R E S E A R C H
Open Access
Approximation of a zero point of
monotone operators with nonsummable
errors
Takanori Ibaraki
**Correspondence: [email protected]
Department of Mathematics Education, Yokohama National University, Tokiwadai, Hodogaya-ku, Yokohama, 240-8501, Japan
Abstract
In this paper, we study an iterative scheme for two different types of resolvents of a monotone operator defined on a Banach space. These resolvents are generalizations of resolvents of a monotone operator in a Hilbert space. We obtain iterative
approximations of a zero point of a monotone operator generated by the shrinking projection method with errors in a Banach space. Using our result, we discuss some applications.
MSC: 47H05; 47H09; 47J25
Keywords: resolvent; monotone operator; metric projection
1 Introduction
LetHbe a real Hilbert space and letA⊂H×Hbe a maximal monotone operator. Then the zero point problem is to findu∈Hsuch that
∈Au. (.)
Such au∈His called a zero point (or a zero) ofA. The set of zero points ofAis denoted byA–. This problem is connected with many problems in Nonlinear Analysis and Op-timization, that is, convex minimization problems, variational inequality problems, equi-librium problems and so on. A well-known method for solving (.) is the proximal point algorithm:x∈Hand
xn+=Jrnxn, n= , , . . . , (.)
where{rn} ⊂],∞[ andJrn= (I+rnA)–. This algorithm was first introduced by Martinet []. In , Rockafellar [] proved that iflim infnrn> andA–=∅, then the sequence
{xn}defined by (.) converges weakly to a solution of the zero point problem. Later, many
researchers have studied this problem; see [–] and others.
On the other hand, Kimura [] introduced the following iterative scheme for finding a fixed point of nonexpansive mappings by the shrinking projection method with error in a Hilbert space:
Theorem .(Kimura []) Let C be a bounded closed convex subset of a Hilbert space H with D=diamC=supx,y∈Cx–y<∞,and let T:C→H be a nonexpansive mapping having a fixed point.Let{n}be a nonnegative real sequence such that=lim supnn<∞.
For a given point u∈H,generate an iterative sequence{xn}as follows:x∈C such that x–u<,C=C,
Cn+=
z∈C:z–Txn ≤ z–xn
∩Cn,
xn+∈Cn+ such that u–xn+≤d(u,Cn+)+n+
for all n∈N.Then
lim sup
n→∞ xn
–Txn ≤.
Further,if= ,then{xn}converges strongly to PF(T)u∈F(T).
We remark that the original result of the theorem above deals with a family of nonexpan-sive mappings, and the shrinking projection method was first introduced by Takahashiet al.[]. This result was extended to more general Banach spaces by Kimura [] (see also Ibaraki and Kimura []).
In this paper, we study the shrinking projection method with error introduced by Kimura [] (see also [, ]). We obtain an iterative approximation of a zero point of a monotone operator generated by the shrinking projection method with errors in a Banach space. Using our result, we discuss some applications.
2 Preliminaries
LetEbe a real Banach space with its dualE∗. The normalized duality mappingJfromE intoE∗is defined by
Jx=x∗∈E∗:x,x∗=x=x∗
for eachx∈E. We also know the following properties: see [, ] for more details. () Jx=∅for eachx∈E;
() ifEis reflexive, thenJis surjective;
() ifEis smooth, then the duality mappingJis single valued.
() ifEis strictly convex, thenJis one-to-one and satisfies thatx–y,x∗–y∗> for eachx,y∈Ewithx=y,x∗∈Jxandy∗∈Jy;
() ifEis reflexive, smooth, and strictly convex, then the duality mappingJ∗:E∗→Eis the inverse ofJ, that is,J∗=J–;
() ifEuniformly smooth, then the duality mappingJis uniformly norm to norm continuous on each bounded set ofE.
LetEbe a reflexive and strictly convex Banach space and letCbe a nonempty closed convex subset ofE. It is well known that for eachx∈Ethere exists a unique pointz∈C such thatx–z=min{x–y:y∈C}. Such a pointzis denoted byPCxandPCis called
Lemma . Let E be a reflexive,smooth, and strictly convex Banach space, let C be a nonempty closed convex subset of E,let PCbe the metric projection of E onto C,let x∈E
and let x∈C.Then x=PCx if and only if
x–y,J(x–x)
≥
for all y∈C.
LetC be a nonempty closed convex subset of a smooth Banach spaceE. A mapping T:C→Eis said to be of type (P) [] if
Tx–Ty,J(x–Tx) –J(y–Ty)≥
for eachx,y∈C. A mappingT:C→Eis said to be of type (Q) [, ] if
Tx–Ty, (Jx–JTx) – (Jy–JTy)≥
for eachx,y∈C. We denote byF(T) the set of fixed points ofT. A pointpinCis said to be an asymptotic fixed point ofT ifCcontains a sequence{xn}such thatxnpand
xn–Txn→. The set of all asymptotic fixed points ofTis denoted byF(Tˆ ). It is clear that
ifT:C→Eis of type (P) andF(T) is nonempty, then
Tx–p,J(x–Tx)≥ (.)
for eachx∈Candp∈F(T). LetE be a reflexive, smooth, and strictly convex Banach space and letCbe a nonempty closed convex subset ofE. It is well known that the metric projectionPCofEontoCis a mapping of type (P). We also know that ifT:C→Eis of
type (Q) andF(T) is nonempty, then
Tx–p,Jx–JTx ≥ (.)
for eachx∈Candp∈F(T).
The following results describe the relation between the set of fixed points and that of asymptotic fixed points for each type of mapping.
Lemma .(Aoyama-Kohsaka-Takahashi []) Let E be a smooth Banach space,let C be a nonempty closed convex subset of E and let T:C→E be a mapping of type(P).If F(T) is nonempty,then F(T)is closed and convex and F(T) =F(Tˆ ).
Lemma .(Kohsaka-Takahashi []) Let E be a strictly convex Banach space whose norm is uniformly Gâteaux differentiable,let C be a nonempty closed convex subset of E and let T:C→E be a mapping of type(Q).If F(T)is nonempty,then F(T)is closed and convex and F(T) =F(Tˆ ).
Theorem .(Tsukada []) Let E be a reflexive and strictly convex Banach space and let{Cn}be a sequence of nonempty closed convex subsets of E.If C= M-limnCnexists and
is nonempty,then for each x∈E,{PCnx}converges weakly to PCx,where PCnis the metric projection of E onto Cn.Moreover,if E has the Kadec-Klee property,the convergence is in
the strong topology.
One of the simplest example of the sequence{Cn}satisfying the condition in this
theo-rem above is a decreasing sequence with respect to inclusion;Cn+⊂Cnfor eachn∈N.
In this case, M-limCn=∞n=Cn(see [, , , ] for more details).
LetEbe a smooth Banach space and consider the following functionV :E×E→R defined by
V(x,y) =x– x,Jy+y (.)
for eachx,y∈E. We know the following properties: () (x–y)≤V(x,y)≤(x+y)for eachx,y∈E; () V(x,y) +V(y,x) = x–y,Jx–Jyfor eachx,y∈E;
() V(x,y) =V(x,z) +V(z,y) + x–z,Jz–Jyfor eachx,y,z∈E;
() ifEis additionally assumed to be strictly convex, thenV(x,y) = if and only ifx=y.
Lemma .(Kamimura-Takahashi []) Let E be a smooth and uniformly convex Banach
space and let{xn}and{yn}be sequences in E such that either{xn}or{yn}is bounded.If
limnV(xn,yn) = ,thenlimnxn–yn= .
The following results show the existence of mappingsg
r andgr, related to the convex
structures of a Banach spaceE. These mappings play important roles in our result.
Theorem .(Xu []) Let E be a Banach space,r∈],∞[and Br={x∈E:x ≤r}.
Then
(i) ifEis uniformly convex,then there exists a continuous,strictly increasing,and convex functiong
r: [, r]→[,∞[withgr() = such that
αx+ ( –α)y≤αx+ ( –α)y–α( –α)g
r
x–y
for allx,y∈Brandα∈[, ];
(ii) ifEis uniformly smooth,then there exists a continuous,strictly increasing,and convex functiongr: [, r]→[,∞[withgr() = such that
αx+ ( –α)y≥αx+ ( –α)y–α( –α)grx–y
for allx,y∈Brandα∈[, ].
Theorem .(Kimura []) Let E be a uniformly smooth and uniformly convex Banach
space and let r> .Then the function g
rand grin Theorem.satisfies
g
r
x–y ≤V(x,y)≤grx–y
3 Approximation theorem for the resolvents of type (P)
In this section, we discuss an iterative scheme of resolvents of a monotone operator de-fined on a Banach space. LetEbe a reflexive, smooth, and strictly convex Banach space. An operatorA⊂E×E∗with domainD(A) ={x∈E:Ax=∅}and rangeR(A) ={Ax:x∈ D(A)}is said to be monotone ifx–y,x∗–y∗ ≥ for any (x,x∗), (y,y∗)∈A. A monotone operatorAis said to be maximal ifA=BwheneverB⊂E×E∗is a monotone operator such thatA⊂B. We denote byA– the set{z∈D(A) : ∈Az}.
LetCbe a nonempty closed convex subset ofE, letr> and letA⊂E×E∗be a mono-tone operator satisfying
D(A)⊂C⊂RI+rJ–A (.)
forr> . It is well known that ifAis maximal monotone operator, thenR(I+rJ–A) =E; see [–]. Hence, ifAis maximal monotone, then (.) holds forC=D(A). We also know thatD(A) is convex; see []. IfAsatisfies (.) forr> , we can define the resolvent (of type (P))Pr:C→D(A) ofAby
Prx=
z∈E: ∈J(z–x) +rAz (.)
for allx∈C. In other words,Prx= (I+rJ–A)–xfor allx∈C. The Yosida approximation
Ar:C→E∗is also definedArx=J(x–Prx)/rfor allx∈C. We know the following; see, for
instance, [, , ]:
() Pris mapping of type (P) fromCintoD(A);
() (Prx,Arx)∈Afor allx∈C;
() Arx ≤ |Ax|:=inf{x∗:x∗∈Ax}for allx∈D(A);
() F(Pr) =A–.
We obtain an approximation theorem for a zero point of a monotone operator in a smooth and uniformly convex Banach space by using the resolvent of type (P).
Theorem . Let E be a smooth and uniformly convex Banach space and let A⊂E×
E∗ be a monotone operator with A–=∅.Let{r
n}be a positive real sequence such that
lim infnrn> ,let C be a nonempty bounded closed convex subset of E satisfying
D(A)⊂C⊂RI+rnJ–A
for all n∈Nand let r∈],∞[such that C⊂Br.Let{δn}be a nonnegative real sequence
and letδ=lim supnδn.For a given point u∈E,generate a sequence{xn} by x=x∈C, C=C,and
yn=Prnxn,
Cn+=
z∈C:yn–z,J(xn–yn)
≥∩Cn,
xn+∈
z∈C:u–z≤d(u,Cn+)+δn+
∩Cn+,
for all n∈N.Then
lim sup
n→∞ xn–yn ≤g –
r (δ).
Proof SinceCn includesA–=∅for alln∈N, {Cn}is a sequence of nonempty closed
convex subsets and, by definition, it is decreasing with respect to inclusion. Letpn=PCnu for alln∈N. Then, by Theorem ., we see that{pn}converges strongly top=PCu,
whereC= ∞
n=Cn. Sincexn∈Cnandd(u,Cn) =u–pn, we see that
u–xn≤ u–pn+δn
for everyn∈N\ {}. From Theorem .(i), we see that forα∈], [,
pn–u≤αpn+ ( –α)xn–u
≤αpn–u+ ( –α)xn–u–α( –α)gr
pn–xn
and thus
αg
r
pn–xn ≤ xn–u–pn–u≤δn.
Asα→, we see thatg
r(pn–xn)≤δnand thuspn–xn ≤g
–
r (δn). Using the definition
ofpn, we see thatpn+∈Cn+and thus
yn–pn+,J(xn–yn)
≥,
or equivalently,
xn–pn+,J(xn–yn)
≥ xn–yn.
Hence we obtain
xn–yn ≤ xn–pn+ ≤ xn–pn+pn–pn+ ≤g–
r (δn) +pn–pn+
for everyn∈N\ {}. Sincelimnpn=pandlim supnδn=δ, we see that
lim sup
n→∞ xn
–yn ≤g–r (δ).
For the latter part of the theorem, suppose thatδ= . Then we see that
lim sup
n→∞ xn–yn ≤g –
r () =
and
lim sup
n→∞ gr
xn–pn ≤lim sup n→∞ δn= .
Therefore, we obtain
lim
Hence, we also obtain
lim
n→∞xn=p and nlim→∞yn=p. (.)
So, from
yn–Pryn=rAryn ≤r|Ayn| ≤r
J(xn–yn)
rn
=r
xn–yn
rn
.
andlim infnrn> , we see thatlimnyn–Pryn= . Then, by Lemma . and (.), we
ob-tainxn→p∈ ˆF(Pr) =F(Pr) =A
–. SinceA–⊂C
, we getp=PCu=PA–u, which
completes the proof.
4 Approximation theorem for the resolvents of type (Q)
We next consider an iterative scheme of resolvents of a monotone operator which is dif-ferent type of Section , in a Banach space. LetCbe a nonempty closed convex subset of a reflexive, smooth, and strictly convex Banach spaceE, letr> and letA⊂E×E∗be a monotone operator satisfying
D(A)⊂C⊂J–R(J+rA) (.)
forr> . It is well known that ifAis maximal monotone operator, thenJ–R(J+rA) =E; see [–]. Hence, ifAis maximal monotone, then (.) holds forC=D(A). We also know thatD(A) is convex; see []. IfAsatisfies (.) forr> , then we can define the resolvent (of type (Q))Qr:C→D(A) ofAby
Qrx={z∈E:Jx∈Jz+rAz} (.)
for allx∈C. In other words,Qrx= (J+rA)–Jxfor allx∈C. We know the following; see,
for instance, [, ]:
() Qris mapping of type (Q) fromCintoD(A);
() (Jx–JQrx)/r∈AQrxfor allx∈C;
() F(Qr) =A–.
Before our result, we need the following lemma.
Lemma . Let E be a reflexive,smooth,and strictly convex Banach space,and let A⊂ E×E∗be a monotone operator.Let r> and C be a closed convex subset of E satisfying (.)for r> .Then the following holds:
V(x,Qrx) +V(Qrx,x)≤r
x–Qrx,x∗
for all(x,x∗)∈A.
Proof Let (x,x∗)∈A. Since (Jx–JQrx)/r∈AQrx, we see that
≤
x–Qrx,x∗–
Jx–JQrx
r
x–Qrx,
Jx–JQrx
r
≤x–Qrx,x∗
,
x–Qrx,Jx–JQrx ≤r
x–Qrx,x∗
.
From the property ofV, we see that
V(x,Qrx) +V(Qrx,x) = x–Qrx,Jx–JQrx ≤r
x–Qrx,x∗
for all (x,x∗)∈A.
We obtain an approximation theorem for a zero point of a monotone operator in a smooth and uniformly convex Banach space by using the resolvent of type (Q).
Theorem . Let E be a uniformly smooth and uniformly convex Banach space and let
A⊂E×E∗be a monotone operator with A–=∅.Let{r
n}be a positive real numbers such
thatlim infnrn> ,let C be a nonempty bounded closed convex subset of E satisfying
D(A)⊂C⊂J–R(J+rnA)
for all n∈Nand let r∈],∞[such that C⊂Br.Let{δn}be a nonnegative real sequence
and letδ=lim supnδn.For a given point u∈E,generate a sequence{xn} by x=x∈C, C=C,and
yn=Qrnxn,
Cn+=
z∈C:yn–z,Jxn–Jyn ≥
∩Cn,
xn+∈
z∈C:u–z≤d(u,Cn+)+δn+
∩Cn+,
for all n∈N.Then
lim sup
n→∞ xn
–yn ≤g–r
grg–
r (δ)
.
Moreover,ifδ= ,then{xn}converges strongly to PA–u.
Proof SinceCn includesA–=∅for alln∈N, {Cn}is a sequence of nonempty closed
convex subsets and, by definition, it is decreasing with respect to inclusion. Letpn=PCnu for alln∈N. Then, by Theorem ., we see that{pn}converges strongly top=PCu,
whereC= ∞
n=Cn. Sincexn∈Cnandd(u,Cn) =u–pn, we see that
u–xn≤ u–pn+δn
for everyn∈N\ {}. From Theorem .(i), we see that forα∈], [,
pn–u≤αpn+ ( –α)xn–u
≤αpn–u+ ( –α)xn–u–α( –α)gr
and thus
αg
r
pn–xn ≤ xn–u–pn–u≤δn.
Asα→, we see thatg
r(pn–xn)≤δnand thuspn–xn ≤g
–
r (δn). Using the definition
ofpn, we see thatpn+∈Cn+and thus
yn–pn+,Jxn–Jyn ≥.
From the property of the functionV, we see that
≤yn–pn+,Jxn–Jyn
= pn+–yn,Jyn–Jxn
=V(pn+,xn) –V(pn+,yn) –V(yn,xn)
≤V(pn+,xn) –V(yn,xn).
By Theorem ., we obtain
V(yn,xn)≤V(pn+,xn)
=V(pn+,pn) +V(pn,xn) + pn+–pn,Jpn–Jxn
≤V(pn+,pn) +gr
pn–xn + pn+–pn,Jpn–Jxn
≤V(pn+,pn) +gr
g–
r (δn) + pn+–pn,Jpn–Jxn.
Sincelim supnδn=δandpn→p, we see that
lim sup
n→∞
V(yn,xn)≤gr
g–
r (δ) .
Therefore, by Theorem ., we see that
lim sup
n→∞ xn
–yn ≤lim sup n→∞
g–
r
V(yn,xn) ≤g–r
grg–
r (δ)
.
For the latter part of the theorem, suppose thatδ= . Then we see that
lim sup
n→∞ xn–yn ≤g –
r
grg–
r ()
=
and
lim sup
n→∞
g
r
xn–pn ≤lim sup n→∞
δn= .
Therefore, we obtain
lim
Hence, we also obtain
lim
n→∞xn=p and nlim→∞yn=p. (.)
SinceEis uniformly smooth, the duality mappingJis uniformly norm-to-norm continu-ous on each bounded subset onE. Therefore, we obtain
lim
n→∞Jxn–Jyn= . (.)
From Lemma . we see that
V(yn,Qryn)≤V(yn,Qryn) +V(Qryn,yn)≤r
yn–Qryn,x
∗
for allx∗∈Ayn. Fromyn,Qryn∈D(A)⊂C⊂Brand (Jxn–Jyn)/rn∈Ayn, we see that
V(yn,Qryn)≤r
yn–Qryn,
Jxn–Jyn
rn
≤ryn–Qryn
Jxn–Jyn
rn
≤r
yn+Qryn
Jxn–Jyn
rn
= rr
Jxn–Jyn
rn
.
Sincelim infnrn> and (.), we obtain
lim sup
n→∞
V(yn,Qryn)≤.
This implieslimnV(yn,Qryn) = . From Theorem ., we see that
lim
n→∞yn–Qryn= .
Then, by Lemma . and (.), we see thatxn→p∈ ˆF(Qr) =F(Qr) =A–. SinceA–⊂
C, we getp=PCu=PA–u, which completes the proof.
5 Applications
In this section, we give some applications of Theorems . and .. We first study the convex minimization problem: LetEbe a reflexive, smooth, and strictly convex Banach space with its dualE∗and letf:E→]–∞,∞] be a proper lower semicontinuous convex function. Then the subdifferential∂f off is defined as follows:
∂f(x) =x∗∈E∗:f(x) +y–x,x∗≤f(y),∀y∈E
D(∂f)⊂D(f)⊂D(∂f). (.)
As a direct consequence of Theorems . and ., we can show the following corollaries.
Corollary . Let E be a smooth and uniformly convex Banach space,let f :E→]–∞,∞] be a proper lower semicontinuous convex function with D(f)being bounded,and let r∈ ],∞[such that D(f)⊂Br.Let{δn}be a nonnegative real sequence and letδ=lim supnδn.
For a given point u∈E,generate a sequence{xn}by x=x∈D(f),C=D(f),and
yn= argmin y∈E
f(y) + rny
–xn
,
Cn+=
z∈D(f) :yn–z,J(xn–yn)
≥∩Cn,
xn+∈
z∈D(f) :u–z≤d(u,Cn+)+δn+
∩Cn+,
for all n∈N,where{rn} ⊂],∞[such thatlim infnrn> .If(∂f)–is nonempty,then
lim sup
n→∞ xn–yn ≤g –
r (δ).
Moreover,ifδ= ,then{xn}converges strongly to P(∂f)–u.
Proof PutC=D(f). Since the subdifferential∂f⊂E×E∗is maximal monotone, we have E=R(I+r∂f) for allr> and hence, from (.), we see that
D(∂f)⊂D(∂f) =D(f) =C⊂E=R(I+r∂f)
for allr> .
Fixr> andz∈C. LetPrbe the resolvent (of type (P)) of∂f, then we also know that
Prz= argmin y∈E
f(y) +
ry–z
.
Therefore, we obtain the desired result by Theorem ..
Corollary . Let E be a uniformly smooth and uniformly convex Banach space,let f : E→]–∞,∞]be a proper lower semicontinuous convex function with D(f)being bounded and let r∈],∞[such that D(f)⊂Br.Let{δn}be a nonnegative real sequence and letδ=
lim supnδn.For a given point u∈E,generate a sequence{xn}by x=x∈D(f),C=D(f),and
yn= argmin y∈E
f(y) + rn
y– rn
y,Jxn
,
Cn+=
z∈D(f) :yn–z,Jxn–Jyn ≥
∩Cn,
xn+∈
z∈D(f) :u–z≤d(u,Cn+)+δn+
∩Cn+,
for all n∈N,where{rn} ⊂],∞[such thatlim infnrn> .If(∂f)–is nonempty,then
lim sup
n→∞ xn–yn ≤g –
r
grg–
r (δ)
.
Proof Fixr> andz∈C. LetQrbe the resolvent (of type (Q)) of∂f, then we also know
that
Qrz= argmin y∈E
f(y) +
ry –
ry,Jz
.
In the same way as Corollary ., we obtain the desired result by Theorem ..
Next, we study the approximation of fixed points for mappings of type (P) and (Q). Be-fore show our applications, we need the following results.
Lemma .([]) Let E be a reflexive,smooth,and strictly convex Banach space,let C be a nonempty subset of E,let T:C→E be a mapping,and let AT⊂E×E∗be an operator
defined by AT=J(T––I).Then T is of mapping of type(P)if and only if ATis monotone.
In this case T= (I+J–AT)–.
Lemma .([]) Let E be a reflexive,smooth,and strictly convex Banach space,let C be a nonempty subset of E and let T:C→E be a mapping,and let AT⊂E×E∗be an operator
defined by AT=JT––J.Then T is a mapping of type(Q)if and only if ATis monotone.In
this case T= (J+AT)–J.
As a direct consequence of Theorems . and ., we can show the following corollaries.
Corollary . Let E be a smooth and uniformly convex Banach space,let C be a bounded closed convex subset of E.Let T:C→C be a mapping of type(P)with F(T)being nonempty and let r∈],∞[such that C⊂Br.Let{δn}be a nonnegative real sequence and letδ=
lim supnδn.For a given point u∈E,generate a sequence{xn}by x=x∈C,C=C,and Cn+=
z∈C:Txn–z,J(xn–Txn)
≥∩Cn,
xn+∈
z∈C:u–z≤d(u,Cn+)+δn+
∩Cn+,
for all n∈N,where{rn} ⊂(,∞)such thatlim infnrn> .Then
lim sup
n→∞ xn–Txn ≤g –
r (δ).
Moreover,ifδ= ,then{xn}converges strongly to PF(T)u.
Proof PutAT=J(T––I) andrn= for alln∈N. From Lemma ., we see thatT is the
resolvent (of type (P)) ofATfor and
D(AT) =R(T)⊂C=D(T) =R
I+J–AT .
Therefore, we obtain the desired result by Theorem ..
letδ=lim supnδn.For a given point u∈E,generate a sequence{xn}by x=x∈C,C=C, and
Cn+=
z∈C:Txn–z,Jxn–JTxn ≥
∩Cn,
xn+∈
z∈C:u–z≤d(u,Cn+)+δn+
∩Cn+,
for all n∈N.Then
lim sup
n→∞ xn–Txn ≤g –
r
grg–
r (δ)
.
Moreover,ifδ= ,then{xn}converges strongly to PF(T)u.
Proof In the same way as Corollary ., we obtain the desired result by Lemma . and
Theorem ..
Competing interests
The author declares to have no competing interests.
Acknowledgements
The author is supported by Grant-in-Aid for Young Scientific (B) No. 24740075 from the Japan Society for the Promotion of Science.
Received: 30 November 2015 Accepted: 29 March 2016 References
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