R E S E A R C H
Open Access
Impulsive quantum difference systems
with boundary conditions
Jessada Tariboon
1*, Sotiris K Ntouyas
2,3and Phollakrit Thiramanus
1*Correspondence: [email protected] 1Nonlinear 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
This article studies the existence and uniqueness of solutions for coupled systems of nonlinear impulsive quantum difference equations with coupled and uncoupled boundary conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. Examples illustrating our results are also presented.
MSC: 26A33; 39A13; 34A37
Keywords: quantum calculus; impulsive quantum difference equation; existence; uniqueness; fixed point theorems
1 Introduction and preliminaries
In this paper, we concentrate on the study of the existence and uniqueness of solutions for a coupled system of nonlinear impulsive quantum difference equations,
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
Dqkx(t) =f(t,x(t),y(t)), t∈J:= [,T],t=tk,
Dpky(t) =g(t,x(t),y(t)), t∈J,t=tk,
x(tk) =Ik(x(tk)), y(tk) =I∗k(y(tk)), k= , , . . . ,m,
ax() +by(T) =λ, ay() +bx(T) =λ,
(.)
where =t<t<t <· · ·<tk <· · ·<tm<tm+ =T,f,g :J×R→Rare continuous
functions,Ik,Ik∗∈C(R,R),u(tk) =u(t+
k) –u(tk),u(t+k) =limh→+u(tk+h),u∈ {x,y}, for
k= , , . . . ,m, and <pk,qk< fork= , , , . . . ,mare given quantum numbers,ai,bi,λi,
i= , are real constants withaa=bb.
The notions of quantum calculus on finite intervals, qk-derivatives, and qk-integrals were introduced in []. For a fixedk∈N∪ {} letJk:= [tk,tk+]⊂Rbe an interval and
<qk< ,k= , , . . . ,mbe a constant. We define theqk-derivative of a functionf:Jk→R at 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→tkDqkf(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 . (.)
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=
qnkfqnkt+ –qkn 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=
qnkfqnkt+ –qnk tk
– ( –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 bytf(s)dqs= ( –q)t
∞
n=qnf(qnt) fort∈[,∞).
For the basic properties of theqk-derivative and theqk-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 [–] and the references cited therein.
Impulsive differential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states. The 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 [–]. Moreover, the interested reader is referred to [–] for some recent results on impulsiveqk-difference equations.
In this paper we prove existence and uniqueness results for the impulsive boundary value problem (.) by using Banach’s contraction mapping principle and Leray-Schauder’s non-linear alternative. The rest of this paper is organized as follows: In Section we present an auxiliary lemma which is used to convert the impulsive boundary value problem (.) into an equivalent integral equation. In Section , we establish an existence and unique-ness result via Banach’s contraction principle, and an existence result by applying Leray-Schauder’s alternative. Results on uncoupled integral boundary conditions case are in Sec-tion . Examples illustrating our results are also presented.
2 An auxiliary lemma
LetJ= [,T],J= [t,t],Jk= (tk,tk+] fork= , , . . . ,m. To define the solutions of problem
Againq-integrating (.) fromttot, wheret∈J, then
Repeating the above process, fort∈J, we obtain
x(t) =x() +
In the same way, we can obtain
y(t) =y() +
Applying the boundary conditions of (.), we get the system
ax() +by() +b
from which we have
Substituting the values ofx() andy() in (.) and (.), we obtain the solutions (.) and
For the sake of convenience, we set
M=
||
TL|a||b|+K|b||b|+K|| +mL|a||b|
, (.)
M=
||
TN|a||b|+N|b||b|+N||
+mN|a||b|+N|b||b|+N|| +|a||λ|+|b||λ|
, (.)
M=
||
TK|a||b|+L|b||b|+L|| +mK|a||b|
, (.)
M=
||
TK|a||b|+L|b||b|+L|| +mL
|b||b|+|| , (.)
M=
||
TN|a||b|+N|b||b|+N||
+mN|a||b|+N|b||b|+N|| +|a||λ|+|b||λ|
. (.)
The first result is concerned with the existence and uniqueness of solutions for the prob-lem (.) and is based on Banach’s contraction mapping principle.
Theorem . Assume that:
(H) The functionsf,g: [,T]×R→Rare continuous and there exist constantsKi,Li> ,
i= , such that for allt∈[,T]andui,vi∈R,i= , ,
f(t,u,u) –f(t,v,v)≤K|u–v|+K|u–v|
and
g(t,u,u) –g(t,v,v)≤L|u–v|+L|u–v|.
(H) The functionsIk,Ik∗:R→Rare continuous and there exist constantsK,L> such that for allt∈[,T]andu,v∈R,k= , , . . . ,m,
Ik(u) –Ik(v)≤K|u–v|
and
Ik∗(u) –I∗k(v)≤L|u–v|.
In addition,assume that
M+M+M+M< ,
where Mi, i= , , , ,are given by(.)-(.)and(.)-(.).Then the boundary value
problem(.)has a unique solution.
Proof Define supt∈[,T]f(t, , ) =N <∞,supt∈[,T]g(t, , ) =N <∞, sup{|Ik()|:k= , , . . . ,m}=N<∞andsup{|Ik∗()|:k= , , . . . ,m}=N<∞such that
r≥max
M
– (M+M)
, M – (M+M)
,
+K x +K y +N (T–tm)
= ||
|a||λ|+|a||b|
m+
k=
L x +L y +N (tk–tk–) +|a||b|mL y
+|a||b|mN+|b||λ|+|b||b|
m+
k=
K x +K y +N (tk–tk–)
+|b||b|mK x +|b||b|mN
+ m+
k=
K x +K y +N (tk–tk–)
+mK x +mN
= x
||
m+
k=
(tk–tk–)
L|a||b|+K|b||b|+K|| +mK
|b||b|+||
+ y
||
m+
k=
(tk–tk–)
L|a||b|+K|b||b|+K|| +mL|a||b|
+ ||
m+
k=
(tk–tk–)
N|a||b|+N|b||b|+N||
+mN|a||b|+N|b||b|+N|| +|a||λ|+|b||λ|
=M x +M y +M
≤(M+M)r+M≤r.
In the same way, we can obtain
T(x,y)(t)
≤ x
||
m+
k=
(tk–tk–)
K|a||b|+L|b||b|+L|| +mK|a||b|
+ y
||
m+
k=
(tk–tk–)
K|a||b|+L|b||b|+L||
+mL
|b||b|+||
+ ||
m+
k=
(tk–tk–)
N|a||b|+N|b||b|+N||
+mN|a||b|+N|b||b|+N|| +|a||λ|+|b||λ|
=M x +M y +M
≤(M+M)r+M≤r.
Now for (x,y), (x,y)∈X×Yand for anyt∈[,T], we get
and consequently we obtain
SinceM+M+M+M< , therefore,T is a contraction operator. So, by Banach’s fixed
point theorem, the operatorT has a unique fixed point, which is the unique solution of problem (.). This completes the proof.
In the next result, we prove the existence of solutions for problem (.) by applying the Leray-Schauder alternative.
For the sake of convenience, we set
M=
||
TB|a||b|+A|b||b|+A|| +mA
|b||b|+|| , (.)
M=
||
TB|a||b|+A|b||b|+A|| +mB|a||b|
, (.)
M=
||
TB|a||b|+A|b||b|+A||
+mB|a||b|+A|b||b|+A|| +|a||λ|+|b||λ|
, (.)
M=
||
TA|a||b|+B|b||b|+B|| +mA|a||b|
, (.)
M=
||
TA|a||b|+B|b||b|+B|| +mB
|b||b|+|| , (.)
M=
||
TA|a||b|+B|b||b|+B||
+mA|a||b|+B|b||b|+B|| +|a||λ|+|b||λ|
, (.)
and
M=min
– (M+M), – (M+M)
. (.)
Lemma .(Leray-Schauder alternative) ([], p.) Let F:E→E be a completely contin-uous operator(i.e.,a map that is restricted to any bounded set in E is compact).Let
E(F) =x∈E:x=λF(x)for some <λ< .
Then either the setE(F)is unbounded,or F has at least one fixed point.
Theorem . Assume that:
(H) The functionsf,g: [,T]×R→Rare continuous and there exist constantsAi,Bi≥ (i= , )andA,B> such that∀xi∈R(i= , )
f(t,x,x)≤A+A|x|+A|x|
and
(H) The functionsIk,Ik∗:R→Rare continuous and there exist constantsA,B≥and
A,B> such that∀x∈R,k= , , . . . ,m
Ik(x)≤A+A|x|
and
Ik∗(x)≤B+B|x|.
In addition it is assumed that
M+M< and M+M< ,
where M, M, M,M are given by(.)-(.)and(.)-(.).Then there exists at
least one solution for the boundary value problem(.).
To prove the theorem we use the following lemma.
Lemma . Assume that(H)and(H)hold.Then the operatorT :X×Y →X×Y is
completely continuous.
Proof By continuity of functionsf andg, the operatorT is continuous.
Let⊂X×Y be bounded. Then there exist positive constantsP,P,P, andPsuch
that
ft,x(t),y(t) ≤P, g
t,x(t),y(t) ≤P, ∀(x,y)∈,
Ik
x(t) ≤P, Ik∗
y(t) ≤P, k= , , . . . ,m.
Then for any (x,y)∈, we have
T(x,y)
≤| |
|a||λ|+|a||b|
m+
k=
tk
tk–
gs,x(s),y(s) dpk–s+ m
k=
Ik∗y(tk)
+|b||λ|+|b||b|
m+
k=
tk
tk–
fs,x(s),y(s) dqk–s+ m
k=
Ik
x(tk)
+ m+
k=
tk
tk–
fs,x(s),y(s) dqk–s+ m
k=
Ik
x(tk)
≤| |
|a||λ|+|a||b|(PT+mP) +|b||λ|+|b||b|(PT+mP)
+PT+mP
Similarly, we get
T(x,y)≤ ||
|a||λ|+|a||b|(PT+mP) +|b||λ|+|b||b|(PT+mP)
+PT+mP
:=D.
Thus, it follows from the above inequalities that the operatorT is uniformly bounded. Next, we show thatT is equicontinuous. Letν,ν∈(tl,tl+) for somel= , , . . . ,mwith
ν<ν. Then we have
T
x(ν),y(ν) –T
x(ν),y(ν)
=
ν
tl
fs,x(s),y(s) dqls–
ν
tl
fs,x(s),y(s) dqls
≤P|ν–ν|.
Analogously, we can obtain
T
x(ν),y(ν) –T
x(ν),y(ν)
=
ν
tl
gs,x(s),y(s) dpls–
ν
tl
gs,x(s),y(s) dpls
≤P|ν–ν|.
Therefore, the operatorT(x,y) is equicontinuous, and thus the operatorT(x,y) is com-pletely continuous.
Proof of Theorem. By Lemma . the operatorT(x,y) is completely continuous. Now, it will be verified that the set E={(x,y)∈X×Y|(x,y) =λT(x,y), ≤λ≤}is bounded. Let (x,y)∈E, then (x,y) =λT(x,y). For anyt∈[,T], we have
x(t) =λT(x,y)(t), y(t) =λT(x,y)(t).
Then
x(t)≤ x
||
m+
k=
(tk–tk–)
B|a||b|+A|b||b|+A||
+mA
|b||b|+||
+ y
||
m+
k=
(tk–tk–)
B|a||b|+A|b||b|+A|| +mB|a||b|
+ ||
m+
k=
(tk–tk–)
B|a||b|+A|b||b|+A||
+mB|a||b|+A|b||b|+A|| +|a||λ|+|b||λ|
and
y(t)≤ x ||
m+
k=
(tk–tk–)
A|a||b|+B|b||b|+B|| +mA|a||b|
+ y
||
m+
k=
(tk–tk–)
A|a||b|+B|b||b|+B||
+mB
|b||b|+||
+ ||
m+
k=
(tk–tk–)
A|a||b|+B|b||b|+B||
+mA|a||b|+B|b||b|+B|| +|a||λ|+|b||λ|
.
Hence we have
x ≤M x +M y +M
and
y ≤M x +M y +M,
which imply that
x + y ≤(M+M) x + (M+M) y +M+M.
Consequently,
(x,y)≤M+M
M
,
for anyt∈[,T], whereMis defined by (.), which proves thatEis bounded. Thus, by
Lemma ., the operatorT has at least one fixed point. Hence the boundary value problem (.) has at least one solution. The proof is complete.
3.1 Examples
Example . Consider the following coupled system of impulsive quantum difference equations with coupled boundary conditions
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
D k+ k+k+x(t) =
tcos(πt)
(et+) | x(t)| |x(t)|++
t+ (t+)
|y(t)| |y(t)|++
, t∈[, ],t=tk,
D√k+ ek+
y(t) = (t++) sinx(t) +e –(t+)
cosy(t) +
t+
, t∈[, ],t=tk,
x(tk) = (k+)+|x(tk|)x|(tk)|, y(tk) =(k+)+|y(tk|)y|(tk)|, tk=k,k= , , , x() + y() = , y() – x() = –.
Hereqk= (k+ )/(k+k+ ),pk= ( √
k+ )/(ek+ ),k= , , , ,m= ,T= ,a
= ,
a= ,b= ,b= –,λ= ,λ= –,f(t,x,y) = (tcos(πt)|x|)/(((et+ ))(|x|+ )) + ((t+
)|y|)/(((t+ ))(|y|+ )) + /,g(t,x,y) = (sinx)/(t++ )+ (e–(t+)cosy)/ + (t+ )/,
Ik(x) =|x|/((k+ ) +|x|), andIk∗(y) =|y|/((k+ ) +|y|). We have|f(t,x,y) –f(t,x,y)| ≤
((/)|x–x|+ (/)|y–y|),|g(t,x,y) –g(t,x,y)| ≤((/)|x–x|+ (/(e))|y–
y|),|Ik(x) –Ik(y)| ≤(/)|x–y|, and|Ik∗(x) –Ik∗(y)| ≤(/)|x–y|. We can find
=aa–bb= = .
With the given values, it is found thatK= /,K= /,K= /,L= /,L=
/(e),L
= /,M.,M.,M.,M.„ and
M+M+M+M. < .
Thus all the conditions of Theorem . are satisfied. Therefore, by the conclusion of The-orem ., problem (.) has a unique solution on [, ].
Example . Consider the following coupled system of impulsive quantum difference equations with coupled boundary conditions:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
D√k+ k+ek+
x(t) = +(t+) sinx(t) +πtan–y(t), t∈[, ],t=tk,
D
sin(k+π)y(t) = t+
e +
x(t)cosy(t) +
t+y(t), t∈[, ],t=tk, x(tk) = tan–(x(tk)
) + , tk=
k
,k= , , . . . , ,
y(tk) =sin(y(tk)) + , tk=k,k= , , . . . , , –x() + y() = –, y() + x() = .
(.)
Hereqk= (k+ )/( √
k+ek+ ),p
k= (sin(((k+ )π)/))/,k= , , , . . . , ,m= ,T= ,
a = –, a = , b = , b = ,λ = –, λ = , f(t,x,y) = (/) + (sinx)/((t+ )) +
(tan–y)/(π),g(t,x,y) = ((t+ )/e) + (xcosy)/ + (y)/(t+ ),Ik(x) = (tan–(x/))/ + , andIk∗(y) = (sin(y/))/ + . We get
=aa–bb= –= .
Since|f(t,x,y)| ≤A+A|x|+A|y|,|g(t,x,y)| ≤B+B|x|+B|y|, whereA= /,A=
/,A= /(π),B= /e,B= /,B= /, it is found thatM.,M
.,M.,M.. Furthermore,
M+M≈. <
and
M+M≈. < .
4 Uncoupled boundary conditions case
In this section, we consider again the system
where
:=a+b= .
We remark thatT depends only onf andT only ong. We call the above system, for
convenience, a ‘coupled system with uncoupled boundary conditions’. In the sequel, we set the constants
M=
||(TK+mK)
|b|+|| , (.)
M=
||TK
|b|+|| , (.)
M=
||
(TN+mN)
|b|+|| +|λ|
, (.)
M=
||TL
|b|+|| , (.)
M=
||(TL+mL)
|b|+|| , (.)
M=
||
(TN+mN)
|b|+|| +|λ|
. (.)
Now we present the existence and uniqueness result for problem (.). We do not pro-vide the proof of this result as it is similar to the one for Theorem ..
Theorem . Assume that:
(H) The functionsf,g: [,T]×R→Rare continuous and there exist constantsKi,Li> ,
i= , such that for allt∈[,T]andui,vi∈R,i= , ,
f(t,u,u) –f(t,v,v)≤K|u–v|+K|u–v|
and
g(t,u,u) –g(t,v,v)≤L|u–v|+L|u–v|.
(H) The functionsIk,Ik∗:R→Rare continuous and there exist constantsK,L> such that for allt∈[,T]andu,v∈R,k= , , . . . ,m
Ik(u) –Ik(v)≤K|u–v|
and
Ik∗(u) –Ik∗(v)≤L|u–v|.
In addition,assume that
M+M+M+M< ,
where M, M, M, M are given by (.)-(.) and (.)-(.), respectively. Then the
Example . Consider the following coupled system of impulsive quantum difference equations with uncoupled boundary conditions
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
D( )kx(t) =
sin(πt) (et+) |
x(t)| |x(t)|++
πt
(t+)
|y(t)|
|y(t)|++ , t∈[, ],t=tk,
D(+k +k)ky(t) =
(t+)cosx(t) +π(t+)|y(t)|+ , t∈[, ],t=tk, x(tk) = (k+)+|x(tk|x)|(t
k)|, y(tk) =
|y(tk)|
(k+)+|y(tk)|, tk= k
,k= , , , ,
x() – x() = , y() + y() = .
(.)
Hereqk= (/)k,pk= (( +k)/( + k))k,k= , , , , ,m= ,T= ,a= ,a= ,b=
–,b= ,λ= ,λ= ,f(t,x,y) = (sin(πt)|x|)/(((et+ ))(|x|+ )) + (πt|y|)/(((t+ ))(|y|+
)) + ,g(t,x,y) = (cosx)/((t+ )) + (|y|)/(π(t+ )) + ,Ik(x) =|x|/((k+ ) +|x|), and
Ik∗(y) =|y|/((k+ ) +|y|). Since|f(t,x,y) –f(t,x,y)| ≤((/)|x–x|+ (π/)|y–y|),
|g(t,x,y) –g(t,x,y)| ≤((/)|x–x|+ (/(π))|y–y|),|Ik(x) –Ik(y)| ≤(/)|x–y|, and|Ik∗(x) –Ik∗(y)| ≤(/)|x–y|. We can find
=a+b= –= and =a+b= = .
With the given values, it is found thatK= /,K=π/,K= /,L= /,L=
/(π),L= /,M.,M.,M.,M., and
M+M+M+M. < .
Thus all the conditions of Theorem . are satisfied. Therefore, by the conclusion of The-orem ., problem (.) has a unique solution on [, ].
The second result dealing with the existence of solutions for the problem (.) is analo-gous to Theorem . and is given below.
In the sequel, we set constants
M=
||(TA+mA)
|b|+|| , (.)
M=
||TA
|b|+|| , (.)
M=
||
(TA+mA)
|b|+|| +|λ|
, (.)
M=
||TB
|b|+|| , (.)
M=
||(TB+mB)
|b|+|| , (.)
M=
||
(TB+mB)
|b|+|| +|λ|
. (.)
Theorem . Assume that:
(H) The functionsf,g: [,T]×R→Rare continuous and there exist constantsAi,Bi≥ (i= , )andA,B> such that∀xi∈R(i= , )
and
g(t,x,x)≤B+B|x|+B|x|.
(H) The functionsIk,Ik∗:R→Rare continuous and there exist constantsA,B≥and
A,B> such that∀x∈R,k= , , . . . ,m,
Ik(x)≤A+A|x|
and
Ik∗(x)≤B+B|x|.
In addition it is assumed that
M+M< and M+M< ,
where M,M,M,Mare given by(.)-(.)and(.)-(.),respectively.Then the
boundary value problem(.)has at least one solution.
Proof Setting
M=min
– (M+M), – (M+M)
,
the proof is similar to that of Theorem .. So we omit it.
Example . Consider the following coupled system of impulsive quantum difference equations with uncoupled boundary conditions:
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
D
+kx(t) = +
sin(
x(t) ) +
(t+)tan–(
y(t)
), t∈[, ],t=tk,
D +k
+k+k
y(t) = +(tt+)x(t) + sin(πt)
(t+)y(t), t∈[, ],t=tk, x(tk) = πsin(
πx(tk) ) +
, tk=
k
,k= , , . . . , ,
y(tk) =sin(y(tk)) + , tk=k,k= , , . . . , , x() – x() = –, y() – y() = –.
(.)
Here qk = /( +k), pk= ( +k)/( + k+k), k= , , , . . . , ,m= ,T = , a = ,
a= ,b= –,b= –,λ= –,λ= –,f(t,x,y) = + (sin(x/))/ + (tan–(y/))/(t+ ),
g(t,x,y) = + (tx)/((t+ )) + (sin(πt)y)/((t+ )),Ik(x) = (sin(πx/))/(π) + /, and
Ik∗(y) =y/(e+y) +π/. We get
=a+b= –= and =a+b= –= .
Since|f(t,x,y)| ≤A+A|x|+A|y|,|g(t,x,y)| ≤B+B|x|+B|y|, whereA= ,A= /,
A= /,B= ,B= /,B= /, it is found thatM.,M.,
M.,M.. Furthermore,
and
M+M≈. < .
Thus all the conditions of Theorem . holds true and consequently the conclusion of Theorem .; 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.
Author details
1Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand.2Department of Mathematics, University of Ioannina, Ioannina, 451 10, Greece.3Nonlinear 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
We would like to thank the reviewers for their valuable comments and suggestions on the manuscript. This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-GOV-58-10.
Received: 4 March 2015 Accepted: 17 May 2015
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