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R E S E A R C H

Open Access

Boundary value problems for fractional

differential equations

Zhigang Hu

*

, Wenbin Liu and Jiaying Liu

*Correspondence:

xzhzgya@126.com

Department of Mathematics, China University of Mining and Technology, Xuzhou, 221008, P.R. China

Abstract

In this paper we study the existence of solutions of nonlinear fractional differential equations at resonance. By using the coincidence degree theory, some results on the existence of solutions are obtained.

MSC: 34A08; 34B15

Keywords: fractional differential equations; boundary value problems; resonance; coincidence degree theory

1 Introduction

In recent years, the fractional differential equations have received more and more atten-tion. The fractional derivative has been occurring in many physical applications such as a non-Markovian diffusion process with memory [], charge transport in amorphous semi-conductors [], propagations of mechanical waves in viscoelastic media [],etc. Phenom-ena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science are also described by differential equations of fractional order (see [–]).

Recently boundary value problems (BVPs for short) for fractional differential equations have been studied in many papers (see [–]).

In [], by means of a fixed point theorem on a cone, Agarwalet al.considered two-point boundary value problem at nonresonance given by

+x(t) +f(t,x(t),Dμ+x(t)) = ,

x() =x() = ,

where  <α< ,μ>  are real numbers,αμ≥ and

+is the Riemann-Liouville

frac-tional derivative.

Zhaoet al.[] studied the following two-point BVP of fractional differential equations:

+x(t) =f(t,x(t)), t∈(, ),

x() =x() =x() = ,

where

+ denotes the Riemann-Liouville fractional differential operator of order α,

 <α≤. By using the lower and upper solution method and fixed point theorem, they obtained some new existence results.

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Liang and Zhang [] studied the following nonlinear fractional boundary value prob-lem:

+x(t) =f(t,x(t)), t∈(, ),

x() =x() =x() =x() = ,

where  <α≤ is a real number,+ is the Riemann-Liouville fractional differential

op-erator of orderα. By means of fixed point theorems, they obtained results on the existence of positive solutions for BVPs of fractional differential equations.

In [], Bai considered the boundary value problem of the fractional order differential equation

+x(t) +a(t)f(t,x(t),x(t)), t∈(, ),

x() =x() =x() =x() = ,

where  <α≤ is a real number,

+ is the Riemann-Liouville fractional differential

op-erator of orderα.

Motivated by the above works, in this paper, we consider the following BVP of fractional equation at resonance

+x(t) =f(t,x(t),x(t),x(t),x(t)), t∈(, ),

x() =x() =x() = , x() =x(), (.)

where

+ denotes the Caputo fractional differential operator of orderα,  <α≤.f :

[, ]×R→ ×Ris continuous.

The rest of this paper is organized as follows. Section  contains some necessary no-tations, definitions and lemmas. In Section , we establish a theorem on existence of so-lutions for BVP (.) under nonlinear growth restriction off, basing on the coincidence degree theory due to Mawhin (see []). Finally, in Section , an example is given to illus-trate the main result.

2 Preliminaries

In this section, we introduce notations, definitions and preliminary facts which are used throughout this paper.

LetXandY be real Banach spaces and letL:domLXY be a Fredholm operator with index zero, andP:XX,Q:YYbe projectors such that

ImP=KerL, KerQ=ImL,

X=KerL⊕KerP, Y=ImL⊕ImQ.

It follows that

L|domL∩KerP:domL∩KerP→ImL

is invertible. We denote the inverse byKP.

Ifis an open bounded subset ofX, anddomL= ∅, the mapN:XY will be calledL-compact onifQN() is bounded andKP(I–Q)N:Xis compact, where

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Lemma .([]) Ifis an open bounded set,let L:domLXY be a Fredholm op-erator of index zero and N:XY L-compact on.Assume that the following conditions are satisfied:

() Lx=λNxfor every(x,λ)∈[(domL\KerL)]×(, ); () Nx∈/ImLfor everyx∈KerL;

() deg(QN|KerL,KerL, )= ,whereQ:YYis a projection such that

ImL=KerQ.

Then the equation Lx=Nx has at least one solution indomL.

Definition . The Riemann-Liouville fractional integral operator of order α>  of a functionxis given by

Iα+x(t) =

(α)

t

(t–s)α–x(s)ds,

provided that the right side integral is pointwise defined on (, +∞).

Definition . The Caputo fractional derivative of orderα>  of a functionxwithx(n–) absolutely continuous on [, ] is given by

+x(t) =In–α+

dnx(t)

dtn =

(n–α)

t

(t–s)n–α–x(n)(s)ds,

wheren= –[–α].

Lemma .([]) Letα> and n= –[–α].If x(n–)∈AC[, ],then

I+α +x(t) =x(t) –

n–

k=

x(k)()

k! t

k.

In this paper, we denoteX=C[, ] with the normx

X=max{x∞,x∞,x∞, x∞}andY=C[, ] with the normyY=y∞, wherex∞=maxt∈[,]|x(t)|.

Obvi-ously, bothXandYare Banach spaces. Define the operatorL:domLXYby

Lx=+x, (.)

where

domL=xX|

+x(t)Y,x() =x() =x() = ,x() =x()

.

LetN:XYbe the operator

Nx(t) =ft,x(t),x(t),x(t),x(t), ∀t∈[, ].

Then BVP (.) is equivalent to the operator equation

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3 Main result

In this section, a theorem on existence of solutions for BVP (.) will be given.

Theorem . Let f : [, ]×RRbe continuous.Assume that

(H) there exist nonnegative functionsa,b,c,d,eC[, ]with(α–)–(b+c+d+e) >

such that

f(t,u,v,w,x)a(t) +b(t)|u|+c(t)|v|+d(t)|w|+e(t)|x|, ∀t∈[, ], (u,v,w,x)∈R,

wherea=a∞,b=b∞,c=c∞,d=d∞,e=e∞;

(H) there exists a constantB> such that for allx∈Rwith|x|>Beither

xf(t,u,v,w,x) > ,t∈[, ], (u,v,w)∈R

or

xf(t,u,v,w,x) < ,t∈[, ], (u,v,w)∈R.

Then BVP(.)has at least one solution in X.

Now, we begin with some lemmas below.

Lemma . Let L be defined by(.),then

KerL=

xX x(t) =x ()

t

,t[, ]

, (.)

ImL=

yY

( –s)α–y(s)ds= 

. (.)

Proof By Lemma .,+x(t) =  has solution

x(t) =x() +x()t+x ()

t

+x()

t

.

Combining with the boundary value condition of BVP (.), one sees that (.) holds. Fory∈ImL, there existsx∈domLsuch thaty=LxY. By Lemma ., we have

x(t) =

(α)

t

(t–s)α–y(s)ds+x() +x()t+x ()

t

+x()

t

.

Then we have

x(t) = 

(α– )

t

(t–s)α–y(s)ds+x().

By the conditions of BVP (.), we see thatysatisfies

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Thus we get (.). On the other hand, supposeyY and satisfies( –s)α–y(s)ds= .

Letx(t) =Iα

+y(t), thenx∈domLand+x(t) =y(t). Soy∈ImL. The proof is complete.

Lemma . Let L be defined by(.),then L is a Fredholm operator of index zero,and the

linear continuous projector operators P:XX and Q:YY can be defined as

Px(t) =x ()

t

, t[, ],

Qy(t) = (α– )

( –s)α–y(s)ds,t∈[, ].

Furthermore,the operator KP:ImL→domL∩KerP can be written by

KPy(t) =

(α)

t

(t–s)α–y(s)ds,t∈[, ].

Proof Obviously,ImP=KerLandPx=Px. It follows fromx= (xPx) +Pxthat X=

KerP+KerL. By a simple calculation, we getKerL∩KerP={}. Then we get

X=KerL⊕KerP.

ForyY, we have

Qy=Q(Qy) =Qy·(α– )

( –s)α–ds=Qy.

Lety= (y–Qy) +Qy, whereyQy∈KerQ=ImL,Qy∈ImQ. It follows fromKerQ=ImL andQy=QythatImQImL={}. Then we have

Y=ImL⊕ImQ.

Thus

dim KerL=dim ImQ=codim ImL= .

This means thatLis a Fredholm operator of index zero.

From the definitions ofP,KP, it is easy to see that the generalized inverse ofLisKP. In

fact, fory∈ImL, we have

LKPy=+Iα+y=y. (.)

Moreover, forx∈domL∩KerP, we getx() =x() =x() =x() = . By Lemma ., we obtain

Iα+Lx(t) =Iα++x(t) =x(t) +x() +x()t+

x()  t

+x()

t

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which together withx() =x() =x() =x() =  yields

KPLx=x. (.)

Combining (.) with (.), we know that KP = (L|domL∩KerP)–. The proof is

com-plete.

Lemma . AssumeX is an open bounded subset such thatdomL=∅,then N is L-compact on.

Proof By the continuity off, we can see thatQN() and KP(I–Q)N() are bounded.

So, in view of the Arzelà-Ascoli theorem, we need only prove thatKP(I–Q)N()⊂Xis

equicontinuous.

From the continuity off, there exists constantA>  such that|(I–Q)Nx| ≤A,x, t∈[, ]. Furthermore, denoteKP,Q=KP(I–Q)Nand for ≤t<t≤,x, we have

(KP,Qx)(t) – (KP,Qx)(t)

≤ 

(α)

t

(t–s)α–(I–Q)Nx(s)dst

(t–s)α–(I–Q)Nx(s)ds

A

(α)

t

(t–s)α–– (t–s)α–ds+ t

t

(t–s)α–ds

= A

(α+ )

tα,

(KP,Qx)(t) – (KP,Qx)(t) ≤

A

(α)

tα–tα–,

(KP,Qx)(t) – (KP,Qx)(t) ≤

A

(α– )

tα–tα– ,

and

(KP,Qx)(t) – (KP,Qx)(t)

= 

(α– )

t

(t–s)α–(I–Q)Nx(s)dst

(t–s)α–(I–Q)Nx(s)ds

A

(α– )

t

(t–s)α–– (t–s)α–ds+ t

t

(t–s)α–ds

A

(α– )

tα–tα–+ (t–t)α–

.

Since,tα–, tα–, andtα– are uniformly continuous on [, ], we see thatK

P,Q()⊂

C[, ], (KP,Q)()⊂C[, ], (KP,Q)()⊂C[, ] and (KP,Q)()⊂C[, ] are

equicontin-uous. Thus, we find thatKP,Q:Xis compact. The proof is completed.

Lemma . Suppose(H), (H)hold,then the set

=

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Proof Takex, thenNx∈ImL. By (.), we have

( –s)α–fs,x(s),x(s),x(s),x(s)ds= .

Then, by the integral mean value theorem, there exists a constant ξ ∈(, ) such that f(ξ,x(ξ),x(ξ),x(ξ),x(ξ)) = . Then from (H), we have|x(ξ)| ≤B.

Fromx∈domL, we getx() = ,x() = , andx() = . Therefore

x(t) = x() +

t

x(s)dsx,

x(t) = x() +

t

x(s)ds ≤x,

and

x(t) = x() + t

x(s)dsx.

That is

x∞≤xxx. (.)

ByLx=λNxandx∈domL, we have

x(t) = λ

(α)

t

(t–s)α–fs,x(s),x(s),x(s),x(s)ds+ x

()t.

Then we get

x(t) = λ

(α– )

t

(t–s)α–fs,x(s),x(s),x(s),x(s)ds+x().

Taket=ξ, we get

x(ξ) = λ

(α– )

ξ

(ξs)α–fs,x(s),x(s),x(s),x(s)ds+x().

Together with|x(ξ)| ≤B, (H), and (.), we have x() ≤ x(ξ) + λ

(α– )

ξ

(ξs)α– fs,x(s),x(s),x(s),x(s) dsB+ 

(α– )

ξ

(ξs)α–a(s) +b(s) x(s) +c(s) x(s) +d(s) x(s) +e(s) x(s) ds

B+ 

(α– )

ξ

(ξs)α–a+bx∞+cx

+dx+ex

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B+ 

(α– )

ξ

(ξs)α–a+ (b+c+d+e)x

ds

B+ 

(α– )

a+ (b+c+d+e)x

.

Then we have

x≤ 

(α– )

t

(t–s)α– fs,x(s),x(s),x(s),x(s) ds+ x()

≤ 

(α– )

t

(t–s)α–a(s) +b(s) x(s) +c(s) x(s)

+d(s) x(s) +e(s) x(s) ds+x()

≤ 

(α– )

t

(t–s)α–a+bx∞+cx

+dx+ex

ds+ x()

≤ 

(α– )

t

(t–s)α–a+ (b+c+d+e)x

ds+ x()

≤ 

(α– )

a+ (b+c+d+e)x

+ x()

B+ 

(α– )

a+ (b+c+d+e)x

.

Thus, from(α– ) – (b+c+d+e) > , we obtain x≤ a+(α– )B

(α– ) – (b+c+d+e)

:=M.

Thus, together with (.), we get

x∞≤xxxM.

Therefore,

xXM.

Sois bounded. The proof is complete.

Lemma . Suppose(H)holds,then the set

={x|x∈KerL,Nx∈ImL}

is bounded.

Proof Forx, we havex(t) =x ()

tandNx∈ImL. Then we get

( –s)α–f

s,x ()

s

,x()

s

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which together with (H) implies|x()| ≤B. Thus, we have

xXB.

Hence,is bounded. The proof is complete.

Lemma . Suppose the first part of(H)holds,then the set

=

x|x∈KerL,λx+ ( –λ)QNx= ,λ∈[, ]

is bounded.

Proof Forx, we havex(t) =x ()

tand

λx

()

t

+ ( –λ)(α– )

×

( –s)α–f

s,x ()

s

,x()

s

,x()s,x()

ds= . (.)

Ifλ= , then|x()| ≤Bbecause of the first part of (H). Ifλ∈(, ], we can also obtain

|x()| ≤B. Otherwise, if|x()|>B, in view of the first part of (H), one has

λ[x

()]

t

+ ( –λ)(α– )

×

( –s)α–x()f

s,x ()

s

,x()

s

,x()s,x()

ds> ,

which contradicts (.).

Therefore,is bounded. The proof is complete.

Remark . Suppose the second part of (H) hold, then the set

=x|x∈KerL, –λx+ ( –λ)QNx= ,λ∈[, ]

is bounded.

Proof of Theorem. Set={xX| xX<max{M,B}+ }. It follows from Lemmas .

and . thatLis a Fredholm operator of index zero andNisL-compact on. By Lem-mas . and ., we see that the following two conditions are satisfied:

() Lx=λNxfor every(x,λ)∈[(domL\KerL)]×(, ); () Nx∈/ImLfor everyx∈KerL.

Take

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According to Lemma . (or Remark .), we know thatH(x,λ)=  forx∈KerL. Therefore

deg(QN|KerL,∩KerL, ) =deg

H(·, ),∩KerL,  =degH(·, ),∩KerL,  =deg(±I,∩KerL, )= .

So the condition () of Lemma . is satisfied. By Lemma ., we find thatLx=Nxhas at least one solution indomL. Therefore, BVP (.) has at least one solution. The proof

is complete.

4 An example

Example . Consider the following BVP:

D

 

+x(t) = (x– ) +t  e–|x

||x|

+t sin(x), t[, ],

x() =x() =x() = , x() =x(). (.)

Here

f(t,u,v,w,x) =

(x– ) + t

e

–|v|–|w|+ t

sin

u.

Choosea(t) =,b(t) = ,c(t) = ,d(t) = ,e(t) =,B= . We getb= ,c= ,d= ,

e=, and

 – 

– (b+c+d+e) > .

Then all conditions of Theorem . hold, so BVP (.) has at least one solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors typed, read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees very much for their helpful comments and suggestions. This research was supported by the Fundamental Research Funds for the Central Universities (2013QNA33).

Received: 11 December 2013 Accepted: 1 July 2014 References

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doi:10.1186/s13661-014-0176-5

Cite this article as:Hu et al.:Boundary value problems for fractional differential equations.Boundary Value Problems

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