Solutions for a fractional difference boundary value problem

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R E S E A R C H

Open Access

Solutions for a fractional difference boundary

value problem

Wei Dong

1*

_{, Jiafa Xu}

2

_{and Donal O’Regan}

3

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Hebei University of Engineering, Handan, Hebei 056038, China Full list of author information is available at the end of the article

Abstract

Using a variational approach and critical point theory, we investigate the existence of solutions for a fractional diﬀerence boundary value problem.

MSC: 26A33; 35A15; 39A12; 44A55

Keywords: fractional diﬀerence boundary value problem; variational approach; critical point theory; solution

1 Introduction

In this work, using variational methods and critical point theory, we study the fractional diﬀerence boundary value problem

⎧ ⎨ ⎩

Tνt–(tνν–x(t)) =f(x(t+ν– )), t∈[,T]N,

x(ν– ) = [tνν–x(t)]t=T= ,

(.)

whereν∈(, ),tνν–andTνt are, respectively, the left fractional diﬀerence and the right

fractional diﬀerence operators,t∈[,T]N:={, , , . . . ,T}, andf :R→Ris continuous.

Fractional calculus has a long history, and there is renewed interest in the study of both fractional calculus and fractional diﬀerence equations. In [, ], the authors discussed properties of the generalized falling function, a corresponding power rule for fractional delta-operators and the commutativity of fractional sums. A number of papers have ap-peared which build the theoretical foundations of discrete fractional calculus (for more details, we refer the reader to [–] and the references therein).

Atici and Eloe [] considered the existence of positive solutions for the following two-point boundary value problem for a nonlinear ﬁnite fractional diﬀerence equation:

⎧ ⎨ ⎩

–ν_y₍_t_{) =}_f₍_t₊_ν_{– ,}_y₍_t₊_ν_{– )),} _t_{= , , . . . ,}_b_{+ ,}

y(ν– ) = , y(ν+b+ ) = . (.)

In [], the authors used the mountain pass theorem, a linking theorem, and Clark’s the-orem to establish the existence of multiple solutions for a fractional diﬀerence boundary value problem with a parameter. Under some suitable assumptions, they obtained some results which ensure the existence of a precise interval of parameters for which the prob-lem admits multiple solutions. We note that there are many papers in the literature [–] which discuss discrete problems via variational and critical point theory.

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In [], Tian and Henderson studied the nth order nonlinear diﬀerence equation

nr(t–n)nx(t–n)+ft,x(t)= , t∈Z, (.)

and established some existence results for anti-periodic solutions under various assump-tions on the nonlinearity. In [], Ye and Tang considered the second-order discrete Hamil-tonian system

u(t– ) +d(t)u(t)μ–u(t) +∇Ht,u(t)= , ∀t∈Z,

and obtained an existence theorem for a nonzeroT-periodic solution.

In the literature on discrete problem via critical point theory, the authors are interested in the existence of at least one solution or inﬁnitely many solutions. The existence of a unique solution is not usually studied. In this paper, using Browder’s theorem, ﬁrst we present a uniqueness result in Section . Then a linking theorem is used to establish exis-tence. Finally, assuming an Ambrosetti-Rabinowitz type condition, we show that problem (.) has many solutions if the nonlinearity is odd.

2 Preliminaries

For convenience, throughout this paper, we arrangem_i₌_jx(i) =  form<j. We present some deﬁnitions and lemmas for discrete fractional operators.

For any integerβ, letNβ:={β,β+ ,β+ , . . .}andt(ν):=(t+ )/(t+  –ν), wheret andνare determined by (.). We also appeal to the convention that ift+  –νis a pole of the gamma function andt+  is not a pole, thent(ν)_{= .}

Deﬁnition .(see [, ]) Theνth fractional sum off forν>  is deﬁned by

–ν

a f(t) =



(ν)

t–ν

s=a

(t–s– )(ν–)f(s) fort∈Na–ν. (.)

We also deﬁne theνth fractional diﬀerence forν>  byν_f₍_t_{) :=}Nν–N_f₍_t_{), where}_t_∈

Na+N–νandN∈Nis chosen so that ≤N–  <ν≤N.

Deﬁnition .(see [, ]) Letf be any real-valued function andν∈(, ). The left discrete

fractional difference and the right discrete fractional difference operators are, respectively, defined as

tνaf(t) =t–(–ν)a f(t)

= 

( –ν)

t+ν–

s=a

(t–s– )(–ν)f(s), t≡a–ν+ (mod),

bνtf(t) = –b–(–ν)t f(t)

= 

( –ν)(–)

b

s=t+–ν

(s–t– )(–ν)f(s), t≡b+ν– (mod).

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Deﬁnition . Suppose thatXis a Banach space andI:X→Ris a functional deﬁned onX. For givenx,y∈X, assume that

lim ε→

I(x+εy) –I(x)

ε (.)

exists. ThenIis Gateaux diﬀerentiable atx, the limit in (.) is called the Gateaux diﬀer-ential ofIatxin directiony, and is denoted by (I(x),y),i.e.,

I(x),y=lim ε→

I(x+εy) –I(x)

ε . (.)

Deﬁnition .(see [, p.]) LetXbe a reﬂexive real Banach space andX∗its dual. The operatorL:X→X∗is said to be demicontinuous ifLmaps strongly convergent sequences inXto weakly convergent sequences inX∗.

Lemma .(Browder theorem, see [, Theorem ..]) Let X be a reﬂexive real Banach space.Moreover,let L:X→X∗be an operator satisfying the conditions

(i) Lis bounded and demicontinuous, (ii) Lis coercive,i.e.,lim_x→∞(L_x(x),x)= +∞,

(iii) Lis monotone on the spaceX,i.e.,for allx,y∈X,we have

L(x) –L(y),x–y≥. (.)

Then the equation L(x) =f∗has at least one solution x∈X for every f∗∈X∗.If,moreover,

the inequality(.)is strict for all x,y∈X,x= y,then the equation L(x) =f∗has precisely one solution x∈X for all f∗∈X∗.

Deﬁnition .(see [–]) LetXbe a real Banach space,I∈C(X,R) andc∈R. We say thatIsatisﬁes the (PS)ccondition if any sequence{xn} ⊂Xsuch thatI(xn)→cand I(xn)→ asn→ ∞has a convergent subsequence.

Lemma .(Linking theorem, Rabinowitz, see [–]) Let X=Y⊕Z be a Banach space with Z closed in X anddimY<∞.Letρ>r> ,and let z∈Z be such thatz=r.Deﬁne

M:=u=y+λz:u ≤ρ,λ≥,y∈Y, N:=u∈Z:u=r, M:=u=y+λz:y∈Y,u=ρandλ≥, oru ≤ρandλ= .

Let I∈C₍_X_,_R₎_{be such that}

b:= inf

u∈NI(u) >a:=u∈Mmax I(u).

If I satisﬁes the(PS)ccondition with

c:=inf γ∈max_u∈MI

γ(u), where:=γ ∈C(M,X) :γ|_M=id,

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Deﬁnition .(see [–]) LetXbe a real Banach space andI∈C₍_X_,_R_{). We say that}_I satisﬁes the Cerami condition ((C) condition for short) if any sequence{xn} ⊂Xsuch that I(xn) is bounded and ( +xn)I(xn) → asn→ ∞, there exists a subsequence of{xn}

which is convergent inX.

Lemma .(Mountain pass theorem, see [–]) Let X be a real Banach space,and let I∈C₍_X_,_R₎_{satisfy the}_(C)_condition_._{If I}₍_θ_{) = }_{and the following conditions hold}_:

(i) there are two positive constantsρ,ηand a closed linear subspaceXofXsuch that codimX=landI|X∩∂Bρ≥η,whereBρis an open ball of radiusρwith centerθ;

(ii) there is a subspaceXwithdimX=m,m>l,such that

I(x)→–∞ asx → ∞,x∈X.

Then I possesses at least m–l distinct pairs of nontrivial critical points.

In what follows, we establish the variational framework for (.). Let

X:=x=x(ν– ),x(ν), . . . ,x(ν+T– )+:x(ν+i– )∈R,i= , , . . . ,T. (.) ThenXis theT+ -dimensional Hilbert space with the usual inner product and the usual norm

(x,z) =

T+ν–

t=ν–

x(t)z(t), x= _T_+ν–

t=ν– x(t)

 

, x,z∈X. (.)

Forα> , we deﬁne theα-norm onX:xα= (T_t_=ν–+ν–|x(t)|α₎_α_{. Since}_dim_X_<_∞_{, we see} that there existcα> ,cα>  such that

cαx ≤ xα≤cαx (.)

for allxbelonging toX(or its subspace).

In view of [, (.)], we can deﬁne an energy functional onXby

I(x) =  

T

t=–

tνν–x(t) 

–

T

t=–

Fx(t+ν– ), x∈X, (.)

where

Fx(t+ν– )=

x(t+ν–)



f(s) ds,

x(ν– ) = , tνν–x(t)

t=T=

–ν ( –ν)

T+ν

s=ν–

(T+s– )(–ν–)x(s) = .

Clearly,I(θ) = . Let

E:=χ=x(ν– ),x(ν– ), . . . ,x(ν+T)+

∈RT+_:_x₍_ν_{– ) = ,}

tνν–x(t)

t=T= 

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Then, from the boundary conditions of (.), it is easy to see thatEis isomorphic toX. In the following, when we sayx∈X, we always imply thatxcan be extended toχ∈Eif it is necessary. Now we claim that ifx= (x(ν– ),x(ν), . . . ,x(ν+T– ))+_∈_X_{is a critical point of}

I, thenχ= (x(ν– ),x(ν– ), . . . ,x(ν+T))+_∈_E_{is precisely a solution of (.). Indeed, since}_I can be viewed as a continuously differentiable functional defined on the finite dimensional Hilbert spaceX, the Fréchet derivativeI(x) is zero if and only if∂I(x)/∂x(i) =  for all

i=ν– ,ν, . . . ,ν+T– . From the relation between the Fréchet derivative and the Gateaux derivative, we obtain

I(x),y= lim ε→

I(x+εy) –I(x)

ε =



ε

 

T

t=–

tνν–

x(t) +εy(t)–tνν–x(t) 

–

T

t=–

F(x+εy)(t+ν– )–Fx(t+ν– )

=

T

t=–

tνν–x(t)tνν–y(t) –

T

t=–

fx(t+ν– )y(t+ν– ). (.)

Therefore, in order to obtain the existence of solutions for (.), we only need to study the existence of critical points of the energy functionalIonX.

Next, noting Deﬁnition ., fort∈[–,T]N–, we let

tνν–x(t) = 

( –ν)

t–(–ν)

s=ν–

(t–s– )(–ν)x(s) :=z(t+ν– ). (.)

Then we have

z(ν– ) = ,

z(ν– ) = 

( –ν) –(–ν)

s=ν–

(–s– )(–ν)x(s) =x(ν– ),

z(ν) = 

( –ν) –(–ν)

s=ν–

( –s– )(–ν)x(s) = ( –ν)x(ν– ) +x(ν),

z(ν+ ) = 

( –ν) –(–ν)

s=ν–

( –s– )(–ν)x(s)

=( –ν)( –ν)

! x(ν– ) + ( –ν)x(ν) +x(ν+ ), ..

.

z(ν+T– ) = 

( –ν)

T–(–ν)

s=ν–

(T–s– )(–ν)x(s)

=(T–ν)(T–  –ν)· · ·( –ν)

T! x(ν– )

+(T–  –ν)(T–  –ν)· · ·( –ν)

(T– )! x(ν) +· · ·

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i.e.,z=Bx, wherez= (z(ν– ),z(ν), . . . ,z(ν+T– ))+_,_x_{= (}_x₍_ν_{– ),}_x₍_ν_{), . . . ,}_x₍_ν₊_T_{– ))}+_, tive. Letλminandλmaxdenote respectively the minimum and the maximum eigenvalues

of (B–₎+_B–_{. Since}_x₌_B–_z_{, we have}

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Theorem . Let(H)hold.Then(.)has precisely one solution for c∈(, λ

λmax).

Proof We shall apply Lemma . to prove the result. From (.), we deﬁne the operator

L(x),y=

T

t=–

tνν–x(t)tνν–y(t) –

T

t=–

fx(t+ν– )y(t+ν– ), ∀x,y∈X. (.)

Clearly, if for ally∈X, there existsx∈Xsuch that (L(x),y) = , thenxis a solution of (.). Let

L(x),y

=

T

t=–

tνν–x(t)tνν–y(t),

L(x),y=

T

t=–

fx(t+ν– )y(t+ν– ), ∀x,y∈X.

We sketch the properties ofLandL. It is clear thatLis a linear operator, and further-more,Lis bounded. Indeed, the Cauchy-Schwarz inequality enables us to obtain, notice (.) and (.),

L(x),y≤ T

t=– _tν

ν–x(t)tνν–y(t)

≤

_T

t=– _tν

ν–x(t) 

  T

t=– _tν

ν–y(t) 

 

≤λT+

λmin

xy<∞, x,y∈X. (.)

Consequently,Lis continuous onX. Next, we show thatLis bounded and continuous. Lety=θ in (H) and|f(θ)|=c> . Then we have from (H)

_f₍_x₎_≤_c_|_x_|₊_c_, _∀_x_∈_X_. _(.)

From (.), the deﬁnition ofL, and the Cauchy-Schwarz inequality, we obtain

L(x),y≤ T

t=

fx(t+ν– )y(t+ν– )

≤

T

t=

cx(t+ν– )+cy(t+ν– )

≤cxy+c √

T+ y<∞, ∀x,y∈X, (.)

L(x) –L(x),y≤ T

t=

fx(t+ν– )–fx(t+ν– )y(t+ν– )

≤

T

t=

cx(t+ν– ) –x(t+ν– )y(t+ν– )

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Therefore,Lis bounded and continuous, as required. Hence,Lis bounded and

continu-Finally, we prove thatLis strictly monotone. Indeed, from (H), we have

All the conditions of Lemma . are satisﬁed, as claimed. Hence, (.) has precisely one

solution. This completes the proof.

Theorem . Let(H)-(H)hold.Then(.)has at least one solution.

Proof From (H), there existsδ>  with

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By virtue of the inequality  <μ< , there existsr>  such that

b:= inf

x=r,x∈ZI(x) > . (.)

From (H), forx∈Y, we see

I(x) =  

T

t=–

tνν–x(t) 

–

T

t=–

Fx(t+ν– )

= 

T–

t=–

z(t+ν– )–

T

t=

Fx(t+ν– )

≤ λl

λmin

x–

T

t=

Fx(t+ν– )

=

T

t=

λl

λmin

x(t+ν– ) –Fx(t+ν– )≤. (.)

From (H), we see that there existc,c>  such that x



f(s) ds≥c|x|α–c, ∀x∈R. (.)

Hence, forx∈X, we ﬁnd

I(x) =  

T

t=–

tνν–x(t) 

–

T

t=–

Fx(t+ν– )

= 

T–

t=–

z(t+ν– )–

T

t=

Fx(t+ν– )

≤ λT+ λmin

x–c

T

t=

x(t+ν– )α+c(T+ ). (.)

Setz:=r ηl+

ηl+withr>  is given in (.). ForY⊕Rz⊂X, (.) holds true. This, together with (.), implies

I(x)≤ λT+ λmin

x–ccααxα+c(T+ ).

Sinceα> , we obtain

lim

x→∞,x∈Y⊕Rz

I(x) = –∞. (.)

Deﬁne

M:=x=y+λz:x ≤ρ,λ≥,y∈Y,

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Sincez∈Zand thenI(z)≥b> , (.) and (.) guarantee that there isρ>rsuch that

a:= max

x∈MI(x)≤.

It remains to prove thatIsatisﬁes the (PS)ccondition. This will be the case if we show

that any sequence{xn}∞n=⊂Xsuch that

d:=sup

n

I(xn) <∞, I(xn)→,

contains a convergent subsequence. Note thatdimX<∞, so we only need to show the boundedness of{xn}∞n=. Takeβ>  such thatβ–∈(,α) fornlarge enough, and (H),

Thus the functionalIsatisﬁes all the conditions of Lemma ., and thenIhas a critical point, and (.) has at least one solution. This completes the proof.

Theorem . Let(H), (H)-(H)hold.Then(.)has at least m–l solutions.

Thus (i) of Lemma . holds true.

ChooseX:=span{η, . . . ,ηm}, wherem>l, anddimX=m. From (H), we see that there existc,c>  such that

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Therefore, from (.) and (.), we arrive at

–I(x)≥– λm λmin

x+c

T

t=

x(t+ν– )α–c(T+ )

≥– λm λmin

x+ccαα xα–c(T+ ).

Sinceα> ,I(x)→–∞asx → ∞,x∈X. Thus (ii) of Lemma . holds true.

Finally, we prove thatIsatisﬁes the (C) condition. Let{xn} ⊂Xbe such that for some M> ,

I(xn)≤M,

 +xnI(xn)→ asn→ ∞.

We claim thatxnis bounded. Otherwise, suppose thatxn → ∞asn→ ∞. It is easy

to see that for anyn∈N, there existsMsuch that I(xn) –

I(xn),xn

≤M.

On the other hand, from (H), there existc,c>  such that

f(x)x– F(x)≥c|x|γ –c, ∀x∈R. Consequently, from (.),

I(xn) –

I(xn),xn

=

T

t=

fxn(t+ν– )

xn(t+ν– ) – F

xn(t+ν– )

≥

T

t=

cxn(t+ν– )

γ –c

≥ccγγxnγ–c(T+ ).

Letn→ ∞, and we get a contradiction.

It is easy to see thatI is even andI(θ) = . Thus all the conditions of Lemma . are satisﬁed, and (.) has at leastm–lsolutions. The proof is complete.

Examples

. Letf(x) =ηx+η, whereη∈(,_λλ_max )andη= . Clearly, (H) holds.

. Letf(x) = λl λmin(x+x

₎_ex_{+ }_x_{. Then}_F₍_x_{) =} λl λminx

_ex₊_x_{. Thus, (H) and (H) hold}

automatically. ForR≥,α= , we see

 < λl

λmin

xex+ x≤xf(x) = λl

λmin

x+xex+ x for all|x|> .

Therefore, (H) holds.

. Letf(x) = x+ x. ThenF(x) =x+xand (H), (H) hold. Chooseα= ,γ= , and we see

lim inf

|x|→∞ x+x

|x| =  > , lim inf_|x|→∞

f(x)x– F(x)

|x| =lim inf_|x|→∞ x

|x|=  > .

(12)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

WD and JX carried out the main results of this article and drafted the manuscript. DO directed the study and helped with the inspection. All the authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Hebei University of Engineering, Handan, Hebei 056038, China.}2_{School of Mathematics,}

Shandong University, Jinan, Shandong 250100, China.3_{School of Mathematics, Statistics and Applied Mathematics,}

National University of Ireland, Galway, Ireland.

Acknowledgements

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript. Research is supported by the NNSF-China (10971046 and 11371117), Shandong and Hebei Provincial Natural Science Foundation (ZR2012AQ007 and A2012402036), GIIFSDU (yzc12063) and IIFSDU (2012TS020).

Received: 30 July 2013 Accepted: 16 October 2013 Published:11 Nov 2013

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