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Wu et al. Fixed Point Theory and Applications 2014, 2014:118
http://www.fixedpointtheoryandapplications.com/content/2014/1/118
RESEARCH
Open Access
Some results on zero points of m-accretive
operators in reﬂexive Banach spaces
Chang Qun Wu1 , Songtao Lv2* and Yunpeng Zhang3
*
Correspondence: [email protected]
School of Mathematics and
Information Science, Shangqiu
Normal University, Shangqiu,
Henan, China
Full list of author information is
available at the end of the article
2
Abstract
A modiﬁed proximal point algorithm is proposed for treating common zero points of
a ﬁnite family of m-accretive operators. A strong convergence theorem is established
in a reﬂexive, strictly convex Banach space with the uniformly Gâteaux diﬀerentiable
norm.
Keywords: accretive operator; nonexpansive mapping; resolvent; ﬁxed point; zero
point
1 Introduction and preliminaries
Let E be a Banach space and let E∗ be the dual of E. Let ·, · denote the pairing between
∗
E and E∗ . The normalized duality mapping J : E → E is deﬁned by
J(x) = f ∈ E∗ : x, f = x = f  ,
∀x ∈ E.
A Banach space E is said to strictly convex if and only if x = y = ( – λ)x + λy for
x, y ∈ E and  < λ <  implies that x = y. Let UE = {x ∈ E : x = }. The norm of E is said
exists for each x, y ∈ UE . In this
to be Gâteaux diﬀerentiable if the limit limt→ x+ty–x
t
case, E is said to be smooth. The norm of E is said to be uniformly Gâteaux diﬀerentiable
if for each y ∈ UE , the limit is attained uniformly for all x ∈ UE . The norm of E is said to be
Fréchet diﬀerentiable if for each x ∈ UE , the limit is attained uniformly for all y ∈ UE . The
norm of E is said to be uniformly Fréchet diﬀerentiable if the limit is attained uniformly
for all x, y ∈ UE . It is well known that (uniform) Fréchet diﬀerentiability of the norm of E
implies (uniform) Gâteaux diﬀerentiability of the norm of E.
Let ρE : [, ∞) → [, ∞) be the modulus of smoothness of E by
x + y – x – y
–  : x ∈ UE , y ≤ t .
ρE (t) = sup

A Banach space E is said to be uniformly smooth if ρEt(t) →  as t → . It is well known
that if the norm of E is uniformly Gâteaux diﬀerentiable, then the duality mapping J is
single valued and uniformly norm to weak∗ continuous on each bounded subset of E.
Recall that a closed convex subset C of a Banach space E is said to have a normal structure if for each bounded closed convex subset K of C which contains at least two points,
there exists an element x of K which is not a diametral point of K , i.e., sup{x – y : y ∈
K} < d(K), where d(K) is the diameter of K .
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Wu et al. Fixed Point Theory and Applications 2014, 2014:118
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Let D be a nonempty subset of a set C. Let ProjD : C → D. Q is said to be
() sunny if for each x ∈ C and t ∈ (, ), we have ProjD (tx + ( – t)ProjD x) = ProjD x;
() a contraction if ProjD = ProjD ;
() a sunny nonexpansive retraction if ProjD is sunny, nonexpansive, and a contraction.
D is said to be a nonexpansive retract of C if there exists a nonexpansive retraction from
C onto D. The following result, which was established in [–], describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Let E be a smooth Banach space and let C be a nonempty subset of E. Let ProjC : E → C
be a retraction and Jϕ be the duality mapping on E. Then the following are equivalent:
() ProjC is sunny and nonexpansive;
() x – ProjC x, Jϕ (y – ProjC x) ≤ , ∀x ∈ E, y ∈ C;
() ProjC x – ProjC y ≤ x – y, Jϕ (ProjC x – ProjC y), ∀x, y ∈ E.
It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction ProjC is
coincident with the metric projection from E onto C. Let C be a nonempty closed convex
subset of a smooth Banach space E, let x ∈ E, and let x ∈ C. Then we have from the above
that x = ProjC x if and only if x – x , Jϕ (y – x ) ≤  for all y ∈ C, where ProjC is a sunny
nonexpansive retraction from E onto C. For more additional information on nonexpansive
retracts, see [] and the references therein.
Let C be a nonempty closed convex subset of E. Let T : C → C be a mapping. In this
paper, we use F(T) to denote the set of ﬁxed points of T. Recall that T is said to be an
α-contractive mapping iﬀ there exists a constant α ∈ [, ) such that Tx – Ty ≤ αx –
y, ∀x, y ∈ C. The Picard iterative process is an eﬃcient method to study ﬁxed points of
α-contractive mappings. It is well known that α-contractive mappings have a unique ﬁxed
point. T is said to be nonexpansive iﬀ Tx – Ty ≤ x – y, ∀x, y ∈ C. It is well known that
nonexpansive mappings have ﬁxed points if the set C is closed and convex, and the space
E is uniformly convex. The Krasnoselski-Mann iterative process is an eﬃcient method for
studying ﬁxed points of nonexpansive mappings. The Krasnoselski-Mann iterative process
generates a sequence {xn } in the following manner:
x ∈ C,
xn+ = αn Txn + ( – αn )xn ,
∀n ≥ .
It is well known that the Krasnoselski-Mann iterative process only has weak convergence
for nonexpansive mappings in inﬁnite-dimensional Hilbert spaces; see [–] for more details and the references therein. In many disciplines, including economics, image recovery, quantum physics, and control theory, problems arise in inﬁnite-dimensional spaces.
In such problems, strong convergence (norm convergence) is often much more desirable
than weak convergence, for it translates the physically tangible property that the energy
xn – x of the error between the iterate xn and the solution x eventually becomes arbitrarily small. To improve the weak convergence of a Krasnoselski-Mann iterative process,
so-called hybrid projections have been considered; see [–] for more details and the
references therein. The Halpern iterative process was initially introduced in []; see []
for more details and the references therein. The Halpern iterative process generates a sequence {xn } in the following manner:
x ∈ C,
xn+ = αn u + ( – αn )Txn ,
∀n ≥ ,
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where x is an initial and u is a ﬁxed element in C. Strong convergence of Halpern iterative
process does not depend on metric projections. The Halpern iterative process has recently
been extensively studied for treating accretive operators; see [–] and the references
therein.
Let I denote the identity operator on E. An operator A ⊂ E × E with domain D(A) = {z ∈
E : Az = ∅} and range R(A) = {Az : z ∈ D(A)} is said to be accretive if for each xi ∈ D(A)
and yi ∈ Axi , i = , , there exists j(x – x ) ∈ J(x – x ) such that y – y , j(x – x ) ≥ . An
accretive operator A is said to be m-accretive if R(I + rA) = E for all r > . In this paper,
we use A– () to denote the set of zero points of A. For an accretive operator A, we can
deﬁne a nonexpansive single valued mapping Jr : R(I + rA) → D(A) by Jr = (I + rA)– for
each r > , which is called the resolvent of A.
Now, we are in a position to give the lemmas to prove main results.
Lemma . [] Let {an }, {bn }, {cn }, and {dn } be four nonnegative real sequences satisfying an+ ≤ ( – bn )an + bn cn + dn , ∀n ≥ n , where n is some positive integer, {bn } is a
number sequence in (, ) such that ∞
n=n bn = ∞, {cn } is a number sequence such that
lim supn→∞ cn ≤ , and {dn } is a positive number sequence such that ∞
n=n dn < ∞. Then
limn→∞ an = .
Lemma . [] Let C be a closed convex subset of a strictly convex Banach space E.
Let N ≥  be some positive integer and let Ti : C → C be a nonexpansive mapping for
each i ∈ {, , . . . , N}. Let {δi } be a real number sequence in (, ) with N
i= δi = . Suppose
N
F(T
)
is
nonempty.
Then
the
mapping
T
is
deﬁned
to
be
nonexpansive
with
that N
i
i= i
N i= N
F( i= Ti ) = i= F(Ti ).
Lemma . [] Let {xn } and {yn } be bounded sequences in a Banach space E and let βn
be a sequence in [, ] with  < lim infn→∞ βn ≤ lim supn→∞ βn < . Suppose that xn+ =
( – βn )yn + βn xn for all n ≥  and
lim sup yn+ – yn – xn+ – xn ≤ .
n→∞
Then limn→∞ yn – xn = .
Lemma . [] Let E be a real reﬂexive Banach space with the uniformly Gâteaux diﬀerentiable norm and the normal structure, and let C be a nonempty closed convex subset of E.
Let f : C → C be α-contractive mapping and let T : C → C be a nonexpansive mapping
with a ﬁxed point. Let {xt } be a sequence generated by the following: xt = tf (xt ) + ( – t)Txt ,
where t ∈ (, ). Then {xt } converges strongly as t →  to a ﬁxed point x∗ of T, which is the
unique solution in F(T) to the following variational inequality: f (x∗ ) – x∗ , j(x∗ – p) ≥ ,
∀p ∈ F(T).
2 Main results
Theorem . Let E be a real reﬂexive, strictly convex Banach space with the uniformly
Gâteaux diﬀerentiable norm. Let N ≥  be some positive integer. Let Am be an m-accretive
operator in E for each m ∈ {, , . . . , N}. Assume that C := N
m= D(Am ) is convex and has
the normal structure. Let f : C → C be an α-contractive mapping. Let {αn }, {βn }, and {γn }
be real number sequences in (, ) with the restriction αn + βn + γn = . Let {δn,i } be a real
number sequence in (, ) with the restriction δn, + δn, + · · · + δn,N = . Let {rm } be a positive
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real numbers sequence and {en,i } a sequence in E for each i ∈ {, , . . . , N}. Assume that
N –
i= Ai () is not empty. Let {xn } be a sequence generated in the following manner:
x ∈ C,
xn+ = αn f (xn ) + βn xn + γn
N
∀n ≥ ,
δn,i Jri (xn + en,i ),
i=
where Jri = (I + ri Ai )– . Assume that the control sequences {αn }, {βn }, {γn }, and {δn,i } satisfy
the following restrictions:
(a) limn→∞ αn = , ∞
n= αn = ∞;
(b)  < lim infn→∞ βn ≤ lim supn→∞ βn < ;
∞
(c)
n= en,m < ∞;
(d) limn→∞ δn,i = δi ∈ (, ).
Then the sequence {xn } converges strongly to x¯ , which is the unique solution to the following
–
variational inequality: f (¯x) – x¯ , J(p – x¯ ) ≤ , ∀p ∈ N
i= Ai ().
Proof Put yn =
N
i= δn,i Jri (xn
N
yn – p ≤
+ en,i ). Fixing p ∈
N
–
i= Ai (),
we have
δn,i Jri (xn + en,i ) – p
i=
N
≤
δn,i (xn + en,i ) – p
i=
≤ xn – p +
N
en,i .
i=
Hence, we have
xn+ – p ≤ αn f (xn ) – p + βn xn – p + γn yn – p
N
≤ αn αxn – p + αn f (p) – p + βn xn – p + γn xn – p + γn
en,i i=
f (p) – p
≤  – αn ( – α) xn – p + αn ( – α)
+
–α
N
i=
N
en,i ≤ max xn – p, f (p) – p +
i=
..
.
N
∞ ≤ max x – p, f (p) – p +
ej,i .
j= i=
This proves that the sequence {xn } is bounded, and so is {yn }. Since
yn – yn– =
N
δn,i Jrm (xn + en,i ) – Jri (xn– + en–,i )
i=
+
N
(δn,i – δn–,i )Jri (xn– + en–,i ),
i=
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we have
yn – yn– ≤
N
δn,i Jri (xn + en,i ) – Jri (xn– + en–,i )
i=
+
N
|δn,i – δn–,i |Jri (xn– + en–,i )
i=
≤ xn – xn– +
N
en,i +
i=
+
N
N
en–,i i=
|δn,i – δn–,i |Jri (xn– + en–,i )
i=
≤ xn – xn– +
N
en,i +
i=
N
en–,i + M
i=
N
|δn,i – δn–,i |,
i=
where M is an appropriate constant such that
M = max supJr (xn + en, ), supJr (xn + en, ), . . . , supJrN (xn + en,N ) .
n≥
n≥
Deﬁne a sequence {zn } by zn :=
xn+ –βn xn
,
–βn
n≥
that is, xn+ = βn xn + ( – βn )zn . It follows that
αn f (xn ) – yn + αn– f (xn– ) – yn– + yn – yn–  – βn
 – βn–
αn f (xn ) – yn + αn– f (xn– ) – yn– + xn – xn– ≤
 – βn
 – βn–
yzn – zn– ≤
+
N
|δn,i – δn–,i |Jri xn– i=
≤
αn f (xn ) – yn + αn– f (xn– ) – yn– + xn – xn–  – βn
 – βn–
N
N
|δn,i – δi | +
|δi – δn–,i | ,
+ M
i=
i=
where M is an appropriate constant such that
M = max sup Jr xn , sup Jr xn , . . . , sup JrN xn .
n≥
n≥
n≥
This implies that
zn – zn– – xn – xn– αn f (xn ) – yn + αn– f (xn– ) – yn– ≤
 – βn
 – βn–
N
N
|δn,i – δi | +
|δi – δn–,i | .
+ M
i=
i=
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From the restrictions (a), (b), (c), and (d), we ﬁnd that
lim sup zn – zn– – xn – xn– ≤ .
n→∞
Using Lemma ., we ﬁnd that limn→∞ zn – xn = . This further shows that
δi Jr . It follows from Lemma . that T is nonexlim supn→∞ xn+ – xn = . Put T = N
N i=– i
N
pansive with F(T) = i= F(Jri ) = i= Ai (). Note that
xn – Txn ≤ xn – xn+ + xn+ – Txn ≤ xn – xn+ + αn f (xn ) – Txn + βn xn – Txn + γn yn – Txn N
|δn,i – δi |.
≤ xn – xn+ + αn f (xn ) – Txn + βn xn – Txn + M
i=
This implies that
N
|δn,i – δi |.
( – βn )xn – Txn ≤ xn – xn+ + αn f (xn ) – Txn + M
i=
It follows from the restrictions (a), (b), and (d) that
lim Txn – xn = .
n→∞
Now, we are in a position to prove that lim supn→∞ f (¯x) – x¯ , J(xn – x¯ ) ≤ , where x¯ =
limt→ xt , and xt solves the ﬁxed point equation
xt = tf (xt ) + ( – t)Txt ,
∀t ∈ (, ).
It follows that
xt – xn  = t f (xt ) – xn , J(xt – xn ) + ( – t) Txt – xn , j(xt – xn )
= t f (xt ) – xt , J(xt – xn ) + t xt – xn , J(xt – xn )
+ ( – t) Txt – Txn , J(xt – xn ) + ( – t) Txn – xn , J(xt – xn )
≤ t f (xt ) – xt , J(xt – xn ) + xt – xn  + Txn – xn xt – xn , ∀t ∈ (, ).
This implies that

xt – f (xt ), J(xt – xn ) ≤ Txn – xn xt – xn ,
t
∀t ∈ (, ).
Since limn→∞ Txn – xn = , we ﬁnd that lim supn→∞ xt – f (xt ), J(xt – xn ) ≤ . Since J is
strong to weak∗ uniformly continuous on bounded subsets of E, we ﬁnd that
f (¯x) – x¯ , J(xn – x¯ ) – xt – f (xt ), J(xt – xn ) ≤ f (¯x) – x¯ , J(xn – x¯ ) – f (¯x) – x¯ , J(xn – xt ) Wu et al. Fixed Point Theory and Applications 2014, 2014:118
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+ f (¯x) – x¯ , J(xn – xt ) – xt – f (xt ), J(xt – xn ) ≤ f (¯x) – x¯ , J(xn – x¯ ) – J(xn – xt ) + f (¯x) – x¯ + xt – f (xt ), J(xn – xt ) ≤ f (xt ) – x¯ J(xn – x¯ ) – J(xn – xt ) + ( + α)¯x – xt xn – xt .
Since xt → x¯ , as t → , we have
lim f (¯x) – x¯ , J(xn – x¯ ) – f (xt ) – xt , J(xn – xt ) = .
t→
For > , there exists δ >  such that ∀t ∈ (, δ), we have
f (¯x) – x¯ , J(xn – x¯ ) ≤ f (xt ) – xt , J(xn – xt ) + .
This implies that lim supn→∞ f (¯x) – x¯ , J(xn – x¯ ) ≤ .
Finally, we show that xn → x¯ as n → ∞. Since ·  is convex, we see that
N

δn,i Jri (xn + en,i ) – x¯ yn – x¯  = i=
≤
N

δn,i Jri (xn + en,i ) – x¯ i=
≤ xn – x¯  +
N
en,i en,i + xn – x¯ .
i=
It follows that
xn+ – x¯  = αn f (xn ) – x¯ , J(xn+ – x¯ ) + βn xn – x¯ , J(xn+ – x¯ )
+ γn yn – x¯ , J(xn+ – x¯ )
≤ αn αxn – x¯ xn+ – x¯ + αn f (¯x) – x¯ , J(xn+ – x¯ )
+ βn xn – x¯ xn+ – x¯ + γn yn – x¯ xn+ – x¯ αn α xn – x¯  + xn+ – x¯  + αn f (¯x) – x¯ , J(xn+ – x¯ )
≤

γn
βn +
xn – x¯  + xn+ – x¯  + xn – x¯ 


+
N
i=
γn
en,i en,i + xn – x¯ + xn+ – x¯  .

Hence, we have
xn+ – x¯  ≤  – αn ( – α) xn – x¯  + αn f (¯x) – x¯ , J(xn+ – x¯ )
+
N
en,i en,i + xn – x¯ .
i=
Using Lemma ., we ﬁnd xn → x¯ as n → ∞. This completes the proof.
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Remark . There are many spaces satisfying the restriction in Theorem ., for example
Lp , where p > .
Corollary . Let E be a Hilbert space and let N ≥  be some positive integer. Let Am be a
maximal monotone operator in E for each m ∈ {, , . . . , N}. Assume that C := N
m= D(Am )
is convex and has the normal structure. Let f : C → C be an α-contractive mapping. Let
{αn }, {βn }, and {γn } be real number sequences in (, ) with the restriction αn + βn + γn = .
Let {δn,i } be a real number sequence in (, ) with the restriction δn, + δn, + · · · + δn,N = . Let
{rm } be a positive real numbers sequence and {en,i } a sequence in E for each i ∈ {, , . . . , N}.
–
Assume that N
i= Ai () is not empty. Let {xn } be a sequence generated in the following
manner:
x ∈ C,
xn+ = αn f (xn ) + βn xn + γn
N
δn,i Jri (xn + en,i ),
∀n ≥ ,
i=
where Jri = (I + ri Ai )– . Assume that the control sequences {αn }, {βn }, {γn }, and {δn,i } satisfy
the following restrictions:
(a) limn→∞ αn = , ∞
n= αn = ∞;
(b)  < lim infn→∞ βn ≤ lim supn→∞ βn < ;
∞
(c)
n= en,m < ∞;
(d) limn→∞ δn,i = δi ∈ (, ).
Then the sequence {xn } converges strongly to x¯ , which is the unique solution to the following
–
variational inequality: f (¯x) – x¯ , p – x¯ ≤ , ∀p ∈ N
i= Ai ().
3 Applications
In this section, we consider a variational inequality problem. Let A : C → E∗ be a single
valued monotone operator which is hemicontinuous; that is, continuous along each line
segment in C with respect to the weak∗ topology of E∗ . Consider the following variational
inequality:
ﬁnd x ∈ C such that y – x, Ax ≥ ,
∀y ∈ C.
The solution set of the variational inequality is denoted by VI(C, A). Recall that the normal
cone NC (x) for C at a point x ∈ C is deﬁned by
NC (x) = x∗ ∈ E∗ : y – x, x∗ ≤ , ∀y ∈ C .
Now, we are in a position to give the convergence theorem.
Theorem . Let E be a real reﬂexive, strictly convex Banach space with the uniformly
Gâteaux diﬀerentiable norm. Let N ≥  be some positive integer and let C be nonempty
closed and convex subset of E. Let Ai : C → E∗ a single valued, monotone and hemicontinu
ous operator. Assume that N
i= VI(C, Ai ) is not empty and C has the normal structure. Let
f : C → C be an α-contractive mapping. Let {αn }, {βn }, and {γn } be real number sequences
in (, ) with the restriction αn + βn + γn = . Let {δn,i } be a real number sequence in (, )
with the restriction δn, + δn, + · · · + δn,N = . Let {rm } be a positive real numbers sequence
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and {en,i } a sequence in E for each i ∈ {, , . . . , N}. Let {xn } be a sequence generated in the
following manner:
x ∈ C,
xn+ = αn f (xn ) + βn xn + γn
N
i=

δn,i VI C, Ai + (I – xn ) ,
ri
∀n ≥ .
Assume that the control sequences {αn }, {βn }, {γn }, and {δn,i } satisfy the following restrictions:
(a) limn→∞ αn = , ∞
n= αn = ∞;
(b)  < lim infn→∞ βn ≤ lim supn→∞ βn < ;
∞
(c)
n= en,m < ∞;
(d) limn→∞ δn,i = δi ∈ (, ).
Then the sequence {xn } converges strongly to x¯ , which is the unique solution to the following
variational inequality: f (¯x) – x¯ , J(p – x¯ ) ≤ , ∀p ∈ N
i= VI(C, Ai ).
Proof Deﬁne a mapping Ti ⊂ E × E∗ by
⎧
⎨A x + N x, x ∈ C,
i
C
Ti x :=
⎩∅,
x ∈/ C.
From Rockafellar [], we ﬁnd that Ti is maximal monotone with Ti– () = VI(C, Ai ). For
each ri > , and xn ∈ E, we see that there exists a unique xri ∈ D(Ti ) such that xn ∈ xri +
ri Ti (xri ), where xri = (I + ri Ti )– xn . Notice that

yn,i = VI C, Ai + (I – xn ) ,
ri
which is equivalent to
y – yn,i , Ai yn,i +

(yn,i – xn ) ≥ ,
ri
∀y ∈ C,
that is, –Ai yn,i + ri (xn – yn,i ) ∈ NC (yn,i ). This implies that yn,i = (I + ri Ti )– xn . Using Theorem ., we ﬁnd the desired conclusion immediately.
From Theorem ., the following result is not hard to derive.
Corollary . Let E be a real reﬂexive, strictly convex Banach space with the uniformly
Gâteaux diﬀerentiable norm. Let C be nonempty closed and convex subset of E. Let A : C →
E∗ a single valued, monotone and hemicontinuous operator with VI(C, A). Assume that C
has the normal structure. Let f : C → C be an α-contractive mapping. Let {αn }, {βn }, and
{γn } be real number sequences in (, ) with the restriction αn + βn + γn = . Let {xn } be a
sequence generated in the following manner:
x ∈ C,

xn+ = αn f (xn ) + βn xn + γn VI C, A + (I – xn ) ,
r
∀n ≥ .
Assume that the control sequences {αn }, {βn }, and {γn } satisfy the following restrictions:
Wu et al. Fixed Point Theory and Applications 2014, 2014:118
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(a) limn→∞ αn = , ∞
n= αn = ∞;
(b)  < lim infn→∞ βn ≤ lim supn→∞ βn < .
Then the sequence {xn } converges strongly to x¯ , which is the unique solution to the following
variational inequality: f (¯x) – x¯ , J(p – x¯ ) ≤ , ∀p ∈ VI(C, Ai ).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this manuscript. All authors read and approved the ﬁnal manuscript.
Author details
1
School of Business and Administration, Henan University, Kaifeng, Henan, China. 2 School of Mathematics and
Information Science, Shangqiu Normal University, Shangqiu, Henan, China. 3 Vietnam National University, Hanoi, Vietnam.
Acknowledgements
The authors are grateful to the editor and the reviewers for useful suggestions which improved the contents of the article.
Received: 16 January 2014 Accepted: 30 April 2014 Published: 14 May 2014
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Cite this article as: Wu et al.: Some results on zero points of m-accretive operators in reﬂexive Banach spaces. Fixed
Point Theory and Applications 2014, 2014:118
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