Nonlocal boundary value problems for second order nonlinear Hahn integro difference equations with integral boundary conditions

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R E S E A R C H

Open Access

Nonlocal boundary value problems for

second-order nonlinear Hahn

integro-difference equations with integral

boundary conditions

Umaphon Sriphanomwan

1

_{, Jessada Tariboon}

1,4*

_{, Nichaphat Patanarapeelert}

1

_{, Sotiris K Ntouyas}

2,3

_and

Thanin Sitthiwirattham

5

*_{Correspondence:}

[email protected]

1_{Nonlinear Dynamic Analysis}

Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand

4_{Centre of Excellence in}

Mathematics, CHE, Sri Ayutthaya Rd., Bangkok, 10400, Thailand Full list of author information is available at the end of the article

Abstract

In this paper, we study a boundary value problem for second-order nonlinear Hahn integro-difference equations with nonlocal integral boundary conditions. Our problem contains two Hahn difference operators and a Hahn integral. The existence and uniqueness of solutions is obtained by using the Banach fixed point theorem, and the existence of at least one solution is established by using the Leray-Schauder nonlinear alternative and Krasnoselskii’s fixed point theorem. Illustrative examples are also presented to show the applicability of our results.

MSC: 39A10; 39A13; 39A70

Keywords: Hahn diﬀerence equations; boundary value problems; existence; uniqueness; ﬁxed point theorems

1 Introduction

A quantum calculus substitute the classical derivative by a difference operator, which al-lows one to deal with sets of non-differentiable functions. There are many different types of quantum difference operators such ash-calculus,q-calculus, Hahn’s calculus, forward quantum calculus and backward quantum calculus. These operators are also found in many applications of mathematical areas such as orthogonal polynomials, basic hyper-geometric functions, combinatorics, the calculus of variations and the theory of relativity. Some recent results in quantum calculus can be found in [–] and the references cited therein.

Hahn [] introduced his difference operatorDq,ω(see Definition .) whereq∈(, ) and ω>  are fixed, which unifies (in the limit) the two best-known and most-used quantum

diﬀerence operators: the Jacksonq-diﬀerence derivativeDq, whereq∈(, ) (cf.[–]);

and the forward diﬀerenceDωwhereω>  (cf.[, , ]). The Hahn diﬀerence operator

is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems (cf.[–]).

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The right inverse of the Hahn diﬀerence operator was introduced by Aldwoah [, ] who deﬁned the right inverse ofDq,ωin the terms of both the Jacksonq-integral containing

the right inverse ofDqand the Nörlund sum involving the right inverse ofω.

Malinowska and Torres [, ] introduced the Hahn quantum variational calculus, while Malinowska and Martins [] studied the generalized transversality conditions for the Hahn quantum variational calculus. Hamzaet al.[, ] studied the theory of lin-ear Hahn diﬀerence equations, and investigated the existence and uniqueness results of the initial value problems with Hahn diﬀerence equations using the method of successive approximations.

Recently, Sitthiwirattham [] initiated the study of boundary value problems for Hahn diﬀerence equations by considering the boundary value problem consisting of the nonlin-ear Hahn diﬀerence equation supplemented with nonlocal three-point boundary condi-tions of the form:

D_q,ωx(t) +f

t,x(t),Dp,θx(pt+θ)

= , t∈[ω,T]q,ω,

x(ω) =ϕ(x),

x(T) =λx(η), η∈(ω,T)q,ω,

(.)

where  <q< ,  <ω<T, ω := _–qω , ≤λ< T–η–ωω, p=q

m_, _m_∈_N_, _θ ₌_ω₍–p –q), f :

[ω,T]q,ω×R×R→Ris a given function, andϕ:C([ω,T]q,ω,R)→Ris a given

func-tional. He proved existence results for (.) by using the Banach and Krasnoselskii ﬁxed point theorems and also gave some numerical examples.

In this paper, motivated by the above papers, we continue the study of boundary value problems for Hahn diﬀerence equations by considering the nonlinear boundary value problem for Hahn integro-diﬀerence equations with nonlocal integral boundary condi-tions of the form

D_q,ωx(t) =f

t,x(t),Dp,θx(pt+θ),Ψp,θx(pt+θ)

, t∈[ω,T]q,ω,

x(ω) =x(T),

x(η) =μ T

ω

g(s)x(s)dq,ωs, η∈(ω,T)q,ω,

(.)

where  <q< ,  <ω<T,ω:=_–qω ,μ T

ωg(r)dq,ωr= ,μ∈R,p=q

m_,_m_∈_N_,_θ₌_ω₍–p –q),

f ∈C([ω,T]q,ω×R×R×R,R), andg∈C([ω,T]q,ω,R+) are given functions, and for ϕ∈C([ω,T]q,ω×[ω,T]q,ω, [,∞))

Ψp,θx(t) := t

ω

ϕ(t,ps+θ)x(ps+θ)dp,θs. (.)

Existence and uniqueness results are proved by using ﬁxed point theorems. Also many special cases and examples are presented.

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2 Preliminaries

In the following, there are notations, deﬁnitions, and lemmas which are used in proving the main results.

Deﬁnition .([]) For  <q< ,ω>  andf deﬁned on an intervalI⊆Rcontaining

ω:=_–qω , the Hahn diﬀerence off is deﬁned by

Dq,ωf(t) =

f(qt+ω) –f(t)

t(q– ) +ω fort=ω,

and Dq,ωf(ω) =f(ω), provided thatf is diﬀerentiable at ω. We callDq,ωf the q,ω

-derivative off, and say thatf isq,ω-diﬀerentiable onI.

This operator unifies and generalizes two well-known difference operators. The first is Jacksonq-difference operator defined by

Dq,f(t) = ⎧ ⎨ ⎩

f(t)–f(qt)

t(–q) , t= ,

f(), t= , (.)

provident thatf() exists. Heref is supposed to be deﬁned on aq-geometric setA⊂R, for whichqt∈Awhenevert∈A.

The second operator is the forward diﬀerence operator

D,ωf(t) =

f(t+ω) –f(t)

ω , (.)

whereω>  is ﬁxed.

Lettinga,b∈I⊆Rwitha<ω<band [k]q=–q

k

–q,k∈N:=N∪ {}, we deﬁne theq, ω-interval by

[a,b]q,ω:= qka+ω[k]q:k∈N

∪ qkb+ω[k]q:k∈N

∪ {ω}

= [a,ω]q,ω∪[ω,b]q,ω

= (a,b)q,ω∪ {a,b}= [a,b)q,ω∪ {b}= (a,b]q,ω∪ {a}.

Example . The interval [

, ]_,can be expressed by

 , 

 ,

=

 ,

  ,

 ,

 ,

,  ,

, ,, . . .

∪

, , , ,

 ,

 , . . .

∪ {}.

An essential function which plays an important role in Hahn’s calculus ish(t) =qt+ω. This function is normally taken to be deﬁned on an intervalI, which contains the number

ω=_–qω . Note thathis a contraction,h(I)⊆I,h(t) <tfort>ω,h(t) >tfort<ω, and

h(ω) =ω. One can see that thekth-order iteration ofh(t) is given byhk(t) =qkt+ω[k]q,

t∈I. Observe that, for eachs∈[a,b]q,ω, the sequence{qks+ω[k]q}∞k=is uniformly

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Iff isq,ω-diﬀerentiablentimes onq,ω-intervalIq,ω, we deﬁne the higher-order

deriva-tives by

Dn_q,ωf(s) :=Dq,ωDn–_q,ωf(s),

whereD

q,ωf(s) :=f(s),s∈Iq,ω⊂R.

Next, we introduce the right inverse of the operatorDq,ω, the so-called q, ω-integral

operator.

Deﬁnition .([]) LetIbe any closed interval ofRcontaininga,bandω. Assuming

thatf :I→Ris a given function, we deﬁne theq,ω-integral off fromatobby

b

a

f(t)dq,ωt:= b

ω

f(t)dq,ωt– a

ω

f(t)dq,ωt,

where

t

ω

f(s)dq,ωs=Iq,ωf(t) =

t( –q) –ω ∞

k=

qkftqk+ω[k]q

, x∈I,

is the convergent series atx=aandx=b.f is calledq,ω-integrable on [a,b] and the sum to the right hand side of the above equation will be called the Jackson-Nörlund sum.

We note that the actual domain of the functionf is [a,b]q,ω⊂I.

The following lemma is thefundamental theorem of Hahn calculus.

Lemma .([]) Let f :I→Rbe continuous atω.Deﬁne

F(x) :=

x

ω

f(t)dq,ωt, x∈I,

then F is continuous atω.Furthermore,Dq,ωF(x)exists for every x∈I and

Dq,ωF(x) =f(x).

Conversely,

b

a

Dq,ωF(t)dq,ωt=F(b) –F(a) for all a,b∈I.

Next, we give some auxiliary lemmas for simplifying calculations.

Lemma .([]) Let <q< ,ω> and x:I→Rbe continuous atω.Then t

ω r

ω

x(s)dq,ωs dq,ωr= t

ω t

qs+ω

x(s)dq,ωr dq,ωs.

Remark . Observe that

t

ω t

qs+ω

x(s)dq,ωr dq,ωs= t

ω

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Lemma .([]) Let <q< andω> ,then

t

ω

dq,ωs=t–ω and t

ω

t– (qs+ω)dq,ωs=

(t–ω)

 +q .

Lemma . Let <q< andω> ,then the following equation holds:

t

ω

· · ·

t

ω

ntimes

dq,ωs· · ·dq,ωr=

(t–ω)n

[n]q!

, (.)

where[n]q! = n

k= –qk

–q.

Proof Mathematical induction will be used in our proof as follows. Forn= , we have

t

ω

dq,ωs=

t( –q) –ω ∞

k=

qk= (t–ω),

which means that equation (.) is true forn= .

Suppose that equation (.) holds forn=k. Hence, forn=k+ , we have

t

ω

· · ·

t

ω k+ times

dq,ωs· · ·dq,ωr= t

ω

(r–ω)k

[k]q!

dq,ωr

=[t( –q) –ω] [k]q!

∞

k=

qktqk+ω[k]q–ω k

=( –q)(t–ω) [k]q!

∞

k=

qktqk–ωqk k

=( –q)(t–ω) [k]q!

∞

k=

qkqk(t–ω)k

=( –q)(t–ω)

k+

[k]q!( –qk+)

=(t–ω)

k+

[k+ ]q!

.

Thus, equation (.) holds forn=k+ . By the principle of induction, equation (.) is true

for alln∈N.

The following lemma deals with the linear variant of problem (.) and gives a presen-tation of the solution.

Lemma . Letμ_ωT

g(r)dq,ωr= and h∈C([ω,T]q,ω,R)be a given function.Then the

function x is a solution of the problem

D_q,ωx(t) =h(t), t∈[ω,T]q,ω, (.)

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if and only if

Proof By Lemma . and Remark ., a general solution for (.) can be written as

x(t) =

From the boundary conditions of (.), we obtain

C= –

On the other hand, by taking the second-orderq,ω-derivative to (.), we have (.). It

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3 Existence and uniqueness results

In this section, we present the existence and uniqueness of solutions for problem (.). Let

X= x|x∈C[ω,T]q,ω,R

andDp,θx∈C

[ω,T]q,ω,R

be the Banach space of all continuous functionsx: [ω,T]q,ω→Rwith the norm deﬁned

by

xX=x+Dp,θx,

wherex=max{|x(t)|:t∈[ω,T]q,ω}andDp,θx=max{|Dp,θx(pt+θ)|:t∈[ω,T]q,ω}.

We deﬁne an operatorF:X→Xby

(Fx)(t)

=

t

ω

t– (qs+ω)fs,x(s),Dp,θx(ps+θ),Ψp,θx(ps+θ)

dq,ωs

–(t–ω) T–ω

T

ω

T– (qs+ω)fs,x(s),Dp,θx(ps+θ),Ψp,θx(ps+θ)

dq,ωs

+ 

Ω

μ T

ω r

ω

g(r)r– (qs+ω)fs,x(s),Dp,θx(ps+θ),Ψp,θx(ps+θ)

dq,ωs dq,ωr

–

η

ω

η– (qs+ω)fs,x(s),Dp,θx(ps+θ),Ψp,θx(ps+θ)

dq,ωs

+ 

T–ω T

ω

T– (qs+ω)fs,x(s),Dp,θx(ps+θ),Ψp,θx(ps+θ)

dq,ωs

×

η–ω–μ T

ω

g(r)(r–ω)dq,ωr

, (.)

whereΩ=  is deﬁned by (.),p=qm,m∈Nandθ=ω(–p_–q).

Obviously, problem (.) has solutions if and only if the operatorFhas ﬁxed points.

3.1 Existence and uniqueness result via Banach’s ﬁxed point theorem

Our ﬁrst result concerns existence and uniqueness of solutions of problem (.) and is based on Banach’s ﬁxed point theorem.

Theorem . Let

ϕ=max ϕ(t,ps+θ) : (t,ps+θ)∈[ω,T]q,ω×[ω,T]q,ω

. (.)

Assume that:

(H) there exist functionshi∈C([ω,T]q,ω,R+),i= , , such that

f(t,x,x,x) –f(t,y,y,y)≤h(t)|x–y|+h(t)|x–y|+h(t)|x–y|

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≤

This implies, by (H), thatFis a contraction. Therefore, by Banach’s ﬁxed point theorem, F has a unique ﬁxed point, which is the unique solution of problem (.) on [ω,T]q,ω.

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Corollary . If hi=L,i= , , with L<_[+ϕ(T–ω)](Φ+Φ) in Theorem.,then problem

(.)has a unique solution on[ω,T]q,ω.

Example . Consider the following boundary value problem for second-order Hahn integro-diﬀerence equation:

⎧ ⎨ ⎩

D

/,/x(t) =f(t,x(t),Dp,θx(pt+θ),Ψp,θx(pt+θ)),

x() =x(), x(,/) = (/)_ecos(πs)_x₍_s₎_d /,/s,

(.)

where

ft,x(t),Dp,θx(pt+θ),Ψp,θx(pt+θ)

= 

( +t_{)( +}_|_x₍_t₎_|₎

e–sin(πt)x+ |x|+e–cos(πt)|D/,/x|

+e–t|Ψ/,/x|

(.)

and

Ψ/,/x(t) = t



e–((/)s+/–t)

 x

(/)s+ /d/,/s.

Hereq= /,ω= /,ω= ,p= /,m= ,θ = /,T= ,η= ,/,μ=

/,g(t) =ecos(πt)_,_ϕ₍_t_,_ps₊_θ_{) =}e–((/)s+/–t)

 . By direct computation, we ﬁnd that

f(t,x,Dp,θx,Ψp,θx) –f(t,y,Dp,θy,Ψp,θy)

≤e–sin

_(_π_t)

 +t x–y+

e–cos(πt)

 +t Dp,θx–Dp,θy+

e–t

 +tΨp,θx–Ψp,θy,

ϕ= /, and (H) is satisﬁed withh(t) = e –sin(πt)

+t , h(t) = e –cos(πt)

+t ,h(t) = e –t

+t. So, Φ= .. From  <g(t) <e, (H) is satisﬁed withN=e. By the given data we ﬁnd

Φ= .,Φ= .. Therefore, we can compute that

S=Φ(Φ+Φ) = ..

Hence, by Theorem ., problem (.) with (.) has a unique solution on [, ]/,/.

3.2 Existence result via 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)andλ∈(, )withu=λF(u).

Theorem . Let f : [ω,T]q,ω×R×R×R→Rbe a continuous function.In addition,

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(H) there exist a functionm∈C([ω,T]q,ω,R),and continuous nondecreasing functions

Firstly, we show that the operatorFmaps bounded sets into bounded sets in the spaceX. LetBr={x∈X:xX≤r},r> . For anyx∈Br, putting

and using the inequalities

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and

is equicontinuous and uniformly bounded. The Arzelá-Ascoli theorem implies thatFis completely continuous.

The result will follow from the Leray-Schauder nonlinear alternative (Lemma .) once we have proved the boundedness of the set of all solutions to equationsx=λFxforλ∈

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In view of (H), there existsMsuch thatxX=M. Let us set

U= x∈C[ω,T]q,ω,R

:xX<M

.

Note that the operatorF:U→C([ω,T]q,ω,R) is continuous and completely

continu-ous. From the choice ofU, there is nox∈∂Usuch thatx=λFxfor someλ∈(, ). Con-sequently, by the nonlinear alternative of Leray-Schauder type (Lemma .), we deduce thatF has a ﬁxed pointx∈Uwhich is a solution of problem (.). This completes the

proof.

Corollary . Suppose that a continuous function f satisﬁes|f(t,x,y,z)| ≤K(|x|+|y|+|z|), K≥.If K[ +ϕ(T–ω)](Φ+Φ) < ,then problem(.)has at least one solution on

[ω,T]q,ω.

Example . Consider the following boundary value problem for second-order Hahn integro-diﬀerence equation:

⎧ ⎨ ⎩

D

/,/x(t) =f(t,x(t),Dp,θx(pt+θ),Ψp,θx(pt+θ)),

x() =x(), x(,/) = (/)_ecos(πs)_x₍_s₎_d /,/s,

(.)

where

= 

(t+ )

 e

–|x|_sin_|_x_|₊ |D/,/x|

( +|D/,/x|)

+ |Ψ/,/x|



( +|Ψ/,/x|)

+ 

(.)

and

Ψ/,/x(t) = t



e–((/)s+/–t)

 x

(/)s+ /d/,/s.

Hereq= /,ω= /,ω= ,p= /,m= ,θ= /,T= ,η= ,/,μ= /,

g(t) =ecos(πt)_,_ϕ₍_t_,_ps₊_θ_{) =}e–((/)s+/–t)

 . Since

f(t,x,Dp,θx,Ψp,θx)

≤ 

(t+ )

 |x|+



|D/,/x|+ 

|Ψ/,/x|+ 

,

ϕ = /, (H) is satisﬁed with m(t) = _(t+) , ψ(|x|) = _|x|, ψ(|D/,/x|) = 

|D/,/x| + , ψ(|Ψ/,/x|) = 

|Ψ/,/x| + . In addition, we see that

 <g(t) <e, then (H) is satisﬁed withN=e. From the above information, we ﬁnd that Φ= ., Φ = .. Therefore, there exists a constant M> .

satisfy-ing (H). Hence, by Theorem ., problem (.) with (.) has at least one solution on

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3.3 Existence result via Krasnoselskii’s ﬁxed point theorem

The ﬁnal existence result is based on Krasnoselskii’s ﬁxed point theorem.

Lemma . (Krasnoselskii’s ﬁxed point theorem []) Let S be a closed, convex and nonempty subset of a Banach space X.Let A,B be the operators such that(a)Ax+By∈S whenever x,y∈S; (b)A is compact and continuous; (c)B is a contraction mapping.Then there exists z∈S such that z=Az+Bz.

Theorem . Assume that(H)and(H)hold.In addition we assume that:

(H) |f(t,x,y,z)| ≤μ(t),for eacht∈[ω,T]q,ω,x,y,z∈Randμ∈C([ω,T],R+).

Then problem(.)has at least one solution on[ω,T]q,ω,provided

T–ω

 +q ( +T–ω)Φ< , (.)

whereΦis deﬁned by(.).

Proof Consider the operatorF:X→Xdeﬁned by (.) as

(Fx)(t) = (Fx)(t) + (Fx)(t), t∈[ω,T]q,ω, (.)

where

(Fx)(t) = –(t–ω) T–ω

T

ω

T– (qs+ω)fˆx(s)dq,ωs

and

(Fx)(t)

=

t

ω

t– (qs+ω)fˆx(s)dq,ωs

+ 

Ω

μ T

ω r

ω

g(r)r– (qs+ω)fˆx(s)dq,ωs dq,ωr– η

ω

η– (qs+ω)fˆx(s)dq,ωs

+ 

T–ω T

ω

T– (qs+ω)fˆx(s)dq,ωs

η–ω–μ T

ω

g(r)(r–ω)dq,ωr

.

Setting max{μ(t) :t∈[ω,T]q,ω}=μ and choosingρ≥ μ(Φ+Φ) we consider

Bρ={x∈C([ω,T]q,ω,R) :xX≤ρ}. For anyx,y∈Bρwe have (Fx)(t) + (Fy)(t)

≤

t

ω

t– (qs+ω)fˆy(t)dq,ωs+

(t–ω)

T–ω T

ω

T– (qs+ω)fˆx(t)dq,ωs

+ 

|Ω|

μ T

ω r

ω

g(r)r– (qs+ω)fˆy(t)dq,ωs dq,ωr

–

η

ω

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+ 

It is easy to prove that

Fx–Fy ≤(T–ω) which implies, by (.), thatFis a contraction mapping.

Continuity of f implies that the operator F is continuous. Also, F is uniformly bounded onBρand equicontinuous, as proved in Theorem .. SoFis relatively compact

onBρ. Hence, by the Arzelá-Ascoli theorem,Fis compact onBρ. Thus all the

assump-tions of Lemma . are satisﬁed. So the conclusion of Lemma . implies that problem

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Corollary . Suppose that a continuous function f satisﬁes(H)with hi=L,i= , , ,

and|f(t,x,y,z)| ≤K,K> .IfT–ω

+q ( +T–ω)( +ϕ(T–ω))L< then problem(.)has

at least one solution on[ω,T]q,ω.

Example . Consider the boundary value problem for second-order Hahn

integro-diﬀerence equation in Example . with

= 

( +t_{)( +}_|_x₍_t₎_|₎

e–sin(πt)x+ |x|+e–cos(πt)|D/,/x|

+e–t|Ψ/,/x|

. (.)

Now, we see that

f(t,x,Dp,θx,Ψp,θx) –f(t,y,Dp,θy,Ψp,θy)

≤e–sin

_(_π_t)

 +t x–y+

e–cos(πt)

 +t Dp,θx–Dp,θy+

e–t

 +tΨp,θx–Ψp,θy.

Then (H) is satisﬁed withh(t) =e –sin(πt)

+t ,h(t) =e –cos(πt)

+t ,h(t) = e –t

+t. Therefore, we

can ﬁnd thatΦ= ., and

S=Φ(Φ+Φ) = . > .

Thus, Theorem . cannot be applied in this case. However, (H) is satisﬁed withμ(t) = 

+t, by|f(t,x,Dp,θx,Ψp,θx)| ≤_+t . Indeed, we ﬁnd that

T–ω

( +q)( +T–ω)Φ= . < .

Then, by using Theorem ., problem (.) with (.) has at least one solution on [/, ]/,/.

Acknowledgements

This research is partially supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in this article. They read and approved the ﬁnal manuscript.

Author details

1_{Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s}

University of Technology North Bangkok, Bangkok, 10800, Thailand.2_{Department of Mathematics, University of Ioannina,}

Ioannina, 451 10, Greece.3_{Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of}

Mathematices, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 4_{Centre of}

Excellence in Mathematics, CHE, Sri Ayutthaya Rd., Bangkok, 10400, Thailand.5_{Mathematics Department, Faculty of}

Science and Technology, Suan Dusit University, Bangkok, 10300, Thailand.

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References

1. Annaby, MH, Mansour, ZS:q-Fractional Calculus and Equations. Springer, Berlin (2012) 2. Jagerman, DL: Diﬀerence Equations with Applications to Queues. Dekker, New York (2000)

3. Aldowah, KA, Malinowska, AB, Torres, DFM: The power quantum calculus and variational problems. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms19, 93-116 (2012)

4. Birto da Cruz, AMC, Martins, N, Torres, DFM: Symmetric diﬀerentiation on time scales. Appl. Math. Lett.26(2), 264-269 (2013)

5. Cruz, B, Artur, MC: Symmetric quantum calculus. PhD thesis, Aveiro University (2012)

6. Wu, GC, Baleanu, D: New applications of the variational iteration method from differential equations toq-fractional difference equations. Adv. Differ. Equ.2013, 21 (2013)

7. Tariboon, J, Ntouyas, SK: Quantum calculus on finite intervals and applications to impulsive difference equations. Adv. Differ. Equ.2013, 282 (2013)

8. Álvarez-Nodarse, R: On characterization of classical polynomials. J. Comput. Appl. Math.196, 320-337 (2006) 9. Hahn, W: Über Orthogonalpolynome, dieq-Diﬀerenzenlgleichungen genügen. Math. Nachr.2, 4-34 (1949) 10. Kac, V, Cheung, P: Quantum Calculus. Springer, New York (2002)

11. Gasper, G, Rahman, M: Basic Hypergeometric Series. Cambridge University Press, Cambridge (2004) 12. Jackson, FH: Basic integration. Q. J. Math.2, 1-16 (1951)

13. Bird, MT: On generalizations of sum formulas of the Euler-Maclaurin type. Am. J. Math.58, 487-503 (1936) 14. Jordan, C: Calculus of Finite Diﬀerences, 3rd edn. Introduction by Carver. Chelsea, New York (1965)

15. Costas-Santos, RS, Marcellán, F: Second structure relation forq-semiclassical polynomials of the Hahn Tableau. J. Math. Anal. Appl.329, 206-228 (2007)

16. Foupouagnigni, M: Laguerre-Hahn orthogonal polynomials with respect to the Hahn operator: fourth-order diﬀerence equation for therth associated and the Laguerre-Freud equations recurrence coeﬃcients. PhD thesis, Université Nationale du Bénin, Bénin (1998)

17. Kwon, KH, Lee, DW, Park, SB, Yoo, BH: Hahn class orthogonal polynomials. Kyungpook Math. J.38, 259-281 (1998) 18. Aldwoah, KA: Generalized time scales and associated diﬀerence equations. PhD thesis, Cairo University (2009) 19. Annaby, MH, Hamza, AE, Aldwoah, KA: Hahn diﬀerence operator and associated Jackson-Nörlund integrals. J. Optim.

Theory Appl.154, 133-153 (2012)

20. Malinowska, AB, Torres, DFM: The Hahn quantum variational calculus. J. Optim. Theory Appl.147, 419-442 (2010) 21. Malinowska, AB, Torres, DFM: Quantum Variational Calculus. Spinger Briefs in Electrical and Computer

Engineering-Control, Automation and Robotics. Springer, Berlin (2014)

22. Malinowska, AB, Martins, N: Generalized transversality conditions for the Hahn quantum variational calculus. Optimization62(3), 323-344 (2013)

23. Hamza, AE, Ahmed, SM: Theory of linear Hahn diﬀerence equations. J. Adv. Math.4(2), 441-461 (2013)

24. Hamza, AE, Ahmed, SM: Existence and uniqueness of solutions of Hahn diﬀerence equations. Adv. Diﬀer. Equ.2013, 316 (2013)

25. Sitthiwirattham, T: Nonlocal three-point boundary value problems for nonlinear second-order Hahn difference equations with two differentq,ω-derivatives. Adv. Differ. Equ.2016, 116 (2016)

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