Existence results for impulsive fractional q-difference equations with anti-periodic boundary conditions

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R E S E A R C H

Open Access

Existence results for impulsive fractional

q

-difference equations with anti-periodic

boundary conditions

Bashir Ahmad

1*

_{, Jessada Tariboon}

2

_{, Sotiris K Ntouyas}

1,3

_{, Hamed H Alsulami}

1

_{and Shatha Monaquel}

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

This paper studies a Caputo type anti-periodic boundary value problem of impulsive fractionalq-diﬀerence equations involving aq-shifting operator of the form

a

q(m) =qm+ (1 –q)a. Concerning the existence of solutions for the given problem,

two theorems are proved via Schauder’s ﬁxed point theorem and the Leray-Schauder nonlinear alternative, while the uniqueness of solutions is established by means of Banach’s contraction mapping principle. Finally, we discuss some examples illustrating the main results.

MSC: 26A33; 39A13; 34A37

Keywords: quantum calculus; impulsive fractionalq-diﬀerence equations; existence; uniqueness; ﬁxed point theorem

1 Introduction

The quantum calculus (calculus without limits orq-calculus) is concerned with differ-ence operators and allows one to deal with sets of nondifferentiable functions. Quantum difference operators appear in several branches of mathematics, such as orthogonal poly-nomials, basic hyper-geometric functions, combinatorics, the calculus of variations, me-chanics, and the theory of relativity. For the fundamental concepts of quantum calculus, we refer the reader to the text by Kac and Cheung [].

In recent years, the topic ofq-calculus has attracted the attention of several researchers and a variety of new results can be found in [–] and the references cited therein.

In [] and [], the authors studied the problems involving nonlinear impulsiveq k-dif-ference equations. In [], some new concepts of fractional quantum calculus were intro-duced in terms of aq-shifting operatoraq(m) =qm+ ( –q)a, and existence results for initial value problems of impulsive fractionalq-diﬀerence equations were obtained.

In this paper we study the following anti-periodic boundary value problem of an impul-sive fractionalq-diﬀerence equation:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ c tkD

αk

qkx(t) =f(t,x(t)), t∈Jk⊆[,T],t=tk,

x(tk) :=x(t+_k) –x(tk) =ϕk(tk–I βk–

qk–x(tk)), k= , , . . . ,m, tkDqkx(t

+

k) –tk–Dqk–x(tk) =ϕ ∗ k(tk–I

γk–

qk–x(tk)), k= , , . . . ,m,

x() = –x(T), Dqx() = –tmDqmx(T),

(.)

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where  =t<t<· · ·<tm<tm+=T,ctkD

αk

qk denotes the Caputoqk-fractional derivative

of orderαk onJk,  <αk≤,  <qk< ,Jk= (tk,tk+],J= [,t],k= , , . . . ,m,J= [,T],

f ∈C(J×R,R),ϕk,ϕ_k∗∈C(R,R),k= , , . . . ,m,tkI

βk

qk,tkI

γk

qk denotes the Riemann-Liouville

qk-fractional integral of ordersβk,γk>  onJk,k= , , , . . . ,m– .

The rest of the paper is organized as follows. In Section , we present some background material for the problem at hand and prove an auxiliary lemma for the linear variant of problem (.). Section  contains the main results which are established by means of Schauder’s ﬁxed point theorem, the Leray-Schauder nonlinear alternative, and Banach’s ﬁxed point theorem. Section  deals with some illustrative examples for the existence re-sults obtained in Section .

2 Preliminaries

First of all, we recall some basic concepts ofq-calculus []. Theq-derivative of a functionf on the interval [a,b] is deﬁned by

(aDqf)(t) =

f(t) –f(aq(t))

( –q)(t–a) , t=a and (aDqf)(a) =t→alim(aDqf)(t), (.) andq-derivative of higher order is given by

aDkqf

(t) =aDkq–(aDqf)(t),

aDqf

(t) =f(t), k∈N.

Theq-derivatives of a product and ratio of functionsf andgon [a,b] are

aDq(fg)(t) =f(t)aDqg(t) +g

aq(t)

aDqf(t) =g(t)aDqf(t) +f

aq(t)

aDqg(t) (.)

and

aDq

f g (t) =

g(t)aDqf(t) –f(t)aDqg(t)

g(t)g(aq(t)) , (.)

whereg(t)g(aq(t))= .

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

(aIqf)(t) = t

a

f(s)ads= ( –q)(t–a) ∞

i=

qifaqi(t)

, t∈[a,b], (.)

with

aIqkf

(t) =aIqk–(aIqf)(t),

aIqf

(t) =f(t), k∈N. (.) The fundamental theorem of calculus applies to the operatorsaDqandaIqas follows:

(aDqaIqf)(t) =f(t). (.)

Iff is continuous att=a, then

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

a

f(s)aDqg(s)adqs= (fg)(t)|ba– b

a

gaq(s)

aDqf(s)adqs. (.) Now we enlist some useful properties of theq-shifting operatoraq(m) =qm+ ( –q)a as follows:

(i) ak_q(m) =ak_q–(aq(m))witha_q(m) =m, for any positive integerk. (ii) a(n–m)()q = ,a(n–m)(qk)=

k–

i=(n–aqi(m)),k∈N∪ {∞}. (iii) a(n–m)q(γ)=n(γ)∞i=

–a n

i q(m/n)

–a n

γ+i q (m/n)

,γ ∈R.

Next we turn to the deﬁnitions of the Riemann-Liouville fractionalq-derivative and the

q-integral on the interval [a,b].

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

aDνqf

(t) =aDlqaIl– ν q f

(t), ν> , (.)

wherelis the smallest integer greater than or equal toν.

Deﬁnition .[] Letα≥ andf be a function deﬁned on [a,b]. The fractionalq -inte-gral of Riemann-Liouville type is given by (aIqf)(t) =f(t) and

aIqαf

(t) =  q(α)

t

a a

t–aq(s) (α–)

q f(s)adqs, α> ,t∈[a,b]. (.) From [], we have the following formulas:

aDαq(t–a) β

= q(β+ )

q(β–α+ )(t–a) β–α

, (.)

aIqα(t–a) β

= q(β+ ) q(β+α+ )

(t–a)β+α. (.)

Lemma . [] Let α,β ∈R+ _{and f be a continuous function on} _[_a_,_b_], _a_≥_._The

Riemann-Liouville fractional q-integral has the following semi-group property:

aIq aβ I α

qf(t) =aIq aα I β

qf(t) =aIqα+βf(t). (.)

Lemma . [] Let f be a q-integrable function on[a,b].Then the following equality holds:

aDαq aI α

qf(t) =f(t), forα> ,t∈[a,b]. (.)

Lemma . [] Letα>  and p be a positive integer. Then for t∈[a,b]the following equality holds:

aIq aα Dpqf(t) =aDpq aI α qf(t) –

p–

k=

(t–a)α–p+k q(α+k–p+ )aD

k

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In this paper we ﬁrst give the deﬁnition of Caputo fractionalq-derivative as follows.

Deﬁnition . The fractionalq-derivative of Caputo type of orderα≥ on the interval [a,b] is deﬁned by (c

aDqf)(t) =f(t) and _c

aD α qf

(t) =aIqn–αaDnqf

(t), α> , (.)

wherenis the smallest integer greater than or equal toα.

Lemma . Letα> and n be the smallest integer greater than or equal toα.Then for t∈[a,b],the following equality holds:

aIqαcaD α

qf(t) =f(t) – n–

k=

(t–a)k q(k+ )aD

k

qf(a). (.)

Proof From Lemma ., forα=p=m, wheremis a positive integer, we have

aIq am Dmqf(t) =aDmq aIqmf(t) – m–

k=

(t–a)k q(k+ )aD

k

qf(a) =f(t) – m–

k=

(t–a)k q(k+ )aD

k qf(a).

Then by Deﬁnition ., we have

aIqαcaD α

qf(t) =aIq aα In– α

q aDnqf(t) =aIq an Dnqf(t) =f(t) – n–

k=

(t–a)k q(k+ )aD

k

qf(a).

Lemma . Let h∈C(J,R).Then the unique solution of

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ c tkD

αk

qkx(t) =h(t), t∈Jk⊆[,T],t=tk,

x(tk) =ϕk(tk–I βk–

qk–x(tk)), k= , , . . . ,m, tkDqkx(t

+

k) –tk–Dqk–x(tk) =ϕ∗k(tk–I γk–

qk–x(tk)), k= , , . . . ,m,

x() = –x(T), Dqx() = –tmDqmx(T),

(.)

is given by

x(t) = – 

m

i=

ti–I αi–

qi–h(ti) +ϕi

ti–I βi– qi–x(ti)

– 

m

i= (T–ti)

ti–I

αi–– qi– h(ti) +ϕ

∗ i

ti–I γi– qi–x(ti)

– tmI

αm

qmh(T) +

t–T

 –

 

m

i=

ti–I αi–– qi– h(ti)

+ϕ∗_iti–I γi– qi–x(ti)

–

tmI

αm–

qm h(T)

+ k

i=

ti–I αi–

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+ k

i=

(t–ti)ti–I αi–– qi– h(ti) +ϕ

∗ i

ti–I

γi– qi–x(ti)

+tkI

αk

qkh(t), (.)

where_(·) = .

Proof Applying the Riemann-Liouville fractionalq-integral operator of orderαon both sides of the ﬁrst equation of (.) fort∈Jand using Lemma ., we obtain

tIqαctD α

qx(t) =x(t) –x() –

Dqx() q()

t=tIqαh(t),

which yields

x(t) =C+Ct+tIqαh(t), (.) whereC=x() andC=Dqx(). In particular, fort=t, we have

x(t) =C+Ct+tIqαh(t) and tDqx(t) =C+tIqα–h(t). (.) Fort∈J, on application of the Riemann-Liouville fractionalq-integral operator of order αto (.) and using the above arguments, we get

x(t) =xt+_+ (t–t)tDqx

t+_+tIqαh(t). (.)

Using the impulsive conditionsx(t_+) =x(t) +ϕ(tIqβx(t)) andtDqx(t+) =tDqx(t) + ϕ∗_(tIqγx(t)), we obtain

x(t) =C+Ct+

tIqαh(t) +ϕ

tIqβx(t)

+ (t–t)

tIqα–h(t) +ϕ∗

tIqγx(t)

+tIqαh(t). In a similar manner, fort∈J, we have

x(t) =C+Ct+

tIqαh(t) +ϕ

tIqβx(t)

+tIqαh(t) +ϕ

tIqβx(t)

+ (t–t)

tIqα–h(t) +ϕ∗

tIqγx(t)

+ (t–t)

tIqα–h(t) +ϕ∗

tIqγx(t)

+tIqαh(t).

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

x(t) =C+Ct+ k

i=

ti–I αi–

+ k

i=

(t–ti)ti–I αi–– qi– h(ti) +ϕ

∗ i

ti–I

+tkI

αk

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where_(·) = . Notice thatx() =Cand

x(T) =C+CT+ m

i=

ti–I αi–

+ m

i= (T–ti)

ti–I

∗ i

ti–I

γi– qi–x(ti)

+tmI

αm

qmh(T).

On the other hand, we have

tkDqkx(t) =C+

k

i=

ti–I αi–– qi– h(ti) +ϕ

∗ i

ti–I

+tkI

αk–

qk h(t),

which impliestDqx() =Cand

tmDqmx(T) =C+

m

i=

ti–I αi–– qi– h(ti) +ϕ

∗ i

ti–I

+tmI

αm–

qm h(T).

Now making use of the boundary conditions given by (.), we ﬁnd that

C= –  CT–

  m i=

ti–I αi–

–  m i= (T–ti)

ti–I

∗ i

ti–I

γi– qi–x(ti)

–

tmI αm

qmh(T)

and

C= –   m i=

ti–I αi–– qi– h(ti) +ϕ

∗ i

ti–I γi– qi–x(ti)

–

tmI

αm– qm h(T).

Substituting the valuesCandCin (.) yields the solution (.).

3 Main results

LetPC(J,R) ={x:J→R:x(t) is continuous everywhere except for sometkat whichx(t+_k) andx(t–

k) exist andx(tk–) =x(tk),k= , , . . . ,m}. Observe thatPC(J,R) is a Banach space equipped with the normxPC=sup{|x(t)|:t∈J}.

In view of Lemma ., we deﬁne an operatorA:PC(J,R)→PC(J,R) by

Ax(t) = – 

m

i=

ti–I αi– qi–f

ti,x(ti)+ϕiti–I βi– qi–x(ti)

–  m i=

(T–ti)ti–I αi–– qi– f

ti,x(ti)+ϕ_i∗ti–I γi– qi–x(ti)

– tmI

αm

qmf

T,x(T)+

t–T

 –   m i=

ti–I αi–– qi– f

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+ϕ∗_iti–I γi– qi–x(ti)

– 

tmI αm– qm f

T,x(T)

+ k

i=

ti,x(ti)+ϕi

+ k

i= (t–ti)

ti–I

αi–– qi– f

ti,x(ti)

+ϕ∗_iti–I γi– qi–x(ti)

+tkI

αk

qkf

t,x(t), (.)

where

aIqpf

u,x(u)=  q(p)

u

a a

u–aq(s) (p–)

q f

s,x(s)adqs,

p∈ {α, . . . ,αm,α– , . . . ,αm– ,β, . . . ,βm–,γ, . . . ,γm–},q∈ {q, . . . ,qm},a∈ {t, . . . ,tm}, andu∈ {t,t,t, . . . ,tm,T}.

For computational convenience, we set

=  

m+

i=

(ti–ti–)αi– q_i_–(αi–+ )

+ 

m

i=

(T–ti)(ti–ti–)αi–– q_i_–(αi–)

+T 

m+

i=

(ti–ti–)αi–– qi–(αi–)

, (.)

=  mM+

 M

m

i=

(T–ti) +T

mM. (.)

Now we present our ﬁrst existence result for the problem (.), which is based on the Schauder ﬁxed point theorem.

Theorem . Assume that

(H) there exist continuous functionsa(t),b(t),and nonnegative constantsM,Msuch that f(t,x)≤a(t) +b(t)|x|, (t,x)∈J×R (.)

withsup_t∈J|a(t)|=a,supt∈J|b(t)|=b,and

ϕk(x)≤M, ϕ∗k(x)≤M, ∀x∈R,k= , , . . . ,m. (.)

Then the anti-periodic boundary value problem(.)has at least one solution on J if

b< . (.)

Proof Let us deﬁne a closed ballBR={x∈PC(J,R) :xPC≤R}with

R>a+  –b

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wherea,bare deﬁned in (H) and,are, respectively, given by (.) and (.). Clearly

BRis a bounded, closed, and convex subset ofPC(J,R). Now we show that the operator

A:PC(J,R)→PC(J,R) deﬁned by (.) has a ﬁxed point in the following two steps.

Step.A:BR→BR.

For anyx∈BR, using (.), we have _Ax(t)≤ 

 m

i=

ti,x(ti)+ϕiti–I βi– qi–x(ti)

+ 

m

i=

(T–ti)ti–I αi–– qi– f

ti,x(ti)+ϕ_i∗ti–I γi– qi–x(ti)

+ tmI

αm

qmf

T,x(T)+T    m i=

ti–I αi–– qi– f

ti,x(ti)

+ϕ_i∗ti–I γi–

qi–x(ti)+  tmI

αm–

qm f

T,x(T) + k i=

ti,x(ti)+ϕi

ti–I βi– qi–x(ti)

+ k

i= (t–ti)

ti–I

αi–– qi– f

ti,x(ti)+ϕi∗

ti–I γi– qi–x(ti) +tkI

αk

qkf

t,x(t)

≤   m i=

a+bxPC

ti–I αi–

qi–(ti) +M +  m i=

(T–ti)a+bxPC

ti–I αi––

qi– (ti) +M

+ 

a+bxPC

tmI

αm

qm(T)

+T    m i=

a+bxPC

qi– (ti) +M

+ 

a+bxPC

tmI

αm– qm (T)

+ m i=

a+bxPC

ti–I αi–

qi–(ti) +M

+ m

i=

(T–ti)a+bxPC

qi– (ti) +M

+a+bxPC

tmI

αm

qm(T)

=  

m+

i=

(ti–ti–)αi– qi–(αi–+ )

a+bxPC

+ mM

+ 

m

i= (T–ti)

(ti–ti–)αi–– qi–(αi–)

a+bxPC

(9)

+T 

m+

i=

(ti–ti–)αi–– qi–(αi–)

a+bxPC

+T Mm =a++bxPC≤R,

which impliesAxPC≤R. Therefore,A:BR→BR.

Step.The operatorA:PC(J,R)→PC(J,R)is completely continuous on BR. Letsup₍_t_,_x₎_∈J×B

R|f(t,x)|=F. For anyτ,τ∈Jk,k= , , . . . ,m, withτ<τ, we have

_Ax(τ) –Ax(τ)≤ |τ–τ|

 

m+

i=

(ti–ti–)αi–– qi–(αi–)

F+

mM 

+|τ–τ| k

i=

(ti–ti–)αi–– qi–(αi–)

F+M

+ F qk(αk)

_tτ

k

tk(τ–tkqk)

(α_k–) qk tkdqks

– τ

tk

tk(τ–tkqk)

(αk–)

qk tkdqks

,

which is independent ofxand tends to zero asτ–τ→. ThereforeAis equicontinuous. ThusABRis relatively compact asABR⊂BRis uniformly bounded. In view of the conti-nuity off,ϕkandϕ_k∗,k= , , . . . ,m, it is clear that the operatorAis continuous. Hence the operatorA:PC(J,R)→PC(J,R) is completely continuous onBR. Applying the Schauder ﬁxed point theorem, we deduce that the operatorAhas at least one ﬁxed point inBR. This shows that the problem (.) has at least one solution onJ.

In the next existence result, we make use of Leray-Schauder’s nonlinear alternative.

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 a continuous,compact(that is,F(U)is a relatively compact subset of C)map.Then either

(i) Fhas a ﬁxed point inU,or

(ii) there is au∈∂U(the boundary ofUinC)andθ∈(, )withu=θF(u). In the sequel, we set

=  

m

i=

(ti–ti–)βi– q_i_–(βi–+ )

, (.)

=  

m

i=

(T–ti)(ti–ti–)γi– qi–(γi–+ )

+T 

m

i=

(ti–ti–)γi– qi–(γi–+ )

. (.)

Theorem . Assume that

(H) there exist a continuous nondecreasing functionψ: [,∞)→(,∞), a continuous

functionp:J→R+_with_p∗₌_sup

(10)

and

ϕk(x)≤M|x|, ϕk∗(x)≤M|x|, ∀x∈R,k= , . . . ,m; (.)

(H) there exists a constantN> such that ( –M–M)N

p∗ψ(N)

> , M+M< , (.)

where,are,respectively,given by(.)and(.).

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

Proof We shall show that the operatorAdeﬁned by (.) has a ﬁxed point. To accomplish this, for a positive numberρ, letBρ={x∈PC(J,R) :xPC≤ρ}denote a closed ball in

PC(J,R). Then forx∈Bρ,t∈J, and using (.), we have _A_x(t)≤ 

 m

i=

p∗ψ(ρ)ti–I αi–

qi–(ti) +ρMti–I βi– qi–(ti)

+ 

m

i=

(T–ti)p∗ψ(ρ)ti–I αi––

qi– (ti) +ρMti–I γi– qi–(ti)

+ p

∗_ψ_(ρ) tmI

αm

qm(T) +

T



 

m

i=

p∗ψ(ρ)ti–I αi–– qi– (ti)

+ρMti–I γi– qi–(ti)

+

p ∗_ψ(ρ)t

mI

αm– qm (T)

+ m

i=

p∗ψ(ρ)ti–I αi–

qi–(ti) +ρMti–I βi– qi–(ti)

+ m

i=

(T–ti)p∗ψ(ρ)ti–I αi––

qi– (ti) +ρMti–I γi– qi–(ti)

+p∗ψ(ρ)tmI

αm

qm(T)

=p∗ψ(ρ)+ρM+ρM:=K, which implies thatAxPC≤K.

To show that the operatorAmaps bounded sets into equicontinuous sets ofPC(J,R), we takeτ,τ∈Jkfor somek∈ {, , , . . . ,m}withτ<τandx∈Bρ. Then we have

_Ax(τ) –Ax(τ)

≤ |τ–τ|

p∗ψ(ρ) 

m+

i=

(ti–ti–)αi–– q_i_–(αi–)

+ρM 

m

i=

(ti–ti–)γi– q_i_–(γi–+ )

+|τ–τ| k

i=

(ti–ti–)αi–– qi–(αi–)

p∗ψ(ρ) +ρM

(ti–ti–)γi– qi–(γi–+ )

+p ∗_ψ(ρ)

qk(αk)

τ

tk

tk(τ–tkqk)

(αk–)

qk tkdqks–

τ

tk

tk(τ–tkqk)

(αk–)

qk tkdqks

(11)

which tends to zero independent ofxasτ→τ. Thus, by the Arzelá-Ascoli theorem, the operatorA:PC(J,R)→PC(J,R) is completely continuous.

Finally, forλ∈(, ), letx=λAx. Then, as in the ﬁrst step, we can get

xPC≤p∗ψ

xPC

+xPCM+xPCM, which can alternatively be written as

( –M–M)xPC

p∗ψ(xPC)

≤.

In view of (H), there existsNsuch thatxPC=N. We deﬁneU={x∈PC(J,R) :xPC<

N}. Note that the operatorA:U→PCis continuous and completely continuous. From the choice ofU, there is nox∈∂U such thatx=λAxfor someλ∈(, ). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma .), we deduce thatAhas a ﬁxed pointx∈U which is a solution of the problem (.) onJ. This completes the proof.

In the last theorem, we apply Banach’s contraction principle to establish the uniqueness of solutions for the problem (.).

Theorem . Assume that there exist a functionW(t)∈C(J,R+₎_{with W}₌_sup

t∈J|W(t)|

and positive constants M,Msuch that

f(t,x) –f(t,y)≤W(t)|x–y|, ∀(t,x)∈J×R (.)

and

ϕk(x) –ϕk(y)≤M|x–y|,

ϕ_k∗(x) –ϕ∗_k(y)≤M|x–y|, x,y∈R,

(.)

for k= , , . . . ,m.If

W+M+M< , (.)

then the problem(.)has a unique solution on J.

Proof For anyx,y∈PC(J,R), we have _Ax(t) –Ay(t)

≤

 m

i=

ti–I αi–

qi–f(ti,x) –f(ti,y)+Mti–I βi–

qi–|x–y|(ti)

+ 

m

i=

(T–ti)ti–I αi––

qi– f(ti,x) –f(ti,y)+Mti–I γi–

+ tmI

αm

qmf(T,x) –f(T,y)+

T



 

m

i=

(12)

+Mti–I γi–

+ tmI

αm–

qm f(T,x) –f(T,y)

+ k

i=

ti–I αi–

qi–f(ti,x) –f(ti,y)+Mti–I βi–

+ k

i= (t–ti)

ti–I

αi––

qi– f(ti,x) –f(ti,y)+Mti–I γi–

qi–|x–y|(ti)

+tkI

αk

qkf(t,x) –f(t,y)

≤(W+M+M)x–yPC,

which yields

Ax–AyPC≤(W+M+M)x–yPC.

By (.), we conclude that Ais a contraction. Thus, by Banach’s contraction mapping principle, the problem (.) has a unique solution onJ. This completes the proof.

4 Examples

In this section, we present three examples to illustrate our results.

Example . Consider the following anti-periodic boundary value problem for impulsive

Caputo fractionalq-diﬀerence equations: ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ c tkD

k+

k+ 

k–k+

x(t) = t+  +_sint_+x_|x(₍t_t)₎_|, t∈[, ]\ {t,t,t},

x(tk) =_kk_+e –(_tk_–I

k– 



k–k+ x(tk))

, tk=k_,k= , , , tkD 

k–k+x(t +

k) –tk–D 

k–k+x(tk) =k_cos₍_log_{( +}_|

tk–I k+

 

k–k+

x(tk)|)), tk=k_,

x() = –x(), D

x() = –Dx().

(.)

Hereαk= (k+ )/(k+ ),qk= /(k– k+ ),k= , , , ,βk–= (k– )/,γk–= (k+ )/,tk=k/,k= , , ,m= ,T = . With the given information, it is found that= .. Also, we have

f(t,x)=t+  + sin

_t x  +|x|

≤t+  + sin

_t_|_x_|_, ϕk(y)=

_kk_{+ }e–y≤e, ϕ_k∗(z)=kcoslog +|z|≤, k= , , .

(13)

Example . Consider the anti-periodic impulsive boundary value problem of fractional

q-diﬀerence equations given by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ c tkD

k+ k+ 

k–k+

x(t) = _(+_t₎(loge(|x(t)|+ ))

_, _t_∈_{[, /]}_{\ {}_t

, . . . ,t}, x(tk) =_+_ksin(tk–I

+(–)k–  

k–k+

x(tk)), tk=k_,k= , . . . , , tkD 

k–k+

x(t+

k) –tk–D 

k–k+

x(tk) = _+_{k t}k–I +(–)k–

 

k–k+

x(tk), tk=k_,

x() = –x(_), D

x() = –Dx(  ).

(.)

Hereαk= (k+ )/(k+ ),qk= /(k– k+ ),k= , , , , ,βk–= ( + (–)k–)/, γk–= ( + (–)k–)/,tk=k/,k= , , , ,m= ,T= /. Using the above data, we ﬁnd that= .,= .,= ., and

f(t,x)=  ( +t)

log_e _| x|  +   ≤ 

( +t) _|

x|

 +  , ϕk(y)= 

 +k|siny| ≤



|y|, ϕ ∗ k(z)=

  +k|z| ≤



|z|, k= , , , .

Setting ψ(x) = (x/) + ,p∗ =sup_t∈_{[,/]}|/( +t)|= /,M= /, and M= /, we ﬁnd thatM+M= . < . Also, there exists a constantN such that

N> . satisfying (.). Clearly the hypothesis of Theorem . holds true. Thus the conclusion of Theorem . implies that the problem (.) has at least one solution on [, /].

Example . Consider the following impulsive anti-periodic problem of a fractional

q-diﬀerence equation: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ c tkD

k+k+

k+ 

k–k+

x(t) =e–t_tsin_+(t+)

x(t)+|x(t)| +|x(t)| +



, t∈[, /]\ {t, . . . ,t}, x(tk) =_ktan–(tk–I

k+ 



k–k+

x(tk)) +_, tk=_k,k= , . . . , ,

tkD  k–k+

x(t+

k) –tk–D 

k–k+

x(tk) = |_tk_–I

k–k+  

k–k+ x(tk)|

k(+|_tk–I k–k+

 

k–k+ x(tk)|)

+_, tk=_k,

x() = –x(_), D

x() = –Dx(  ).

(.)

Hereαk= (k+k+ )/(k+ ),qk= /(k– k+ ),k= , , , , , ,βk–= (k+ )/, γk–= (k– k+ )/,tk=k/,k= , , , , ,m= ,T= /. With the given values, we ﬁnd that= .,= ., and= .. Also, we have

f(t,x) –f(t,x)≤

e–tsin(t+ )

t_{+ }

|x–x|, ϕk(y) –ϕk(y)=

k

tan –_y

–tan–y≤ 

|y–y|, ϕ_k∗(z) –ϕ∗k(z)=

 k

|z|

 +|z| – |z|

 +|z| ≤

(14)

It is easy to see thatW= /. Hence,W+M+M= . < . Thus all the conditions of Theorem . are satisﬁed. Hence it follows by the conclusion of Theo-rem . that the problem (.) has a unique solution on [, /].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, BA, JT, SKN, HHA, and SM, 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_{Nonlinear Dynamic Analysis Research Center,}

Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand.3_{Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece.}

Acknowledgements

The project was funded by the Deanship of Scientiﬁc Research (DSR), King Abdulaziz University, under grant no. 14-130-36-RG. The authors, therefore, acknowledge with thanks DSR’s technical and ﬁnancial support.

Received: 13 September 2015 Accepted: 4 January 2016

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