R E S E A R C H
Open Access
Applications of quantum calculus on finite
intervals to impulsive difference inclusions
Sotiris K Ntouyas
1and Jessada Tariboon
2,3**Correspondence:
2Department of Mathematics,
Faculty of Applied Science, Nonlinear Dynamic Analysis Research Center, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand
3Centre of Excellence in
Mathematics, CHE, Si Ayutthaya Rd., Bangkok, 10400, Thailand Full list of author information is available at the end of the article
Abstract
Recently Tariboon and Ntouyas (Adv. Differ. Equ. 2013:282, 2013) introduced the notions ofqk-derivative andqk-integral of a function on finite intervals. As
applications existence and uniqueness results for initial value problems for first- and second-order impulsiveqk-difference equations was proved. In this paper, continuing
the study of Tariboon and Ntouyas (Adv. Differ. Equ. 2013:282, 2013), we apply the quantum calculus to initial value problems for impulsive first- and second-order qk-difference inclusions. We establish new existence results, when the right hand side
is convex valued, by using the nonlinear alternative of Leray-Schauder type. Some illustrative examples are also presented.
MSC: 34A60; 26A33; 39A13; 34A37
Keywords: qk-derivative;qk-integral; impulsiveq-difference inclusion
1 Introduction and preliminaries
In [] the notions ofqk-derivative and qk-integral of a functionf :Jk := [tk,tk+]→R,
have been introduced and their basic properties was proved. As applications, existence and uniqueness results for initial value problems for first- and second-order impulsiveqk -difference equations was proved.
We recall 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 →Ris 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.
The properties ofqk-derivative are discussed in [].
Definition . Assumef :Jk→Ris a continuous function. Then theqk-integral is de-fined by
t
tk
f(s)dqks= ( –qk)(t–tk)
∞
n=
qnkfqnkt+ –qkntk
(.)
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=
qnkfqnkt+ –qnktk
– ( –qk)(a–tk) ∞
n=
qn kf
qn
ka+
–qn k
tk
.
Note that iftk= andqk=q, then (.) reduces toq-integral of a functionf(t), defined by t
f(s)dqs= ( –q)t
∞
n=qnf(qnt) fort∈[,∞).
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, that is, differential equations involving the impulse ef-fect, appear as a natural description of observed evolution phenomena of several real world problems. For some monographs on the impulsive differential equations we refer to [–].
Here, we remark that the classicalq-calculus cannot be considered in problems with impulses as the definition of q-derivative fails to work when there are impulse points tk∈(qt,t) for somek∈N. On the other hand, this situation does not arise for impul-sive problems on aq-time scale as the pointstandqt=ρ(t) are consecutive points, where ρ:T→Tis the backward jump operator; see []. In [], quantum calculus on finite in-tervals, the pointstandqkt+ ( –qk)tkare considered only in an interval [tk,tk+].
There-fore, the problems with impulses at fixed times can be considered in the framework of qk-calculus.
In this paper, continuing the study of [], we apply qk-calculus to establish existence results for initial value problems for impulsive first- and second-order qk-difference in-clusions. In Section , we consider the following initial value problem for the first-order qk-difference inclusion:
Dqkx(t)∈F
t,x(t), t∈J:= [,T],t=tk,
x(tk) =Ik
x(tk)
, k= , , . . . ,m,
x() =x,
where x∈R, =t<t<t<· · ·<tk <· · ·<tm<tm+=T, f : [,T]×R→P(R) is
a multivalued function,P(R) is the family of all nonempty subjects ofR, Ik∈C(R,R), x(tk) =x(t+k) –x(tk),k= , , . . . ,mand <qk< fork= , , , . . . ,m.
In Section , we study the existence of solutions for the following initial value problem for second-order impulsiveqk-difference inclusion:
Dq
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, Dqx() =β,
(.)
whereα,β∈RandIk,Ik∗∈C(R,R).
We establish new existence results, when the right hand side is convex valued by using the nonlinear alternative of Leray-Schauder type.
The paper is organized as follows. In Section , we recall some preliminary facts that we need in the sequel. In Section we establish the existence result for first-orderqk -difference inclusions, while the existence result for second-orderqk-difference inclusions is presented in Section . Some illustrative examples are also presented.
2 Preliminaries
In this section we recall some basic concepts of multivalued analysis [, ].
For a normed space (X, · ), letPcl(X) ={Y∈P(X) :Yis closed},Pcp(X) ={Y∈P(X) : Yis compact}, andPcp,c(X) ={Y∈P(X) :Yis compact and convex}.
A multivalued mapG:X→P(X) isconvex (closed) valuedifG(x) is convex (closed) for allx∈X; isboundedon bounded sets ifG(B) =x∈BG(x) is bounded inXfor allB∈Pb(X) (i.e.supx∈B{sup{|y|:y∈G(x)}}<∞); is calledupper semicontinuous (u.s.c.)onXif for each x∈X, the setG(x) is a nonempty closed subset ofX, and if for each open setN ofX
containingG(x), there exists an open neighborhoodNofx such thatG(N)⊆N; is
said to becompletely continuousifG(B) is relatively compact for everyB∈Pb(X). In the sequel, we denote byC=C([,T],R) the space of all continuous functions from [,T]→Rwith norm x =sup{|x(t)|:t∈[,T]}. ByL([,T],R) we denote the space of
all functionsf defined on [,T] such that x L=T
|x(t)|dt<∞.
For eachy∈C, define the set of selections ofFby
SF,y:= v∈C:v(t)∈F
t,y(t)on [,T].
Definition . A multivalued mapF:J×R→P(R) is said to be Carathéodory (in the sense ofqk-calculus) ifx−→F(t,x) is upper semicontinuous onJ. Further a Carathéodory function Fis called L-Carathéodory if there exists ϕα∈L(J,R+) such that F(t,x) =
sup{|v|:v∈F(t,x)} ≤ϕα(t) for all x ≤αonJfor eachα> .
We recall the well-known nonlinear alternative of Leray-Schauder for multivalued maps and a useful result regarding closed graphs.
is a upper semicontinuous compact map.Then either
(i) Fhas a fixed point inU,or
(ii) there is au∈∂Uandλ∈(, )withu∈λF(u).
Lemma . ([, ]) Let X be a Banach space. Let F :J×R→Pcp,c(X) be an L -Carathéodory multivalued map and let be a linear continuous mapping from L(J,R) to C(J,R).Then the operator
◦SF:C(J,R)→Pcp,c
C(J,R), x→( ◦SF)(x) = (SF,x)
is a closed graph operator in C(J,R)×C(J,R).
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(tk–) 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}.
3 First-order impulsiveqk-difference inclusions
In this section, we study the existence of solutions for the first-order impulsive qk -difference inclusion (.).
The following lemma was proved in [].
Lemma . If y∈PC(J,R),then for any t∈Jk,k= , , , . . . ,m,the solution of the problem
Dqkx(t) =y(t), t∈J,t=tk,
x(tk) =Ik
x(tk)
, k= , , . . . ,m,
x() =x
(.)
is given by
x(t) =x+
<tk<t
tk
tk–
y(s)dqk–s+
<tk<t
Ik
x(tk)
+ t
tk
y(s)dqks, (.)
with<(·) = .
Before studying the boundary value problem (.) let us begin by defining its solution.
Definition . A functionx∈PC(J,R) is said to be a solution of (.) ifx() =x,x(tk) = Ik(x(tk)),k= , , . . . ,m, and there existsf ∈L(J,R) such thatf(t)∈F(t,x(t)) onJand
x(t) =x+
<tk<t
tk
tk–
f(s)dqk–s+
<tk<t
Ik
x(tk)
+ t
tk
f(s)dqks.
Theorem . Assume that:
(H) there exist a continuous nondecreasing functionψ: [,∞)→(,∞)and a function
Then the initial value problem(.)has at least one solution on J.
Proof Define the operatorH:PC(J,R)→P(PC(J,R)) by
We will show thatHsatisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show thatHis convex for each x∈PC(J,R).This step is obvious sinceSF,xis convex (Fhas convex values), and therefore we omit the proof.
Now we show that H maps bounded sets into equicontinuous sets of PC(J,R). Let
Obviously the right hand side of the above inequality tends to zero independently ofx∈ Bρasτ–τ→. Therefore it follows by the Arzelá-Ascoli theorem thatH:PC(J,R)→
P(PC(J,R)) is completely continuous.
Thus, it follows by Lemma . that ◦SF is a closed graph operator. Further, we have hn(t)∈ (SF,xn). Sincexn→x∗, therefore, we have
h∗(t) =x+
<tk<t
tk
tk–
f∗(s)dqk–s+
<tk<t
Ik
x∗(tk)
+ t
tk
f∗(s)dqks,
for somef∗∈SF,x∗.
Finally, we show there exists an open setU⊆C(J,R) withx∈/H(x) for anyλ∈(, ) and allx∈∂U. Letλ∈(, ) andx∈λH(x). Then there existsv∈L(J,R) withf ∈S
F,xsuch that, fort∈J, we have
x(t) =x+
<tk<t
tk
tk–
f(s)dqk–s+
<tk<t
Ik
x(tk)
+ t
tk
f(s)dqks.
Repeating the computations of the second step, we have
x(t)≤ |x|+Tψ
x p + m
k=
ck.
Consequently, we have
x |x|+Tψ
x p +mk=ck ≤.
In view of (H), there existsMsuch that x =M. Let us set
U= x∈PC(J,R) : x <M.
Note that the operatorH:U→P(PC(J,R)) is upper semicontinuous and completely con-tinuous. From the choice ofU, there is nox∈∂Usuch thatx∈λH(x) for someλ∈(, ). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma .), we de-duce thatHhas a fixed pointx∈Uwhich is a solution of the problem (.). This completes
the proof.
Example . Let us consider the following first-order initial value problem for impulsive qk-difference inclusions:
D
+kx(t)∈F
t,x(t), t∈J= [, ],t=tk= k ,
x(tk) = | x(tk)| +|x(tk)|
, k= , , . . . , ,
x() = .
(.)
Hereqk= /( +k),k= , , , . . . , ,m= ,T = , andIk(x) =|x|/( +|x|). We find that |Ik(x) –Ik(y)| ≤(/)|x–y|and|Ik(x)| ≤.
(a) LetF: [, ]×R→P(R) be a multivalued map given by
x→F(t,x) =
|x|
|x|+sinx+ +t+ ,e
–x
+ t
+
Forf∈F, we have
|f| ≤max
|x|
|x|+sinx+ +t+ ,e
–x+t+
≤, x∈R.
Thus,
F(t,x)P:=sup |y|:y∈F(t,x)≤ =p(t)ψ x , x∈R,
withp(t) = ,ψ( x ) = . Further, using the condition (H) we find thatM> . Therefore,
all the conditions of Theorem . are satisfied. So, problem (.) withF(t,x) given by (.) has at least one solution on [, ].
(b) IfF: [, ]×R→P(R) is a multivalued map given by
x→F(t,x) =
(t+ )x
x+ ,
t|x|(cosx+ )
(|x|+ )
. (.)
Forf∈F, we have
|f| ≤max
(t+ )x
x+ ,
t|x|(cosx+ )
(|x|+ )
≤t+ , x∈R.
Here F(t,x) P :=sup{|y|:y∈F(t,x)} ≤(t+ ) =p(t)ψ( x ), x∈R, with p(t) =t+ , ψ( x ) = . It is easy to verify thatM> .. Then, by Theorem ., the problem (.) withF(t,x) given by (.) has at least one solution on [, ].
4 Second-order impulsiveqk-difference inclusions
In this section, we study the existence of solutions for the second-order impulsive qk -difference inclusion (.).
We recall the following lemma from [].
Lemma . If y∈C(J,R),then for any t∈J,the solution of the problem
Dq
kx(t) =y(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, Dqx() =β,
(.)
is given by
x(t) =α+βt
+
<tk<t
tk
tk–
tk–qk–s– ( –qk–)tk–
y(s)dqk–s+Ik
x(tk)
+t
<tk<t
tk
tk–
fy(s)dqk–s+I
∗ k
–
(A) there exists a constantM> such that
M
Then the initial value problem(.)has at least one solution on J.
–
We will show thatHsatisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show thatHis convex for each x∈PC(J,R).This step is obvious sinceSF,xis convex (Fhas convex values), and therefore we omit the proof.
+
Now we show that H maps bounded sets into equicontinuous sets of PC(J,R). Let τ,τ∈J,τ<τ withτ∈Ju,τ∈Jv,u≤vfor someu,v∈ {, , , . . . ,m}andx∈Bρ. For
Obviously the right hand side of the above inequality tends to zero independently ofx∈ Bρasτ–τ→. Therefore it follows by the Arzelá-Ascoli theorem thatH:PC(J,R)→
+T
Thus, it follows by Lemma . that ◦SF is a closed graph operator. Further, we have hn(t)∈ (SF,xn). Sincexn→x∗, therefore, we have
Repeating the computations of the second step, we have
Consequently, we have
x |α|+|β|T+ p ψ( x )+
m
k=[ck+c∗k(T+tk)]≤ .
In view of (A), there existsMsuch that x =M. Let us set
U= x∈PC(J,R) : x <M.
Note that the operatorH:U→P(PC(J,R)) is upper semicontinuous and completely con-tinuous. From the choice ofU, there is nox∈∂Usuch thatx∈λH(x) for someλ∈(, ). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma .), we de-duce thatHhas a fixed pointx∈Uwhich is a solution of the problem (.). This completes
the proof.
Example . Let us consider the following second-order impulsiveqk-difference inclu-sion with initial conditions:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
D +k
x(t)∈F(t,x(t)), t∈J= [, ],t=tk=k,
x(tk) = (+|x(|txk()t|k)|), k= , , . . . , , D
+kx(t
+
k) –D+k–x(tk) = |x(tk)|
(+|x(tk)|), k= , , . . . , ,
x() = , D
x() = .
(.)
Hereqk= /( +k),k= , , , . . . , ,m= ,T = ,α= ,β= ,Ik(x) =|x|/(( +|x|)), andIk∗(x) =|x|/(( +|x|)). We find that|Ik(x) –Ik(y)| ≤(/)|x–y|,|Ik∗(x) –Ik∗(y)| ≤ (/)|x–y|, andIk(x)≤/,Ik∗(x)≤/; and we have
=
m+
k=
(tk–tk–)
+qk–
+ m
k=
(T+tk)(tk–tk–)≈..
(a) LetF: [, ]×R→P(R) be a multivalued map given by
x→F(t,x) =
|x|
|x|+sinx+ +t+ ,e
–x+ t
+
. (.)
Forf∈F, we have
|f| ≤max
|x|
|x|+sinx+ +t+ ,e
–x+t+
≤, x∈R.
Thus,
F(t,x)P:=sup |y|:y∈F(t,x)≤ =p(t)ψ x , x∈R,
withp(t) = ,ψ( x ) = . Further, using the condition (A) we find
M +
k=[ +
( +tk)]
which impliesM> .. Therefore, all the conditions of Theorem . are satisfied. So, problem (.) withF(t,x) given by (.) has at least one solution on [, ].
(b) IfF: [, ]×R→P(R) is a multivalued map given by
x→F(t,x) =
(t+ )x
x+ ,
t|x|(cosx+ )
(|x|+ )
. (.)
Forf∈F, we have
|f| ≤max
(t+ )x
x+ ,
t|x|(cosx+ )
(|x|+ )
≤t+ , x∈R.
Here F(t,x) P :=sup{|y|:y∈F(t,x)} ≤(t+ ) =p(t)ψ( x ), x∈R, with p(t) =t+ , ψ( x ) = . It is easy to verify thatM> .. Then, by Theorem ., the problem (.) withF(t,x) given by (.) has at least one solution on [, ].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this article. They read and approved the final manuscript.
Author details
1Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece.2Department of Mathematics, Faculty of
Applied Science, Nonlinear Dynamic Analysis Research Center, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand.3Centre of Excellence in Mathematics, CHE, Si Ayutthaya Rd., Bangkok, 10400, Thailand.
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 is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
Received: 16 June 2014 Accepted: 30 September 2014 Published:13 Oct 2014 References
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