞. It is easy to show that there existsr>  such thatyn ≤r,n= , , , . . . , and

yn–y → asn→+∞. Then there exist constantsMr,f,M,r,I,M,r,J≥ such that Fyn(t)≤Mr,ft

k_{( –t)}l_{, t∈(ti,ti}

+),i∈Nm,n= , , , . . . ,

I

ti,yn(ti),Dθ_+yn(ti),Dβ_+–yn(ti)≤M,r,I, i∈Nm,n= , , , . . . , ()

J

Using the definition in (), one sees fort∈(ti,ti+] that

(t–ti)–α(Tyn)(t) – (Ty)(t)

≤(t–ti)–α t



(t–s)α–

(α) Fyn(s) –Fy(s)ds

+ (t–ti)–α 



( –s)α–

(α) Fyn(s) –Fy(s)ds+ m

j=

( –tj)α–Iyn(tj) –Iy(tj)

+ m

j=

( –tj)α–

(α) Jyn(tj) –Jy(tj)

tα–_{+ (t–t}

i)–α i

j=

(t–tj)α–Iyn(tj) –Iy(tj)

+ (t–ti)–α i

j=

|Jyn(tj) –Jy(tj)| (α) (t–tj)

α–

≤Mr,f(t–ti)–α t



(t–s)α–

(α) s

k_{( –s)}l_ds

+ Mr,f 



( –s)α–

(α) s

k_{( –s)}l_{ds+ M}

,r,I m

j=

( –tj)α–+ M,r,J m

j=

( –tj)α–

(α)

+ mM,r,I+ M,r,J m

j=

tj+–tj

(α)

≤Mr,f(t–ti)–αtα+k+l 



( –w)α+l–

(α) w k_dw

+ Mr,f

B(α+l,k+ )

(α) + M,r,I m

j=

( –tj)α–+ M,r,J m

j=

( –tj)α– (α)

+ mM,r,I+ M,r,J m

j=

tj+–tj

(α)

≤Mr,f

B(α+l,k+ )

(α) + M,r,I _m

j=

( –tj)α–+m

+ M,r,J m

j=

( –tj)α–

(α) + tj+–tj

(α)

.

Similarly we can prove fort∈(ti,ti+] that

(t–ti)+δ–αDδ_+(Tyn)(t) –Dδ+(Ty)(t)

≤Mr,f

B(α+l–δ,k+ )

(α–δ) +

(α– ) (α–δ– )

B(α+l,k+ )

(α)

+ M,r,J

(α– ) (α–δ– )

m

j=

( –tj)α–

(α) + (α) (α–δ)

m

j=

tj+–tj

(α)

+ M,r,I

(α– ) (α–δ– )

m

j=

( –tj)α–+

(α– ) (α–δ– )m

and

Dα–

+ (T_y_n)(t) –D_α–+ (T_y_)(t)

≤

t



Fyn(s) –Fy(s)ds+ i

j=

Jyn(tj) –Jy(tj)≤Mr,fB(l+ ,k+ ) + mM,r,J.

By Lebesgue’s dominated convergence theorem, we can show that

lim

n→∞_t∈(sup_t i,ti+]

(t–ti)–α(Txn)(t) – (Tx)(t)= ,

lim

n→∞_t∈(sup_t

i,ti+]

(t–ti)+δ–αDδ+(Txn)(t) –Dδ+(Tx)(t)= ,

lim

n→∞_t∈(sup_t i,ti+]

Dα_+–(Txn)(t) –D_α+–(Tx)(t)= .

HenceTxn–Tx → asn→ ∞. ThenT is continuous.

Let⊆Xand⊆Ybe bounded sets ofXandY, respectively. Then there existsr> 

such thatx,y ≤r,x∈,y∈. So there exist constantsMr,f,Mr,g,M,r,I,M,r,I,M,r,J,

M,r,J≥ such that () and () hold for allx∈,y∈.

Step. Prove that{Ty:y∈}is bounded.

We have similarly to Step  fort∈(ti,ti+]

(t–ti)–α(Ty)(t)≤Mr,f

B(α+l,k+ )

(α) +M,r,I _m

j=

( –tj)α–+m

+M,r,J m

j=

( –tj)α–

(α) + tj+–tj

(α)

,

(t–ti)+δ–αDδ_+(Ty)(t)

≤Mr,f

B(α+l–δ,k+ )

(α–δ) +

(α– ) (α–δ– )

B(α+l,k+ )

(α)

+M,r,J

(α– ) (α–δ– )

m

j=

( –tj)α– (α) +

(α) (α–δ)

m

j=

tj+–tj (α)

+M,r,I

(α– ) (α–δ– )

m

j=

( –tj)α–+ (α– ) (α–δ– )m

,

Dα_+–(Ty)(t)≤Mr,fB(l+ ,k+ ) +mM,r,J.

It follows that there exists a constantM>  such thatTy ≤Mfor ally∈. Similarly

we see that there exists a constantM>  such thatTx ≤Mfor allx∈. From the

above discussion, we see that{T(x,y) :x∈,y∈}is bounded.

Step. Prove that

t→(t–ti)–α(Ty)(t) :y∈

,

t→(t–t)+δ–αDδ_+(Ty)(t) :x∈

t→Dα_+–(Ty)(t) :y∈

are equi-continuous on each (ti,ti+] (i∈Nm).

SinceT:X→X, we can take

(t–ti)–α(Tx)(t)|t=ti=limt→t

+

i(t–ti)–α(Tx)(t), (t–ti)+δ–αDδ+(Tx)(t)|t=ti=lim

t→t+_i(t–ti)

+δ–α_Dδ

+(Tx)(t),

Dα_+–(Tx)(t)|t=ti=lim t→t+i

Dα_+–(Tx)(t).

Then

(t–ti)–α(Tx)(t), (t–ti)+δ–αDδ_+(Tx)(t),Dα_+–(Tx)(t)

are continuous on [ti,ti+] for eachx∈. So

t–α_(Tx)(t),t+δ–α_Dδ

+(Tx)(t),Dα_–+ (Tx)(t)

are uniformly continuous, for anyε> , there existsδ> , whens,s∈[ti,ti+],|s–s|<

δ, andx∈, we can get

(s–ti)–α(Tx)(s) – (s–ti)–α(Tx)(s)<ε,

(s–ti)+δ–αDδ+(Tx)(s) – (s–ti)–α(Tx)(s)<ε,

Dα_+–(Tx)(s) –Dα+–(Tx)(s)<ε.

This shows us that

t→t–α_{(Tx)(t) :x∈,}

t→t+δ–α_Dδ

+(Tx)(t) :x∈

,

t→Dα–

+ (Tx)(t) :x∈

are equi-continuous on any closed subinterval of [ti,ti+] (i∈Nm).

It follows from Steps - thatTis completely continuous. The proof is ended.

3 Main results

In this section, we prove the main theorem. Suppose thatσj,τj≥ (j= , , ) are constants. We need the following assumptions:

(H) there exist nonnegative constantsAj,Bj(j= , , ) and two functionsφ,ψsuch

thatpψis anα-well integrable function andqφaβ-well integrable function and

f

t, y

(t–ti)–β,

y

(t–ti)–β+θ,y

–ψ(t)

≤ 

j=

Aj|yj|σj_{, t}∈_(ti,ti