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R E S E A R C H

Open Access

Multi-point boundary value problems for

a coupled system of nonlinear fractional

differential equations

Chengbo Zhai

*

and Mengru Hao

*Correspondence:

cbzhai@sxu.edu.cn

School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, P.R. China

Abstract

In this paper, we investigate the existence and uniqueness of solutions to the coupled system of nonlinear fractional differential equations

–Dν1

0+y1(t) =

λ

1a1(t)f(y1(t),y2(t)), –Dν2

0+y2(t) =

λ

2a2(t)g(y1(t),y2(t)),

where0+ is the standard Riemann-Liouville fractional derivative of order

ν

,t∈(0, 1),

ν

1,

ν

2∈(n– 1,n] forn> 3 andnN, and

λ

1,

λ

2> 0, with the multi-point boundary value conditions:y1(i)(0) = 0 =y2(i)(0), 0≤in– 2;0+y1(1) =

m–2

i=1 biDβ0+y1(

ξ

i);

0+y2(1) =

m–2

i=1 biDβ0+y2(

ξ

i), wheren– 2 <

β

<n– 1, 0 <

ξ

1<

ξ

2<· · ·<

ξ

m–2< 1, bi≥0 (i= 1, 2,. . .,m– 2) with

ρ

1:=

m–2

i=1 bi

ξ

1–β–1< 1, and

ρ

2:=

m–2

i=1 bi

ξ

2–β–1< 1.

Our analysis relies on the Banach contraction principle and Krasnoselskii’s fixed point theorem.

MSC: 26A33; 34B18; 34B27

Keywords: existence and uniqueness; solutions; Riemann-Liouville fractional derivative; multi-point boundary value problems; Banach contraction principle; Krasnoselskii’s fixed point theorem

1 Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbi-trary non-integer order. The first definition of fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer derivative and integral, as a generalization of the traditional integer order differential and integral calculus, was mentioned already in  by Leibniz and L’Hospital. With the help of fractional calculus, we can describe natural phenomena and mathematical models more accurately. The fractional differential equations play an important role in various fields of engineering, physics, economics and biological sciences,etc.(see [–] for example). In consequence, the subject of fractional differential equations is gaining much importance and attention. For more details on basic theory of fractional differential equations, one can see the monographs of Diethelm [], Kilbaset al.[], Miller and Ross [], Podlubny [] and Tarasov [], and the papers [–] and the references therein.

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As is known to all, the initial and boundary value problems for nonlinear fractional dif-ferential equations arise from the study of models of control, porous media, electrochem-istry, viscoelasticity, electromagnetics,etc.Recently, the existence and uniqueness of solu-tions of initial and boundary value problems for nonlinear fractional equasolu-tions have been extensively studied (see [–]), and some are coupled systems of nonlinear fractional differential equations (see [, , , , ]).

In [], Mophou studied the mild solutions to impulsive fractional differential equations ⎧

⎪ ⎨ ⎪ ⎩

tx(t) =Ax(t) +f(t,x(t)), tI= [,T],t=tk,

x() =x∈X,

x|t=tk=Ik(x(t

k)), k= , , . . . ,m,

where  <α< , the operatorA:D(A)⊂XXis a generator ofC-semigroup (T(t))t≥

on a Banach space X,

t is the Caputo fractional derivative,f :I×XX is a given

continuous function,Ik:XX,  =t<t<· · ·<tm<tm+=T,x|t=tk=x(t

+

k) –x(tk–).

Some existence and uniqueness results for the equations were established by means of Krasnoselskii’s fixed point theorem.

By using the same fixed point theorem, Goodrich [] considered the existence of a positive solution to the following system of differential equations of fractional order:

+y(t) =λa(t)f(y(t),y(t)),

+y(t) =λa(t)g(y(t),y(t)),

()

where

+ is the standard Riemann-Liouville fractional derivative of order ν, t∈(, ), ν,ν∈(n– ,n] forn>  andnN, andλ,λ> , with the following boundary value

conditions:

y(i)() =  =y(i)(), ≤in– ,

+y(t)

t==  = D

α

+y(t)

t=, ≤αn– ,

under the assumptions thata,a,f,gare nonnegative and continuous.

Very recently, Sun et al.[] considered the coupled system of multi-term nonlinear fractional differential equations

u(t) =f(t,v(t),Dβv(t), . . . ,DβNv(t)), Dαiu() = , i= , , . . . ,n

,

v(t) =g(t,u(t),Dρu(t), . . . ,DρNu(t)), Dσjv() = , j= , , . . . ,n

,

wheret∈(, ],α>β>β>· · ·>βN> ,σ>ρ>ρ· · ·>ρN > ,n= [α] + ,n= [σ] + ,

βq,ρq<  andq∈ {, , . . . ,N}. By using the Schauder fixed point theorem and the Banach

contraction principle, some results of existence and uniqueness of solutions for the cou-pled system are obtained.

However, to our knowledge, there are few works that deal with multi-point boundary value problems for a coupled system of nonlinear fractional differential equations. The purpose of this article is to investigate the solutions for the coupled system of nonlinear fractional differential equations () with the multi-point boundary conditions:

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+y() = m–

i=

biDβ+y(ξi), ()

+y() = m–

i=

biDβ+y(ξi), ()

wheren–  <β <n– ,  <ξ<ξ<· · ·<ξm–< ,bi≥ (i= , , . . . ,m– ) withρ:=

m– i= biξ

ν–β–

i < , andρ:=

m– i= biξ

ν–β–

i < .

Motivated by the above-mentioned works and recent works on coupled systems of frac-tional differential equations, we consider the existence and uniqueness of solutions of cou-pled system ()-() by means of the Banach contraction principle and Krasnoselskii’s fixed point theorem. In our paper, we do not suppose thata,a,f,gare nonnegative.

With this context in mind, the outline of this paper is as follows. In Section  we recall certain results from the theory of continuous fractional calculus. In Section  we provide some conditions under which problem ()-() will have a unique solution or at least one solution.

2 Preliminaries

For the convenience of the reader, we present here some definitions, lemmas and basic results that will be used in the proofs of our theorems.

Definition .(see []) Letν>  withνR. Suppose thaty: [a, +∞)→R. Then theνth Riemann-Liouville fractional integral is defined to be

Dν

a+y(t) :=  (ν)

t

a

y(s)(ts)ν–ds,

whenever the right-hand side is defined. Similarly, withν>  andνR, we define theνth Riemann-Liouville fractional derivative to be

a+y(t) :=  (nν)

dn

dtn

t

a

y(s) (ts)ν+–nds,

wherenNis the unique positive integer satisfyingn– ≤ν<nandt>a.

Lemma .(see []) LetαR.Then DnDα

a+y(t) =Dna++αy(t),for each nN,where y(t)is

assumed to be sufficiently regular so that both sides of the equality are well defined. More-over,ifβ∈(–∞, ]andγ ∈[, +∞),then Dγa+Daβ+y(t) =Daγ++βy(t).

Lemma .(see []) The general solution to Dν

a+y(t) = ,where n–  <νn andν> , is the function y(t) =c–+c–+· · ·+cntνn,where ciRfor each i.

Lemma . Let hCn([, ])be given.Then the unique solution to problemDν

+y(t) = h(t)together with the boundary conditions y(i)() = and Dβ

+y() = m–

i= biDβ+y(ξi),where

n–  <β<n– and≤in– ,is

y(t) = 

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where

G(t,s) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

–(ts()νν)– +(ν)(––ρ)[( –s)νβ–m–

i= bi(ξis)νβ–],

st,ξi–<sξi,i= , , . . .m– , –

(ν)(–ρ)[( –s)

νβ–m–

i= bi(ξis)νβ–],

ts,ξi–<sξi,i= , , . . .m– ,

()

is the Green function for this problem,whereρ=mi=–biξ

νβ–

i < andξ= ,ξm–= .

Proof We know that the general solution to our problem is

y(t) =c–+c–+· · ·+cntνnD–+νh(t),

we immediately observe that the boundary value conditiony(i)() = , in–, implies

thatc=c=· · ·=cn= . On the other hand,D

β

+y() = m–

i= biD

β

+y(ξi) implies that

– 

(νβ) 

( –s)νβ–h(s)ds+c

(ν) (νβ)

=

m–

i=

bi

– 

(νβ) ξi

(ξis)νβ–h(s)ds+c

(ν) (νβ)ξ

νβ– i

.

That is to say,

c=

(ν)( –ρ)

( –s)νβ–h(s)ds

m–

i=

bi

ξi

(ξis)νβ–h(s)ds

.

Therefore, the unique solution is

y(t) = – t

(ts)ν–

(ν) h(s)ds

+ t

ν–

(ν)( –ρ) 

( –s)νβ–h(s)ds m–

i=

bi

ξi

(ξis)νβ–h(s)ds

= 

G(t,s)h(s)ds.

The proof is complete.

3 Main results

This section deals with the existence and uniqueness of solutions to problem ()-(). LetErepresent the Banach space ofC[, ] when equipped with the usual supremum norm · . Then putX:=E×E, whereXis equipped with the norm

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for (y,y)∈X. Observe thatXis also a Banach space (see []). In addition, define two

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(H) The parametersλ,λsatisfyλ,λ<, where

We are ready to state the existence and uniqueness result.

Theorem . Suppose that conditions(H)-(H)are satisfied.Then the boundary value

problem()-()has a unique solution.

Proof Let us set

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×fs,y(s),y(s)

fs,u(s),u(s)ds

+

m–

i=

bi

ξi

(ξis)νβ–λa(sf

s,y(s),y(s)

fs,u(s),u(s)ds

λD(y–u,y–u)+λ

(ν–β)

(ν)( –ρ)

×

m–

i=

biD–(ν–β)L(ξi) +D–(ν–β)L()

(y–u,y–u)

≤ 

(y–u,y–u).

That is to say, T(y,y) –T(u,u) ≤ (y–u,y–u) = (y,y) – (u,u) .

Hence, we find thatT:rBr

 andTis a contraction, whereBr ={yB: y

r }.

Similarly, we haveT:rBr

 andTis a contraction. Consequently, for any (y,y)∈ r,

S(y,y)=T(y,y),T(y,y)≤

r +

r  ≤r,

i.e.,S(r)⊂r. And, for (y,y), (u,u)∈rand for eacht∈[, ],

S(y,y) –S(u,u)≤

(y–u,y–u)= 

(y,y) – (u,u).

So,S:rrandSis a contraction. Thus, the conclusion of the theorem follows from

the contraction mapping principle.

Our next result is based on the following well-known fixed point theorem due to Kras-noselskii.

Lemma .(Krasnoselskii []) Let K be a closed convex and nonempty subset of a Banach space E.Let T,S be the operators such that:

(i) Tx+SyKwheneverx,yK; (ii) Tis compact and continuous;

(iii) Sis a contraction mapping.

Then there exists zK such that z=Tz+Sz.

Now we are ready to state and prove the following existence result.

Theorem . Suppose that conditions(H)-(H), (H), (H)are satisfied.Then there exists

at least one solution of the boundary value problem()-().

Proof Let us fix

r≥max

 λμ L

(ν+ )

+ 

(ν)( –ρ)(ν–β)

m–

i=

biξiνβ+ 

,

λμ L

(ν+ )

+ 

(ν)( –ρ)(ν–β)

m–

i=

biξiν–β+ 

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and considerr={(y,y)∈X: (y,y) ≤r}. We define the operatorsQandQonr

as

T(y,y) :=Q(y,y) +Q(y,y),

where

Q(y,y)

(t) = –

t

(ts)ν– (ν)

λa(s)f

s,y(s),y(s)

ds,

Q(y,y)

(t) = t ν–

(ν)( –ρ)

( –s)ν–β–λ

a(s)f

s,y(s),y(s)

ds

m–

i=

bi

ξi

(ξis)νβ–λa(s)f

s,y(s),y(s)

ds

.

For all (y,y), (u,u)∈r, we find that

Q(y,y) +Q(u,u)

μ Lλ

(ν+ )

+ 

(ν)( –ρ)(ν–β)

m–

i=

biξiνβ+ 

r

.

Thus,Q(y,y) +Q(u,u)∈r

 for all (y,y), (u,u)∈r. From assumption (H), we have

Q(y,y) –Q(u,u)≤λ

(ν–β)

(ν)( –ρ)

m–

i=

biD–(ν–β)L(ξi) +D–(ν–β)L()

×(y–u,y–u)

≤ 

(y–u,y–u)

= 

(y,y) – (u,u),

where (y,y), (u,u)∈X,t∈[, ]. So,Q is a contraction mapping. We next consider

the operatorQ. Evidently, the continuity off implies that the operatorQis continuous.

Also,Qis uniformly bounded onras

Q(y,y)≤λ

μ L(ν+ )

≤r.

Now, we show thatQ(y,y)(t) is equicontinuous. In fact, sincea,f are bounded on the

compact set [, ] and [, ]×r, respectively, we can define

M=max

sup (t,y,y)∈[,]×r

a(t)f(t,y,y), sup (t,y,y)∈[,]×r

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and we have, for anyt,t∈[, ],

Arzela-Ascoli theorem,Qis compact onr. Similarly, we set

T(y,y) :=P(y,y) +P(y,y),

And we can obtain a similar conclusion to the operatorT. Now, we let

A(y,y) :=

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(I) For all (y,y), (u,u)∈r, we find that

A(y,y) +B(u,u)=Q(y,y) +Q(u,u)+P(y,y) +P(u,u)≤r.

Thus,A(y,y) +B(u,u)∈rfor all (y,y), (u,u)∈r.

(II) For (y,y), (u,u)∈X,t∈[, ],

B(y,y) –B(u,u)=Q(y,y) –Q(u,u)+P(y,y) –P(u,u)

≤ 

(y–u,y–u)= 

(y,y) – (u,u).

So,Bis a contraction mapping.

(III) The continuity ofQandPimplies that the operatorAis continuous. Also,Ais

uniformly bounded onras

A(y,y)=Q(y,y)+P(y,y)≤r.

Moreover, for anyt,t∈[, ],

A(y,y)(t) –A(y,y)(t)

=Q(y,y)(t) –Q(y,y)(t),P(y,y)(t) –P(y,y)(t)

Q(y,y)(t) –Q(y,y)(t)+P(y,y)(t) –P(y,y)(t)

λ

M

(ν+ )

 –t

ν

 +λ

M

(ν+ )

 –t

ν

 ,

which is independent of (y,y). Therefore,A is equicontinuous onr. Hence, by the

Arzela-Ascoli theorem,Ais compact onr.

Thus all the assumptions of Lemma . are satisfied, soS(y,y) =A(y,y) +B(y,y) has

at least one fixed point. Hence, we obtain that ()-() has at least one solution.

4 Conclusions

There are few works that deal with multi-point boundary value problems for a coupled system of nonlinear fractional differential equations. In this article, we study multi-point boundary value problems for a coupled system of nonlinear fractional differential equa-tions ()-(). By using Green’s function, the Banach contraction principle and Krasnosel-skii’s fixed point theorem, we establish some new existence, uniqueness theorems of solu-tions for multi-point boundary value problems for a coupled system of nonlinear fractional differential equations ()-().

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

The research was supported by the Youth Science Foundation of China (11201272) and the Science Foundation of Shanxi Province (2010021002-1).

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