Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 453452, 14 pages
doi:10.1155/2012/453452
Research Article
Viscosity Methods of
Asymptotically Pseudocontractive and
Asymptotically Nonexpansive Mappings for
Variational Inequalities
Xionghua Wu,1 Yeong-Cheng Liou,2
Zhitao Wu,1 and Pei-Xia Yang1
1
2
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
Correspondence should be addressed to Xionghua Wu, wuxionghua2003@163.com
Received 27 June 2012; Accepted 21 August 2012
Academic Editor: Yonghong Yao
Copyright q 2012 Xionghua Wu et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Let {tn } ⊂ 0, 1 be such that tn → 1 as n → ∞, let α and β be two positive numbers such that
α β 1, and let f be a contraction. If T be a continuous asymptotically pseudocontractive selfmapping of a nonempty bounded closed convex subset K of a real reflexive Banach space with a
uniformly Gateaux differentiable norm, under suitable conditions on the sequence {tn }, we show
the existence of a sequence {xn }n satisfying the relation xn 1 − tn /kn fxn tn /kn T n xn and
prove that {xn } converges strongly to the fixed point of T , which solves some variational inequality
provided T is uniformly asymptotically regular. As an application, if T be an asymptotically
nonexpansive self-mapping of a nonempty bounded closed convex subset K of a real Banach space
with a uniformly Gateaux differentiable norm and which possesses uniform normal structure,
we prove that the iterative process defined by z0 ∈ K, zn1 1 − tn /kn fzn αtn /kn T n zn βtn /kn zn converges strongly to the fixed point of T .
1. Introduction
Let E be a real Banach space with dual E∗ and K a nonempty closed convex subset of E. Recall
that a mapping T : K → K is said to be asymptotically pseudocontractive if, for each n ∈ N
and x, y ∈ K, there exist j ∈ Jx − y and a constant kn ≥ 1 with limn → ∞ kn 1 such that
2
T n x − T n y ≤ kn x − y ,
1.1
2
Abstract and Applied Analysis
∗
where J : E → 2E denote the normalized duality mapping defined by
Jx x∗ ∈ E∗ : x, x∗ x2 , x∗ x, x ∈ E .
1.2
The class of asymptotically pseudocontractive mappings is essentially wider than the class of
asymptotically nonexpansive mappings. A mapping T is called asymptotically nonexpansive
if there exists a sequence {kn } ⊂ 1, ∞ with limn → ∞ kn 1 such that
n
T x − T n y ≤ kn x − y
1.3
for all integers n ≥ 0 and all x, y ∈ K. A mapping f : K → K is called a contraction if there
exists a constant γ ∈ 0, 1 such that
fx − f y ≤ γ x − y,
∀x, y ∈ K.
1.4
It is clear that every contraction is nonexpansive, every nonexpansive mapping is asymptotically nonexpansive, and every asymptotically nonexpansive mapping is asymptotically
pseudocontractive. The converses do not hold. The asymptotically nonexpansive mappings
are important generalizations of nonexpansive mappings. For details, you may see 1.
The mapping T is called uniformly asymptotically regular in short u.a.r. if for each
> 0 there exists n0 ∈ N such that
n1
T x − T n x ≤ ,
1.5
for all n ≥ n0 and x ∈ K and it is called uniformly asymptotically regular with sequence {n }
in short u.a.r.s. if
n1
T x − T n x ≤ n ,
1.6
for all integers n ≥ 1 and all x ∈ K, where n → 0 as n → ∞.
The viscosity approximation method of selecting a particular fixed point of a given
nonexpansive mapping was proposed by Moudafi 2 who proved the strong convergence of
both the implicit and explicit methods in Hilbert spaces, see 2, Theorems 2.1 and 2.2. Xu 3
studied the viscosity approximation methods proposed by Moudafi 2 for a nonexpansive
mapping in a uniformly smooth Banach space.
Very recently, Shahzad and Udomene 4 obtained fixed point solutions of variational
inequalities for an asymptotically nonexpansive mapping defined on a real Banach space with
uniformly Gateaux differentiable norm possessing uniform normal structure. They proved
the following theorem.
Theorem 1.1. Let E be a real Banach space with a uniformly Gateaux differentiable norm possessing
uniform normal structure, let K be a nonempty closed convex and bounded subset of E, let T :
K → K be an asymptotically nonexpansive mapping with sequence {kn }n ⊂ 1, ∞, and let
f : K → K be a contraction with constant α ∈ 0, 1. Let {tn }n ⊂ 0, ξn be such that
Abstract and Applied Analysis
3
limn → ∞ tn 1, ∞
n1 tn 1 − tn ∞, and limn → ∞ kn − 1/kn − tn 0, where ξn min{1 −
αkn /kn − α, 1/kn }. For an arbitrary z0 ∈ K let the sequence {zn } be iteratively defined by
zn1 1−
tn
tn
fzn T n zn ,
kn
kn
n ∈ N.
1.7
Then
i for each integer n ≥ 0, there is a unique xn ∈ K such that
xn 1−
tn
tn
fxn T n xn ;
kn
kn
1.8
if in addition
lim xn − T xn 0,
n→∞
lim zn − T zn 0,
n→∞
1.9
then
ii the sequence {zn }n converges strongly to the unique solution of the variational inequality:
p ∈ FT such that
I − f p, j p − x∗ ≤ 0,
∀x∗ ∈ FT .
1.10
Remark 1.2. We note that T n1 x − T n x ≤ kn T x − x, then the condition 1.9 limn → ∞ xn −
T xn 0 and limn → ∞ zn − T zn 0 imply that
lim T n1 xn − T n xn 0,
n→∞
lim T n1 zn − T n zn 0,
1.11
n→∞
respectively. In other words, if an asymptotically nonexpansive mapping T satisfies the
condition 1.9 then T must be u.a.r.s.
Inspired by the works in 4–8, in this paper, we suggest and analyze a modification
of the iterative algorithm.
Let {tn } ⊂ 0, 1, let α and β be two positive numbers such that α β 1, and let f be a
contraction on K, a sequence {zn } iteratively defined by: z0 ∈ K,
zn1
βtn
tn
αtn n
fzn 1−
T zn zn .
kn
kn
kn
1.12
Remark 1.3. The algorithm 1.12 includes the algorithm 1.7 of Chidume et al. 5 and
Shahzad and Udomene 4 as a special case.
4
Abstract and Applied Analysis
We show the convergence of the proposed algorithm 1.12 to the unique solution of
some variational inequality some related works on VI, please see 9–12. In this respect, our
results can be considered as a refinement and improvement of the known results of Chidume
et al. 5, Shahzad and Udomene 4, and Lim and Xu 13.
2. Preliminaries
Let S {x ∈ E : x 1} denote the unit sphere of the Banach space E. The space E is said to
have a Gateaux differentiable norm if the limit
x ty − x
2.1
lim
n→∞
t
exists for each x, y ∈ S, and we call E smooth; E is said to have a uniformly Gateaux
differentiable norm if for each y ∈ S the limit 2.1 is attained uniformly for x ∈ S. Further,
E is said to be uniformly smooth if the limit 2.1 exists uniformly for x, y ∈ S × S. It is
well known 14 that if E is smooth then any duality mapping on E is single-valued, and if
E has a uniformly Gateaux differentiable norm then the duality mapping is norm-to-weak∗
uniformly continuous on bounded sets.
Let K be a nonempty closed convex and bounded subset of the Banach space E and
let the diameter of K be defined by dK sup{x − y : x, y ∈ K}. For each x ∈ K, let
rx, K sup{x − y : y ∈ K} and let rK inf{rx, K : x ∈ K} denote the Chebyshev
radius of K relative to itself. The normal structure coefficient NE of E is defined by
dK
: K is a closed convex and bounded subset of E with dK > 0 . 2.2
NE inf
rK
A space E such that NE > 1 is said to have uniform normal structure. It is known that
every space with a uniform normal structure is reflexive, and that all uniformly convex and
uniformly smooth Banach spaces have uniform normal structure see 13.
We will let LIM be a Banach limit. Recall that LIM ∈ l∞ ∗ such that LIM 1,
lim infn → ∞ an ≤ LIMn an ≤ lim supn → ∞ an , and LIMn an LIMn an1 for all {an }n ∈ l∞ . Let
{xn } be a bounded sequence of E. Then we can define the real-valued continuous convex
function g on E by gz LIMn xn − z2 for all z ∈ K.
Let T : K → K be a nonlinear mapping and M {x ∈ K : gx minz∈K gz}. T is
said to satisfy the property S if for any bounded sequence {xn } in K, limn → ∞ xn − T xn 0
implies M ∩ FT /
∅.
Lemma 2.1 see 15. Let E be a Banach space with the uniformly Gateaux differentiable norm and
u ∈ E. Then
gu inf gz
2.3
LIMz, Jxn − u
≤ 0
2.4
z∈E
if and only if
for all z ∈ E.
Abstract and Applied Analysis
5
Lemma 2.2 see 16. Assume {an } is a sequence of nonnegative real numbers such that
an1 ≤ 1 − γn an δn γn ,
2.5
where {γn } is a sequence in 0, 1 and {δn } is a sequence such that
1 ∞
n1 γn ∞;
2 lim supn → ∞ δn ≤ 0 or ∞
n1 |δn γn | < ∞.
Then limn → ∞ an 0.
Lemma 2.3 see 17. Let {xn } and {yn } be bounded sequences in a Banach space X and let {βn }
be a sequence in 0, 1 with 0 < lim infn → ∞ βn ≤ lim supn → ∞ βn < 1. Suppose that
xn1 1 − βn yn βn xn
2.6
for all n ≥ 0 and
lim sup yn1 − yn − xn1 − xn ≤ 0.
n→∞
2.7
Then limn → ∞ yn − xn 0.
Lemma 2.4. Let E be an arbitrary real Banach space. Then
x y2 ≤ x2 2 y, j x y ,
2.8
for all x, y ∈ E and for all jx y ∈ Jx y.
Lemma 2.5 see 5. Let E be a Banach space with uniform normal structure, K a nonempty closed
convex and bounded subset of E, and T : K → K an asymptotically nonexpansive mapping. Then T
has a fixed point.
3. Main Results
Theorem 3.1. Let E be a real reflexive Banach space with a uniformly Gateaux differentiable norm,
K a nonempty closed convex and bounded subset of E, T : K → K a continuous asymptotically
pseudocontractive mapping with sequence {kn }n ⊂ 1, ∞, and f : K → K a contraction with
constant γ ∈ 0, 1. Let {tn } ⊂ 0, 1 − γkn /kn − γ be such that limn → ∞ tn 1 and limn → ∞ kn −
1/kn − tn 0. Suppose T satisfies the property (S). Then
i for each integer n ≥ 0, there is a unique xn ∈ K such that
xn tn
tn
fxn T n xn ;
1−
kn
kn
3.1
if T is u.a.r.s., then
ii the sequence {xn }n converges strongly to the unique solution of the variational inequality:
p ∈ FT such that I − f p, j p − x∗ ≤ 0, ∀x∗ ∈ FT .
3.2
6
Abstract and Applied Analysis
Proof. By the conditions on {tn }, tn < 1 − γkn /kn − γ implies 1 − tn /kn γ tn < 1 for
each integer n ≥ 0, then the mapping Sn : K → K defined for each x ∈ K by Sn x 1 −
tn /kn fx tn /kn T n x is a strictly pseudocontractive mapping.
Indeed, for all x, y ∈ K, we have
Sn x − Sn y, j x − y 1−
tn fx − f y , j x − y
kn
tn n
T x − T n y, j x − y
kn
tn fx − f y x − y tn x − y2
≤ 1−
kn
2
tn
≤
1−
γ tn x − y .
kn
3.3
It follows 18, Corollary 1 that Sn possesses exactly one fixed point xn in K such that Sn xn xn .
From 3.1, we have
tn
tn
n
1
−
−
1
T
x
fx
xn − T n xn n
n
kn
kn
tn fxn − T n xn −→ 0 as n −→ ∞.
1−
kn
3.4
Notice that
tn n
tn 1
−
x
−
T
x
fx
−
T
x
T
xn − T xn n
n
n
n
kn
kn
tn fxn − T xn tn ≤ 1−
T n xn − T n1 xn kn
kn
tn T n1 xn − T xn kn
tn fxn − T xn tn ≤ 1−
T n xn − T n1 xn kn
kn
3.5
tn
k1 xn − T n xn .
kn
Therefore, from 3.4, 3.5, and T which is u.a.r.s., we obtain xn − T xn → 0 as n → ∞.
Define a function g : K → R by
gz LIMn xn − z2
3.6
for all z ∈ K. Since g is continuous and convex, gz → ∞ as z → ∞, and E is
reflexive, g attains it infimum over K. Let z0 ∈ K such that gz0 minz∈K gz and let
Abstract and Applied Analysis
7
M {x ∈ K : gx minz∈K gz}. Then M is nonempty because z0 ∈ M. Since T satisfies
the property S, it follows that M∩FT /
∅. Suppose that p ∈ M∩FT . Then, by Lemma 2.1,
we have
3.7
LIMn x − p, j xn − p ≤ 0
for all x ∈ K. In particular, we have
LIMn f p − p, j xn − p ≤ 0.
3.8
On the other hand, from 3.1, we have
1 − tn /kn tn fxn − T n xn xn − T n xn 1 −
fxn − xn .
kn
tn /kn
Now, for any p ∈ FT , we have
xn − T n xn , J xn − p xn − p T n p − T n xn , J xn − p
2
≥ − kn − 1xn − p
3.9
3.10
≥ − kn − 1B2
for some B > 0 and it follows from 3.9 that
which implies that
tn kn − 1 2
B ,
xn − fxn , j xn − p ≤
kn − tn
lim sup xn − fxn , j xn − p ≤ 0.
3.11
3.12
n→∞
Consequently, similar to the lines of the proof of 4, Theorem 3.1, Theorem 3.1 is easily
proved. This completes the proof.
Corollary 3.2. Let E be a real Banach space with a uniformly Gateaux differentiable norm possessing
uniform normal structure, K a nonempty closed convex and bounded subset of E, T : K → K
be an asymptotically nonexpansive mapping with sequence {kn }n ⊂ 1, ∞, and f : K → K a
contraction with constant γ ∈ 0, 1. Let {tn } ⊂ 0, 1 − γkn /kn − γ be such that limn → ∞ tn 1
and limn → ∞ kn − 1/kn − tn 0. Then
i for each integer n ≥ 0, there is a unique xn ∈ K such that
tn
tn
xn 1 −
3.13
fxn T n xn ;
kn
kn
if T is u.a.r.s., then
ii the sequence {xn }n converges strongly to the unique solution of the variational inequality:
p ∈ FT such that
I − f p, j p − x∗ ≤ 0,
∀x∗ ∈ FT .
3.14
8
Abstract and Applied Analysis
Theorem 3.3. Let E be a real Banach space with a uniformly Gateaux differentiable norm possessing
uniform normal structure, K a nonempty closed convex and bounded subset of E, T : K → K an
asymptotically nonexpansive mapping with sequence {kn }n ⊂ 1, ∞, and f : K → K a contraction
with constant γ ∈ 0, 1. Let {tn } ⊂ 0, ξn be such that limn → ∞ tn 1, ∞
n1 tn 1 − tn ∞, and
limn → ∞ kn − 1/kn − tn 0, where ξn min{1 − γkn /kn − γ, 1/kn }. For an arbitrary
z0 ∈ K, let the sequence {zn }n be iteratively defined by 1.12. Then
i for each integer n ≥ 0, there is a unique xn ∈ K such that
xn tn
tn
1−
fxn T n xn ;
kn
kn
3.15
if T is u.a.r.s., then
ii the sequence {zn }n converges strongly to the unique solution of the variational inequality:
p ∈ FT such that
I − f p, j p − x∗ ≤ 0,
∀x∗ ∈ FT .
3.16
Proof. Set αn tn /kn , then αn → 1 as n → ∞. Define
zn1 βαn zn 1 − βαn yn .
3.17
Observe that
yn1 − yn zn2 − βαn1 zn1 zn1 − βαn zn
−
1 − βαn1
1 − βαn
1 − αn1 fzn1 ααn1 T n1 zn1
1 − βαn1
−
1 − αn fzn ααn T n zn
1 − βαn
1 − αn1 fzn1 − fzn 1 − βαn1
1 − αn
1 − αn1
−
fzn 1 − βαn1 1 − βαn
ααn1 n1
T zn1 − T n1 zn
1 − βαn1
ααn1 n1
T zn − T n zn
1 − βαn1
ααn1
ααn
T n zn .
−
1 − βαn1 1 − βαn
3.18
Abstract and Applied Analysis
9
It follows that
yn1 − yn − zn1 − zn 1 − αn1
1 − αn1
1 − αn fzn ≤
γzn1 − zn −
1 − βαn1
1 − βαn1 1 − βαn
ααn1 n1
T zn1 − T n1 zn 1 − βαn1
ααn1 n1
T zn − T n zn 1 − βαn1
ααn1
ααn n
−
T zn − zn1 − zn 1 − βαn1 1 − βαn 1 − αn1
1 − αn n1
n
fzn ααn1 ≤ −
z
−
T
z
T
n
n
1 − βα
1 − βα 1 − βα
n1
ααn1
1 − βα
n1
n
ααn n
−
T zn 1 − βαn 3.19
n1
1 − αn1
ααn1
γ
kn1 − 1 zn1 − zn .
1 − βαn1
1 − βαn1
We note that
kn1 − γ − αkn1 β − γ 1 − αkn1 − β
≥ 1 − α − β 0.
3.20
It follows that
1 − γ kn1
1 − γ kn1
≤
,
≤
kn1 − γ
αkn1 β − γ
tn1
3.21
which implies that
kn1 tn1 α tn1 β − tn1 γ ≤ 1 − γ kn1
⇒ αkn1 αn1 αn1 β − αn1 γ ≤ 1 − γ
⇒ αkn1 αn1 1 − αn1 γ ≤ 1 − αn1 β
⇒
3.22
αkn1 αn1 1 − αn1 γ
≤ 1.
1 − αn1 β
From 3.19 and 3.22, we obtain
lim sup yn1 − yn − zn1 − zn ≤ 0.
n→∞
3.23
10
Abstract and Applied Analysis
Hence, by Lemma 2.3 we know
lim yn − zn 0,
n→∞
3.24
lim zn1 − zn 0.
3.25
zn − T n zn ≤ zn1 − zn zn1 − T n zn ≤ zn1 − zn 1 − αn fzn − T n zn 3.26
consequently
n→∞
On the other hand,
βαn zn − T n zn ,
which implies that
lim zn − T n zn 0.
3.27
n→∞
Hence, we have
zn − T zn ≤ zn − T n zn T n zn − T n1 zn T n1 zn − T zn ≤ zn − T n zn T n zn − T n1 zn k1 zn − T n zn 1 k1 zn − T n zn T n zn − T n1 zn −→ 0
3.28
n −→ ∞.
From 3.15, xm −zn 1−αm fxm −zn αm T m xm −zn . Applying Lemma 2.4, we estimate
as follows:
xm − zn 2 ≤ α2m T m xm − zn 2 21 − αm fxm − zn , jxm − zn ≤ α2m T m xm − T m zn T m zn − zn 2
21 − αm fxm − xm , jxm − zn xm − zn 2
≤ α2m km xm − zn T m zn − zn 2
2
21 − αm fxm − xm , jxm − zn km
xm − zn 2
2
α2m km
xm − zn 2 2km xm − zn T m zn − zn T m zn − zn 2
21 − αm 2
fxm − xm , jxm − zn km
xm − zn 2
Abstract and Applied Analysis
11
2
1 − 1 − αm 2 km
xm − zn 2
T m zn − zn 2km xm − zn T m zn − zn 2
21 − αm fxm − xm , jxm − zn km
xm − zn 2
2
≤ 1 1 − αm 2 km
xm − zn 2
T m zn − zn 2km xm − zn T m zn − zn 21 − αm fxm − xm , jxm − zn .
3.29
Since K is bounded, for some constant M > 0, it follows that
2
2
km
− 1 km
1 − αm 2
lim sup fxm − xm , jzn − xm ≤
M
1 − αm
n→∞
3.30
Mzn − T m zn lim sup
,
1 − αm
n→∞
so that
lim sup fxm − xm , jzn − xm ≤
n→∞
2
2
km
− 1 km
1 − αm 2
1 − αm
M.
3.31
By Corollary 3.2, xm → p ∈ FT , which solve the variational inequality 3.16. Since j is
norm to weak∗ continuous on bounded sets, in the limit as m → ∞, we obtain that
lim sup f p − p, j zn − p ≤ 0.
n→∞
From Lemma 2.4, we estimate as follows:
zn1 − p2 1 − αn fzn − p ααn T n zn − p βαn zn − p2
2
≤ ααn T n zn − p βαn zn − p
21 − αn fzn − p, j zn1 − p
2
≤ α2 α2n T n zn − p 2αβα2n T n zn − pzn − p
2
β2 α2n zn − p 21 − αn fzn − f p , j zn1 − p 21 − αn f p − p, j zn1 − p
3.32
12
Abstract and Applied Analysis
2
≤ α2 kn2 2αβkn β2 α2n zn − p
21 − αn γ zn − pzn1 − p
21 − αn f p − p, j zn1 − p
2
2 2 ≤ α2n kn2 zn − p γ1 − αn zn − p zn1 − p
21 − αn f p − p, j zn1 − p ,
3.33
so that
2
t 1 − αn γ zn1 − p2 ≤ n
zn − p2
1 − 1 − αn γ
1 − αn f p − p, j zn1 − p
1 − 1 − αn γ
1 − 21 − αn γ − t2n
zn − p2
1−
1 − 1 − αn γ
2
2
Let
3.34
1 − αn f p − p, j zn1 − p .
1 − 1 − αn γ
1 − 21 − αn γ − t2n
.
λn 1 − 1 − αn γ
3.35
Consequently, following the lines of the proof of 4, Theorem 3.3, Theorem 3.3 is easily
proved.
From the lines of the proof of Theorem 3.3, we can obtain the following corollary.
Corollary 3.4. Let E be a real Banach space with a uniformly Gateaux differentiable norm possessing
uniform normal structure, K a nonempty closed convex and bounded subset of E, T : K → K an
asymptotically nonexpansive mapping with sequence {kn }n ⊂ 1, ∞, and f : K → K a contraction
with constant γ ∈ 0, 1. Let {tn } ⊂ 0, ξn be such that limn → ∞ tn 1, ∞
n1 tn 1 − tn ∞, and
limn → ∞ kn − 1/kn − tn 0, where ξn min{1 − γkn /kn − γ, 1/kn }. For an arbitrary
z0 ∈ K, let the sequence {zn }n be iteratively defined by 1.12. Then
i for each integer n ≥ 0, there is a unique xn ∈ K such that
tn
tn
fxn T n xn ;
xn 1 −
kn
kn
3.36
if T satisfies limn → ∞ T n1 xn − T n xn 0 and limn → ∞ T n1 zn − T n zn 0 then
ii the sequence {zn }n converges strongly to the unique solution of the variational inequality:
p ∈ FT such that
I − f p, j p − x∗ ≤ 0,
∀x∗ ∈ FT .
3.37
Abstract and Applied Analysis
13
Remark 3.5. Since every nonexpansive mapping is asymptotically nonexpansive, our
theorems hold for the case when T is simply nonexpansive. In this case, the boundedness
requirement on K can be removed from the above results.
Remark 3.6. Our results can be viewed as a refinement and improvement of the corresponding
results by Shahzad and Udomene 4, Chidume et al. 5, and Lim and Xu 13.
Example 3.7. Let T : C → C be a nonexpansive mapping. Let the iterative sequence {xn } be
defined by
xn1 1
1
u 1−
T xn .
n
n
3.38
It is easy to see that {xn } converges strongly to some fixed point of T .
In particular, let H R2 and define T : R2 → R2 by
T reiθ reiθπ/2 ,
3.39
and take that u eiπ is a fix element in C. It is obvious that T is a nonexpansive mapping
with a unique fixed point x∗ 0. In this case, 3.38 becomes
xn1 1 iπ
1
e 1−
rn eiθn π/2 .
n
n
3.40
It is clear that the complex number sequence {xn } converges strongly to a fixed point x∗ 0.
Acknowledgment
Y.-C. Liou was partially supported by NSC 101-2628-E-230-001-MY3.
References
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Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
3 H.-K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical
Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
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