Three point boundary value problems for nonlinear second order impulsive q difference equations

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R E S E A R C H

Open Access

Three-point boundary value problems for

nonlinear second-order impulsive

q

-difference equations

Jessada Tariboon

1*

_{and Sotiris K Ntouyas}

2

*_{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

The quantum calculus on ﬁnite intervals was studied recently by the authors in Adv. Diﬀer. Equ. 2013:282, 2013, where the concepts ofqk-derivative andqk-integral of a functionf:Jk:= [tk,tk+1]→Rhave been introduced. In this paper, we prove existence

and uniqueness results for nonlinear second-order impulsiveqk-diﬀerence three-point boundary value problems, by using Banach’s contraction mapping principle and Krasnoselskii’s ﬁxed-point theorem.

MSC: 26A33; 39A13; 34A37

Keywords: qk-derivative;qk-integral; impulsiveqk-diﬀerence equation; existence; uniqueness; three-point boundary conditions; ﬁxed-point theorems

1 Introduction

In this article, we investigate the nonlinear second-order impulsiveqk-diﬀerence equation with three-point boundary conditions

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

D

qkx(t) =f(t,x(t)), t∈J:= [,T],t=tk,

x(tk) =Ik(x(tk)), k= , , . . . ,m,

Dqkx(t +

k) –Dqk–x(tk) =I

∗

k(x(tk)), k= , , . . . ,m,

x() = , x(T) =x(η),

(.)

where  =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,x(t_k+) =limh→x(tk+h),η∈(tj,tj+)

a constant for somej∈ {, , , . . . ,m}and  <qk<  fork= , , , . . . ,m.

The theory of quantum calculus on finite intervals was developed recently by the authors in []. In [] the concepts ofqk-derivative andqk-integral of a functionf :Jk:= [tk,tk+]→ R, are defined and their basic properties proved. As applications, existence and unique-ness results for initial value problems for first- and second-order impulsiveqk-difference equations are proved.

The book by Kac and Cheung [] covers many of the fundamental aspects of the quan-tum 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 refer-ences cited therein.

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Impulsive differential equations, that is, differential equations involving an impulse ef-fect, appear as a natural description of observed evolution phenomena of several real-world problems. For some monographs on impulsive differential equations we refer to [–].

In the present paper we prove existence and uniqueness results for the impulsive bound-ary value problem (.) by using Banach’s contraction mapping principle and Krasnosel-skii’s ﬁxed-point theorem. The rest of this paper is organized as follows: In Section  we present the notions of qk-derivative andqk-integral on ﬁnite intervals and collect their properties. The main results are proved in Section , while examples illustrating the re-sults are presented in Section .

2 Preliminaries

In this section we present the notions ofqk-derivative andqk-integral on finite intervals. For a fixedk∈N∪ {}letJk:= [tk,tk+]⊂Rbe an interval and  <qk<  be a constant. We defineqk-derivative of a functionf :Jk→Rat a pointt∈Jkas follows.

Deﬁnition . Assumef :Jk→Ris a continuous function and lett∈Jk. Then the ex-pression

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 functionf att.

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

tk=  andqk=qin (.), thenDqkf=Dqf, whereDqis the well-knownq-derivative of the

functionf(t) deﬁned by

Dqf(t) =

f(t) –f(qt)

( –q)t . (.)

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

Deﬁnition . Let f :Jk →R is a continuous function, we call the second-order q k-derivativeD

qkf providedDqkf isqk-diﬀerentiable onJkwithD 

qkf =Dqk(Dqkf) :Jk→R.

Similarly, we deﬁne the higher-orderqk-derivativeDn

qk :Jk→R.

The properties of theqk-derivative are summarized in the following theorem.

Theorem . Assume 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

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(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) .

Deﬁnition . Assumef :Jk→Ris a continuous function. Then theqk-integral is de-ﬁned by

t

tk

f(s)dqks= ( –qk)(t–tk) ∞

n=

qn_kfqn_kt+ –q_kntk

, (.)

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

t

a

f(s)dqks= t

tk

f(s)dqks– a

tk

f(s)dqks

= ( –qk)(t–tk)

∞

n=

qn_kfqn_kt+ –qn_ktk

– ( –qk)(a–tk)

∞

n=

qn_kfqn_ka+ –qn_ktk

.

Note that iftk=  andqk=q, then (.) reduces to theq-integral of a functionf(t), deﬁned by_tf(s)dqs= ( –q)t∞n=qnf(qnt) fort∈[,∞).

Theorem . For t∈Jk,the following formulas hold:

(i) Dqk t

tkf(s)dqks=f(t); (ii) _tt

kDqkf(s)dqks=f(t);

(iii) _atDqkf(s)dqks=f(t) –f(a)fora∈(tk,t).

3 Main results

Let J= [,T], J= [t,t], Jk = (tk,tk+] for k= , , . . . ,m. Let PC(J,R) = {x:J →R:

x(t) is continuous everywhere except for sometkat whichx(t+k) andx(tk–) exist andx(tk–) =

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

Lemma . The unique solution of problem(.)is given by

x(t) = –t

j

k= tk

tk–

fs,x(s)dqk–s+I

∗

k

x(tk)

– t

T–η m

k=j+ tk

tk–

s

tk–

fσ,x(σ)dqk–σdqk–s+Ik

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– t

T–η m

k=j+ tk

tk–

∗

k

x(tk) (T–tk)

+ t

T–η

η

tj s

tj

fσ,x(σ)dqjσdqjs

– t

T–η

T

tm s

tm

fσ,x(σ)dqmσdqms

+

<tk<t tk

tk–

s

tk–

fσ,x(σ)dqk–σdqk–s+Ik

x(tk)

+

<tk<t tk

tk–

∗

k

x(tk) (t–tk)

+

t

tk s

tk

fσ,x(σ)dqkσdqks, (.)

with_<(·) = .

Proof Fort∈J, taking theq-integral for the ﬁrst equation of (.), we get

Dqx(t) =Dqx() +

t



fs,x(s)dqs, (.)

which yields

Dqx(t) =Dqx() +

t



fs,x(s)dqs. (.)

Fort∈Jwe obtain byq-integrating (.),

x(t) =x() +Dqx()t+

t

 s



fσ,x(σ)dqσdqs

:=A+Bt+

t

 s



fσ,x(σ)dqσdqs

x() =A,Dqx() =B

.

In particular, fort=t

x(t) =A+Bt+ t

 s



fσ,x(σ)dqσdqs. (.)

Fort∈J= (t,t],q-integrating (.), we have

Dqx(t) =Dqx

t+_+

t

t

fs,x(s)dqs.

Using the third condition of (.) with (.), it follows that

Dqx(t) =B+

t



fs,x(s)dqs+I∗

x(t)

+

t

t

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Taking theq-integral to (.) fort∈J, we obtain

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

x(t) =A+Bt+

Repeating the above process, fort∈J, we get

x(t) =A+Bt

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+

Substituting the constantBinto (.), we obtain (.) as required.

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

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For convenience, we set

k=

(tk–tk–)(T–tk) +

(tk–tk–)

 +qk–

M+M+ (T–tk)M, (.)

k=

(tk–tk–)(T–tk) +

(tk–tk–)

 +qk–

L+L+ (T–tk)L, (.)

fork= , . . . ,m.

Theorem . Assume that:

(H) The functionf : [,T]×R→Ris a continuous and there exists a constantL> such

that|f(t,x) –f(t,y)| ≤L|x–y|,for eacht∈Jandx,y∈R.

(H) The functionsIk,I_k∗:R→Rare continuous and there exist constantsL,L> such

that|Ik(x) –Ik(y)| ≤L|x–y|and |I∗_k(x) –I_k∗(y)| ≤L|x–y| for eachx,y∈R,k= , , . . . ,m.

If

:=T

j

k=

(tk–tk–)L+L

+ T

T–η m

k=j+

k+

TL

T–η

(η–tj)  +qj

+(T–tm)



 +qm

+ m

k=

k+

(T–tm)  +qm

L≤δ< , (.)

then the impulsive qk-diﬀerence boundary value problem(.)has a unique solution on J.

Proof First, we transform the problem (.) into a ﬁxed-point problem,x=Ax, where the operatorAis deﬁned by (.). By using Banach’s contraction principle, we shall show that

Ahas a ﬁxed point which is the unique solution of problem (.).

Set sup_t_∈_J|f(t, )|=M <∞, sup{|Ik()|:k= , , . . . ,m}=M <∞, sup{|I_k∗()|:k=

, , . . . ,m}=M<∞and a constant

ρ=T

j

k=

(tk–tk–)M+M

+ T

T–η m

k=j+

k

+ TM

T–η

(η–tj)

 +qj

+(T–tm)



 +qm +

m

k=

k+

(T–tm)  +qm

M. (.)

Choosingr≥ _–ρ_ε, whereδ≤ε< , we show thatABr⊂Br, whereBr={x∈PC(J,R) :

x ≤r}. Forx∈Br, we have

Ax

≤sup

t∈J

t

j

k= tk

tk–

∗

k

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(10)

+T x–y

T–η

(η–tj)

 +qj

+(T–tm)



 +qm

L

+ m

k=

(tk–tk–)

 +qk–

L+L x–y +

m

k=

(tk–tk–)L+L

(T–tk) x–y

+(T–tm)



 +qm

L x–y

= x–y .

As< ,Ais a contraction. Hence, by Banach’s contraction mapping principle, we ﬁnd

thatAhas a ﬁxed point which is the unique solution of problem (.).

Our next result is based on Krasnoselskii’s ﬁxed-point theorem.

Lemma .(Krasnoselskii’s ﬁxed-point theorem) [] Let M be a closed,bounded,convex and nonempty subset of a Banach space X.Let A,B be the operators such that(a)Ax+By∈ M whenever x,y∈M; (b)A is compact and continuous; (c)B is a contraction mapping.

Then there exists z∈M such that z=Az+Bz. Further, we use the notation

θ=T

j

k=

(tk–tk–) +

T T–η

m

k=j+

(tk–tk–)

 +qk–

+ T

T–η m

k=j+

(T–tk)(tk–tk–) +

T(η–tj) (T–η)( +qj)

+ T(T–tm)



(T–η)( +qm) +

m+

k=

(tk–tk–)

 +qk–

+ m

k=

(T–tk)(tk–tk–), (.)

and

θ=jTN+

(m–j)TN

T–η +mN+N m

k=

(T–tk) + TN

T–η m

j+

(T–tk). (.)

Theorem . Let f:J×R→Rbe a continuous function.Assume that(H)holds and in

addition suppose that:

(H) |f(t,x)| ≤μ(t),∀(t,x)∈J×R,andμ∈C(J,R+).

(H) There exist constantsN,N> such that|Ik(x)| ≤Nand|I∗k(x)| ≤Nfor allx∈R,

fork= , , . . . ,m.

Then the impulsive qk-diﬀerence boundary value problem(.)has at least one solution

on J provided that

jTL+mL+

T(m–j)L

T–η +L m

k=

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To show thatSis a contraction, forx,y∈PC(J,R), we have

From (.), it follows thatSis a contraction.

Next, the continuity off implies that the operatorS is continuous. Further,Sis uni-formly bounded onBRby

Sx ≤ μ θ.

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subsets, it follows thatSis relative compact onBR. Hence, by the Arzelá-Ascoli theorem,

Sis compact onBR. Thus all the assumptions of Lemma . are satisﬁed. Hence, by the conclusion of Lemma ., the impulsiveqk-diﬀerence boundary value problem (.) has at

least one solution onJ.

4 Examples

Example . Consider the following nonlinear second-order impulsive qk-diﬀerence equation with three-point boundary condition:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

D

 +k

x(t) = e–cos _t

|x(t)|

(+t)_(+|_x₍_t_)|), t∈J= [, ],t=tk=_k, x(tk) = |x(tk)|

(+|x(tk)|), k= , , . . . , ,

D  +kx(t

+

k) –D_+_kx(tk) =_tan–(_x(tk)), k= , , . . . , ,

x() = , x() =x(_).

(.)

Hereqk= /( +k) fork= , , , . . . , ,m= ,T= ,η= /,j= ,f(t,x) = (e–cos _t

|x|)/ (( +t)_{( +}_|_x_|_)),_I_k(_x_{) =}_|_x_|_{/(( +}_|_x_|_{)) and}_I∗

k(x) = (/)tan–(x/). Since

f(t,x) –f(t,y)≤(/)|x–y|,

Ik(x) –Ik(y)≤(/)|x–y| and I_k∗(x) –I_k∗(y)≤(/)|x–y|,

then (H) and (H) are satisﬁed withL= (/),L= (/),L= (/). We can show that

≈. < .

Hence, by Theorem ., the three-point impulsiveqk-diﬀerence boundary value problem (.) has a unique solution on [, ].

Example . Consider the following nonlinear second-order impulsive qk-diﬀerence equation with three-point boundary condition:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

D +k

x(t) = sin₍_t_+)(πt) | x(t)|

(+|x(t)|), t∈J= [, ],t=tk=_k, x(tk) = |x(tk)|

(+|x(tk)|), k= , , . . . , ,

D  +kx(t

+

k) –D_+_kx(tk) =

|x(tk)|

(+|x(tk)|), k= , , . . . , ,

x() = , x() =x(_).

(.)

Setqk= /( +k) fork= , , , . . . , ,m= ,T= ,η= /,j= ,f(t,x) = (sin(πt)|x|)/ ((t+ )_{( +}_|_x_|_)),_I_k(_x_{) =}_|_x_|_{/(( +}_|_x_|_{)) and}_I∗

k(x) =|x|/(( +|x|)). Since

Ik(x) –Ik(y)≤(/)|x–y| and Ik∗(x) –Ik∗(y)≤(/)|x–y|,

then (H) is satisﬁed withL= (/),L= (/). It is easy to verify that|f(t,x)| ≤μ(t)≡,

Ik(x)≤N= / andIk∗(x)≤N= / for allt∈[, ],x∈R,k= , . . . ,m. Thus (H) and

(H) are satisﬁed. We can show that

jTL+mL+

T(m–j)L

T–η +L m

k=

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Hence, by Theorem ., the three-point impulsiveqk-diﬀerence boundary value problem (.) has at least one 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 of J Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand.

Received: 11 November 2013 Accepted: 7 January 2014 Published:27 Jan 2014

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