Nonlocal boundary value problems for impulsive fractional \(q {k}\) difference equations

(1)

R E S E A R C H

Open Access

Nonlocal boundary value problems for

impulsive fractional

q

_k

-difference equations

Bashir Ahmad

1*

_{, Ahmed Alsaedi}

1

_{, Sotiris K Ntouyas}

1,2

_{, Jessada Tariboon}

3

_{and Faris Alzahrani}

1

*_{Correspondence:}

[email protected]

1_{Nonlinear Analysis and Applied}

Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

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

Abstract

In this paper, we investigate the existence and uniqueness of solutions for a nonlocal boundary value problem of impulsive fractionalqk-diﬀerence equations involving a

newqk-shifting operatora

qk(m) =qkm+ (1 –qk)a. Our main results rely on Banach’s contraction mapping principle, Leray-Schauder nonlinear alternative, and Rothe ﬁxed point theorem. Examples illustrating the obtained results are also presented.

MSC: 26A33; 39A13; 34A37

Keywords: quantum calculus; impulsive fractionalqk-diﬀerence equations;

existence; uniqueness; ﬁxed point theorem

1 Introduction

The main purpose of this manuscript is to study the existence and uniqueness of solutions for impulsive boundary value problems of fractionalqk-diﬀerence equations of the form

⎧ ⎪ ⎨ ⎪ ⎩

tkD

αk

qkx(t) =f(t,x(t)), t∈Jk⊆[,T],t=tk,

tkI

–αk

qk x(t

+

k) –x(tk) =ϕk(x(tk)), k= , , . . . ,m,

atI

–α

q x() =bx(T) +

m l=cltlI

γl

qlx(tl+),

(.)

where  =t<t<· · ·<tm<tm+=T,tkD

αk

qk denotes the Riemann-Liouvilleqk-fractional

derivative of orderαkonJk,  <αk≤,  <qk< ,Jk= (tk,tk+],J= [,t],k= , , . . . ,m, J= [,T],f ∈C(J×R,R),ϕk∈C(R,R),k= , , . . . ,m,tkI

αk

qk denotes the Riemann-Liouville

qk-fractional integral of orderαk>  onJk,a,b,cl∈R,γl> ,l= , , , . . . ,m.

The quantum calculus is known as the calculus without limits and provides a descent approach to deal with sets of nondifferentiable functions by considering difference opera-tors. Quantum difference operators play an important role in several mathematical areas such as orthogonal polynomials, basic hyper-geometric functions, combinatorics, the cal-culus of variations, mechanics, and the theory of relativity. The book by Kac and Cheung [] covers many fundamental aspects of the quantum calculus.

In recent years, the topic ofq-calculus has attracted the attention of several researchers, and a variety of new results can be found in the papers [–] and the references therein. In [], the notions ofqk-derivative andqk-integral for a functionf :Jk:= [tk,tk+]→

R, were introduced, and several their properties were obtained. Also, the existence and uniqueness results for initial value problems of ﬁrst- and second-order impulsive

(2)

qk-diﬀerence equations were studied. qk-calculus analogues of some classical inte-gral inequalities such as Hölder, Hermite-Hadamard, trapezoid, Ostrowski, Cauchy-Bunyakovsky-Schwarz, Grüss and Grüss-Čebyšev were proved in [].

In [], new concepts of fractional quantum calculus were deﬁned by introducing a new

q-shifting operatoraq(m) =qm+ ( –q)a. After giving the basic properties of the new

q-shifting operator, theq-derivative andq-integral were defined. New definitions of the Riemann-Liouville fractionalq-integral andq-difference on an interval [a,b] were given, and their basic properties were discussed. As applications of the new concepts, exis-tence and uniqueness results for first- and second-order initial value problems for impul-sive fractionalq-difference equations were presented. Recently, the existence of solutions for impulsive fractional q-difference equations with antiperiodic boundary conditions was discussed in [], whereas the existence results for a nonlinear impulsiveqk-integral boundary value problem were obtained in [].

In this paper, we consider a boundary value problem of impulsive fractional qk -dif-ference equations (.) by introducing a newqk-shifting operatoraqk(m) =qkm+ ( –qk)a

and establish some existence results for the new problem. The rest of this paper is orga-nized as follows: In Section , we recall some known facts about fractionalqk-calculus, present an auxiliary lemma, which is used to convert problem (.) into a ﬁxed point prob-lem, and a lemma dealing with useful bounds. Section  contains the main results, whereas some illustrative examples are presented in Section .

2 Preliminaries

For any positive integerk, theqk-shifting operator:aqk(m) =qkm+ ( –qk)a[] satisﬁes

the relation

akqk(m) =a

k–

qk

aqk(m)

withaqk(m) =m.

We deﬁne the power ofqk-shifting operator as

a(n–m)()qk = , a(n–m)

(k)

qk =

k–

i=

n–aiqk(m)

, k∈N∪ {∞}.

Ifγ ∈R, then

a(n–m)(γqk)=n

(γ)

∞

i=

 –a n

i qk(m/n)

 –a n

γ+i qk (m/n)

, n= .

Theqk-derivative of a functionf on interval [a,b] is deﬁned by

(aDqkf)(t) =

f(t) –f(aqk(t))

( –qk)(t–a)

, t=a and (aDqkf)(a) =lim_t_→_a(aDqkf)(t),

and theqk-derivative of higher order is given by

aDkqkf

(t) =aDkq–k (aDqkf)(t),

aDqkf

(3)

Theqk-integral of a functionf deﬁned on the interval [a,b] is given by

(aIqkf)(t) =

t

a

f(s)ads= ( –qk)(t–a) ∞

i=

qi_kfaqi_k(t)

, t∈[a,b]

and

aIqkkf

(t) =aIkqk–(aIqkf)(t),

aIqkf

(t) =f(t), k∈N.

The fundamental theorem ofqk-calculus applies to the operatoraDqk andaIqk as follows:

(aDqkaIqkf)(t) =f(t).

Iff is continuous att=a, then

(aIqkaDqkf)(t) =f(t) –f(a).

The formula ofqk-integration by parts on the interval [a,b] is

b

a

f(s)aDqkg(s)adqks= (fg)(t)

b a–

b

a

gaqk(s)

aDqkf(s)adqks.

Now we recall the deﬁnitions of the Riemann-Liouville fractionalqk-integral andqk -dif-ference on interval [a,b].

Deﬁnition . Let ν ≥ , and let f be a function deﬁned on [a,b]. The fractional

qk-integral of Riemann-Liouville type is given by (aIqkf)(t) =h(t) and

aIqνkf

(t) = 

qk(ν)

t

a a

t–aqk(s)

(ν–)

qk f(s)adqks, ν> ,t∈[a,b].

Deﬁnition . The fractionalqk-derivative of Riemann-Liouville type of orderν≥ on the interval [a,b] is deﬁned by (aDqkf)(t) =f(t) and

aDνqkf

(t) =aDlqkaI

l–ν

qk f

(t), ν> ,

wherelis the smallest integer greater than or equal toν.

Lemma . Letα,β∈R+,and let f be a continuous function on[a,b],a≥.The Riemann-Liouville fractional qk-integral has the following semigroup property:

aIqβkaI

α

qkf(t) =aI

α

qkaI

β

qkf(t) =aI

α+β

qk f(t).

Lemma . Let f be a qk-integrable function on[a,b].Then

aDαqkaI

α

(4)

Lemma . Letα> ,and let p be a positive integer.Then,for t∈[a,b],

aIqαkaD

p

qkf(t) =aD

p qkaI

α

qkf(t) –

p–

k=

(t–a)α–p+k

qk(α+k–p+ )

aDkqkf(a).

From [] we have the following formulas

aDαqk(t–a)

β₌ qk(β+ )

qk(β–α+ )

(t–a)β–α, (.)

aIqαk(t–a)

β

= qk(β+ )

qk(β+α+ )

(t–a)β+α. (.)

In the sequel, we deﬁnePC(J,R) ={x:J→R,x(t) is continuous everywhere except for some tk at whichx(tk+) andx(t–k) exist andx(tk–) =x(tk), k= , , , . . . ,m}. For β∈R+,

we introduce the spaceCβ,k(Jk,R) ={x:Jk→R: (t–tk)βx(t)∈C(Jk,R)}with the norm

xCβ,k =supt∈Jk|(t–tk)

β_x₍_t₎_|_and_PC

β={x:J→R: for eacht∈Jk, (t–tk)βx(t)∈C(Jk,R),

k= , , , . . . ,m}with the norm

xPCβ =max

sup t∈Jk

(t–tk)βx(t):k= , , , . . . ,m

.

Clearly,PCβis a Banach space.

Lemma . Let y∈AC(J,R).Then x∈PC(J,R)is a solution of

⎧ ⎪ ⎨ ⎪ ⎩

tkD

αk

qkx(t) =y(t), t∈J,t=tk,

tkI

–αk

qk x(t

+

k) –x(tk) =ϕk(x(tk)), k= , , . . . ,m,

atI

–α

q x() =bx(T) +

m l=cltlI

γl

qlx(tl+),

(.)

if and only if

x(t) = (t–tk)

αk–

qk(αk)

_k_–

j=

(tj+–tj)αj–

qj(αj)

bm–

j=

j<i≤m

(ti+–ti)αi–

qi(αi)

×tjI

αj

qjy(tj+) +ϕj+

x(tj+)

+ btmI

αm

qmy(T)

+ m

l=

cl(tl+–tl)αl+γl– ql(αl+γl)

_l_–

j=

j<i≤l–

qi(αi)

×tjI

αj

x(tj+)

+ m

l= cl

tlI

αl+γl

ql y(tl+)

+(t–tk)

αk–

qk(αk)

_k_–

j=

j<i≤k–

qi(αi)

tjI

αj

x(tj+)

+tkI

αk

(5)

whereb_a(·) = ,b_a(·) = for a>b,_a_<_a(·) = ,and the nonzero constant is deﬁned by

=a–b _m

j=

qj(αj)

– m

l=

cl(tl+–tl)αl+γl–

ql(αl+γl)

_l_–

j=

qj(αj)

. (.)

Proof Applying the Riemann-Liouville fractionalq-integral of orderαto both sides of

the ﬁrst equation of (.) fort∈J, we obtain

tI

α

qtD

α

qx(t) =tI

α

qtDqtI

–α

q x(t) =tI

α

qy(t). (.)

From Lemmas ., ., and . fort∈J, we have

x(t) = t

α– q(α)

tI

–α

q x() +tI

α

qy(t).

Fort∈J, applying the Riemann-Liouville fractionalq-integral of orderα again to the

ﬁrst equation in (.) and using the previous process, we get

x(t) = (t–t)

α–

q(α)

tI

–α

q x

t_++tI

α

qy(t). (.)

The impulsive condition implies that

x(t) = (t–t)

α–

q(α)

tα–

 q(α)

tI

–α

q x() +tI

α

qy(t) +ϕ

x(t)

+tI

α

qy(t).

Similarly, fort∈J, we have

x(t) = (t–t)

α–

q(α)

(t–t)α– q(α)

tα–

 q(α)

tI

–α

q x() +tI

α

qy(t) +ϕ

x(t)

+tI

α

qy(t) +ϕ

x(t)

+tI

α

qy(t).

Repeating this process fort∈Jk⊆J,k= , , , . . . ,m, we obtain

x(t) = (t–tk)

α_k– qk(αk)

_k_–

j=

qj(αj)

tI

–α

q x()

+(t–tk)

αk–

qk(αk)

_k_–

j=

j<i≤k–

qi(αi)

tjI

αj

x(tj+)

+tkI

αk

(6)

In particular, fort=T, we get

Taking the Riemann-Liouville fractionalql-integral of orderγlon (.) fromtltotl+and

using (.), we have

By the boundary condition of (.) we ﬁnd that

tI

Substituting the value oftI

–α

q x() into (.) yields (.). The converse follows by direct

computation. This completes the proof.

Lemma . Assume that all conditions of Lemma . hold. In addition, assume that

sup_t_∈_J|y(t)|=Nand there exists a constant N such that|ϕk(x)| ≤N for k= , , . . . ,m and x∈R.Then the following inequality holds:

(7)

(8)

×

This completes the proof.

3 Main results

Now we present our ﬁrst result, which deals with the existence and uniqueness of solu-tions for problem (.) and is based on the Banach contraction mapping principle.

Theorem . Assume that there exist a functionM∈C(J,R+₎_{and a positive constant M}  such that

(9)

Then problem(.)has a unique solution on J if

(M+M)Tβ< , (.)

where M=supt∈J|M(t)|,the constants,are deﬁned in Lemma.,andβ> .

Proof Consider the operatorL:PC(J,R)→PC(J,R) deﬁned by (.) and show thatL∈

PCβ. For this, letτ,τ∈Jk. Then we have

(τ–tk)βLx(τ) – (τ–tk)βLx(τ)

≤(τ–tk)β+αk–– (τ–tk)β+αk–

qk(αk)

Kx

+(τ–tk)βtkI

αk

qkf

τ,x(τ)

– (τ–tk)βtkI

αk

qkf

τ,x(τ),

where

Kx:=

_k_–

j=

qj(αj)

|b| | |

_m_–

j=

j<i≤m

qi(αi)

×tjI

αj

qjf

tj+,x(tj+)+ϕj+

x(tj+)

+ |b|

| |tmI

αm

qmf

T,x(T)

+ m

l=

|cl|(tl+–tl)αl+γl–

| |ql(αl+γl)

_l_–

j=

j<i≤l–

qi(αi)

×tjI

αj

qjf

x(tj+)

+ m

l= |cl|

| |tlI

αl+γl

ql f

tl+,x(tl+)

+

_k_–

j=

j<i≤k–

qi(αi)

×tjI

αj

qjf

x(tj+)

. (.)

Asτ→τ, we get|(τ–tk)βLx(τ) – (τ–tk)βLx(τ)| → for eachk= , , . . . ,m. Thus,

Lx(t)∈PCβ.

Now we deﬁne the ballBr={x∈PCβ(J,R) :xPCβ ≤r}. We will show thatLBr⊂Br. Letsup_t_∈_J|f(t, )|=A,max{|ϕ()|:k= , . . . ,m}=Aand choose a constantrsuch that

r≥ (A+A)T β

 – (M+M)Tβ

.

Then, for anyx∈Brandt∈J, we have

(t–tk)βLx(t)≤

(t–tk)β+αk–

qk(αk)

Kx+ (t–tk)βtkI

α_k

qkf

(10)

whereKxis given by (.). Using the inequalities

f(s,x)≤f(s,x) –f(s, )+f(s, )≤Mr+A, ϕ(x)≤ϕ(x) –ϕ()+ϕ()≤Mr+A

in (.) forx∈Brands∈Jand the computational details of Lemma ., together with

Kx≤

_k_–

j=

qj(αj)

|b| | |

_m_–

j=

j<i≤m

qi(αi)

×

(Mr+A)

(tj+–tj)αj

(αj+ )

+ (Mr+A)

+ (Mr+A) |b| | |

(T–tm)αm

(αm+ )

+ m

l=

| |ql(αl+γl)

_l_–

j=

j<i≤l–

qi(αi)

×

(Mr+A)

(tj+–tj)αj

(αj+ )

+ (Mr+A)

+ m

l= |cl|

| |

(tl+–tl)αl+γl

ql(αl+γl+ )

+

_k_–

j=

j<i≤k–

qi(αi)

(Mr+A)

(tj+–tj)αj

(αj+ )

+ (Mr+A)

,

we obtain

(t–tk)βLx(t)≤(t–tk)β

(Mr+A) +(Mr+A)

≤r(+M)Tβ+ (A+A)Tβ ≤r.

This implies thatLxPCβ≤rand, consequently,LBr⊂Br. For allx,y∈PCβ(J,R) andt∈J, as in Lemma ., we get

_Lx(t) –Ly(t)≤(M+M)x–yPCβ.

Multiplying both sides of this inequality by (t–tk)βfor eacht∈Jk, we have

(t–tk)βLx(t) –Ly(t)≤(t–tk)β(M+M)x–yPCβ

≤Tβ(M+M)x–yPCβ,

which leads toLx–LyPCβ≤T

β₍_M_₊_M_₎_x_–_y

PCβ. In view of condition (.), it follows by the Banach contraction mapping principle that the operatorLis a contraction. Hence,Lhas a ﬁxed point, which is a unique solution of problem (.) onJ.

(11)

Lemma .(Nonlinear alternative for single valued maps []) Let E be a Banach space,

C a closed,convex subset of E,U an open subset of C,and∈U.Suppose that F:U→C is continuous and compact(that is,F(U)is a relatively compact subset of C)map.Then either

(i) Fhas a ﬁxed point inU,or

(ii) there areu∈∂U(the boundary ofUinC)andθ∈(, )withu=θF(u).

Theorem . Assume that

(H) there exist continuous nondecreasing functionsQ,V: [,∞)→(,∞)and a

continu-ous functionp:J→R+_{such that}

f(t,x)≤p(t)Q|x| and ϕk(x)≤V

|x| (.)

for all(t,x)∈(J×R)andk= , , . . . ,m;

(H) there exists a constantM∗> such that such that

M∗

(p∗Q(M∗)+V(M∗))Tβ

> , (.)

wherep∗=sup_t_∈_J|p(t)|,β> ,and the constants,are deﬁned in Lemma..

Then problem(.)has at least one solution on J.

Proof First, we show that the operatorLdeﬁned by(.)maps bounded sets(balls)into bounded sets in PCβ. To accomplish this, for a positive number ρ, letBρ={x∈PCβ : xPCβ ≤ρ}be a ball inPCβ. Then, forx∈Bρandt∈J, using the method of proof used in Lemma ., we obtain

_Lx(t)≤(t–tk)

αk–

qk(αk)

Kx+tkI

αk

qkf

t,x(t),

whereKxis deﬁned by (.). From (H) we have

Kx≤

_k_–

j=

qj(αj)

|b| | |

_m_–

j=

j<i≤m

qi(αi)

×

p∗Q(ρ)

(tj+–tj)αj

(αj+ )

+V(ρ)

+p∗Q(ρ)|b|

| |

(T–tm)αm

(αm+ )

+ m

l=

| |ql(αl+γl)

_l_–

j=

j<i≤l–

qi(αi)

×

p∗Q(ρ)

(tj+–tj)αj

(αj+ )

+V(ρ)

+ m

l= |cl|

| |

(tl+–tl)αl+γl

ql(αl+γl+ )

+

_k_–

j=

j<i≤k–

qi(αi)

p∗Q(ρ)

(tj+–tj)αj

(αj+ )

+V(ρ)

(12)

and thus

_Lx(t)≤p∗Q(ρ)+V(ρ).

Therefore, (t–tk)β|Lx(t)| ≤(t–tk)β(p∗Q(ρ)+V(ρ)), which means thatLxPCβ ≤

Tβ₍_p∗_Q₍_ρ₎

+V(ρ)).

Next we show thatLmaps bounded sets into equicontinuous sets of PCβ.

Lettingτ,τ∈Jkfor somek∈ {, , , . . . ,m}withτ<τandx∈Bρ, whereBρis a ball

inPCβ, we have

_Lx(τ) –Lx(τ)≤

(τ–tk)αk–– (τ–tk)αk–

qk(αk)

Kx

+tkI

αk

qkf

τ,x(τ)

–tkI

αk

qkf

τ,x(τ)

≤(τ–tk)αk–– (τ–tk)αk–

qk(αk)

Kx

+p∗Q(ρ)(τ–tk)

αk_{– (}_τ

–tk)αk

qk(αk+ )

. (.)

Asτ→τ, the right-hand side of inequality (.) tends to zero independently ofx, that

is,

(τ–tk)βLx(τ) – (τ–tk)βLx(τ)→ as|τ–τ| →.

Therefore, by the Arzelà-Ascoli theorem,L:PCβ→PCβis completely continuous.

Our result will follow from the Leray-Schauder nonlinear alternative once we show the boundedness of the set of all solutions to the equationx(t) =λLx(t) for  <λ< . Letxbe a solution. For anyt∈Jandx∈PCβ, following the method of proof used in the ﬁrst step

together with condition (H), we get

xPCβ ≤

p∗QxPCβ

+V

xPCβ



Tβ.

In consequence, we have

xPCβ

(p∗Q(xPCβ)+V(xPCβ))T

β ≤.

By condition (H) there exists M∗ such that xPCβ = M∗. We deﬁne U={x∈PCβ :

xPCβ <M∗}. Note that the operatorL:U→PCβ is continuous and completely con-tinuous. By the choice ofUthere is nox∈∂Usuch thatx=λLxfor someλ∈(, ). Con-sequently, by the nonlinear alternative of Leray-Schauder type (Lemma .) we deduce thatLhas a ﬁxed pointx∈U, which is a solution of problem (.) onJ. This completes

the proof.

A key to prove the ﬁnal result is based on the following ﬁxed point theorem.

(13)

(i) (Altman)Ax–x_≥_Ax_–_x_{for all}_x_∈_∂_,

(ii) (Rothe)Ax ≤ xfor allx∈∂ ,

(iii) (Petryshyn)Ax ≤ Ax–xfor allx∈∂ .

Thendeg(I–A, ,θ) = ,and hence A has at least one ﬁxed point in .

Theorem . Assume that

(H) the continuous functionsf :J×R→Randϕk:R→R,k= , , . . . ,m,satisfy

lim x→

f(t,x)

x =  and xlim→ ϕk(x)

x = , k= , , . . . ,m. (.)

Then problem(.)has at least one solution on J.

Proof Letx∈PCβ. Takingεsuﬃciently small, we can choose two positive constantsδand δsuch that|f(t,x)|<ε|x|wheneverxPCβ<δandϕk(x) <ε|x|wheneverxPCβ <δfor

k= , , . . . ,m. Settingδ=min{δ,δ}, we deﬁne the open ballBδ={x∈PCβ:xPCβ <δ}. As in Theorem ., it is clear that the operatorL:PC→PCis completely continuous. For anyx∈∂Bδ, we have

_Lx(t)= (t–tk)

αk–

qk(αk)

_k_–

j=

qj(αj)

|b| | |

_m_–

j=

j<i≤m

qi(αi)

×tjI

αj

qjf

x(tj+)

+ |b|

| |tmI

αm

qmf

T,x(T)

+ m

l=

| |ql(αl+γl)

_l_–

j=

j<i≤l–

qi(αi)

×tjI

αj

qjf

x(tj+)

+ m

l= |cl|

| |tlI

αl+γl

ql f

tl+,x(tl+)

+(t–tk)

αk–

qk(αk)

_k_–

j=

j<i≤k–

qi(αi)

×tjI

αj

qjf

x(tj+)

+tkI

αk

qkf

t,x(t)

≤(+)ε|x|.

Settingε≤(+)–, we deduce that

|Lx| ≤ |x|.

Multiplying both sides of this inequality by (t–tk)β, we haveLxPCβ≤ xPCβ. It follows from Lemma .(ii) that problem (.) has at least one solution onJ.

4 Examples

(14)

Example . Consider the following nonlocal boundary value problem for impulsive frac-tionalq-diﬀerence equations:

⎧

hold. Therefore, by the conclusion of Theorem ., problem (.) has a unique solution on [, /].

(15)

SettingQ(x) = (x/) + ,V(x) = (x/) + (/),p∗= /, andβ= , there exists a constant

M∗> . satisfying (.). Thus, the hypothesis of Theorem . is satisﬁed. In consequence, the conclusion of Theorem . applies, and problem (.) has at least one solution on [, ].

Example . Consider the problem of impulsive fractionalq-diﬀerence equations given

by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tkD

(k+k+

k+k+)

(k+k+

k+k+)

x(t) =__t_+t (sinx(t) –x(t))ex(t)cosx(t), t∈[, /]\tk,

tkI

( k+k+

k+k+)

(k+k+

k+k+)

x(t+_k) –x(tk) =kx

₍_t

k)+kx(tk) log(|x₍_t

k)|+) , tk=k,k= , , , ,

  I

   x

() = x(

 ) +



l=(l

_+_l_+

l_+_l_+)tlI

(l_+) (l+l+

l+l+)

x(tl+).

(.)

Hereαk= (k+k+ )/(k+ k+ ),qk= (k+ k+ )/(k+ k+ ),γk= (k+ )/,

ck = (k + k+ )/(k+ k+ ), k = , , , , ,a= /, b= /,T = /, tk =k/,

k= , , , . With this data, we ﬁnd that| |= .= . The functionsf(t,x) = ((t)/(t+ ))(sinx–x)excosx_and_ϕ

k(x) = (kx+ kx)/(log(|x|+ )),k= , , , , satisfy

lim x→

f(t,x)

x =xlim→

t

t+ 

_sin x x – 

excosx= 

and

lim x→

ϕk(x)

x =xlim→

kx_{+ }_kx

log(|x|_{+ )}= , k= , , , .

Thus, condition (H) of Theorem . holds. Therefore, by applying Theorem . we

con-clude that problem (.) has at least one solution on [, /].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, BA, AA, SKN, JT, and FA contributed to each part of this work equally and read and approved the ﬁnal version of the manuscript.

Author details

1_{Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science,}

King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 2_{Department of Mathematics, University of}

Ioannina, Ioannina, 451 10, Greece.3_{Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of}

Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand.

Acknowledgements

This paper was funded by the Deanship of Scientiﬁc Research (DSR) at King Abdulaziz University. The authors, therefore, acknowledge with thanks DSR for technical and ﬁnancial support. The authors thank the reviewers for their useful comments.

Received: 4 March 2016 Accepted: 27 April 2016

References

1. Kac, V, Cheung, P: Quantum Calculus. Springer, New York (2002)

2. Bangerezako, G: Variationalq-calculus. J. Math. Anal. Appl.289, 650-665 (2004)

(16)

4. Ferreira, RAC: Nontrivial solutions for fractionalq-diﬀerence boundary value problems. Electron. J. Qual. Theory Diﬀer. Equ.2010, 70 (2010)

5. Ahmad, B, Ntouyas, SK: Boundary value problems forq-diﬀerence inclusions. Abstr. Appl. Anal.2011, Article ID 292860 (2011)

6. Ahmad, B: Boundary-value problems for nonlinear third-orderq-diﬀerence equations. Electron. J. Diﬀer. Equ.2011, 94 (2011)

7. Annaby, MH, Mansour, ZS:q-Fractional Calculus and Equations. Lecture Notes in Mathematics, vol. 2056. Springer, Berlin (2012)

8. Graef, JR, Kong, L: Positive solutions for a class of higher order boundary value problems with fractionalq-derivatives. Appl. Math. Comput.218, 9682-9689 (2012)

9. Ahmad, B, Ntouyas, SK, Purnaras, IK: Existence results for nonlocal boundary value problems of nonlinear fractional q-diﬀerence equations. Adv. Diﬀer. Equ.2012, 140 (2012)

10. Zhou, W, Liu, H: Existence solutions for boundary value problem of nonlinear fractionalq-diﬀerence equations. Adv. Diﬀer. Equ.2013, 113 (2013)

11. Ferreira, RAC, Goodrich, CS: On positive solutions to fractional difference inclusions. Analysis35, 73-83 (2015) 12. Tariboon, J, Ntouyas, SK: Quantum calculus on finite intervals and applications to impulsive difference equations.

Adv. Diﬀer. Equ.2013, 282 (2013)

13. Tariboon, J, Ntouyas, SK: Quantum integral inequalities on ﬁnite intervals. J. Inequal. Appl.2014, 121 (2014) 14. Tariboon, J, Ntouyas, SK, Agarwal, P: New concepts of fractional quantum calculus and applications to impulsive

fractionalq-diﬀerence equations. Adv. Diﬀer. Equ.2015, 18 (2015)

15. Ahmad, B, Tariboon, J, Ntouyas, SK, Alsulami, H, Monaquel, S: Existence results for impulsive fractionalq-diﬀerence equations with anti-periodic boundary conditions. Bound. Value Probl.2016, 16 (2016)

16. Zhang, L, Ahmad, B, Wang, G: Existence results for nonlinear impulsiveqk-integral boundary value problems. Publ. Inst. Math. (Belgr.)99(113), 227-235 (2016)

17. Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2003)