The existence of solutions for some fractional finite difference equations via sum boundary conditions

R E S E A R C H

Open Access

The existence of solutions for some fractional

finite difference equations via sum boundary

conditions

Ravi P Agarwal

1,2

_{, Dumitru Baleanu}

3,4*

_{, Shahram Rezapour}

5

_{and Saeid Salehi}

5

*_{Correspondence:}

dumitru@cankaya.edu.tr

3_{Department of Mathematics,}

Cankaya University, Ogretmenler Cad. 14, Balgat, Ankara, 06530, Turkey

4_{Institute of Space Sciences,}

Magurele, Bucharest, Romania Full list of author information is available at the end of the article

Abstract

In this manuscript we investigate the existence of the fractional finite difference equation (FFDE)

μ_μ–2x(t) =g(t+

μ

– 1,x(t+

μ

– 1),

x(t+

μ

– 1)) via the boundary conditionx(μ– 2) = 0 and the sum boundary conditionx(μ+b+ 1) =α_k=μ–1x(k) for order 1 <

μ

≤2, whereg:N_μμ+_–1b+1×R×R→R,

α

∈N_μμ_–1+b, andt∈Nb₀+2. Along the same lines, we discuss the existence of the solutions for the following FFDE:

μ_μ–3x(t) =g(t+

μ

– 2,x(t+

μ

– 2)) via the boundary conditionsx(μ– 3) = 0 and

x(μ+b+ 1) = 0 and the sum boundary conditionx(α) =_kβ_=γx(k) for order 2 <

μ

≤3, whereg:Nμ_μ+_–2b+1×R→R,b∈N0,t∈Nb+3

0 , and

α

,β,γ ∈N μ+b

μ–2 with

γ

<

β

<

α.

MSC: 34A08

Keywords: fractional finite difference equation; fixed point

1 Introduction

By the late th century, combined efforts made by several mathematicians led to a fairly solid understanding of fractional calculus in the continuous setting but significantly less is still known about discrete fractional calculus (see for example [, ] and [] and the references therein). Recently, there has been a strong interest in this subject but still little progress was made in developing the theory of fractional finite difference equations (see [–] and [] and the references therein).

Discrete fractional calculus is a powerful tool for the processes which appears in na-ture,e.g.biology, ecology and other areas (see for example [] and [] and the references therein), where the discrete models have to be considered in order to describe properly the complexity of the dynamical processes with memory effect. We notice that the exis-tence of solutions for fractional finite difference equations is a hot topic of the fractional calculus with direct implications in modeling of some real world phenomena which have only discrete behaviors.

Motivated by the above mentioned results, in this paper we investigate the fractional finite difference equation

μ_μ–x(t) =gt+μ– ,x(t+μ– ),x(t+μ– )

via the boundary conditionsx(μ– ) =  andx(μ+b+ ) =_kα₌μ–x(k), whereb∈N,

t∈Nb+

 ,  <μ≤,g:N

b+μ+

μ– ×R×R→R, andα∈N

b+μ μ–.

Moreover, we investigate the FFDE given by

μ_μ–x(t) =gt+μ– ,x(t+μ– )

via the boundary conditionsx(μ– ) = ,x(μ+b+ ) = , andx(α) =β_k=γx(k), where b∈N,t∈Nb+,  <μ≤,g:N

μ+b+

μ– ×R→Randα,β,γ ∈N

μ+b

μ–withγ <β<α.

In the following we present the basic definitions and theorems used in this manuscript. In Section  we present the main result. The manuscript ends with our conclusions.

2 Preliminaries

As you know, the gamma function is defined by

(z) =

∞



e–ttz–dt,

which converges in the right half of the complex planeRe(z) > . It is well known that

(z+ ) =z(z) and(n) = (n– )! for alln∈N. Now, we define

tμ:= (t+ )

(t+  –μ)

for allt,μ∈R[]. Ift+  –μis a pole of the gamma function andt+  is not a pole, then we definetμ=  []. For example, we have (μ– )μ–= . Also, one can verify that

μμ=μμ–=(μ+ ) andtμ_tμ+=t–μ.

In this paper, we use the notationsNp={p,p+ ,p+ , . . .}for allp∈RandNqp={p,p+

,p+ , . . . ,q}for all real numberspandqwheneverq–pis a natural number.

Letμ>  withm–  <μ<mfor some natural numberm. Theμth fractional sum off based atais defined as []

–_aμf(t) = 

(μ)

t–μ

r=a

t–σ(r)μ–f(r)

for allt∈Na+μ, whereσ(r) =r+  is the forward jump operator. Similarly, we define

μ_af(t) = 

(–μ)

t+μ

r=a

t–σ(r)–μ–f(r)

for allt∈Na+m–μ[]. Note that the domain ofraf isNa+m–rforr>  andNa–rforr< .

Also, for the natural numberμ=m, we have the known formula []

μ_af(t) =mf(t) =

m

i=

(–)i

m i

f(t+m–i).

We define

Lemma .[] Let g:Na→Rbe a mapping and m a natural number.Then the general

solution of the equationμa+μ–mx(t) =g(t)is given by

x(t) =

m

i=

Ci(t–a)μ–i+–aμg(t)

for all t∈Na+μ–m,where C, . . . ,Cmare arbitrary constants.

Letg:Nbμ++–μ×R×R→Rbe a mapping andma natural number. By using a similar

proof, one can check that the general solution of the equationμμ–mx(t) =g(t+μ–m+

,x(t+μ–m+ ),x(t+μ–m+ )) is given by

x(t) =

m

i=

Citμ–i+–μg

t+μ–m+ ,x(t+μ–m+ ),x(t+μ–m+ )

for allt∈Nμ–m. In particular, the general solution has the following representation:

x(t) =

m

i=

Citμ–i+



(μ)

t–μ

r=

t–σ(r)μ–

×gr+μ–m+ ,x(r+μ–m+ ),x(r+μ–m+ ) (.)

for allt∈Nμ–m. The next theorem plays an important role in our main results.

Theorem .[] Every continuous function from a compact,convex,nonempty subset of

a Banach space to itself has a fixed point.

3 Main results

In the following, we are ready to provide the main results. First, we investigate the FFDE

μ_μ–x(t) =gt+μ– ,x(t+μ– ),x(t+μ– )

via the boundary conditionsx(μ– ) =  andx(μ+b+ ) =_kα₌μ–x(k), whereb∈N,

t∈Nb+

 ,  <μ≤,g:N

b+μ+

μ– ×R×R→R, andα∈N

b+μ μ–.

Lemma . Let b∈N,t∈Nb+,  <μ≤,g:N

b+μ+

μ– ×R×R→R,andα∈N

b+μ μ–.Then

xis a solution of the problem

μ_μ–x(t) =gt+μ– ,x(t+μ– ),x(t+μ– )

via the boundary conditions x(μ+b+ ) =α_k=μ–x(k)and x(μ– ) = if and only if xis

a solution of the fractional sum equation

x(t) =

b+

r=

where

G(t,r,α) = t μ–α

k=r+μ(k–σ(r)) μ–

((μ+b+ )μ––_μ(α+ )μ)(μ)

– (μ+b+  –σ(r)) μ– 

μ(α+ ) μ_tμ–

(μ+b+ )μ–(μ)((μ+b+ )μ––_μ(α+ )μ)

–(μ+b+  –σ(r)) μ–_tμ–

(μ)(μ+b+ )μ– +

(t–σ(r))μ–

(μ)

whenever r<t–μ≤α–μor r≤α–μ<t–μ,

G(t,r,α) = – (μ+b+  –σ(r)) μ– 

μ(α+ ) μ

tμ–

(μ+b+ )μ–_{(μ)((μ+b+ )}μ–_– 

μ(α+ ) μ₎

–(μ+b+  –σ(r)) μ–_tμ–

(μ)(μ+b+ )μ– +

(t–σ(r))μ– (μ)

wheneverα–μ<r≤t–μ,

G(t,r,α) = t μ–α

k=r+μ(k–σ(r)) μ–

((μ+b+ )μ––_μ(α+ )μ)(μ)

– (μ+b+  –σ(r)) μ– 

μ(α+ ) μ_tμ–

(μ+b+ )μ–(μ)((μ+b+ )μ––_μ(α+ )μ)

–(μ+b+  –σ(r)) μ–_tμ–

(μ)(μ+b+ )μ–

whenever t–μ≤r≤α–μand

G(t,r,α) = – (μ+b+  –σ(r)) μ– 

μ(α+ ) μ

tμ–

(μ+b+ )μ–_{(μ)((μ+b+ )}μ–_– 

μ(α+ ) μ₎

–(μ+b+  –σ(r)) μ–_tμ–

(μ)(μ+b+ )μ–

whenever t–μ≤α–μ<r orα–μ≤t–μ<r.

Proof Letxbe a solution of the problem

μ_μ–x(t) =gt+μ– ,x(t+μ– ),x(t+μ– )

via the boundary conditionsx(μ+b+ ) =α_k=μ–x(k) andx(μ– ) = . By using

Lem-ma ., we get

x(t) =Ctμ–+Ctμ–+–μg

t+μ– ,x(t+μ– ),x(t+μ– )

=Ctμ–+Ctμ–+



(μ)

t–μ

r=

t–σ(r)μ–

×gr+μ– ,x(r+μ– ),x(r+μ– )

Hence,

and so by interchanging the order of summations, we have

α

Now, letxbe a solution of the fractional sum equation

x(t) =

b+

r=

Thenxis a solution of the equation

x(t) =  (μ+b+ )μ–

α

k=μ–

x(k)

– 

(μ)

b+

r=

μ+b+  –σ(r)μ–gr+μ– ,x(r+μ– ),x(r+μ– )

tμ–

+ 

(μ)

t–μ

r=

t–σ(r)μ–gr+μ– ,x(r+μ– ),x(r+μ– )

= 

(μ+b+ )μ– α

k=μ–

x(k) – 

(μ)

b+

r=

μ+b+  –σ(r)μ–

×gr+μ– ,x(r+μ– ),x(r+μ– )

tμ–

+–μgt+μ– ,x(t+μ– ),x(t+μ– ).

It is easy to check thatx(μ– ) = . Also, we have

x(μ+b+ )

= 

(μ+b+ )μ–

α

k=μ–

x(k) –



(μ)

b+

r=

μ+b+  –σ(r)μ–

×gr+μ– ,x(r+μ– ),x(r+μ– )

(μ+b+ )μ–

+ 

(μ)

b+

r=

μ+b+  –σ(r)μ–gr+μ– ,x(r+μ– ),x(r+μ– )

= α

k=μ–

x(k).

Moreover, we haveμ_μ–x(t) =g(t+μ– ,x(t+μ– ),x(t+μ– )). This completes

the proof.

Some authors tried to find the maximum or exact value ofb_r=+|G(t,r,α)|in some pa-pers (see for example [, ] and []). Now, we show thatb_r=+G(t,r,α) is bounded, whereG(t,r,α) is the Green function of the last result.

Lemma . For each t∈Nb_μ++_–μandα∈Nb_μ+_–μ,we have

b+

r=

G(t,r,α)

≤

b+

r=

G(t,r,α)≤MG

Proof Since(α) >  for allα> , we have

(μ+b+ )μ–=(μ+b+ ) (b+ )! > 

for allμ>  andb> . Thus,G(t,r,α) is a (finite) real number, for allt∈Nbμ++–μ,α∈N

b+μ μ–

andr∈Nb_+. Consequently, both sums in the statement are finite, becauseNb_+is finite.

Theorem . Let g:Nμb+–μ+×R×R→Rbe bounded and continuous in its second and

third variables.Then the fractional finite difference equationμ_μ–x(t) =g(t+μ– ,x(t+

μ– ),x(t+μ– ))via the boundary conditions x(μ+b+ ) =α_k=μ–x(k)and x(μ– ) =  has a solution xwith x(t)∈[–MG,MG],for all admissible t.

Proof Sinceg is bounded, there exists a constantC such that|g(u,v,w)| ≤Cfor allu∈ Nb+μ+

μ– andv,w∈R. LetXbe the Banach space of real valued functions defined onN

μ+b+

μ–

via the norm

x=maxx(t):t∈Nμ_μ+_–b+

andK={x∈X :x ≤CMG}. One can check easily thatK is a compact, convex, and

nonempty subset ofX. Now, define the mapTonKby

Tx(t) =

b+

r=

G(t,r,α)gr+μ– ,x(r+μ– ),x(r+μ– )

for allt∈Nμμ+–b+. First, we show thatT(K)⊆K. Letx∈Kandt∈N

μ+b+

μ– . Then

Tx(t)=

b+

r=

G(t,r,α)gr+μ– ,x(r+μ– ),x(r+μ– )

≤

b+

r=

G(t,r,α)gr+μ– ,x(r+μ– ),x(r+μ– )

≤CMG.

Sincet∈Nμ_μ+_–b+ was arbitrary,Tx ≤CMGand soT(K)⊆K. Now, we show thatT is

continuous. Let>  be given. Sincegis continuous in its second and third variables, it is uniformly continuous in its second and third variables on [–CMG,CMG] and so there

existsδ>  such that|g(t,u,u) –g(t,v,v)|< _MG for allt∈Nμμ+–b+ andu,u,v,v∈

[–CMG,CMG] with|u–v|<δand|u–v|<δ. Thus, we get

_{Ty(t) –Tx(t)}

=

b+

r=

G(t,r,α)gr+μ– ,y(r+μ– ),y(r+μ– )

–

b+

r=

G(t,r,α)gr+μ– ,x(r+μ– ),x(r+μ– )

≤

b+

r=

G(t,r,α)gr+μ– ,y(r+μ– ),y(r+μ– )

–gr+μ– ,x(r+μ– ),x(r+μ– )

≤

b+

r=

_G(t,r,α) MG ≤

MG

MG

=

for allt∈Nμ_μ+_–b+. Hence,Tx–Ty<and soTis continuous onK. By using Theorem ., Thas a fixed pointxand so, by using Lemma ., the fractional finite difference equation

μ_μ–x(t) =gt+μ– ,x(t+μ– ),x(t+μ– )

via the boundary conditionsx(μ+b+ ) =α_k=μ–x(k) andx(μ– ) =  has a solution in

[–MG,MG].

Now, we consider the fractional finite difference equationμ_μ–x(t) =g(t+μ– ,x(t+

μ– )) via the boundary conditionsx(μ– ) = ,x(μ+b+ ) =  andx(α) =β_k=γx(k), where  <μ≤ andα,β,γ ∈N_μμ+_–bwithγ<β<α.

Lemma . Let b∈N,t∈Nb+,  <μ≤,g:N

b+μ+

μ– ×R→R,andα,β,γ ∈N

μ+b

μ–with γ <β<α.Then xis a solution of the problemμμ–x(t) =g(t+μ– ,x(t+μ– ))via the

boundary conditions x(μ+b+ ) = ,x(α) =β_k=γx(k),and x(μ– ) = if and only if xis

a solution of the fractional sum equation

x(t) =

b+

r=

G(t,r,β,α)gr+μ– ,x(r+μ– ),

where

G(t,r,β,α)

= (t

μ–_–tμ–_)(μ+b–r)μ–

(αμ–(b–α+μ+ ) –_μ_–((β+ )μ–_–γμ–_{) –} 

μ((β+ )

μ_–γμ₎₎

× (



μ–((β+ )

μ–_–γμ–_{)(α–μ+ ) –} 

μ((β+ ) μ

–γμ))

(b–α+μ+ )(μ+b+ )μ–_(μ)

+(t

μ–_{(α–μ+ ) –t}μ–_)(μ+b–r)μ–

(b–α+μ+ )(μ+b+ )μ–(μ)

+ (t

μ–_–tμ–_{)(α–r– )}μ–

(αμ–(b–α+μ+ ) – (_μ_–((β+ )μ––γμ–) –_μ((β+ )μ–γμ)))

× (



μ((β+ )

μ_–γμ_{) – (b+ )} 

μ–((β+ )

μ–_–γμ–₎₎

αμ–(b–α+μ+ )(μ)

+(t

μ–_{– (b+ )t}μ–_{)(α–r– )}μ–

αμ–(b–α+μ+ )(μ)

+ (t

μ–_–tμ–₎β

k=r+μ(k–σ(r)) μ–

(αμ–(b–α+μ+ ) – (_μ_–((β+ )μ–_–γμ–_{) –} 

μ((β+ )

μ_–γμ_)))(μ)

+(t–σ(r)) μ–

whenever r≤t–μ,r≤β–μ,

G(t,r,β,α)

= (t

μ–_–tμ–_)(μ+b–r)μ–

(αμ–(b–α+μ+ ) –_μ_–((β+ )μ––γμ–) –_μ((β+ )μ–γμ))

× (



μ–((β+ )

μ–_–γμ–_{)(α–μ+ ) –} 

μ((β+ ) μ

–γμ))

(b–α+μ+ )(μ+b+ )μ–(μ)

+(t

μ–_{(α–μ+ ) –t}μ–_)(μ+b–r)μ–

(b–α+μ+ )(μ+b+ )μ–(μ)

+ (t

μ–_–tμ–_{)(α–r– )}μ–

(αμ–(b–α+μ+ ) – (_μ_–((β+ )μ––γμ–) –_μ((β+ )μ–γμ)))

× (



μ((β+ )

μ_–γμ_{) – (b+ )} 

μ–((β+ )

μ–_–γμ–₎₎

αμ–(b–α+μ+ )(μ)

+(t

μ–_{– (b+ )t}μ–_{)(α–r– )}μ–

αμ–(b–α+μ+ )(μ)

+ (t

μ–_–tμ–₎β

k=r+μ(k–σ(r)) μ–

(αμ–(b–α+μ+ ) – (_μ_–((β+ )μ––γμ–) –_μ((β+ )μ–γμ)))(μ)

whenever t–μ<r,r≤β–μ,

G(t,r,β,α)

= (t

μ–_–tμ–_)(μ+b–r)μ–

(αμ–(b–α+μ+ ) –_μ_–((β+ )μ––γμ–) –_μ((β+ )μ–γμ))

× (



μ–((β+ )

μ–_–γμ–_{)(α–μ+ ) –} 

μ((β+ ) μ_–γμ₎₎

(b–α+μ+ )(μ+b+ )μ–(μ)

+(t

μ–_{(α–μ+ ) –t}μ–_)(μ+b–r)μ–

(b–α+μ+ )(μ+b+ )μ–_(μ)

+ (t

μ–_–tμ–_{)(α–r– )}μ–

(αμ–(b–α+μ+ ) – (_μ_–((β+ )μ–_–γμ–_{) –} 

μ((β+ )

μ_–γμ₎₎₎

× (



μ((β+ ) μ

–γμ) – (b+ )_μ_–((β+ )μ––γμ–))

αμ–(b–α+μ+ )(μ)

+(t

μ–_{– (b+ )t}μ–_{)(α–r– )}μ–

αμ–(b–α+μ+ )(μ)

whenever t–μ<r,β–μ<r,r≤α–μ,

G(t,r,β,α)

= (t

μ–_–tμ–_)(μ+b–r)μ–

(αμ–(b–α+μ+ ) –_μ_–((β+ )μ––γμ–) –_μ((β+ )μ–γμ))

× (



μ–((β+ )

μ–_–γμ–_{)(α–μ+ ) –} 

μ((β+ ) μ

–γμ))

(b–α+μ+ )(μ+b+ )μ–_(μ)

+(t

μ–_{(α–μ+ ) –t}μ–_)(μ+b–r)μ–

(b–α+μ+ )(μ+b+ )μ–_(μ) +

(t–σ(r))μ–

whenever r≤t–μ,α–μ<r and

G(t,r,β,α)

= (t

μ–_–tμ–_)(μ+b–r)μ–

(αμ–(b–α+μ+ ) –_μ_–((β+ )μ––γμ–) –_μ((β+ )μ–γμ))

× (



μ–((β+ )

μ–_–γμ–_{)(α–μ+ ) –} 

μ((β+ ) μ

–γμ))

(b–α+μ+ )(μ+b+ )μ–_(μ)

+(t

μ–_{(α–μ+ ) –t}μ–_)(μ+b–r)μ–

(b–α+μ+ )(μ+b+ )μ–_(μ)

whenever t–μ<r,α–μ<r.

Proof Letxbe a solution of the problemμμ–x(t) =g(t+μ– ,x(t+μ– )) via the

bound-ary conditionsx(μ+b+ ) = ,x(α) =β_k=γx(k), andx(μ– ) = . By using Lemma ., we get

x(t) =Ctμ–+Ctμ–+Ctμ–+



(μ)

t–μ

r=

t–σ(r)μ–gr+μ– ,x(r+μ– )

.

Similar to the proof of Lemma ., by using the boundary value conditions we obtainC=

,

C=

–

αμ–(b–α+μ+ ) β

k=γ

x(k)

– 

(b–α+μ+ )(μ+b+ )μ–_(μ)

b+

r=

(μ+b–r)μ–gr+μ– ,x(r+μ– )

+ 

αμ–(b–α+μ+ )(μ) α–μ

r=

(α–r– )μ–gr+μ– ,x(r+μ– )

and

C=

b+ 

αμ–(b–α+μ+ ) β

k=γ

x(k)

+ α–μ+ 

(b–α+μ+ )(μ+b+ )μ–(μ)

b+

r=

(μ+b–r)μ–gr+μ– ,x(r+μ– )

– b+ 

αμ–(b–α+μ+ )(μ) α–μ

r=

(α–r– )μ–_{gr+μ– ,x}

(r+μ– )

.

Thus,

x(t) =

tμ––tμ–

αμ–(b–α+μ+ ) β

k=γ

x(k)

+ t

μ–_{(α–μ+ ) –t}μ–

(b–α+μ+ )(μ+b+ )μ–_(μ)

b+

r=

×gr+μ– ,x(r+μ– )

by replacing (.) in (.), we get

x(t)

=

b+

r=

(tμ–_–tμ–_)(μ+b–r)μ–

(αμ–(b–α+μ+ ) –_μ_–((β+ )μ––γμ–) – _μ((β+ )μ–γμ))

× (



μ–((β+ )

μ–_–γμ–_{)(α–μ+ ) –} 

μ((β+ ) μ_–γμ₎₎

(b–α+μ+ )(μ+b+ )μ–(μ)

+(t

μ–_{(α–μ+ ) –t}μ–_)(μ+b–r)μ–

(b–α+μ+ )(μ+b+ )μ–(μ)

gr+μ– ,x(r+μ– )

+ α–μ

r=

(tμ–_–tμ–_{)(α–r– )}μ–

(αμ–(b–α+μ+ ) – (_μ_–((β+ )μ––γμ–) –_μ((β+ )μ–γμ)))

× (



μ((β+ )

μ_–γμ_{) – (b+ )} 

μ–((β+ )

μ–_–γμ–₎₎

αμ–(b–α+μ+ )(μ)

+(t

μ–_{– (b+ )t}μ–_{)(α–r– )}μ–

αμ–(b–α+μ+ )(μ)

gr+μ– ,x(r+μ– )

+ β–μ

r=

(tμ––tμ–)_kβ_=r+μ(k–σ(r)) μ–

(αμ–(b–α+μ+ ) – (_μ_–((β+ )μ–_–γμ–_{) –} 

μ((β+ )

μ_–γμ_)))(μ)

×gr+μ– ,x(r+μ– )

+

t–μ

r=

(t–σ(r))μ– (μ) g

r+μ– ,x(r+μ– )

=

b+

r=

G(t,r,β,α)gr+μ– ,x(r+μ– )

.

Now, letxbe a solution of the fractional sum equation

x(t) =

b+

r=

G(t,r,β,α)gr+μ– ,x(r+μ– ).

Similar to proof of the Lemma ., we conclude thatxis a solution to the problem

μ_μ–x(t) =gt+μ– ,x(t+μ– )

via the boundary conditionsx(μ+b+ ) = , x(α) =β_k=γx(k), andx(μ– ) = . This

completes the proof.

By using similar proofs of Lemma . and Theorem ., we obtain the next results.

Lemma . For each t∈Nbμ++–μandα,β∈N

b+μ

μ–,we have

b+

r=

G(t,r,β,α)

≤

b+

r=

G(t,r,β,α)≤M_G

Theorem . Assume that g:Nbμ+–μ+×R→Ris continuous and bounded in its second

variable.Then the fractional finite difference equationμ_μ–x(t) =g(t+μ– ,x(t+μ– )) via the boundary conditions x(μ– ) = ,x(μ+b+ ) = ,and x(α) =β_k=γx(k)has a

solution xwith x(t)∈[–MG,MG],for all admissible t.

4 An example

Now, we provide an example for the first investigated problem.

Example . Consider the equation

via the boundary value conditions x(–

) =  and x(

in the first problem. Thus, we should investigate the fractional finite difference equation

is continuous and bounded in its second and third variables. Now, we show thatMG=

.. Also, the Green function is given by

Table 1 The values of the Green function

Similar calculations give us the values ofGsummarized in Table . Thus,MG≥



r=|G(t,r,α)|= .. Hence by using Theorem ., (.) has a solution

xwithx(t)∈[–., .] for all admissiblet.

5 Conclusions

In this manuscript based on a fixed point theorem we provided the existence results for two fractional finite difference equations in the presence of the sum boundary conditions. One example illustrates our results.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final version of the manuscript.

Author details

1_{Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA.} 2_{Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia.} 3_{Department of Mathematics, Cankaya University, Ogretmenler Cad. 14, Balgat, Ankara, 06530, Turkey.}4_{Institute of Space}

Sciences, Magurele, Bucharest, Romania. 5_{Department of Mathematics, Azarbaijan Shahid Madani University, Azarshahr,}

Tabriz, Iran.

Acknowledgements

Research of the third and fourth authors was supported by Azarbaijan Shahid Madani University.

Received: 16 September 2014 Accepted: 17 October 2014 Published:31 Oct 2014

References

1. Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ: Fractional Calculus Models and Numerical Methods. World Scientific, Singapore (2012)

3. Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)

4. Atici, FM, Eloe, PW: A transform method in discrete fractional calculus. Int. J. Differ. Equ.2, 165-176 (2007)

5. Atici, FM, Eloe, PW: Discrete fractional calculus with the nabla operator. Electron. J. Qual. Theory Differ. Equ.3, special edition I, 1-12 (2009)

6. Baleanu, D, Rezapour, S, Salehi, S: Ak-dimensional system of fractional finite difference equations. Abstr. Appl. Anal.

2014, Article ID 312578 (2014)

7. Elaydi, SN: An Introduction to Difference Equations. Springer, Berlin (1996)

8. Goodrich, CS: Some new existence results for fractional difference equations. Int. J. Dyn. Syst. Differ. Equ.3, 145-162 (2011)

9. Holm, M: Sum and differences compositions in discrete fractional calculus. CUBO13, 153-184 (2011)

10. Miller, KS, Ross, B: Fractional difference calculus. In: Proc. International Symposium on Univalent Functions, Fractional Calculus and Their Applications. Nihon University, Koriyama (1988)

11. Mohan, JJ, Deekshitulu, GVSR: Fractional order difference equations. Int. J. Differ. Equ.2012, Article ID 780619 (2012) 12. Pan, Y, Han, Z, Sun, S, Zhao, Y: The existence of solutions to a system of discrete fractional boundary value problems.

Abstr. Appl. Anal.2012, Article ID 707631 (2012)

13. Wu, GC, Baleanu, D: Discrete fractional logistic map and its chaos. Nonlinear Dyn.75, 283-287 (2014) 14. Wu, GC, Baleanu, D: Discrete chaos in fractional sine and standard maps. Phys. Lett. A378, 484-487 (2013) 15. Alyousef, K: Boundary value problems for discrete fractional equations. PhD thesis, University of Nebraska-Lincoln

(2012)

16. Awasthi, P: Boundary value problems for discrete fractional equations. PhD thesis, University of Nebraska-Lincoln, Ann Arbor, MI (2013)

17. Holm, M: The theory of discrete fractional calculus: development and applications. PhD thesis, University of Nebraska-Lincoln, Ann Arbor, MI (2011)

18. Garling, DJH, Gorenstein, D, Diesk, TT, Walters, P: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

19. Atici, FM, Eloe, PW: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc.137, 981-989 (2009)

10.1186/1687-1847-2014-282