Separated boundary value problems for second order impulsive q integro difference equations

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R E S E A R C H

Open Access

Separated boundary value problems for

second-order impulsive

q

-integro-difference

equations

Chatthai Thaiprayoon

1

_{, 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

This article studies the existence and uniqueness of solutions for a boundary value problem of nonlinear second-order impulsiveqk-integro-diﬀerence equations with

separated boundary conditions. Several new results are obtained by applying a variety of ﬁxed point theorems. Some examples are presented to illustrate the results. MSC: 26A33; 39A13; 34A37

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

uniqueness; ﬁxed point theorems

1 Introduction

In this paper, we study the separated boundary value problem for impulsiveqk

-integro-diﬀerence equation of the following form: ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

D

qkx(t) =f(t,x(t), (Sqkx)(t)), t∈J:= [,T],t=tk,

x(tk) =Ik(x(tk)), k= , , . . . ,m,

Dqkx(tk+) –Dqk–x(tk) =Ik∗(x(tk)), k= , , . . . ,m,

x() +Dqx() = , x(T) +Dqmx(T) = ,

(.)

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

(Sqkx)(t) =

t

tk

φ(t,s)x(s)dqks, t∈(tk,tk+],k= , , , . . . ,m, (.)

φ:J×J→[,∞) is 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) and  <qk<  fork= , , , . . . ,m.

The notions ofqk-derivative andqk-integral on ﬁnite intervals were introduced in [].

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) = lim t→tk

Dqkf(t) (.)

is called theqk-derivative of functionf att.

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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 →Rbe a continuous function, we call the second-order qk

-derivativeD

qkf providedDqkf isqk-diﬀerentiable onJkwithD 

qkf =Dqk(Dqkf) :Jk→R.

Similarly, we deﬁne higher orderqk-derivativeDnqk:Jk→R.

Theqk-integral is deﬁned as follows.

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_kn tk (.)

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_k tk

– ( –qk)(a–tk)

∞

n=

qn_kfqn_ka+ –qn_k tk .

Note that iftk=  andqk=q, then (.) reduces toq-integral of a functionf(t), deﬁned

by_tf(s)dqs= ( –q)t

∞

n=qnf(qnt) fort∈[,∞).

For the basic properties ofqk-derivative andqk-integral we refer to [].

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.

Impulsive differential equations serve as basic models to study the dynamics of processes 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 dynamics and medicine. For some recent works on the theory of impulsive differential equations, we refer the interested reader to the monographs [–].

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value problem (.) into an equivalent integral equation. The main results are given in Section , while examples illustrating the results are presented in Section .

2 An auxiliary lemma

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 norm x PC=sup{|x(t)|;t∈J}.

We now consider the following linear case: ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

D

qkx(t) =h(t), t∈J,t=tk,

x(tk) =Ik(x(tk)), k= , , . . . ,m,

Dqkx(tk+) –Dqk–x(tk) =Ik∗(x(tk)), k= , , . . . ,m,

x() +Dqx() = , x(T) +Dqmx(T) = ,

(.)

whereh:J→Ris a continuous function.

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

x(t) =

 –t T

m

k=

tk

tk– s

tk–

h(r)dqk–r dqk–s+Ik

x(tk)

+

 –t T

m

k=

tk

tk–

h(s)dqk–s+I ∗

k

x(tk) (T–tk+ )

+

 –t T

T

tm

s

tm

h(r)dqmr dqms+

 –t

T

tm

h(s)dqms

+

<tk<t

tk

tk– s

tk–

x(tk)

+

<tk<t

tk

tk–

k

x(tk) (t–tk) +

t

tk

s

tk

h(r)dqkr dqks, (.)

withb_i₌_a(·) = for a>b.

Proof Taking theq-integral for the ﬁrst equation of (.), fort∈J, we have

Dqx(t) =Dqx() + t



h(s)dqs, (.)

which leads to

Dqx(t) =Dqx() + t



h(s)dqs. (.)

Fort∈Jwe get byq-integrating (.),

x(t) =x() +Dqx()t+ t



s



h(r)dqr dqs

:=A+Bt+ t



s



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In particular, fort=t, we obtain

x(t) =A+Bt+

t



s



h(r)dqr dqs. (.)

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

Dqx(t) =Dqx

t+_ + t

t

h(s)dqs.

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

Dqx(t) =B+ t



h(s)dqs+I ∗



x(t) +

t

t

h(s)dqs. (.)

Takingq-integral to (.) fort∈J, we obtain

x(t) =xt+_ +

B+ t



h(s)dqs+I ∗



x(t) (t–t)

+ t

t s

t

h(r)dqr dqs. (.)

Applying the second equation of (.) with (.) and (.), we get

x(t) =A+Bt+

t



s



h(r)dqr dqs+I

x(t)

+

B+ t



h(s)dqs+I ∗



x(t) (t–t) +

t

t s

t

h(r)dqr dqs

=A+Bt+ t



s



h(r)dqr dqs+I

x(t)

+ t



h(s)dqs+I ∗



x(t) (t–t) +

t

t s

t

h(r)dqr dqs.

Repeating the above process, fort∈Jk, we get

x(t) =A+Bt

+

<tk<t

tk

tk– s

tk–

x(tk)

+

<tk<t

tk

tk–

k

x(tk) (t–tk)

+ t

tk

s

tk

h(r)dqkr dqks. (.)

From the ﬁrst boundary condition of (.) (i.e. x() +Dqx() = ) and (.), we have

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Also, the second boundary condition of (.) (i.e. x(T) +Dqmx(T) = ) and (.), yields

Substituting constantsAandBinto (.), we obtain (.) as requested.

3 Main results

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+

<tk<t

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s+I ∗

k

x(tk) (t–tk)

+ t

tk

s

tk

fr,x(r), (Sqkx)(r) dqkr dqks. (.)

It should be noticed that problem (.) has solutions if and only if the operatorAhas ﬁxed points.

Our ﬁrst result is an existence and uniqueness result for the impulsive boundary value problem (.) by using the Banach contraction mapping principle.

Letφ=max{φ(t,s) : (t,s)∈J×J}. Further, for convenience we set

ω= + T T

m+

k=

L(tk–tk–)

 +qk–

+φL(tk–tk–)



 +qk–+q_k_–

+ +T T

m+

k=

L(tk–tk–) +

φL(tk–tk–)

 +qk–

+ + T T

m

k=

(T–tk)

L(tk–tk–) +

 +qk–

+mL( + T)

T +

mL( +T)

T +

L( + T)

T

m

k=

(T–tk), (.)

and

λ=

M( + T)

T

m+

k=

(tk–tk–)

 +qk–

+M( +T) T

m+

k=

(tk–tk–)

+M( + T) T

m

k=

(T–tk)(tk–tk–) +

mM( + T)

T

+mM( +T)

T +

M( + T)

T

m

k=

(T–tk). (.)

Theorem . Assume that:

(H) The functionf :J×R→Ris continuous and there exist constantsL,L> such

that

ft,x, (Sqkx) –f

t,y, (Sqky) ≤L|x–y|+L

Sqk|x–y| ,

for eacht∈Jandx,y∈R,k= , , , . . . ,m.

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

that

_I_k_{(x) –}_I_k_(y)_≤_L__|_x_–_y_| and I_k∗(x) –I_k∗(y)≤L|x–y|,

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If

ω≤δ< , (.)

whereωis deﬁned by(.),andδ> ,then the boundary value problem(.)has a unique solution on J.

Proof We transform the boundary value problem (.) into a fixed point problem,x=Ax, where the operatorAis defined by (.). By using the Banach contraction mapping prin-ciple, we shall show thatAhas a fixed point which is the unique solution of the boundary value problem (.).

Let M, M, and M be nonnegative constants such that supt∈J|f(t, , )| = M,

sup{|Ik()|:k= , , . . . ,m}=M, andsup{|Ik∗()|:k= , , . . . ,m}=M. By choosing a

con-stantRas

R≥ λ  –ε,

whereδ≤ε<  andλdeﬁned by (.), we will show thatABR⊂BR, where a ballBRis

deﬁned byBR={x∈PC(J,R) : x ≤R}. Forx∈BR, we have Ax

≤sup

t∈J

 –t

T

m

k=

tk

tk– s

tk–

fr,x(r), (Sqk–x)(r) dqk–r dqk–s+Ik

x(tk)

+

 –t T

m

k=

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s+I ∗

k

x(tk) (T–tk+ )

+

 –t T

T

tm

s

tm

fr,x(r), (Sqmx)(r) dqmr dqms

+

 –t T

T

tm

fs,x(s), (Sqmx)(s) dqms

+

<tk<t

tk

tk– s

tk–

x(tk)

+

<tk<t

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s+I ∗

k

x(tk) (t–tk)

+ t

tk

s

tk

fr,x(r), (Sqkx)(r) dqkr dqks

≤

 +T

T

m

k=

tk

tk– s

tk–

x(tk)

+

 +T T

m

k=

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s+I ∗

k

x(tk)

(T–tk+ )

+

 +T T

T

tm

s

tm

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+ + T T

m

k=

(LR+M) +

 +T T

m

k=

(LR+M) +

 + T T

m

k=

(LR+M)(T–tk)

=ωR+λ≤(δ+  –ε)R≤R,

which implies thatABR⊂BR.

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

≤ + T

T

m+

k=

tk

tk– s

tk–

fr,x(r), (Sqk–x)(r) –f

r,y(r), (Sqk–y)(r) dqk–r dqk–s

+ +T T

m+

k=

tk

tk–

fs,x(s), (Sqk–x)(s) –f

s,y(s), (Sqk–y)(s) dqk–s

+ + T T

m

k=

(T–tk)

tk

tk–

fs,x(s), (Sqk–x)(s) –f

s,x(s), (Sqk–x)(s) dqk–s

+ + T T

m

k=

Ik

x(tk) –Ik

y(tk) +

 +T T

m

k=

I_k∗x(tk) –Ik∗

y(tk)

+ + T T

m

k=

_I∗

k

x(tk) –Ik∗

y(tk) (T–tk)

≤ + T

T

m+

k=

L(tk–tk–)

 +qk–

+φL(tk–tk–)



 +qk–+qk–

x–y

+ +T T

m+

k=

L(tk–tk–) +

 +qk–

x–y

+ + T T

m

k=

(T–tk)

L(tk–tk–) +

 +qk–

x–y

+m( + T)L

T x–y +

m( +T)L

T x–y +

( + T)L x–y

T

m

k=

(T–tk)

=ω x–y ,

which implies that Ax–Ay ≤ω x–y . Asω< ,Ais a contraction. Therefore, by the Banach contraction mapping principle, we ﬁnd thatAhas a ﬁxed point which is the unique solution of problem (.). This completes the proof.

The second existence result is based on Schaefer’s ﬁxed point theorem.

Theorem . Assume that:

(H) f :J×R→Ris a continuous function and there exists a constantN> such that

ft,x, (Sqkx) ≤N,

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(H) The functionsIk,Ik∗:R→Rare continuous and there exist constantsN,N> such

that

Ik(x)≤N and Ik∗(x)≤N,

for allx∈R,k= , , . . . ,m.

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

Proof We will use Schaefer’s fixed point theorem to prove thatA, defined by (.), has a fixed point. We divide the proof into four steps.

Step :Continuity ofA.

Let{xn}be a sequence such thatxn→xinPC(J,R). Sincef is a continuous function on

J×R_and_I

k,I∗_kare continuous functions onRfork= , , . . . , we have

ft,xn(t), (Sqkxn)(t) →f

t,x(t), (Sqkx)(t) ,

andIk(xn(tk))→Ik(x(tk)),Ik∗(xn(tk))→Ik∗(x(tk)) fork= , , . . . , asn→ ∞.

Then, for eacht∈J, we get (Axn)(t) – (Ax)(t)

=

 –t T

m

k=

tk

tk– s

tk–

fr,xn(r), (Sqk–xn)(r)

–fr,x(r), (Sqk–x)(r) dqk–r dqk–s+Ik

xn(tk) –Ik

x(tk)

+

 –t T

m

k=

tk

tk–

fs,xn(s), (Sqk–xn)(s) –f

s,x(s), (Sqk–x)(s) dqk–s

+I_k∗xn(tk) –Ik∗

x(tk)

(T–tk+ )

+

 –t T

T

tm

s

tm

fr,xn(r), (Sqmxn)(r) –f

r,x(r), (Sqmx)(r) dqmr dqms

+

 –t T

T

tm

fs,xn(s), (Sqmxn)(s) –f

s,x(s), (Sqmx)(s) dqms

+

<tk<t

tk

tk– s

tk–

fr,xn(r), (Sqk–xn)(r) –f

r,x(r), (Sqk–x)(r) dqk–r dqk–s

+Ik

xn(tk) –Ik

x(tk)

+

<tk<t

tk

tk– _f

s,xn(s), (Sqk–xn)(s) –f

s,x(s), (Sqk–x)(s) dqk–s

+I_k∗xn(tk) –Ik∗

x(tk)

(t–tk)

+ t

tk

s

tk

fr,xn(r), (Sqkxn)(r) –f

r,x(r), (Sqkx)(r) dqkr dqks,

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Step :Amaps bounded sets into bounded sets in PC(J,R).

So, let us prove that for anyr> , there exists a positive constantρsuch that for each x∈Br={x∈PC(J,R) : x ≤r}, we have Ax ≤ρ. For anyx∈Br, we have

Step :Amaps bounded sets into equicontinuous sets of PC(J,R).

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+|τ–τ| T

m

k=

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s+I ∗

k

x(tk)

(T–tk+ )

+|τ–τ| T

T

tm

s

tm

+|τ–τ| T

T

tm

+|τ–τ| i

k=

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s+I ∗

k

x(tk)

+ τ

ti

s

ti

fr,x(r), (Sqix)(r) dqir dqis–

τ

ti

s

ti

fr,x(r), (Sqix)(r) dqir dqis

≤|τ–τ|

T

m

k=

N(tk–tk–)

 +qk–

+N

+|τ–τ| T

m

k=

N(tk–tk–) +N

(T–tk+ ) +|

τ–τ|

T

N(T–tm)

 +qm

+|τ–τ| T

N(T–tm)

+|τ–τ| i

k=

N(tk–tk–) +N

+|τ–τ|N  +qi

(τ+τ+ ti).

Asτ→τ, the right-hand side of the above inequality (which is independent ofx) tends

to zero. As a consequence of Steps  to , together with the Arzelá-Ascoli theorem, we deduce thatA:PC(J,R)→PC(J,R) is completely continuous.

Step :We show that the set

E=x∈PC(J,R) :x=θAx for some <θ< 

is bounded.

Letx∈E. Thenx(t) =θ(Ax)(t) for some  <θ < . Thus, for eacht∈J, by using the computations of Step , we have that

Ax ≤ρ.

This shows that the setEis bounded. As a consequence of Schaefer’s ﬁxed point theo-rem, we conclude thatAhas a ﬁxed point which is a solution of the impulsiveqk

-integro-diﬀerence boundary value problem (.).

The third existence result for the impulsive boundary value problem (.) 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

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(c) Bis a contraction mapping.

Then there exists z∈M such that z=Az+Bz. For convenience we put

= + T T

m+

k=

(tk–tk–)

 +qk–

+ +T T

m+

k=

(tk–tk–)

+ + T T

m

k=

(T–tk)(tk–tk–), (.)

and

λ=

m( + T)N

T +

m( +T)N

T +

( + T)N

T

m

k=

(T–tk). (.)

Theorem . Let f :J×R→Rbe a continuous function.Assume that:

(A) |f(t,x,y)| ≤μ(t),∀(t,x,y)∈J×R×Randμ∈C(J,R+).

(A) There exist constantsN,N> such that|Ik(x)| ≤N and|Ik∗(x)| ≤N,∀x∈R,for

k= , , . . . ,m.

(A) There exist constantsK,K> such that|Ik(x) –Ik(y)| ≤K|x–y|and|Ik∗(x) –Ik∗(y)| ≤

K|x–y|,∀x,y∈R,fork= , , . . . ,m.

If

m( + T)K

T +

m( +T)K

T +

( + T)K

T

m

k=

(T–tk) < , (.)

then boundary value problem(.)has at least one solution on J.

Proof We deﬁnesup_t_∈_J|μ(t)|= μ and choose a suitable constantρas

ρ≥ μ +λ,

whereandλare deﬁned by (.) and (.), respectively. We deﬁne the operatorsand

onBρ={x∈PC(J,R) : x ≤ρ}as

(x)(t) =

 –t T

m

k=

tk

tk– s

tk–

fr,x(r), (Sqk–x)(r) dqk–r dqk–s

+

 –t T

m

k=

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s

(T–tk+ )

+

 –t T

T

tm

s

tm

+

 –t T

T

tm

(14)

+

<tk<t

tk

tk– s

tk–

fr,x(r), (Sqk–x)(r) dqk–r dqk–s

+

<tk<t

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s

(t–tk)

+ t

tk

s

tk

fr,x(r), (Sqkx)(r) dqkr dqks

and

(x)(t) =

 –t T

m

k=

Ik

x(tk) +

 –t

T

m

k=

I_k∗x(tk) (T–tk+ )

+

<tk<t

Ik

x(tk) +

<tk<t

I_k∗x(tk) (t–tk).

Forx,y∈Bρ, we have

x+y ≤ μ

 + T T

m+

k=

(tk–tk–)

 +qk–

+ +T T

m+

k=

(tk–tk–)

+ + T T

m

k=

(T–tk)(tk–tk–)

+m( + T)N

T +

m( +T)N

T +

( + T)N

T

m

k=

(T–tk)

= μ +λ

≤ρ. Thus,x+y∈Bρ.

Forx,y∈PC(J,R), from (A), we have

x–y ≤ + T T

m

k=

Ik

x(tk) –Ik

y(tk) +

 +T T

m

k=

I_k∗x(tk) –Ik∗

y(tk)

+ + T T

m

k=

I_k∗x(tk) –Ik∗

y(tk) (T–tk)

≤ m( + T)K x–y

T +

m( +T)K x–y

T

+( + T)K x–y T

m

k=

(T–tk),

which implies, by (.), thatis a contraction mapping.

Continuity off implies that the operatoris continuous. Also,is uniformly bounded onBρas

x ≤ μ .

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We deﬁnesup₍_t_,_x₎_∈_J_×_B_ρ|f(t,x)|=f <∞,τ,τ∈(ti,ti+) for somei∈ {, , . . . ,m}withτ<

τand consequently we get

(x)(τ) – (x)(τ)

≤|τ–τ|

T

m

k=

tk

tk– s

tk– _f

r,x(r), (Sqk–x)(r) dqk–r dqk–s

+|τ–τ| T

m

k=

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s

(T–tk+ )

+|τ–τ| T

T

tm

s

tm

+|τ–τ| T

T

tm

+|τ–τ| i

k=

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s

+ τ

ti

s

ti

fr,x(r), (Sqix)(r) dqir dqis–

τ

ti

s

ti

fr,x(r), (Sqix)(r) dqir dqis

≤|τ–τ|f

T

m

k=

(tk–tk–)

 +qk–

+|τ–τ|f T

m

k=

(tk–tk–)(T–tk+ )

+|τ–τ|f T

(T–tm)

 +qm

+|τ–τ|f

T (T–tm) +|τ–τ|f

i

k=

(tk–tk–)

+|τ–τ|f  +qi

(τ+τ+ ti),

which is independent ofxand tends to zero asτ–τ→. Thus,is equicontinuous. So

is relatively compact onBρ. Hence, by the Arzelá-Ascoli theorem,is compact onBρ.

Thus all the assumptions of Lemma . are satisﬁed. So the boundary value problem (.) has at least one solution onJ. The proof is completed.

Our ﬁnal, fourth existence result is based on the Leray-Schauder 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). Theorem . Assume that:

(A) There exist a continuous nondecreasing functionψ: [,∞)→(,∞)and a continuous

functionp:J→R+_{such that}

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(A) There exist continuous nondecreasing functionsϕ,ϕ: [,∞)→(,∞)such that

Ik(x)≤ϕ

|x| and I_k∗(x)≤ϕ

|x| ,

for allx∈R,k= , , . . . ,m.

(A) There exists a constantM∗> such that

M∗

pψ(M∗)Q+φM∗Q+Q

> ,

wherep=max{p(t) :t∈J},φ=max{φ(t,s) : (t,s)∈J×J}and

Q=

 + T T

m+

k=

(tk–tk–)

 +qk–

+ +T T

m+

k=

(tk–tk–)

+ + T T

m

k=

(T–tk)(tk–tk–),

Q=

 + T T

m+

k=

(tk–tk–)

 +qk–+qk–

+ +T T

m+

k=

(tk–tk–)

 +qk–

+ + T T

m

k=

(T–tk)

(tk–tk–)

 +qk–

,

Q=

m( + T)ϕ(M∗)

T +

m( +T)ϕ(M∗)

T +

( + T)ϕ(M∗)

T

m

k=

(T–tk).

Then the impulsive boundary value problem(.)has at least one solution on J.

Proof First we show thatAmaps bounded sets (balls) into bounded sets in PC(J,R).For a positive numberρ, letBρ={x∈PC(J,R) : x ≤ρ}be a bounded ball inPC(J,R). Then

fort∈Jwe have (Ax)(t)

≤| –t|

T

m

k=

tk

tk– s

tk–

x(tk)

+| –t| T

m

k=

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s+I ∗

k

x(tk)

(T–tk+ )

+| –t| T

T

tm

s

tm

_f

r,x(r), (Sqmx)(r) dqmr dqms

+| –t| T

T

tm

+

m

k=

tk

tk– s

tk–

x(tk)

+

m

k=

tk

tk–

fs,x(s), (Sqk–x)(s) dqk–s+I ∗

k

x(tk)

(17)

+

Next we show thatAmaps bounded sets into equicontinuous sets of PC(J,R).

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+|τ–τ|

The right-hand side of the above inequality, which is independent ofx, tends to zero as τ→τ. From all above and by the Arzelá-Ascoli theoremA:PC(J,R)→PC(J,R) is

com-pletely continuous.

(19)

(20)

+m( + T)ϕ( x )

T +

m( +T)ϕ( x )

T

+( + T)ϕ( x ) T

m

k=

(T–tk)

=pψ

x

 + T

T

m+

k=

(tk–tk–)

 +qk–

+ +T T

m+

k=

(tk–tk–)

+ + T T

m

k=

(T–tk)(tk–tk–)

+φ x

 + T

T

m+

k=

(tk–tk–)

 +qk–+qk–

+ +T T

m+

k=

(tk–tk–)

 +qk–

+ + T T

m

k=

(T–tk)

(tk–tk–)

 +qk–

+m( + T)ϕ( x ) T

+m( +T)ϕ( x )

T +

( + T)ϕ( x )

T

m

k=

(T–tk)

=pψ

x Q+φ x Q+Q.

Consequently, we have

x

pψ( x )Q+φ x Q+Q

≤.

In view of (A), there existsM∗such that x =M∗. Let us set

U=x∈PC(J,R) : x <M∗.

Note that the operatorA:U→PC(J,R) is continuous and completely continuous. From the choice ofU, there is nox∈∂U such thatx=θAxfor someθ ∈(, ). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma .), we deduce thatAhas a ﬁxed pointx∈Uwhich is a solution of the problem (.). This completes the proof. 4 Examples

In this section, we will give examples to illustrate our main results.

Example . Consider the following boundary value problem for the nonlinear second-order impulsiveqk-integro-diﬀerence equation:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

D k+

k+

x(t) = _et₍_tt_+) |x(t)|

|x(t)|++ t 

t tk

sinπ(t–s)

 x(s)dqks, t∈J,t=tk,

x(tk) = _(_k_+)+|x(tk_|)_x|₍_tk₎_|, k= , , . . . , ,

Dk+

k+x(t

+ k) –D k

k+x(tk) = |x(tk)|

(k+)+|x(tk)|, k= , , . . . , ,

x() +D

x() = , x() +Dx() = .

(.)

HereJ= [, ],tk=k/,qk= (k+ )/(k+ ) fork= , , , . . . , ,m= ,T= ,f(t,x,Sqkx) =

(t|x|)/(et_(t_{+ )}₍_|_x_|_{+ )) + (t}_/)t

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I_k∗(x) =|x|/((k+ ) +|x|). Since f(t,x) –f(t,y)≤ 

|x–y|+ (/)

Sqk|x–y| ,

Ik(x) –Ik(y)≤



|x–y| and I ∗

k(x) –Ik∗(y)≤

 |x–y|,

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

show that

ω≈. < .

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

Example . Consider the following boundary value problem for the nonlinear second-order impulsiveqk-integro-diﬀerence equation:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

D

k+ k+

x(t) = t

(+x₎+ t _sin_t

t|_tkt cosπ(_t–s)x(s)d_qks|+, t∈J,t=tk,

x(tk) = _kk₊sin_t_|_x₍π_tt

k)|, k= , , . . . , ,

Dk+ k+x(t

+

k) –D_k_k_++x(tk) =

kcos_t

k+|x(tk)|sint, k= , , . . . , ,

x() +D

x() = , x() +Dx() = .

(.)

HereJ= [, ],tk=k/,qk= (k+)/(k+) fork= , , , . . . , ,m= ,T= ,f(t,x,Sqkx) =

((t_)/(+_x₎_)+(t_sin_t)/(t_|t tk((cos

_π(t_{–s))/)x(s)}_d

qks|+),Ik(x) = (ksinπt)/(k+t|x(tk)|)

andI_k∗(x) = (kcos_t)/(k₊_|_x(t

k)|sint). We can show that

ft,x, (Sqkx) =

_{( +}t_x₎ +

t_sin_t

t|_tkt cosπ(_t–s)x(s)dqks|+ 

≤ =N,

Ik(x)=

_kk₊sin_t_|_x(tπt_k₎_|≤ =N, and Ik∗(x)=

_k₊k_|_x(tcos_k₎_|t_sin_t≤ =N.

Hence, by Theorem ., boundary value problem (.) has at least one solution on [, ].

Example . Consider the following nonlinear second-order impulsive qk-diﬀerence

equation with separated boundary condition: ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Dk+ k+

x(t) =t₍sin_t_+)πt |

x|

|x|++

cosπ_t

|t tk et

–s t+et–sx(s)dqks|

, t∈J,t=tk,

x(tk) = _(_k_+)+|x(tk_|)_x|₍_tk₎_|, k= , , . . . , ,

Dk+ k+x(t

+ k) –D k

k–x(tk) = |x(tk)|

(k+)+|x(tk)|, k= , , . . . , , x() +D

x() = , x() +Dx() = ,

(.)

where J = [, ], tk =k/, qk = (k + )/(k + ) for k = , , , . . . , , m = , T = ,

f(t,x) = (tsinπt/(t+ )₎₍_|_x_|_/(_|_x_|_{+ )) + (cos(πt}_/))/(_|t

tk(et–s/(t+et–s))x(s)dqks|),Ik(x) =

|x(tk)|/((k+ ) +|x(tk)|) andI_k∗(x) =|x(tk)|/((k+ ) +|x(tk)|). Since

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

(tsinπt)/(t+ )₊_e–t_cos(πt_/),_I

k(x)≤N= , andI∗_k(x)≤N=  for allt∈[, ],x∈R,

k= , . . . ,m. Thus (A) and (A) are satisﬁed. We can show that

m( + T)K

T +

m( +T)K

T +

( + T)K

T

m

k=

(T–tk)≈. < .

Hence, by Theorem ., boundary value problem (.) has at least one solution on [, ].

Example . Consider the following nonlinear second-order impulsive qk-diﬀerence

equation with separated boundary condition: ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

D k+

k+

x(t) = _x_+x_et_+++

sinπt et_+ –cost

t tk

(t–s)sint

et–s_+ x(s)dqks, t∈J,t=tk,

x(tk) = sin_ππ₍_kx_+)(tk), k= , , . . . , ,

Dk+

k+x(t

+ k) –Dk+

k+x(tk) =

x(tk)coskt

(k+) , k= , , . . . , ,

x() +D

x() = , x() +Dx() = .

(.)

Here J= [, ],tk=k/,qk= (k+ )/(k+ ) fork= , , , . . . , ,m= , T= , f(t,x) =

(x/(x _{+ e}t _{+ )) + ( +}_sin_πt)/(et _{+ ) –}_cos_tt

tk((t–s)sint/(e

t–s_{+ ))x(s)}_d

qks,Ik(x) =

(sinπx)/(π(k+ )), andI_k∗(x) = (xcos_kt)/((k_{+ )). Clearly,}

f(t,x)= x x_{+ e}t_{+ }+

 +sinπt et_{+ } –cost

t

tk

(t–s)sint et–s_{+ } x(s)dqks

≤

 et_{+ }

_| x+ |

 + _tt

k

(t–s)sint et–s_{+ } x(s)dqks

, Ik(x)=

_π(ksinπx_{+ )}≤|x|

 and I ∗

k(x)=

_(kxcos_{+ )}kt≤|x|

.

Choosingp(t) = 

et_+,ψ(|x|) =|x+_|,ϕ(|u|) =_|x|, andϕ(|x|) = |_x|, we obtain

M∗

.M∗+ .> 

which implies thatM∗> .. Hence, by Theorem ., boundary value problem (.) has at least one 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.

Authors’ information

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

Author details

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

(23)

Acknowledgements

The research of C Thaiprayoon and J Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand.

Received: 3 December 2013 Accepted: 28 February 2014 Published:17 Mar 2014 References

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10.1186/1687-1847-2014-88

Cite this article as:Thaiprayoon et al.:Separated boundary value problems for second-order impulsive