Quantum calculus on finite intervals and applications to impulsive difference equations

R E S E A R C H

Open Access

Quantum calculus on finite intervals and

applications to impulsive difference

equations

Jessada Tariboon

_{and Sotiris K Ntouyas}

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, Thailand Full list of author information is available at the end of the article

Abstract

In this paper we initiate the study of quantum calculus on finite intervals. We define theqk-derivative andqk-integral of a function and prove their basic properties. As an

application, we prove existence and uniqueness results for initial value problems for first- and second-order impulsiveqk-difference equations.

MSC: 26A33; 39A13; 34A37

Keywords: existence;qk-derivative;qk-integral; impulsiveqk-difference equation;

existence; uniqueness

1 Introduction

Quantum calculus is the modern name for the investigation of calculus without limits. The quantum calculus orq-calculus began with FH Jackson in the early twentieth century, but this kind of calculus had already been worked out by Euler and Jacobi. Recently it arose interest due to high demand of mathematics that models quantum computing.q-calculus appeared as a connection between mathematics and physics. It has a lot of applications in different mathematical areas such as number theory, combinatorics, orthogonal poly-nomials, basic hyper-geometric functions and other sciences quantum theory, mechanics and the theory of relativity. The book by Kac and Cheung [] covers many of the funda-mental aspects of quantum calculus. It has been shown that quantum calculus is a subfield of the more general mathematical field of time scales calculus. Time scales provide a uni-fied framework for studying dynamic equations on both discrete and continuous domains. The text by Bohner and Peterson [] collected much of the core theory in the calculus of time scales. In studying quantum calculus, we are concerned with a specific time scale, called theq-time scale, defined as follows:T:=qN:={qt_:t∈N

}, whereq> .

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 cited therein.

In this paper we initiate the study of quantum calculus on finite intervals. We define theqk-derivative of a functionf :Jk:= [tk,tk+]→Rand prove its basic properties such

as the derivative of a sum, of a product or a quotient of two functions. Also, we de-fine theqk-integral and prove its basic properties. As an application, we prove existence and uniqueness results for initial value problems for first- and second-order impulsive

q-difference equations.

The classicalq-calculus cannot be used in problems with impulses because if an im-pulse pointtkfor somek∈Nappears between the pointstandqt, then the definition of

q-derivative does not work. However, this situation does not occur in impulsive problems onq-time scale because the pointstandqt=ρ(t) are consecutive points. In quantum cal-culus on finite intervals, the pointstandqkt+ ( –qk)tkare considered only in an interval [tk,tk+]. Therefore,qk-calculus can be applied to systems with impulses at fixed times.

The rest of the paper is organized as follows. In Section  we recall some basic concepts ofq-calculus. In Section  we give the new notions ofqk-derivative andqk-integral on finite intervals and prove its basic properties. In Section  we apply the results of Section  to impulsiveqk-difference equations and prove existence and uniqueness results. Examples illustrating the abstract results are also presented.

2 Preliminaries

Let us recall some basic concepts ofq-calculus [, ].

Definition . Letf be a function defined on aq-geometric setI,i.e.,qt∈Ifor allt∈I. For  <q< , we define theq-derivative as

Dqf(t) =

f(t) –f(qt)

( –q)t , t∈I\ {}, Dqf() =tlim→Dqf(t).

Note that

lim

q→Dqf(t) =qlim→

f(qt) –f(t) (q– )t =

df(t)

dt

iff is differentiable. The higher-orderq-derivatives are given by

D_qf(t) =f(t), Dn_qf(t) =DqDnq–f(t), n∈N.

It is obvious that theq-derivative of a function is a linear operator. That is, for any con-stantsaandb, we have

Dq

af(t) +bg(t)=aDq

f(t)+bDq

g(t).

The standard rules for differentiation of products and quotients apply in quantum cal-culus. Thus by Definition . we can easily prove that

Dq

f(t)g(t)=f(qt)Dqg(t) +g(t)Dqf(t)

=f(t)Dqg(t) +g(qt)Dqf(t), (.)

Dq

f(t)

g(t)

=g(t)Dqf(t) –f(t)Dqg(t)

g(qt)g(t) . (.)

Fort≥, we setJt={tqn:n∈N∪{}}∪{}and define the definiteq-integral of a function

f :Jt→Rby

Iqf(t) = t



f(s)dqs=

∞

n=

t( –q)qnftqn

Fora,b∈Jt, we set

b

a

f(s)dqs=Iqf(b) –Iqf(a) = ( –q)

∞

n=

qnbfbqn –afaqn .

Note that fora,b∈Jt, we havea=tqn,b=tqn for somen,n∈N, thus the definite

integral_abf(s)dqsis just a finite sum, so no question about convergence is raised. We note that

DqIqf(t) =f(t),

while iff is continuous att= , then

IqDqf(t) =f(t) –f().

Inq-calculus, the integration by parts formula is

t



f(x)Dqg(x)dqx=

f(x)g(x)t_– t



Dqf(x)g(qx)dqx.

Further, reversing the order of integration is given by

t



s



f(r)dqr dqs= t



t

qr

f(r)dqs dqr.

In the limitq→, the above results correspond to their counterparts in standard calculus.

3 Quantum calculus on finite intervals

In this section we extend the notions ofq-derivative andq-integral of the previous section on finite intervals. For a fixedk∈N∪{}, letJk:= [tk,tk+]⊂Rbe an interval and  <qk<  be a constant. We define the qk-derivative of a functionf :Jk→Rat a point t∈Jk as follows.

Definition . Assume thatf :Jk→Ris a continuous function, and lett∈Jk. Then the expression

Dqkf(t) =

f(t) –f(qkt+ ( –qk)tk)

( –qk)(t–tk) , t=tk,

Dqkf(tk) =_tlim_→t k

Dqkf(t),

(.)

is called theqk-derivative of a functionf att.

We say thatf isqk-differentiable onJkprovidedDqkf(t) exists for allt∈Jk. Note that if

tk=  andqk=qin (.), thenDqkf=Dqf, whereDqis theq-derivative of the functionf(t)

Example . Letf(t) =t_{fort∈[, ] andq}

k=_. Now, we consider

Dqkf(t) =

t_{– (q}

kt+ ( –qk)tk) ( –qk)(t–tk)

=( +qk)t

_{– q}

ktkt– ( –qk)t_k

t–tk

=t

_{– t– }

(t– ) , t∈(, ]

andlim_t→tkDqkf(t) = , ift= . In particular,D_f() =  can be interpreted as a difference

quotientf()–_–f().

Example . In classicalq-calculus, we haveDqtn= [n]qtn–, where [n]q=–q

n

–q. However,

qk-calculus givesDqk(t–tk)

n_{= [n]q}

k(t–tk)

n–_{. Indeed,f(t) = (t–tk)}n_,t∈J k, then

Dqkf(t) =

(t–tk)n_{– (q}

kt+ ( –qk)tk–tk)n ( –qk)(t–tk)

=(t–tk) n_–qn

k(t–tk)n ( –qk)(t–tk)

= [n]q_k(t–tk)n–,

where [n]qk=

–qn_k

–qk.

Theorem . Assume that f,g:Jk→Rare qk-differentiable on Jk.Then: (i) The sumf +g:Jk→Risqk-differentiable onJkwith

Dqk

f(t) +g(t) =Dqkf(t) +Dqkg(t).

(ii) For any constantα,αf :Jk→Risqk-differentiable onJkwith

Dqk(αf)(t) =αDqkf(t).

(iii) The productfg:Jk→Risqk-differentiable onJkwith

Dqk(fg)(t) =f(t)Dqkg(t) +g

qkt+ ( –qk)tk Dqkf(t)

=g(t)Dqkf(t) +f

qkt+ ( –qk)tk Dqkg(t).

(iv) Ifg(t)g(qkt+ ( –qk)tk)= ,then f_g isqk-differentiable onJkwith

Dqk

f g

(t) = g(t)Dqkf(t) –f(t)Dqkg(t)

g(t)g(qkt+ ( –qk)tk) .

(iii) From Definition ., we have

Dqk(fg)(t) =

f(t)g(t) –f(qkt+ ( –qk)tk)g(qkt+ ( –qk)tk) ( –qk)(t–tk)

=f(t)g(t) –f(t)gqkt+ ( –qk)tk +f(t)g

qkt+ ( –qk)tk

–fqkt+ ( –qk)tk g

qkt+ ( –qk)tk /( –qk)(t–tk)

=f(t)

g(t) –g(qkt+ ( –qk)tk) ( –qk)(t–tk)

+gqkt+ ( –qk)tk

f(t) –f(qkt+ ( –qk)tk) ( –qk)(t–tk)

=f(t)Dqkg(t) +g

qkt+ ( –qk)tk Dqkf(t).

The proof of the second equation in part (iii) is of a similar manner by interchanging the functionsf andg.

(iv) For theqk-derivative of a quotient, we can find that

Dqk

f g

(t) =

f(t)

g(t)–

f(qkt+(–qk)tk)

g(qkt+(–qk)tk)

( –qk)(t–tk)

=f(t)g(qkt+ ( –qk)tk) –g(t)f(qkt+ ( –qk)tk)

g(t)g(qkt+ ( –qk)tk)( –qk)(t–tk)

=

g(t)

f(t) –f(qkt+ ( –qk)tk) ( –qk)(t–tk)

–f(t)

g(t) –g(qkt+ ( –qk)tk) ( –qk)(t–tk)

g(t)gqkt+ ( –qk)tk

=g(t)Dqkf(t) –f(t)Dqkg(t)

g(t)g(qkt+ ( –qk)tk)

.

Remark . In Example . we recall that inq-difference, iff(t) =tn_{, thenD}

qtn= [n]tn–. We cannot have a simple formula forqk-difference. Using the derivative of a product, we have for somen:

Dqkt= ,

Dqkt

_=D

qk(t·t) = ( +qk)t+ ( –qk)tk,

Dqkt

_=D

qk

t·t = +qk+qk t+

 +qk– qk ttk+ ( –qk)tk,

Dqkt

_=D

qk

t·t

= +qk+qk+qk t+

 +qk+qk– qk tkt + +qk– qk+ qk tkt+ ( –qk)tk.

In addition, we should define the higherqk-derivative of functions.

Definition . Let f : Jk → R be a continuous function. We call the second-order

qk-derivativeD_qkf providedDqkf isqk-differentiable onJkwithD



qkf =Dqk(Dqkf) :Jk→R.

Similarly, we define the higher-orderqk-derivativeDn

For example, iff :Jk→R, then we have

D_qkf(t) =Dqk

Dqkf(t)

=Dqkf(t) –Dqkf(qkt+ ( –qk)tk)

( –qk)(t–tk)

=

f(t)–f(qkt+(–qk)tk)

(–qk)(t–tk) –

f(qkt+(–qk)tk)–f(qkt+(–qk)tk)

(–qk)(t–tk)

( –qk)(t–tk)

=f(t) – f(qkt+ ( –qk)tk) +f(q



kt+ ( –qk)tk)

( –qk)_(t–tk) , t=tk,

andD

qkf(tk) =limt→tkD



qkf(t).

To construct theqk-antiderivativeF(t), we define a shifting operator by

EqkF(t) =F

qkt+ ( –qk)tk .

It is easy to prove by using mathematical induction that

En_qkF(t) =Eqk

E_qn_k–F (t) =Fqn_kt+ –qn_k tk , wheren∈NandE

qkF(t) =F(t).

Then we have by Definition . that

F(t) –F(qkt+ ( –qk)tk) ( –qk)(t–tk) =

 –Eqk

( –qk)(t–tk)F(t) =f(t).

Therefore, theqk-antiderivative can be expressed as

F(t) =   –Eqk

( –qk)(t–tk)f(t) .

Using the geometric series expansion, we obtain

F(t) = ( –qk)

∞

n=

En_qk(t–tk)f(t)

= ( –qk)

∞

n=

qn_kt+ –qn_k tk–tk f

qn_kt+ –qn_k tk

= ( –qk)(t–tk)

∞

n=

qn_kfqn_kt+ –qn_k tk . (.)

It is clear that the above calculus is valid only if the series in the right-hand side of (.) is convergent.

Definition . Assume thatf :Jk→Ris a continuous function. Then theqk-integral is defined by

t

tk

f(s)dqks= ( –qk)(t–tk)

∞

n=

fort∈Jk. Moreover, ifa∈(tk,t), then the definiteqk-integral is defined by

Proof (i) Using Definitions . and ., we get

(ii) By computing directly, we have

(iii) The part (ii) of this theorem implies that t

Proof The results of (i)-(ii) follow from Definition .. (iii) From Theorem . part (iii), we have

f(t)Dqkg(t) =Dqk(fg)(t) –g

qkt+ ( –qk)tk Dqkf(t).

Takingqk-integral for the above equation and applying Theorem . part (ii), we get the

result in (iii) as required.

Theorem .(Reversing the order ofqk-integration) Let f ∈C(Jk,R),then the following formula holds:

Proof By Definition ., we have

= ( –qk)

then we get that

t

4 Impulsiveqk-difference equations

LetJ= [,T],J= [t,t],Jk= (tk,tk+] fork= , , . . . ,m. LetPC(J,R) ={x:J→R:x(t) is

continuous everywhere except for sometkat whichx(t+_k) andx(t_k–) exist andx(t–_k) =x(tk),

k= , , . . . ,m}.PC(J,R) is a Banach space with the norms x PC=sup{|x(t)|;t∈J}.

4.1 First-order impulsiveqk-difference equations

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

Dqkx(t) =f

t,x(t) , t∈J,t=tk,

x(tk) =Ik

x(tk), k= , , . . . ,m,

x() =x,

(.)

where x∈R,  =t<t<t<· · ·<tk <· · ·<tm <tm+ =T, f :J×R→Ris a

con-tinuous function, Ik∈C(R,R), x(tk) =x(tk+) –x(tk), k= , , . . . ,mand  <qk <  for

k= , , , . . . ,m.

Lemma . If x∈PC(J,R)is a solution of(.),then for any t∈Jk,k= , , , . . . ,m,

x(t) =x+

<tk<t

tk

tk–

fs,x(s) dqk–s

+

<tk<t

Ik

x(tk) + t

tk

fs,x(s) dqks, (.)

with_<(·) = ,is a solution of(.).The converse is also true.

Proof Fort∈J,q-integrating (.), it follows

x(t) =x+

t



fs,x(s) dqs,

which leads to

x(t) =x+

t



fs,x(s) dqs.

Fort∈J, takingq-integral to (.), we have

x(t) =xt+_ + t

t

fs,x(s) dqs.

Sincex(t+

) =x(t) +I(x(t)), then we have

x(t) =x+

t



fs,x(s) dqs+ t

t

fs,x(s) dqs+I

Againq-integrating (.) fromttot, wheret∈J, then

x(t) =xt+_ + t

t

fs,x(s) dqs

=x+

t



fs,x(s) dqs+ t

t

fs,x(s) dqs+ t

t

fs,x(s) dqs

+I

x(t) +I

x(t) .

Repeating the above procession, fort∈J, we obtain (.).

On the other hand, assume thatx(t) is a solution of (.). Applying theqk-derivative on (.) fort∈Jk,k= , , , . . . ,m, it follows that

Dqkx(t) =f

t,x(t) .

It is easy to verify thatx(tk) =Ik(x(tk)),k= , , . . . ,mandx() =x. This completes the

proof.

Theorem . Assume that the following assumptions hold:

(H) f :J×R→Ris a continuous function and satisfies

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

(H) Ik:R→R,k= , , . . . ,m,are continuous functions and satisfy Ik(x) –Ik(y)≤M|x–y|, M> ,∀x,y∈R.

If

LT+mM≤δ< ,

then the nonlinear impulsive qk-difference initial value problem(.)has a unique solution

on J.

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

(Ax)(t) =x+

<tk<t

tk

tk–

fs,x(s) dqk–s

+

<tk<t

Ik

x(tk) + t

tk

fs,x(s) dqks,

with_<(·) = . Assume thatsup_t∈J|f(t, )|=N andmax{|Ik()|:k= , , . . . ,m}=N;

we choose a constantrsuch that

r≥ 

 –ε

|x|+NT+mN

where δ ≤ε< . Now, we will show that ABr⊂Br, where a ball Br ={x∈PC(J,R) :

x ≤r}. For anyx∈Brand for eacht∈J, we have

(Ax)(t)≤ |x|+

<tk<t

tk

tk–

fs,x(s) dqk–s

+

<tk<t

_Ik

x(tk) + t

tk

_f

s,x(s) dqks

≤ |x|+

<tk<T

tk

tk–

fs,x(s) –f(s, )+f(s, ) dqk–s

+

<tk<T

_Ik

x(tk) –Ik()+Ik()

+ T

tm

fs,x(s) –f(s, )+f(s, ) dqms

≤ |x|+ (Lr+N)

<tk<T

tk

tk–

dqk–s

+

<tk<T

(Mr+N) + (Lr+N)

T

tm

dqms

≤ |x|+ (Lr+N)T+m(Mr+N)

≤(δ+  –ε)r≤r.

This implies thatABr⊂Br.

Forx,y∈PC(J,R) and for eacht∈J, we have

(Ax)(t) – (Ay)(t)≤

<tk<t

tk

tk–

fs,x(s) –fs,y(s) dqk–s

+

<tk<t

Ik

x(tk) –Ik

y(tk)

+ t

tk

fs,x(s) –fs,y(s) dqks

≤

<tk<T

tk

tk–

Lx(s) –y(s) dqk–s

+

<tk<T

Mx(tk) –y(tk)

+ T

tm

Lx(s) –y(s) dqms

≤(LT+mM) x–y .

It follows that

AsLT+mM< , by the Banach contraction mapping principle,Ais a contraction. There-fore,Ahas a fixed point which is a unique solution of (.) onJ.

Example . Consider the following first-order impulsiveqk-difference initial value prob-lem:

D  +kx(t) =

e–t|x(t)|

(t+√)_{( +}|_x(t)|₎, t∈J= [, ],t=tk=

k

,

x(tk) = |x(tk)|  +|x(tk)|

, k= , , . . . , ,

x() = .

(.)

Hereqk= /( +k),k= , , , . . . , ,m= ,T= ,f(t,x) = (e–t|x|)/((t+

√

)_{( +|x|)) and}

Ik(x) =|x|/( +|x|). Since|f(t,x) –f(t,y)| ≤(/)|x–y|and|Ik(x) –Ik(y)| ≤(/)|x–y|, then (H), (H) are satisfied withL= (/),M= (/). We can show that

LT+mM=

+  =

 < .

Hence, by Theorem ., the initial value problem (.) has a unique solution on [, ].

4.2 Second-order impulsiveqk-difference equations

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

qk-difference equation of the form

D_q

kx(t) =f

t,x(t) , t∈J,t=tk,

x(tk) =Ik

x(tk), k= , , . . . ,m,

Dqkx

t+_k –Dqk–x(tk) =I

∗

k

x(tk) , k= , , . . . ,m,

x() =α, Dqx() =β,

(.)

whereα,β∈R,  =t<t<t<· · ·<tk<· · ·<tm<tm+=T,f :J×R→Ris a continuous

function, Ik,I_k∗∈C(R,R),x(tk) =x(t+_k) –x(tk) fork= , , . . . ,m and  <qk<  fork= , , , . . . ,m.

Lemma . The unique solution of problem(.)is given by

x(t) =α+βt+

<tk<t

tk

tk–

tk–qk–s– ( –qk–)tk– f

s,x(s) dqk–s+Ik

x(tk)

+t

<tk<t

tk

tk–

fs,x(s) dqk–s+I

∗

k

x(tk)

–

<tk<t

tk tk

tk–

fs,x(s) dqk–s+I

∗

k

x(tk)

+ t

tk

t–qks– ( –qk)tk f

s,x(s) dqks, (.)

Proof Fort∈J, takingq-integral for the first equation of (.), we get

which, on changing the order ofq-integral, takes the form

x(t) =α+βt+

Using the third condition of (.) with (.) yields that

Dqx(t) =β+

Applying the second equation of (.) with (.) and (.), we get

=α+βt+

Repeating the above process, fort∈J, we obtain (.) as required.

Next, we prove the existence and uniqueness of a solution to the initial value problem (.). We shall use the Banach fixed point theorem to accomplish this.

Theorem . Assume that(H)and(H)hold.In addition,we suppose that:

then the initial value problem(.)has a unique solution on J.

For anyx,y∈PC(J,R), we have

are satisfied withL= (/),M= (/),M∗= (/). We find that

ν=

m+

k=

(tk–tk–)

 +qk–

=,,

, , ν= m

k=

(tk–tk–) =

 ,

ν=

m

k=

tk(tk–tk–) =



, ν= m

k=

tk=  .

Clearly,

L(ν+Tν+ν) +mM+ (mT+ν)M∗= . < .

Hence, by Theorem ., the initial value problem (.) has a unique solution on [, ].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok,}

Bangkok, Thailand.2_{Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece.}

Authors’ information

Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

Acknowledgements

This research was funded by the Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Thailand. Project No. 5742106.

Received: 15 June 2013 Accepted: 29 August 2013 Published:07 Nov 2013 References

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

2. Bohner, M, Peterson, A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)

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

4. Dobrogowska, A, Odzijewicz, A: Second orderq-difference equations solvable by factorization method. J. Comput. Appl. Math.193, 319-346 (2006)

5. Gasper, G, Rahman, M: Some systems of multivariable orthogonalq-Racah polynomials. Ramanujan J.13, 389-405 (2007)

6. Ismail, MEH, Simeonov, P:q-Difference operators for orthogonal polynomials. J. Comput. Appl. Math.233, 749-761 (2009)

7. Bohner, M, Guseinov, GS: Theh-Laplace andq-Laplace transforms. J. Math. Anal. Appl.365, 75-92 (2010) 8. El-Shahed, M, Hassan, HA: Positive solutions ofq-difference equation. Proc. Am. Math. Soc.138, 1733-1738 (2010) 9. Ahmad, B: Boundary-value problems for nonlinear third-orderq-difference equations. Electron. J. Differ. Equ.2011, 94

(2011)

10. Ahmad, B, Alsaedi, A, Ntouyas, SK: A study of second-orderq-difference equations with boundary conditions. Adv. Differ. Equ.2012, 35 (2012)

11. Ahmad, B, Ntouyas, SK, Purnaras, IK: Existence results for nonlinearq-difference equations with nonlocal boundary conditions. Commun. Appl. Nonlinear Anal.19, 59-72 (2012)

12. Ahmad, B, Nieto, JJ: On nonlocal boundary value problems of nonlinearq-difference equations. Adv. Differ. Equ. 2012, 81 (2012)

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

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

15. Yu, C, Wang, J: Existence of solutions for nonlinear second-orderq-difference equations with first-orderq-derivatives. Adv. Differ. Equ.2013, 124 (2013)

10.1186/1687-1847-2013-282