R E S E A R C H
Open Access
Convergence analysis of an iterative
algorithm for monotone operators
Sun Young Cho
1, Wenling Li
2and Shin Min Kang
3**Correspondence:
3Department of Mathematics and
RINS, Gyeongsang National University, Jinju, 660-701, Korea Full list of author information is available at the end of the article
Abstract
In this paper, an iterative algorithm is proposed to study some nonlinear operators which are inverse-strongly monotone, maximal monotone, and strictly
pseudocontractive. Strong convergence of the proposed iterative algorithm is obtained in the framework of Hilbert spaces.
MSC: 47H05; 47H09
Keywords: inverse-strongly monotone mapping; maximal monotone operator;
resolvent; strictly pseudocontractive mapping; fixed point
1 Introduction
Splitting methods have recently received much attention due to the fact that many non-linear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation, and this operator is decomposed as the sum of two nonlinear operators. Study of fixed (zero) point approximation algorithms for computing fixed (zero) points constitutes now a topic of in-tensive research efforts. Many well-known problems can be studied by using algorithms which are iterative in their nature. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set in which the required solution lies. The problem of finding a point in the intersection of these convex sets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration and radiation therapy treatment planning is to find a point in the intersection of common fixed (zero) point sets of a family of nonlinear mappings; see, for example, [–].
In this paper, we will investigate the problem of finding a common solution to inclusion problems and fixed point problems based on an iterative algorithm. Strong convergence of the proposed iterative algorithm has been obtained in the framework of Hilbert spaces. The organization of this paper is as follows. In Section , we provide some necessary pre-liminaries. In Section , an iterative algorithm is proposed and analyzed. Some subresults of the main results are also discussed in this section.
2 Preliminaries
From now on, we always assume thatHis a real Hilbert space with the inner product·,· and the norm · , respectively. LetCbe a nonempty closed convex subset ofH.
LetS:C→Cbe a mapping.F(S) stands for the fixed point set ofS; that is,F(S) :={x∈ C:x=Sx}.
Recall thatSis said to be nonexpansive iff
Sx–Sy ≤ x–y, ∀x,y∈C.
Sis said to be asymptotically nonexpansive iff there exists a sequence{kn} ⊂[,∞) with
limn→∞kn= such that
Snx–Sny≤k
nx–y, ∀x,y∈C.
Recall thatSis said to be strictly pseudocontractive iff there exits a positive constantκ such that
Sx–Sy≤ x–y+κ(x–Sx) – (y–Sy), ∀x,y∈C.
Sis said to be asymptotically strictly pseudocontractive iff there exits a positive constant κand a sequence{kn} ⊂[,∞) withlimn→∞kn= such that
Snx–Sny≤knx–y+κx–Snx
–y–Sny, ∀x,y∈C.
LetA:C→Hbe a mapping. Recall thatAis said to be monotone iff
Ax–Ay,x–y ≥, ∀x,y∈C.
Ais said to be inverse-strongly monotone iff there exists a constantα> such that
Ax–Ay,x–y ≥αAx–Ay, ∀x,y∈C.
For such a case,Ais also said to beα-inverse-strongly monotone. It is not hard to see that inverse-strongly monotone mappings are Lipschitz continuous.
A multivalued operatorT :H→H with the domainD(T) ={x∈H:Tx=∅}and the
rangeR(T) ={Tx:x∈D(T)}is said to be monotone if forx∈D(T),x∈D(T),y∈Tx andy∈Tx, we havex–x,y–y ≥. A monotone operatorTis said to be maximal if its graph G(T) ={(x,y) :y∈Tx}is not properly contained in the graph of any other monotone operator. LetIdenote the identity operator onHandT:H→Hbe a maximal
monotone operator. Then we can define, for eachλ> , a nonexpansive single-valued mappingJλ:H→HbyJλ= (I+λT)–. It is called theresolventofT. We know thatT– =
F(Jλ) for allλ> andJλis firmly nonexpansive; see [–] and the references therein.
Recently, many authors have investigated the solution problems of nonlinear operator equations or inequalities based on iterative methods; see, for instance, [–] and the references therein. In [], Kamimura and Takahashi investigated the problem of finding zero points of a maximal monotone operator via the following iterative algorithm:
where{αn}is a sequence in (, ),{λn}is a positive sequence,T :H→H is a maximal
monotone andJλn= (I+λnT)–. They showed that the sequence{xn}generated in (.)
converges weakly to somez∈T–() provided that the control sequence satisfies some restrictions.
Recall that the classical variational inequality is to find anx∈Csuch that
Ax,y–x ≥, ∀y∈C. (.)
In this paper, we useVI(C,A) to denote the solution set of (.). It is known thatx∈Cis a solution to (.) iffxis a fixed point of the mappingPC(I–λA), whereλ> is a
con-stant,Istands for the identity mapping, andPCstands for the metric projection fromH
ontoC. IfAisα-inverse-strongly monotone andλ∈(, α], then the mappingPC(I–rA)
is nonexpansive; see [] for more details. It follows thatVI(C,A) is closed and convex. In [], Takahashi an Toyoda investigated the problem of finding a common solution of variational inequality problem (.) and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:
x∈C, xn+=αnxn+ ( –αn)SPC(xn–λnAxn), ∀n≥, (.)
where{αn}is a sequence in (, ),{λn}is a positive sequence,S:C→Cis a nonexpansive
mapping andA:C→His an inverse-strongly monotone mapping. They proved that the sequence{xn}generated in (.) converges weakly to somez∈VI(C,A)∩F(S) provided
that the control sequence satisfies some restrictions.
In [], Tada and Takahashi investigated the problem of finding a common solution of an equilibrium problem and a fixed point problem involving nonexpansive mappings by considering the following iterative algorithm:
⎧ ⎨ ⎩
un∈C such that F(un,u) +rnu–un,un–xn ≥, ∀u∈C,
xn+=αnxn+ ( –αn)Sun
(.)
for eachn≥, where{αn}is a sequence in (, ),{rn}is a positive sequence,S:C→Cis a
nonexpansive mapping andF:C×C→Ris a bifunction. They showed that the sequence
{xn}generated in (.) converges weakly to somez∈EP(F)∩F(S), whereEP(F) stands for
the solution set of the equilibrium problem, provided that the control sequence satisfies some restrictions.
In [], Manaka and Takahashi introduced the following iteration:
x∈C, xn+=αnxn+ ( –αn)SJλn(I–λnA)xn, n≥, (.)
where{αn}is a sequence in (, ),{λn}is a positive sequence,S:C→Cis a nonexpansive
mapping,A:C→His an inversely-strongly monotone mapping,B:D(B)⊂C→His a
maximal monotone operator,Jλn= (I+λnB)–is the resolvent ofB. They showed that the
sequence{xn}generated in (.) converges weakly to somez∈(A+B)–()∩F(S) provided
that the control sequence satisfies some restrictions.
point problems involving asymptotically strictly pseudocontractive mappings based on a one-step iterative method. Weak convergence theorems are established in the framework of Hilbert spaces.
In order to obtain our main results in this paper, we need the following lemmas. Recall that a space is said to satisfy Opial’s property [] if, for any sequence{xn} ⊂H
withxnx, wheredenotes the weak convergence, the inequality
lim inf
n→∞ xn–x<lim infn→∞ xn–y
holds for everyy∈Hwithy=x. Indeed, the above inequality is equivalent to the following:
lim sup
n→∞ xn–x<lim supn→∞ xn–y.
Lemma .[] Let C be a nonempty,closed,and convex subset of H,A:C→H be a mapping,and B:H⇒H be a maximal monotone operator. Then F(Jr(I–λA)) = (A+
B)–().
Lemma . Let H be a real Hilbert space.For any a∈(, )and x,y∈H,the following holds:
ax+ ( –a)y=ax+ ( –a)y–a( –a)x–y.
Lemma .[] Let{an},{bn},and{cn}be three nonnegative sequences satisfying the
fol-lowing condition:
an+≤( +bn)an+cn, ∀n≥n,
where n is some nonnegative integer, ∞
n=bn<∞ and
∞
n=cn<∞. Then the limit
limn→∞anexists.
Lemma .[] Let C be a nonempty closed convex subset of H and S be an asymptotically κ-strictly pseudocontractive mapping.Then we have
(a) Sis uniformly Lipschitz continuous;
(b) I–Sis demiclosed at zero,that is,if{xn}is a sequence inCwithxnxand
xn–Sxn→,thenx∈F(S).
The following lemma can be obtained from [] immediately.
Lemma . Let H be a real Hilbert space.The following holds:
N
i= aixi
=
N
i=
aixi– N
i=j
aiajxi–xj,
where N≥denotes some positive integer,a,a, . . . ,aN are real numbers with
N i=ai=
3 Main results
Theorem . Let C be a nonempty closed convex subset of H.Let N≥be some positive integer and S:C→C be an asymptotically strictly pseudocontractive mapping with the constantκand the sequence{kn}.Let Am:C→H be an inverse-strongly monotone
map-ping with the constantαmand Bmbe a maximal monotone operator on H such that the
do-main of Bmis included in C for each m∈ {, , . . . ,N}.AssumeF=
N
m=(Am+Bm)–()∩
F(S)=∅. Let {αn,},{αn,}, . . . ,{αn,N} and {βn} are real number sequences in (, ). Let {rn,}, . . . ,and{rn,N}be positive real number sequences.Let{xn}be a sequence in C
gen-erated in the following iterative process:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈C,
yn=βnxn+ ( –βn)Snxn,
xn+=αn,yn+
N
m=αn,mJrn,m(xn–rn,mAmxn), n≥,
(.)
where Jrn,m= (I+rn,mBm)
–is the resolvent of B
m.Assume that the sequences{αn,},{αn,}, . . . ,
{αn,N},{βn},{rn,}, . . . ,{rn,N},and{kn}satisfy the following restrictions: (a) Nm=αn,m= and <a≤αn,m< ,∀m∈ {, . . . ,N};
(b) ≤κ≤βn≤b< ;
(c) <c≤rn,m≤d< αm,∀m∈ {, . . . ,N}; (d) ∞n=(kn– ) <∞,
where a,b,c,and d are positive real numbers.Then the sequence{xn}generated in(.)
converges weakly to some point inF.
Proof First, we showI–rn,mAmis nonexpansive. In view of the restriction (c), we find that
(I–rn,mAm)x– (I–rn,mAm)y
=x–y– rn,mx–y,Amx–Amy+rn,mAmx–Amy
≤ x–y–rn,m(αm–rn,m)Amx–Amy ≤ x–y.
This proves thatI–rn,mAmis nonexpansive. Letp∈F. In view of Lemma ., we find that
p=Sp=Jrn,m(p–rn,mAmp).
Puttingun,m=Jrn,m(xn–rn,mAmxn), we find that
un,m–p ≤(xn–rn,mAmxn) – (p–rn,mAmp)
≤ xn–p. (.)
In view of Lemma ., we find from the restriction (b) that
yn–p =βnxn+ ( –βn)Snxn–p
=βnxn–p+ ( –βn)Snxn–p
–βn( –βn)xn–Snxn
≤βnxn–p+ ( –βn)knxn–p+ (κ–βn)xn–Snxn
≤knxn–p. (.)
From (.) and (.), we have
xn+–p=
αn,(yn–p) + N
m=
αn,m(un,m–p)
≤αn,yn–p+ N
m=
αn,mun,m–p
≤αn,knxn–p+ N
m=
αn,mxn–p
≤knxn–p. (.)
We draw the conclusion thatlimn→∞xn–pexists with the aid of Lemma .. This
im-plies that the sequence{xn}is bounded. In view of Lemma ., we find that
xn+–p =
αn,(yn–p) + N
m=
αn,m(un,m–p)
≤αn,yn–p+ N
m=
αn,mun,m–p
–αn,αn,ryn–un,r
≤αn,knxn–p+ N
m=
αn,mxn–p
–αn,αn,ryn–un,r
≤knxn–p–αn,αn,ryn–un,r, ∀r∈ {, , . . . ,N}, (.)
which yields
αn,αn,ryn–un,r≤knxn–p–xn+–p, ∀r∈ {, , . . . ,N}.
In view of the restriction (a), we find that
lim
n→∞yn–un,m= , ∀r∈ {, , . . . ,N}. (.)
On the other hand, we have
un,m–p≤(xn–rn,mAmxn) – (p–rn,mAmp)
=xn–p– rn,mxn–p,Amxn–Amp+rn,mAmxn–Amp
It follows that
xn+–p≤αn,yn–p+ N
m=
αn,mun,m–p
≤αn,knxn–p+ N
m=
αn,mun,m–p
≤knxn–p– N
m=
αn,mrn,m(αm–rn,m)Amxn–Amp.
This in turn implies that
N
m=
αn,mrn,m(αm–rn,m)Amxn–Amp≤knxn–p–xn+–p.
It follows from the restrictions (b) and (d) that
lim
n→∞Amxn–Amp= . (.)
Notice that
un,m–p≤
(xn–rn,mAmxn) – (p–rn,mAmp),un,m–p
=
(xn–rnAmxn) – (p–rnAmp)
+un,m–p
–(xn–rnAmxn) – (p–rnAmp) – (un,m–p)
≤
xn–p+un,m–p–xn–un,m–rn(Amxn–Amp)
≤
xn–p+un,m–p–xn–un,m–rnAmxn–Amp
+ rnxn–un,mAmxn–Amp
≤
xn–p+un,m–p–xn–un,m
+ rnxn–un,mAmxn–Amp
.
It follows that
un,m–p≤ xn–p–xn–un,m+ rn,mxn–un,mAmxn–Amp. (.)
This implies that
xn+–p =
αn,(yn–p) + N
m=
αn,m(un,m–p)
≤αn,yn–p+ N
m=
≤knxn–p– N
m=
αn,mxn–un,m
+
N
m=
αn,mrn,mxn–un,mAmxn–Amp,
which finds that
N
m=
αn,mxn–un,m≤knxn–p–xn+–p
+
N
m=
αn,mrn,mxn–un,mAmxn–Amp.
In view of the restriction (a), we find from (.) that
lim
n→∞xn–un,m= . (.)
Notice that
xn–yn ≤ xn–un,m+un,m–yn.
From (.) and (.), we obtain that
lim
n→∞xn–yn= . (.)
On the other hand, we have
Snxn–xn≤Snxn–
βnxn+ ( –βn)Snxn+βnxn+ ( –βn)Snxn
–xn
=βnSnxn–xn+yn–xn,
which yields
( –βn)Snxn–xn≤ yn–xn.
This implies from the restriction (c) and (.) that
lim
n→∞S nx
n–xn= . (.)
Notice that
xn+–xn ≤αn,yn–xn+ N
m=
αn,mun,m–xn.
This implies from (.) and (.) that
lim
On the other hand, we have
xn–Sxn ≤ xn–xn++xn+–Sn+xn+
+Sn+xn+–Sn+xn+Sn+xn–Sxn.
SinceSis uniformly continuous, we obtain from (.) and (.) that
lim
n→∞Sxn–xn= . (.)
Since{xn}is bounded, there exists a subsequence{xni}of{xn}such thatxni ω∈C. We
find thatω∈F(S) with the aid of Lemma ..
Next, we showω∈(Am+Bm)– for everym∈ {, , . . . ,N}. In view of (.), we can
choose a subsequence{uni,m}of{un,m}such thatuni,m ω. Notice that
un,m=Jrn,m(xn–rn,mAmxn).
This implies that
xn–rn,mAmxn∈(I+rn,mBm)un,m.
That is,
xn–un,m
rn,m
–Amxn∈Bmun,m.
SinceBmis monotone, we get for any (um,vm)∈G(Bm) that
un,m–um,
xn–un,m
rn,m
–Amxn–vm
≥. (.)
Replacingnbyniand lettingi→ ∞, we obtain from (.) that
ω–um, –Amω–vm ≤.
This means –Amωm∈Bmω, that is, ∈(Am+Bm)(ω). Hence we getω∈(Am+Bm)–() for
everym∈ {, , . . . ,N}. This completes the proof thatω∈F.
Suppose there is another subsequence{xnj}of{xn}such thatxnj ω. Then we can show
thatω∈F in the same way. Assumeω=ω. Sincelimn→∞xn–pexits for anyp∈F.
Putlimn→∞xn–ω=d. Since the space satisfies Opial’s condition, we see that
d=lim inf
i→∞ xni–ω
<lim inf
i→∞ xni–ω
= lim
n→∞xn–ω
=lim inf
j→∞ xnj–ω
<lim inf
This is a contradiction. This shows thatω=ω. This proves that the sequence{xn}
con-verges weakly toω∈F. This completes the proof.
IfN= , then we have the following.
Corollary . Let C be a nonempty closed convex subset of H.Let S:C→C be an asymp-totically strictly pseudocontractive mapping with the constant κ and the sequence{kn}.
Let A:C→H be an inverse-strongly monotone mapping with the constantα,and B be a maximal monotone operator on H such that the domain of B is included in C.Assume
F= (A+B)–()∩F(S)=∅.Let{α
n,},{αn,},and{βn}be real number sequences in(, ).
Let{rn}be a positive real number sequence.Let{xn}be a sequence in C generated in the
following iterative process:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈C,
yn=βnxn+ ( –βn)Snxn,
xn+=αn,yn+αn,Jrn(xn–rnAxn), n≥,
where Jrn= (I+rnB)–is the resolvent of B.Assume that the sequences{αn,},{αn,},{βn}, {rn},and{kn}satisfy the following restrictions:
(a) m=αn,m= and <a≤αn,m< ,∀m∈ {, }; (b) ≤κ≤βn≤b< ;
(c) <c≤rn≤d< α; (d) ∞n=(kn– ) <∞,
where a,b,c,and d are positive real numbers.Then the sequence{xn}converges weakly to
some point inF.
If Sis asymptotically nonexpansive, then we find from Theorem . the following by lettingβn= .
Corollary . Let C be a nonempty closed convex subset of H.Let N≥be some positive integer and S:C→C be an asymptotically nonexpansive mapping with the sequence{kn}.
Let Am:C→H be an inverse-strongly monotone mapping with the constantαmand let Bm
be a maximal monotone operator on H such that the domain of Bmis included in C for each
m∈ {, , . . . ,N}.AssumeF=mN=(Am+Bm)–()∩F(S)=∅.Let{αn,},{αn,}, . . . ,{αn,N},
and{βn}be real number sequences in(, ).Let{rn,}, . . . ,and{rn,N}be positive real number
sequences.Let{xn}be a sequence in C generated in the following iterative process:
x∈C, xn+=αn,Snxn+ N
m=
αn,mJrn,m(xn–rn,mAmxn), n≥,
where Jrn,m= (I+rn,mBm)–is the resolvent of Bm.Assume that the sequences{αn,},{αn,}, . . . , {αn,N},{βn},{rn,}, . . . ,{rn,N},and{kn}satisfy the following restrictions:
(a) Nm=αn,m= and <a≤αn,m< ,∀m∈ {, . . . ,N}; (b) <b≤rn,m≤c< αm,∀m∈ {, . . . ,N};
(c) ∞n=(kn– ) <∞,
where a,b and c are positive real numbers.Then the sequence{xn}converges weakly to
IfSis the identity mapping, then we draw from Theorem . the following.
Corollary . Let C be a nonempty closed convex subset of H.Let N≥be some positive integer.Let Am:C→H be an inverse-strongly monotone mapping with the constantαmand
let Bmbe a maximal monotone operator on H such that the domain of Bmis included in C
for each m∈ {, , . . . ,N}.AssumeF=Nm=(Am+Bm)–()=∅.Let{αn,},{αn,}, . . . ,and
{αn,N}be real number sequences in(, ).Let{rn,}, . . . ,and{rn,N}be positive real number
sequences.Let{xn}be a sequence in C generated in the following iterative process:
x∈C, xn+=αn,xn+ N
m=
αn,mJrn,m(xn–rn,mAmxn), n≥,
where Jrn,m= (I+rn,mBm)–is the resolvent of Bm.Assume that the sequences{αn,},{αn,}, . . . , {αn,N},{rn,}, . . . ,and{rn,N}satisfy the following restrictions:
(a) Nm=αn,m= and <a≤αn,m< ,∀m∈ {, . . . ,N}; (b) <b≤rn,m≤c< αm,∀m∈ {, . . . ,N},
where a,b,and c are positive real numbers.Then the sequence {xn} converges weakly to
some point inF.
Letf :H→(–∞,∞] be a proper lower semicontinuous convex function. Define the subdifferential
∂f(x) =z∈H:f(x) +y–x,z ≤f(y),∀y∈H
for allx∈H. Then∂f is a maximal monotone operator ofHinto itself; see [] for more details. LetCbe a nonempty closed convex subset ofHandiCbe the indicator function
ofC, that is,
iCx=
⎧ ⎨ ⎩
, x∈C,
∞, x∈/C.
Furthermore, we define the normal coneNC(v) ofCatvas follows:
NCv=
z∈H:z,y–v ≤,∀y∈H
for anyv∈C. TheniC:H→(–∞,∞] is a proper lower semicontinuous convex function
onHand∂iC is a maximal monotone operator. LetJrx= (I+r∂iC)–xfor anyr> and
x∈H. From∂iCx=NCxandx∈C, we get
v=Jrx ⇔ x∈v+rNCv
⇔ x–v,y–v ≤, ∀y∈C,
⇔ v=PCx,
where PC is the metric projection from H into C. Similarly, we can get thatx∈(A+
Corollary . Let C be a nonempty closed convex subset of H.Let N≥be some posi-tive integer and S:C→C be an asymptotically strictly pseudocontractive mapping with the constantκ and the sequence{kn}.Let Am:C→H be an inverse-strongly monotone
mapping with the constantαm for each m∈ {, , . . . ,N}.AssumeF=
N
m=VI(C,Am)∩
F(S)=∅. Let {αn,},{αn,}, . . . ,{αn,N}, and {βn} be real number sequences in (, ). Let {rn,}, . . . ,and{rn,N}be positive real number sequences.Let{xn}be a sequence in C
gen-erated in the following iterative process:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x∈C,
yn=βnxn+ ( –βn)Snxn,
xn+=αn,yn+
N
m=αn,mPC(xn–rn,mAmxn), n≥.
Assume that the sequences{αn,},{αn,}, . . . ,{αn,N},{βn},{rn,}, . . . ,{rn,N},and{kn} satisfy
the following restrictions:
(a) Nm=αn,m= and <a≤αn,m< ,∀m∈ {, . . . ,N}; (b) ≤κ≤βn≤b< ;
(c) <c≤rn,m≤d< αm,∀m∈ {, . . . ,N}; (d) ∞n=(kn– ) <∞,
where a,b,c,and d are positive real numbers.Then the sequence{xn}converges weakly to
some point inF.
Proof PuttingBm=∂iCfor everym∈ {, , . . . ,N}, we seeJrn,m=PC. We can immediately
draw from Theorem . the desired conclusion.
IfSis the identity mapping, then we find from Corollary . the following.
Corollary . Let C be a nonempty closed convex subset of H.Let N≥be some positive integer.Let Am:C→H be an inverse-strongly monotone mapping with the constantαmfor
each m∈ {, , . . . ,N}.AssumeF=Nm=VI(C,Am)=∅.Let{αn,},{αn,}, . . . ,and{αn,N}be
real number sequences in(, ).Let{rn,}, . . . ,and{rn,N}be positive real number sequences.
Let{xn}be a sequence in C generated in the following iterative process:
x∈C, xn+=αn,xn+ N
m=
αn,mPC(xn–rn,mAmxn), n≥.
Assume that the sequences{αn,},{αn,}, . . . ,{αn,N},{βn},{rn,}, . . . ,{rn,N},and{kn} satisfy
the following restrictions:
(a) Nm=αn,m= and <a≤αn,m< ,∀m∈ {, . . . ,N}; (b) <b≤rn,m≤c< αm,∀m∈ {, . . . ,N};
(c) ∞n=(kn– ) <∞,
where a,b,and c are positive real numbers.Then the sequence {xn} converges weakly to
some point inF.
Competing interests
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Gyeongsang National University, Jinju, 660-701, Korea.2School of Mathematics and
Information Science, Henan Polytechnic University, Jiaozuo, 454000, China.3Department of Mathematics and RINS,
Gyeongsang National University, Jinju, 660-701, Korea.
Acknowledgements
The authors are grateful to the editor and the reviewers’ suggestions which improved the contents of the article.
Received: 17 October 2012 Accepted: 9 February 2013 Published: 22 April 2013 References
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