R E S E A R C H
Open Access
Existence and uniqueness of solutions for
the Schrödinger integrable boundary value
problem
Jianjie Wang
1, Ali Mai
2and Hong Wang
3**Correspondence:
3Faculty of Science and Technology,
Universiti Kebangsaan Malaysia, Bangi, Malaysia
Full list of author information is available at the end of the article
Abstract
This paper is mainly devoted to the study of one kind of nonlinear Schrödinger differential equations. Under the integrable boundary value condition, the existence and uniqueness of the solutions of this equation are discussed by using new Riesz representations of linear maps and the Schrödinger fixed point theorem.
Keywords: Integrable boundary value; Nonlinear Schrödinger differential equation; Schrödinger fixed point theorem
1 Introduction
The nonlinear Schrödinger differential (NSD) equation is one of the most important in-herently discrete models. NSD equations play a crucial role in the modeling of a great variety of phenomena, ranging from solid state and condensed matter physics to biology [1–4]. For example, they have been successfully applied to the modeling of localized pulse propagation optical fibers and wave guides, to the study of energy relaxation in solids, to the behavior of amorphous material, to the modeling of self-trapping of vibrational energy in proteins or studies related to the denaturation of the NSD double strand [5].
In 1961, Gross considered a NSD equation with Dirac distribution defect (see [6]),
iut+ 1
2uxx+qδau+g
|u|2u= 0 in×R+,
where⊂R,u=u(x,t) is the unknown solution maps×R+ intoC,δa is the Dirac distribution at the pointa∈, namely,δa,v=v(a) forv∈H1(), andq∈Rrepresents its intensity parameter. Such a distribution is introduced in order to model physically the defect at the pointx=a(see [7]). The function g represents a generalization of the classical nonlinear Schrödinger equation (see for example [8]). As for other contributions to the analysis of nonlinear Schrödinger equations, we refer to Refs. [9–12] and the references therein.
In this paper, we consider the following NSD equation:
Xs=x+
s
0
b(s,Xs)ds+
s
0
h(s,Xs)dBs+
s
0
σ(s,Xs)dBs, (1)
where 0≤s≤SandBis the quadratic variation of the Brownian motionB.
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RETRACTED
It is worth mentioning that (1) comes from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths (see [13] for details). When the coefficientsb,handσare constants in (1), the Lévy dynamics becomes the Brownian dynamics, and (1) reduces to the classical stochastic differential equation
Ys=ξ+
S
s
f(s,Ys,Zs)ds+
S
s
g(s,Ys,Zs)dBs–
S
s
ZsdBs– (KS–Ks) (2) under standard Lipschitz conditions onf(s,y,z),g(s,y,z) iny,zand theLpG(S) (p> 1) integrability condition onξ. The solution (Y,Z,K) is universally defined in the space of the Schrödinger framework, in which the processes have a strong regularity property. It should be noted thatKis a decreasing Schrödinger martingale.
It is well known that classical stochastic differential equations are encountered when one applies the stochastic maximum principle to optimal stochastic control problems. Such equations are also encountered in the probabilistic interpretation of a general type of systems quasilinear PDEs, as well as in finance (see [13–15] for details).
The rest of this paper is organized as follows. In Sect. 2, we introduce some notions and results. In Sect. 3, the main results and their proofs are presented.
2 Preliminaries
In this section, we introduce some notations and preliminary results in Schrödinger framework which are needed in the following section. More details can be found in [16– 19].
LetS=C0([0,S];R), the space of real valued continuous functions on [0,S] withw0= 0, be endowed with the distance (see [20])
dw1,w2:=
∞
N=1
(max0≤s≤N|w1s–w2s|)∧1
2N (3)
and letBs(w) =wsbe the canonical process. Denote byF:={Fs}0≤s≤Sthe natural filtration generated byBs, letL0(S) be the space of allF-measurable real functions. Let
Lip(S) :=φ(Bs1, . . . ,Bsn) :∀n≥1,s1, . . . ,sn∈[0,S],∀φ∈Cb,Lip
Rn, whereCb,Lip(Rn) denotes the set of bounded Lipschitz functions inRn(see [21]).
In the sequel, we will work under the following assumptions.
(H1) Foru∈R3,ε> 0, (x)∈LG2(S),f(·,u),g(·,u),b(·,u),h(·,u),σ(·,u)∈M2G(0,S);
(H2) Foru1,u2∈R3, there exists a positive constantC1such that
fs,u1–fs,u2∨bs,u1–fs,u2∨As,u1–As,u2≤C1u1–u2
and
x1– x2≤C1x1–x2;
(H3) Foru1,u2∈R3, there exists a positive constantC2such that
As,u1–As,u2,u1–u2≤–C2u1–u2 2
.
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A sublinear functional onLip(S) satisfies: for allX,Y∈Lip(S),
(I) monotonicity:E[X]≥E[Y]ifX≤Y; (II) constant preserving:E[C] =CforC∈R; (III) sub-additivity:E[X+Y]≤E[X] +E[Y]; (IV) positive homogeneity:E[λX] =λE[X]forλ≥0.
The tripe (,Lip(S),E) is called a sublinear expectation space andEis called a sublinear expectation.
Definition 2.1(see [22]) A random variableX∈Lip(S) is the Schrödinger normal dis-tributed with parameters (0, [σ2,σ2]), i.e.,X∼N(0, [σ2,σ2]) if for eachφ∈Cb,Lip(R),
u(s,x) :=Eφ(x+√tX)
is a viscosity solution to the following PDE:
⎧ ⎨ ⎩
∂u
∂s +G
∂2u
∂x2 = 0,
us0=φ(x), onR+×R, where
G(a) :=a
+σ2–a–σ2 2
anda∈R.
Definition 2.2(see [23]) We call a sublinear expectationEˆ :Lip(S)→Ra Schrödinger expectation if the canonical processBis a Schrödinger Brownian motion underEˆ[·], that is, for each 0≤s≤t≤S, the incrementBs–Bs∼N(0, [σ2(s–s),σ2])(s–s) and for all
n> 0, 0≤s1≤ · · · ≤sn≤Sandϕ∈Lip(S)
ˆ
E ϕ(Bs1, . . . ,Bsn–1,Bsn–Bsn–1)
=Eˆ ψ(Bs1, . . . ,Bsn–1)
,
where
ψ(x1, . . . ,xn–1) :=Eˆ ϕ(x1, . . . ,xn–1,√sn–sn–1B1)
.
We can also define the conditional Schrödinger expectationEˆsofξ∈Lip(S) knowing
Lip(t) fort∈[0,S]. Without loss of generality, we can assume thatξ has the representa-tion
ξ=ϕB(s1),B(s2) –B(s1), . . . ,B(sn) –B(sn1)
witht=si, for some 1≤i≤n, and we put
ˆ
Esi ϕ
B(s1),B(s2) –B(s1), . . . ,B(sn) –B(sn–1)
=ϕ˜B(s1),B(s2) –B(s1), . . . ,B(si) –B(si–1)
,
RETRACTED
where
˜
ϕ(x1, . . . ,xi) =Eˆ ϕ
x1, . . . ,xi,B(si+1) –B(si), . . . ,B(sn) –B(sn–1)
.
Forp≥1, we denote byLpG(S) the completion ofLip(S) under the natural norm
Xp,G:=
ˆ
E |X|p 1 p.
ˆ
Eis a continuous mapping onLip(S) endowed with the norm · 1,G. Therefore, it can be extended continuously toL1G(S) under the normX1,G.
Next, we introduce the Itô integral of Schrödinger Brownian motion.
LetM0G(0,S) be the collection of processes in the following form: for a given partition
πS={s0,s1, . . . ,sN}of [0,S], set
ηs(w) = N–1
k=0
ξk(w)I[sk,sk+1)(s),
whereξk∈Lip(tk) andk= 0, 1, . . . ,N– 1 are given.
Forp≥1, we denote byHGp(0,S),MpG(0,S) the completion ofM0
G(0,S) under the norm
ηHp G(0,S)=
ˆ
E
S
0 |
ηs|2ds
p 21p
and
ηMp G(0,S)=
ˆ
E
S
0
|ηs|pds
1
p ,
respectively. It is easy to see that
HG2(0,S) =M2G(0,S).
As in [24], for eachη∈HGp(0,S) withp≥1, we can define Itô integral0SηsdBs. More-over, the followingB–D–Ginequality holds.
Let GαG(0,S) denote the collection of processes (Y,Z,K) such thatY∈Sα
G(0,S),Z∈
Hα
G(0,S),Kis a decreasing Schrödinger martingale withK0= 0 andKS∈LαG(). Lemma 2.1 (see [25]) Assume thatξ ∈LβG(S), f,g∈M
β
G(0,S)and satisfy the Lipschitz
condition for someβ> 1.Then Eq. (2)has a unique solution(Y,Z,K)∈GαG(0,S)for any
1 <α<β.
In [26], the authors also got the explicit solution of the following special type of NSD equation.
Lemma 2.2 Assume that{as}s∈[0,S],{cs}s∈[0,S] are bounded processes in MG1(0,S)andξ ∈
L1
G(S),{ms}s∈[0,S],{ns}s∈[0,S]∈M1G(0,S).Then the NSD equation Ys=Eˆs
ξ+
S
s
(asYs+ms)ds+
S
s
(csYs+ns)dBs
RETRACTED
has an explicit solution, Ys= (Xs)–1Eˆs
XSξ+
S
s
(ms)ds+
S
s
(ns)dBs
,
where
Xs=exp
s
0
asds+
s
0
csdBs
.
Lemma 2.3(see [27]) Suppose that a nonnegative real sequence{ai}∞i=1= 1satisfying 8ai+1≤2ai+ai–1
for any i≥1.Then there exists a positive constant c,such that2ia
i≤c for any i≥0.
3 Main results and their proofs
In this section, we introduce the main results and their proofs.
Letu:= (x,y,z),A(s,u) := (–g(s,u),h(s,u),σ(s,u)). [·,·] denotes the usual inner product in real number space and| · |denotes the Euclidean norm.
Our first main result can be summarized as follows.
Theorem 3.1 Suppose that(H1)–(H3)are satisfied.Then there exists s∈[0,S]such that
(1)has a nontrivial and nonnegative solution.
Proof Let a nonnegative real sequence{u(k)}k∈N⊂Fsuch that{A(s,u(k))}k∈N is bounded Lipschitz functions inRnand
lim
k→∞
1 +u(k)As,u(k)= 0.
So there exists a positive constantC3 such that|A(s,u(k))| ≤C3 (see [28]), which con-cludes that
2C3≥2A
s,u(k)–As,u(k),u(k)
= +∞
n=–∞ γn g
s,u(nk)u(nk)– 2hs,u(nk). (4) It follows from (H1) and (4) that
F(un)≤v–ω 4γ¯ u
2
n (5)
for any|un| ≤η, wheren∈Zandηis a positive real number satisfyingη∈(0, 1). Then (H2) and (5) immediately give
gs,u(nk)un(k)> 2hs,u(nk)≥0, (6)
hs,u(nk)≤ p+qu(nk)μ/2 gs,u(nk)u(nk)– 2hs,u(nk). (7)
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By Lemma 2.3, (6) and (7), we have
1 2u
(k)2
=As,u(k)+τ 2u
(k)2 l2+
n∈Z(|u(k)n |≤η)
nh
s,u(nk)+ n∈Z(|u(k)n |≥η)
nh
s,u(nk)
≤As,u(k)+ τ 2vu
(k)2 +v–τ
4
n∈Z(|u(k)n |≤η)
u(nk)2
+¯
n∈Z(|u(k)n |≥η)
p+qu(nk)μ/2 gs,un(k)u(nk)– 2hs,u(nk)
≤c+ τ 2vu
(k)2+v–τ 4v u
(k)2+ 2c¯p+qvμ/2u(k)μ
,
which gives
v–τ
4v u
(k)2
≤c+ 2c¯p+qvμ/2u(k)μ
.
It is obvious that the nonnegative real sequence{u(k)}
k∈Nis bounded inE, so there exists a positive constantC4such that (see [29])
u(k)≤C4 (8)
for anyk∈N, which givesu(k)u(0)inEask→ ∞.
Letεbe a given number. Then there exists a positive numberζ such that
g(s,u)≤ε|u| (9)
for anyu∈Rfrom (H3), where|u| ≤ζ.
It follows from (H1) that there exists a positive integerC5satisfying
ζ2vn≥C52 (10)
for any|n| ≥C5.
By (8), (9) and (10), we obtain
C52un(k)2=C25vn
u(nk)2≤vnζ2u(k)2≤C52vnζ2 (11) for any|n| ≥C5.
Sinceu(k)u(0)inEask→ ∞, it is obvious thatu(k)
n converges tou(0)n pointwise for all
n∈Z, that is,
lim
k→∞u
(k)
n =u(0)n (12)
for anyn∈Z, which together with (11) gives
u(0)n 2≤ζ2 (13) for any|n| ≥C5.
RETRACTED
It follows from (12), (13) and the continuity ofg(s,u) onuthat there exists a positive integerC6such that
D
n=–D
nf
u(nk)–fu(0)n <ε (14) for anyk≥C6.
Meanwhile, we have
|n|≥D
nf
u(nk)–gs,u(0)n u(nk)–u(0)n
≤
|n|≥D
¯
fu(nk)+gs,u(0)n u(nk)+u(0)n
≤ ¯ε |n|≥D
u(nk)+u(0)n u(nk)+u(0)n
≤2ε¯
+∞
n=–∞
u(nk)2+u(0)n 2
≤2ε¯
v
K12+u(0)2 (15) from (H3), (8), (9) and the Hölder inequality.
Sinceεis arbitrary, we obtain
+∞
n=–∞
ng
s,u(nk)–gs,u(0)n →0 (16) ask→ ∞.
It follows that
As,u(k)–As,u(0),u(k)–u(0)
=u(k)–u(0)2–τu(k)–u(0)2l2– +∞
n=–∞
n
gs,u(nk)–gs,u(0)n u(k)–u(0)
≥v–τ
v u
(k)–u(0)2– +∞
n=–∞
n
gs,u(nk)–gs,u(0)n u(k)–u(0)
from (14), (15) and (16), which gives
v–τ
v u
(k)–u(0)2
≤As,u(k)–As,u(0),u(k)–u(0)
+ +∞
n=–∞ n
gs,u(nk)–gs,u(0)n u(k)–u(0).
SinceA(s,u(k)) –A(s,u(0)),u(k)–u(0) →0 ask→ ∞andv>τ> 0,u(k)→u(0)inE.
So the proof is complete.
The following lemma provides the main mathematical result in the sequel.
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Lemma 3.1 Let E⊂L0(S)andLEbe a mapping from L0(S)onto E.If LE(x) =arg min
y∈c x–y
for any x∈L0(S),thenLEis called the orthogonal projection from L0(S)onto E.
Further-more,we have the following properties:
(I) x–LEx,z–LEx ≤0;
(II) LEx–LEy2≤ LEx–LEy,x–y;
(III) LEx–z2≤ x–z2+LEx–x2
for any x,y∈L0(S)and z∈E. Our main result reads as follows.
Theorem 3.2 Let assumptions (H1)–(H3) hold. Then there exists a unique solution
(X,Y,Z,K)for the NSD equation(1).
Proof Existence. By Lemma 2.1, whenα= 0, for∀β.,.,λ.,ϕ.,ψ.∈M2G(0,S),ξ ∈L2G(), (1) has a solution. Moreover, by Lemma 2.2, we can solve (2) successively for the caseα∈
0,δ0], [δ0, 2δ0], . . . . It turns out that, whenα= 1, for∀β.,.,λ.,ϕ.,ψ.∈M2G(0,S),ξ∈L2G(), the solution of (1) exists, then we deduce that the solution of the NSD equation (1) exists.
Now, we prove the uniqueness.
Let (u,K) = (X,Y,Z,K) and (u,K) = (X,Y,Z,K) be two solutions of the NSD equation (1). We set
(Xˆs,Yˆs,Zˆs,Kˆs) :=Xs–Xs,Ys–Ys,Zs–Zs,Ks–Ks
.
From (H1)–(H2), it is easy to see that
ˆ
E
sup
0≤s≤S| ˆ Xs|2
+Eˆ
sup
0≤s≤S| ˆ Ys|2
<∞. (17)
In view of the property of the projection (see [30]), we infer thatuˆ=LSi(uˆ–tX∗Xuˆ) for anys> 0. Further, we get from condition in (17) that
μn≤ 2
ρ(X∗X)Zn.
It follows thatI–μn
ZnX
∗Xis nonexpansive. Hence,
un+1–uˆ=LSi
un–μnX∗Xvn+Zn(vn–un)
–LSiuˆ–tX∗Xuˆ
=LSi
(1 –Zn)un+Zn
I–μn Zn
X∗X
vn
–LSi
(1 –Zn)uˆ+Zn
I–μn Zn
X∗X
ˆ
u
≤(1 –Zn)un–uˆ+Zn
I–μn
Zn X∗X
vn–
I–μn Zn
X∗X
ˆ
u
≤(1 –Zn)un–uˆ+Znvn–uˆ. (18)
RETRACTED
Sinceα→0 asn→ ∞andKn∈(0,ρ(X2∗X)), it follows from (18) that
α≤1 –Knρ(X
∗X)
2
asn→ ∞, that is, Kn 1 –Yn∈
0,ρ(X
∗X)
2
.
We deduce from (18) that
vn–uˆ=LSi
(1 –Yn)un–KnX∗Xun
–LSiuˆ–tX∗Xuˆ
≤(1 –Yn)
un– Kn 1 –Yn
X∗Xun
+
Ynuˆ+ (1 –Yn)(uˆ– Kn 1 –Yn
X∗Xuˆ
≤–Ynuˆ+ (1 –Yn)
un– Kn 1 –Yn
X∗Xun–uˆ+ Kn 1 –Yn
X∗Xuˆ, which is equivalent to
vn–uˆ ≤Yn–uˆ+ (1 –Yn)un–uˆ. (19)
We obtain from (19)
un–uˆ ≤(1 –Zn)un–uˆ+Zn
Yn–uˆ+ (1 –Yn)un–uˆ
≤(1 –ZnYn)un–uˆ+ZnYn–uˆ
≤maxun–uˆ,–uˆ
.
So
un–uˆ ≤max
un–uˆ,–uˆ
.
Consequently,unis bounded, and so isvn. LetT= 2LSi–I. From Lemma 2.1, one can know that the projection operatorLSiis monotone and nonexpansive, and 2LSi–Iis non-expansive.
So
un+1=
I+T
2
(1 –Zn)un+Zn
1 –μn Zn
X∗X
vn
=I–Zn 2 un+
Zn 2
I–μn Zn
X∗X
vn+
T
2
(1 –Zn)un+Zn
I–μn Zn
X∗X
vn
,
which yields
un+1= 1 –Zn
2 un+ 1 +Zn
2 bn,
RETRACTED
where
bn=
Zn(I–μn
ZnX
∗X)v
n+T[(1 –Zn)un+Zn(I–ZμnnX
∗X)vn]
1 +Zn
.
On the other hand, we have (see [31])
bn+1–bn ≤
λn+1 1 +λn+1
I–μn+1
λn+1 X∗X
vn+1–
I–μn Zn
X∗X
vn
+ λn+1 1 +λn+1
– λn 1 +λn
I–μn Zn
X∗X
vn
+ T
1 +λn+1
(1 –λn+1)un+1+λn+1
I–μn+1
λn+1 X∗X
vn+1
– T
1 +λn+1
(1 –λn)un+λn
I–μn
λn X∗X
vn
+ 1 1 +λn+1
– 1
1 +λn
T
(1 –λn)un+λn
I–μn
λn X∗X
vn
.
For convenience, letcn= (I–μλnnX
∗X)v
n. Using Lemma 2.2, it follows that
I–μn
λn X∗X
is nonexpansive and averaged. Hence,
bn+1–bn ≤
λn+1 1 +λn+1
cn+1–cn+
λn+1
1 +λn+1 – λn
1 +λn
cn
+ T
1 +λn+1
(1 –λn+1)un+1+λn+1cn+1– (1 –λn)un+λncn
+ 1 1 +λn+1
– 1
1 +λn
T (1 –λn)un+λncn
≤ λn+1 1 +λn+1
cn+1–cn+
λn+1
1 +λn+1 – λn
1 +λn
cn +1 –λn+1
1 +λn+1
un+1–un+
λn+1 1 +λn+1
cn+1–cn+
λn–λn+1 1 +λn+1
un +λn+1–λn
1 +λn+1
cn+
1 +1λn+1 – 1 1 +λn
T (1 –λn)un+λncn, which yields
cn+1–cn=
I–μn+1
λn+1 X∗X
vn+1–
I–μn
λn X∗X
vn
≤ vn+1–vn
=LSi (1 –αn+1)un+1–KnX∗Xun+1
–LSi (1 –αn)un–KnX∗Xun
≤I–n+1X∗X
un+1–
I–n+1X∗X
un+ (n–n+1)X∗Xun
RETRACTED
+αn+1–un+1+αnun
≤ un+1–un+|n–n+1|X∗Xun+αn+1–un+1+αnun. So we infer that
bn+1–bn ≤
λn+1
1 +λn+1 – λn
1 +λn
cn+
λn–λn+1 1 +λn+1
un+
λn+1–λn 1 +λn+1
cn +un+1–un+
1 +1λn+1 – 1 1 +λn
T (1 –λn)un+λncn
+|n–n+1|un+αn+1–un+1+αnun. (20) By virtue oflimn→∞(λn+1–Zn) = 0 (see [28]), it follows that
lim
n→∞
λn+1 1 +λn+1
– λn 1 +λn
= 0.
Moreover,{un}and{vn}are bounded, and so is{cn}. Therefore, (20) reduces to
lim
n→∞sup
bn+1–bn–un+1–un
≤0. (21)
Applying (21) and Lemma 2.3, we get
lim
n→∞bn–un= 0. (22)
Combining (21) with (22), we obtain
lim
n→∞xn+1–xn= 0. (23)
Applying theG-Itô formula toXˆsYˆs, then we obtain
NS+XˆS (XS) –
XS–
S
0
A(s,us) –As,us)us–us
dBs =
S
0
ˆ
Xs (–f)(s,us) – (–f)(s,us)
+Yˆs b(s,us) –b
s,usds+MS (24) from (23), where
Ms=
t
0
ˆ
Ys
σ(s,us) –σ
s,us+XˆsZˆs
dBs+
t
0
(Xˆs)+dKs+
t
0
(Xˆs)–dKs and
Ns=
t
0
(Xˆs)+dKs+
t
0
(Xˆs)–dKs.
RETRACTED
By Lemma 2.3 and (24), we know that bothMsandNsare Schrödinger martingale. More-over, we know that (see [32])
NS– (–C)
S
0
us–us 2
dBs
≤NS+C| ˆXS|2+C
S
0
us–us 2
dBs
≤–
S
0
| ˆXs|2+| ˆYs|2ds+MS (25) from (H3).
Taking the Schrödinger expectation on both sides of (25), together with Lemma 2.2 and the property of the Schrödinger expectation, we know that
0≤–σ2Eˆ
–C
S
0
|us–us|2ds
/≤ ˆE
–
S
0
| ˆXs|2+| ˆYs|2
ds
≤0, (26)
which impliesu=uin the space ofM2
G(0,S). It follows from Lemma 2.2 that the NSD equation has a unique solution, thenK=K. Thus (1) has a unique solution.
4 Conclusions
This paper was mainly devoted to the study of one kind of nonlinear Schrödinger differen-tial equations. Under the integrable boundary value condition, the existence and unique-ness of the solutions of this equation were discussed by using new Riesz representations of linear maps and the Schrödinger fixed point theorem.
Acknowledgements
The authors are thankful to the honorable reviewers for their valuable suggestions and comments, which improved the paper. This paper was written during a short stay of the corresponding author at the Faculty of Science and Technology of Universiti Kebangsaan Malaysia as a visiting professor. She would also like to thank the school of Mathematics and their members for their warm hospitality.
Funding
This work was supported by the National Natural Science Foundation of China (No. 11526183) and the Shanxi Province Education Science “13th Five-Year” program (No. 16043).
Availability of data and materials Not applicable.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, China.2Department of Applied
Mathematics, Yuncheng University, Yuncheng, China. 3Faculty of Science and Technology, Universiti Kebangsaan
Malaysia, Bangi, Malaysia.
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Received: 21 February 2018 Accepted: 1 May 2018
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