New concepts of Hahn calculus and impulsive Hahn difference equations

R E S E A R C H

Open Access

New concepts of Hahn calculus and

impulsive Hahn difference equations

Jessada Tariboon

1*

_{, Sotiris K Ntouyas}

2,3

_{and Weerawat Sudsutad}

1

*_{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 Full list of author information is available at the end of the article

Abstract

In this paper, we introduce new concepts of Hahn difference operator, the

qk,

ω

k-Hahn difference operator. We aim to establish a calculus of differences based

on theqk,

ω

k-Hahn difference operator. We construct a right inverse of the

qk,

ω

k-Hahn operator and study some of its properties. As applications, we establish

existence and uniqueness results for first- and second-order impulsiveqk,

ω

k-Hahn

difference equations.

MSC: 34A08; 34A12; 34A60; 39A10; 39A13

Keywords: Hahn difference operator; Jacksonq-difference operator; Jackson q-integral; Nörlund sums; impulsive difference equations

1 Introduction and preliminaries

Many physical phenomena are described by equations involving nondifferentiable func-tions,e.g., generic trajectories of quantum mechanics []. Several different approaches to deal with nondifferentiable functions are followed in the literature, including the time scale approach, the fractional approach, and the quantum approach.

Quantum difference operators are receiving an increase of interest due to their appli-cations see,e.g., [–]. Roughly speaking, a quantum calculus substitutes the classical derivative by a difference operator, which allows one to deal with sets of nondifferentiable functions.

In [], Hahn introduced the quantum difference operator Dq,ω, whereq∈(, ) and

ω>  are fixed. The Hahn operator 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

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

The aim of this paper is to introduce new concepts of Hahn’s difference operator, the

qk,ωk-Hahn difference operator, to establish a calculus based on this operator and to

con-struct the associated integral. The steps are parallel to []. While some properties are straightforward extensions of classical results, some others need special treatments. As applications of theqk,ωk-Hahn difference operator we establish existence and uniqueness

results for first- and second-order impulsive fractional differential equations.

Impulsive differential equations serve as basic models to study the dynamics of pro-cesses that are subject to sudden changes in their states. Recent development in this field

has been motivated by many applied problems, such as control theory, population dynam-ics, and medicine. For some recent works on the theory of impulsive differential equations, we refer the interested reader to the monographs [–]. Impulsive quantum difference equations have been established by Tariboon and Ntouyas in [] by improving the classi-cal quantum classi-calculus which does not work when there exists at least one impulsive point appearing between two different points in the definition of q-derivative. For recent re-sults on the topics of initial and boundary value problems of impulsive quantum difference equations, we refer the reader to [].

We organize this paper as follows. In Section , some basic formulas of Hahn’s differ-ence operator and the associated Jackson-Nörlund integral calculus are briefly reviewed. Our results are formulated and proved in Section . Applications to impulsive fractional difference equations are given in Section .

2 Preliminaries

Letq∈(, ) andω> . Define

ω:=

ω

 –q (.)

and letIbe a real interval containingω.

Definition .(Hahn’s difference operator []) Letf:I→R. The Hahn difference oper-ator off is defined by

Dq,ωf(t) = ⎧ ⎨ ⎩

f(t)–f(qt+ω)

t(–q)–ω , t=ω, f(ω), t=ω,

(.)

provided thatf is differentiable atω.

The functionf is calledq,ω-differentiable onI, ifDq,ωf(t) exists for allt∈I.

Note that whenq→ we obtain the forwardω-difference operator

D,ωf(t) =

f(t+ω) –f(t)

ω , (.)

and whenω=  we obtain the Jacksonq-difference operator

Dq,f(t) =

⎧ ⎨ ⎩

f(t)–f(qt)

t(–q) , t= ,

f(), t= , (.)

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

Hence, we can state that the Dq,ω operator generalizes (in the limit) the forward

ω-difference and the Jacksonq-difference operators [, ]. Notice also that, under appropriate conditions,

lim

q→,ω→Dq,ωf(t) =f _(t).

Lemma .([]) Let f,g:I→Rbe q,ω-differentiable at t∈I.Then the following state-ments are true:

(i) Dq,ω(f +g)(t) =Dq,ωf(t) +Dq,ωg(t),

(ii) Dq,ωfg(t) =g(t)Dq,ωf(t) +f(qt+ω)Dq,ωg(t),

(iii) Dq,ωcf(t) =cDq,ωf(t),for any constantc∈R,

(iv) Dq,ω(f_g)(t) =

g(t)Dq,ωf(t)–f(t)Dq,ωg(t)

g(t)g(qt+ω) ,forg(t)g(qt+ω)= ,

(v) f(tq+ω) =f(t) + ((qt+ω) –t)Dq,ωf(t),t∈I.

Leth(t) =qt+ω,t∈I. Note thathis a contraction,h(I)⊆I,h(t) <tfort>ω,h(t) >t

fort<ω, andh(ω) =ω.

We use the standard notation of theq-number as [α]q=–q

α

–q forα∈R.

Lemma .([]) Let k∈Nand t∈I.Then

hk(t) =h◦h◦ · · · ◦ h(t)

i-times

=qkt+ω[k]q, t∈I. (.)

Next, we define the notion of aq,ω-integral, known as the Jackson-Nörlund integral.

Definition .([]) Letf :I→Rbe a function anda,b,ω∈I. Theq,ω-integral off

fromatobis defined by

b

a

f(s)dq,ωs= b

ω

f(s)dq,ωs– a

ω

f(s)dq,ωs, (.)

where

t

ω

f(s)dq,ωs=

t( –q) –ω

∞

k=

qkftqk+ω[k]q

, t∈I, (.)

provided that the series converges att=aandt=b.

The functionf isq,ω-integrable overIif it isq,ω-integrable over [a,b], for alla,b∈I. Note that in the integral formulas (.) and (.), whenω→, we obtain the Jackson

q-integral

b

a

f(s)dqs=

b



f(s)dqs–

a



f(s)dqs,

where

t



f(s)dqs=t( –q)

∞

k=

qkftqk, t∈I

(see,e.g., []); while ifq→ we obtain the Nörlund sum,

b

a

f(s)ωs= b

+∞f(s)ωs– a

where

t

+∞f(s)ωs= –ω +∞

k=

f(t+kω)

(see,e.g., [, , ]).

The following properties of Jackson-Nörlund integration can be found in [].

Lemma . Let f,g:I→Rbe q,ω-integrable on I,K∈R,and a,b,c∈I.Then the follow-ing formulas hold:

(i) _aaf(t)dq,ωt= ,

(ii) _abKf(t)dq,ωt=K b

a f(t)dq,ωt,

(iii) _abf(t)dq,ωt= – a

b f(t)dq,ωt,

(iv) _abf(t)dq,ωt= b

c f(t)dq,ωt+ c

af(t)dq,ωt,

(v) _ab(f(t) +g(t))dq,ωt= b

a f(t)dq,ωt+ b

a g(t)dq,ωt,

(vi) _abf(t)Dq,ωg(t)dq,ωt= [f(t)g(t)]b_a– b

a Dq,ωf(t)g(qt+ω)dq,ωt.

Property (vi) of the above lemma is known asq,ω-integration by parts. The next result is thefundamental theorem of Hahn calculus.

Lemma .([]) Let f :I→Rbe continuous atωand define F(t) := t

ωf(s)dq,ωs.Then

F is continuous atω.In addition,Dq,ωF(t)exists for every t∈I and

Dq,ωF(t) =f(t). (.)

On the other hand,

b

a

Dq,ωf(s)dq,ωs=f(b) –f(a) for all a,b∈I. (.)

Existence and uniqueness results for first-order abstract Hahn difference equations were studied in [], by using the method of successive approximation.

3 New concepts of Hahn calculus

Let there be a dense intervalJk= [tk,tk+]⊆Rand given constants  <qk< ,ωk>  and

θk=

ωk

 –qk +tk. (.)

Note that iftk= ,qk=q, andωk=ω, thenθk=ω, whereωis defined in (.).

Definition . Letf be a function defined onJk. Theqk,ωk-Hahn difference operatoris

given by

tkDqk,ωkf(t) =

f(t) –f(qkt+ ( –qk)tk+ωk)

( –qk)(t–tk) –ωk

, t=θk, (.)

We say thatf isqk,ωk-differentiable onJkprovidedtkDqk,ωkf(t) exists for allt∈Jk. Note that ifωk=  in (.), thentkDqk,f=tkDqkf, wheretkDqkis theqk-derivative of the function

f(t) which was first established in [] by

tkDqkf(t) =

f(t) –f(qkt+ ( –qk)tk) ( –qk)(t–tk)

. (.)

It is easy to see that iftk=  andqk=q, then (.) is reduced to the Jacksonq-difference

operator in (.).

Example . Letf(t) =t_fort∈J

k= [, ] and constantsqk= /,ωk= . Thenθk= 

and theqk,ωk-Hahn derivative onJkis given by

D ,f(t) =

t_{– (} t+ )

(

)(t– ) – 

=t

_{– t– }

(t– ) , t= ,

andD

,f() = .

It is easy to prove the following results.

Theorem . Let f,g:Jk→Rbe qk,ωk-differentiable at t∈Jk.Then the following formu-las hold:

(i) tkDqk,ωk(f+g)(t) =tkDqk,ωkf(t) +tkDqk,ωkg(t),

(ii) tkDqk,ωkfg(t) =g(t)tkDqk,ωkf(t) +f(qkt+ ( –qk)tk+ωk)tkDqk,ωkg(t),

(iii) tkDqk,ωkcf(t) =ctkDqk,ωkf(t),for any constantc∈R,

(iv) tkDqk,ωk(

f g)(t) =

g(t)_tkD_qk,ω_kf(t)–f(t)_tkD_qk,ω_kg(t)

g(t)g(qkt+(–qk)tk+ωk) ,forg(t)g(qkt+ ( –qk)tk+ωk)= .

Next, we define the higher-orderqk,ωk-derivative of functions.

Definition . Letf be a function defined on Jk. We define the second-order qk,ωk

-derivative tkDqk,ωkf provided tkDqk,ωkf is qk,ωk-differentiable on Jk with tkDqk,ωkf =

tkDqk,ωk(tkDqk,ωkf) :Jk →R. In addition, we define the higher-order qk,ωk-derivative

tkD

n

qk,ωkf:Jk→R, withtkD

n

qk,ωkf =tkDqk,ωk(tkD

n–

qk,ωkf) andtkD

 qk,ωkf =f.

The new definition ofqk,ωk-integral is given as follows.

Definition . Assumef :Jk→Ris a function anda,b∈Jk. We define theqk,ωk-integral

off fromatobby

b

a

f(s)tkdqk,ωks:=

b

θk

f(s)tkdqk,ωks–

a

θk

f(s)tkdqk,ωks, (.)

where

t

θk

f(s)tkdqk,ωks=

(t–tk)( –qk) –ωk ∞

i=

qi_kfqi_kt+ –qi_ktk+ωk[i]qk

fort∈Jk, provided that the series converge att=aandt=b. The functionf is called

qk,ωk-integrable onJkand we say thatf isqk,ωk-integrable over [a,b] for alla,b∈Jk.

Note that iftk= ,qk=q, andωk=ω, then (.) and (.) are reduced to (.) and (.),

respectively.

As customary, the following properties should be to stated. However, the proof is easy and we omit it.

Theorem . Let f,g:Jk→Rbe qk,ωk-integrable on Jk,K∈R,and a,b,c∈Jk.Then the following formulas hold:

(i) _aaf(s)tkdqk,ωks= ,

(ii) _abKf(s)tkdqk,ωks=K

b

a f(s)tkdqk,ωks,

(iii) _abf(s)tkdqk,ωks= –

a

b f(s)tkdqk,ωks,

(iv) _abf(s)tkdqk,ωks=

b

c f(s)tkdqk,ωks+

c

af(s)tkdqk,ωks,

(v) _ab(f(s) +g(s))tkdqk,ωks=

b

a f(s)tkdqk,ωks+

b

a g(s)tkdqk,ωks.

Lemma . Let h be the transformation

h(t) :=qkt+ ( –qk)tk+ωk, t∈Jk, (.)

andθk∈Jkis defined by(.).Then the ith-order iteration of h is given by

hi(t) =h◦h◦ · · · ◦ h(t)

i-times

=qi_kt+ –qi_ktk+ωk[i]qk, t∈Jk. (.)

In addition,the sequence{hi_(t)}∞

i=is an increasing(a decreasing)sequence in i when t<θk

(θk<t)with

lim i→∞h

i_{(t) =θ}

k, t∈Jk. (.)

Proof By directly computation, it is easy to show that (.) holds. Fort∈Jkandi∈N, we have

hi+(t) –hi(t) =q_ki( –qk)(tk–t) +ωk

[i+ ]qk– [i]qk

=qi_k( –qk)(θk–t).

Ift<θk orθk<t, then we see that the sequence{hi(t)}∞i= is increasing or decreasing,

re-spectively. Therefore, equation (.) is true for allt∈Jk.

Now, we will state and prove the fundamental theorem ofqk,ωk-Hahn calculus.

Theorem . Suppose that the function f:Jk→Ris continuous atθk∈Jk.We define

F(t) :=

t

θk

f(s)tkdqk,ωks, t∈Jk. (.)

(i) tkDqk,ωkF(t) =f(t),

(ii) _θt

ktkDqk,ωkf(s)tkdqk,ωks=f(t) –f(θk),

(iii) _{a t}b _kDq_k,ωkf(s)tkdqk,ωks=f(b) –f(a).

Proof From (.), we observe that

Fqkt+ ( –qk)tk+ωk

=qkt+ ( –qk)tk+ωk

–tk( –qk) –ωk

× ∞

i=

qi_kfqkt+ ( –qk)tk+ωk

qi_k+ –qi_ktk+ωk[i]qk

=qk(t–tk) +ωk

( –qk) –ωk ∞

i=

q_kifq_ki+t+ –qi+_k tk+ωk[i+ ]qk

.

Then, by (.), we have

tkDqk,ωkF(t) =

F(t) –F(qkt+ ( –qk)tk+ωk)

( –qk)(t–tk) –ωk

=

∞

i= qi_k

fqi_kt+ –q_kitk+ωk[i]qk

–(qk(t–tk) +ωk)( –q) –ωk ( –qk)(t–tk) –ωk

fqi+_k t+ –q_ki+tk+ωk[i+ ]qk

=

∞

i=

qi_kfqi_kt+ –qi_ktk+ωk[i]qk

–qkfqi+_k t+ –qi+_k tk+ωk[i+ ]qk

=f(t).

This shows that (i) holds.

To prove (ii), by Definitions ., ., and Lemma ., we get

t

θk

tkDqk,ωkf(s)tkdqk,ωks

=(t–tk)( –qk) –ωk ∞

i=

qi_k(tkDqk,ωkf)

qi_kt+ –qi_ktk+ωk[i]qk

=(t–tk)( –qk) –ωk ∞

i= qi_k

×f(qikt+ ( –qik)tk+ωk[i]qk) –f(qk(q

i

kt+ ( –qik)tk+ωk[i]qk) + ( –qk)tk+ωk) ( –qk)(qi_kt+ ( –qi_k)tk+ωk[i]qk–tk) –ωk

=

∞

i=

fqi_kt+ –q_kitk+ωk[i]qk

–fqi+_k t+ –qi+_k tk+ωk[i+ ]qk

Now, we show that (iii) holds. From (ii) for anya,b∈Jk, we obtain

b

a

tkDqk,ωkf(s)tkdqk,ωks=

b

θk

tkDqk,ωkf(s)tkdqk,ωks–

a

θk

tkDqk,ωkf(s)tkdqk,ωks

=f(b) –f(a).

This completes the proof.

Lemma . Let f,g:Jk→Rbe qk,ωk-integrable on Jk.Then the following integration by parts formula holds:

b

a

g(s)tkDqk,ωkf(s)tkdqk,ωks

=f(s)g(s)b_a–

b

a

fqks+ ( –qk)tk+ωk

tkDqk,ωkg(s)tkdqk,ωks.

Proof By Theorem . we have

b

a tk Dqk,ωk

f(s)g(s)tkdqk,ωks= (fg)(b) – (fg)(a).

On the other hand, by (ii) of Theorem . and (v) of Theorem .,

b

a

tkDqk,ωk

f(s)g(s)tkdqk,ωks

=

b

a

g(s)tkDqk,ωkf(s)tkdqk,ωks

+

b

a

fqks+ ( –qk)tk+ωk

tkDqk,ωkg(s)tkdqk,ωks.

Combining these two equalities we get the desired formula.

Lemma . Letθk∈Jk,α∈R,andβ∈R\ {–}.Then for t∈Jkthe following formulas hold:

(i) tkDqk(t–θk)

α_{= [α]}

qk(t–θk)

α–_,

(ii) _θt

k(s–θk)

β

tkdqk,ωks= (

–qk

–qβ_k+)(t–θk) β+_.

Proof From Definition ., fort=θk, we have

tkDqk,ωk(t–θk)

α

=(t–θk)

α_{– (q}

kt+ ( –qk)tk+ωk–θk)α

( –qk)(t–tk) –ωk

=(t–θk)

α_–qα k(t–θk)α

( –qk)(t–θk)

= [α]qk(t–θk)

α–_.

Now, we are going to prove (ii). Forβ∈R\ {–}, Definition . implies

The proof is completed.

Corollary . For a,b∈Jk,the following formula holds:

(i) and (ii) are obvious. To prove (iii), from (i) and (ii) we obtain

b

Theorem . Let f be the qk,ωk-integrable function on Jk.Then we have t

Proof By Definition ., we have

= ( –qk)

4 Impulsiveqk,

ω

k-Hahn difference equations

In this section, we use our results on qk,ωk-Hahn calculus to establish existence and

uniqueness results for impulsiveqk,ωk-Hahn difference equations of the first and second

order. LetJ= [t,t],Jk= (tk,tk+] fork= , , . . . ,mbe subintervals ofJ= [,T] such that

θk∈Jkfork= , , , . . . ,m. LetPC(J,R) ={x:J→R:x(t) is continuous everywhere

ex-cept for sometkat whichx(t+k) andx(tk–) exist andx(t–k) =x(tk),k= , , . . . ,m}.PC(J,R) is

a Banach space with the normxPC=sup{|x(t)|:t∈J}.

4.1 First-order impulsiveqk,

ω

k-Hahn difference equations

In this subsection, we study the existence and uniqueness of solutions for the following initial value problem for first-order impulsiveqk,ωk-Hahn difference equation

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

tkDqk,ωkx(t) =f(t,x(t)), t∈J,t=tk, x(tk) =ϕk(x(tk)), k= , , . . . ,m, x() =α,

(.)

whereα∈R,  =t<t<t<· · ·<tk<· · ·<tm<tm+=T,f :J×R→Ris a continuous function, ϕk∈C(R,R),x(tk) =x(t_k+) –x(tk),k= , , . . . ,m, and quantum numbers  < qk< ,ωk>  such thatθk∈Jkfork= , , , . . . ,m.

Lemma . Let x∈PC(J,R)satisfying(.).The impulsive qk,ωk-Hahn difference initial value problem(.)is equivalent to the integral equation

x(t) =α+

t<tk<t

tk

tk–

fs,x(s)tk–dqk–,ωk–s+

t<tk<t ϕk

x(tk)

+

t

tk

fs,x(s)tkdqk,ωks, (.)

with_t<t= .

Proof Fort∈J, applyingq,ω-integral fromttotin the first equation of (.) and using

Theorem .(iii), we obtain

x(t) =α+

t

t

fs,x(s)tdq,ωs.

Sinceθ∈J, we havet≥θand also, fort=t,

x(t) =α+ t

t

fs,x(s)tdq,ωs.

Fort∈J, taking theq,ω-integral to the first equation of (.) withk=  and applying

Theorem .(iii) again, we have

x(t) =xt+_+

t

t

From the impulsive conditionx(t_+) =x(t) +ϕ(x(t)), we get

x(t) =α+

t

t

fs,x(s)tdq,ωs+

t

t

fs,x(s)tdq,ωs+ϕ

x(t)

.

Fort∈J, theq,ω-integration and impulsive condition imply

x(t) =xt+_+

t

t

fs,x(s)tdq,ωs

=α+

t

t

fs,x(s)tdq,ωs+

t

t

fs,x(s)tdq,ωs+

t

t

fs,x(s)tdq,ωs

+ϕ

x(t)+ϕ

x(t).

From the above process, for anyt∈Jk,k= , , . . . ,m, we obtain the desired result in (.). Conversely, for anyt∈Jk,k= , , . . . ,m, applyingqk,ωk-derivative to (.) and using

Theorem .(i), we have

tkDqk,ωkx(t) =f

t,x(t).

By direct computation, we havex(tk) =ϕk(x(tk)) and alsox() =α. The proof is

com-pleted.

Now, we are in a position to prove an existence and uniqueness result for the problem (.), via Banach contraction mapping principle.

Theorem . Suppose that the following assumptions are fulfilled:

(H) the continuous functionf :J×R→Rsatisfies

f(t,x) –f(t,y)≤L|x–y|, L> ,∀t∈J,x,y∈R;

(H) the continuous functionsϕk:R→R,k= , , . . . ,msatisfy ϕk(x) –ϕk(y)≤L|x–y|, L> ,∀x,y∈R.

If

LT+mL< , (.)

then the impulsive qk,ωk-Hahn difference initial value problem(.)has a unique solution on J.

Proof Let us define an operatorA:PC(J,R)→PC(J,R) by

Ax(t) =α+

t<tk<t

tk

tk–

fs,x(s)tk–dqk–,ωk–s+

t<tk<t ϕk

x(tk)

+

t

tk

with_t<t= . Letsup_t∈J|f(t, )|=Mandmax{|ϕk()|:k= , , . . . ,m}=M. Choosing

a positive constantrsuch that

r≥|α|+MT+mM

 – (LT+mL) ,

and setting a ballBr={x∈PC(J,R) :x ≤r}, we will show thatABr⊂Br. For anyx∈Br

andt∈J, we have

_Ax(t)≤ |α|+

t<tk<t

tk

tk–

fs,x(s)tk–dqk–,ωk–s+

t<tk<t

ϕk

x(tk)

+

t

tk

fs,x(s)tkdqk,ωks

≤ |α|+

t<tk<T

tk

tk–

_f

s,x(s)–f(t, )+f(t, )tk–dqk–,ωk–s

+

t<tk<T

ϕk

x(tk)–ϕk()+ϕk()

+

T

tm

fs,x(s)–f(t, )+f(t, )tmdqm,ωms

≤ |α|+ (Lr+M)

t<tk<T

tk

tk–

tk–dqk–,ωk–s

+m(Lr+M) + (Lr+M)

T

tm

tmdqm,ωms

=|α|+MT+mM+r(LT+mL)≤r.

This means thatAx ≤r, which yieldsABr⊂Br. Forx,y∈PC(J,R) and for eacht∈J, we have

_Ax(t) –Ay(t)≤

t<tk<t

tk

tk–

fs,x(s)–fs,y(s)tk–dqk–,ωk–s

+

t<tk<t

ϕk

x(tk)–ϕk

y(tk)

+

t

tk

fs,x(s)–fs,y(s)tkdqk,ωks

≤Lx–y

t<tk<T

tk

tk–

tk–dqk–,ωk–s

+mLx–y+Lx–y T

tm

tmdqm,ωms

= (LT+mL)x–y,

which leads toAx–Ay ≤(LT+mL)x–y. AsLT +mL< , it follows from the

Example . Consider the first-order impulsiveqk,ωk-Hahn difference initial value

prob-Therefore, by Theorem ., we deduce that the problem (.) has a unique solution on [, ].

4.2 Second-order impulsiveqk,

ω

k-Hahn difference equations

In this subsection, we consider the second-order initial value problem of the impulsive

Proof Fort∈J, takingq,ω-integral for the first equation of (.) and using the second

Applying the first impulsive condition of (.), that is,x(t+

=α+βt+

Repeating the above method, fort∈J, we obtain (.) as desired.

Conversely, it can easily be shown by direct computation that the integral equation (.) satisfies the impulsive initial value problem (.). This completes the proof.

From Example .(iii) withb=tk+, we set the notation

(k) =(tk+–tk)

_–ω

k(tk+–tk)

 +qk .

Also, we use the notations

(U) =U

with_t<t= . By transforming the impulsive initial value problem (.) into a fixed point problemx=Qx, we will show that the operatorQhas a fixed point which is a unique solution of problem (.) via the Banach contraction mapping principle.

Setting sup_t∈J|f(t, )|=N, max{|ϕk()|: k= , , . . . ,m}=N, and max{|ϕ_k∗()|: k=

, , . . . ,m}=N, we will prove thatQBR⊂BR, whereBR={x∈PC(J,R) :x ≤R}and

the positive constantRsatisfies

R≥|α|+|β|T+ (N)

 – (L) . (.)

Forx∈BR, taking into account Example .(iii), we get

_Qx(t)≤ |α|+|β|T

+

t<tk<T

tk

tk–

s

tk–

fu,x(u)–f(u, )

+f(u, )tk–dqk–,ωk–utk–dqk–,ωk–s+ϕk

x(tk)

–ϕk()+ϕk()

+T t<tk<T

tk

tk–

_f

s,x(s)–f(s, )+f(s, )tk–dqk–,ωk–s

+ϕ_k∗x(tk)–ϕ_k∗()+ϕ∗_k()

+

t<tk<T

tk tk

tk–

fs,x(s)–f(s, )+f(s, )tk–dqk–,ωk–s

+ϕ_k∗x(tk)

–ϕ_k∗()+ϕ∗_k()

+

T

tm

s

tm

_f

u,x(u)–f(u, )+f(u, )tmdqm,ωmutmdqm,ωms

≤ |α|+|β|T+

m

k=

(LR+N)(k– ) +LR+N

+T _m

k=

(tk–tk–)(LR+N) +LR+N

+

m

k= tk

(tk–tk–)(LR+N) +LR+N

+ (LR+N)(m)

=|α|+|β|T+R (L) + (N)≤R.

Then we haveQx ≤R, which impliesQBR⊂BR.

Finally, forx,y∈PC(J,R) and for eacht∈J, we get

_Qx(t) –Qy(t)≤

t<tk<T

tk

tk–

s

tk–

fu,x(u)–fu,y(u)tk–dqk–,ωk–utk–dqk–,ωk–s

+ϕk

x(tk)

–ϕk

+T traction mapping principle thatQis a contraction. Therefore, we see that the operatorQ has a fixed point which is a unique solution of the impulsiveqk,ωk-Hahn difference initial

value problem (.) onJ. The proof is completed.

Example . Consider the second-order impulsiveqk,ωk-Hahn difference initial value

problem of the form

respectively. From the above information, we find that

Therefore, by Theorem ., we deduce that the problem (.) has a unique solution on [, ].

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}

Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.

Acknowledgements

This research was funded by Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Thailand. Contract No. 5942107.

Received: 25 February 2016 Accepted: 22 September 2016

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