Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 787419, 13 pages
doi:10.1155/2012/787419
Research Article
Iterative Algorithm for Common
Fixed Points of Infinite Family of Nonexpansive
Mappings in Banach Spaces
Songnian He and Jun Guo
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Songnian He, hesongnian2003@yahoo.com.cn
Received 9 January 2012; Accepted 18 January 2012
Academic Editor: Yonghong Yao
Copyright q 2012 S. He and J. Guo. 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 C be a nonempty closed convex subset of a real uniformly smooth Banach space X, {Tk }∞
k1 :
C → C
an
infinite
family
of
nonexpansive
mappings
with
the
nonempty
set
of
common
fixed
iterative algorithm
points ∞
k1 FixTk , and f : C → C a contraction.
n We introduce anexplicit
n
x
αn fxn 1 − αn Ln xn , where Ln k1 ωk /sn Tk , Sn k1 ωk , and wk > 0 with
n1
∞
k1 ωk 1. Under certain appropriate conditions on {αn }, we prove that {xn } converges strongly
∗
∗
to a common fixed point x
of {Tk }∞
k1 , which solves the following variational inequality: x −
∞
∗
∗
fx , Jx − p ≤ 0, p ∈ k1 FixTk , where J is the normalized duality mapping of X. This
algorithm is brief and needs less computational work, since it does not involve W-mapping.
1. Introduction
Let X be a real Banach space, C a nonempty closed convex subset of X, and X ∗ the dual space
∗
of X. The normalized duality mapping J : X → 2X is defined by
Jx x∗ ∈ X ∗ : x, x∗ x2 , x∗ x ,
∀x ∈ X.
1.1
If X is a Hilbert space, then J I, where I is the identity mapping. It is well known that if X
is smooth, then J is single valued.
Recall that a mapping f : C → C is a contraction, if there exists a constant α ∈ 0, 1
such that
fx − f y ≤ αx − y,
∀x, y ∈ C.
1.2
2
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We use ΠC to denote the collection of all contractions on C, that is,
ΠC f : f is a contraction on C .
1.3
A mapping T : C → C is said to be nonexpansive, if
T x − T y ≤ x − y,
∀x, y ∈ C.
1.4
We use FixT to denote the set of fixed points of T , namely, FixT {x ∈ C : T x x}.
One classical way to study nonexpansive mappings is to use contractions to approximate a
nonexpansive mapping 1–11. Browder 1 first considered the following approximation
in a Hilbert space. Fix u ∈ C and define a contraction Ft from C into itself by
Ft x tu 1 − tT x,
x ∈ C,
1.5
where t ∈ 0, 1. Banach contraction mapping principle guarantees that Ft has a unique fixed
point in C. Denote by zt ∈ C the unique fixed point of Ft , that is,
zt tu 1 − tT zt .
1.6
In the case of T having fixed points, Browder 1 proved the following.
Theorem 1.1. In a Hilbert space, as t → 0, zt defined in 1.6 converges strongly to a fixed point of
T that is closest to u, that is, the nearest point projection of u onto FixT .
Halpern 3 introduced an iteration process discretization of 1.6 in a Hilbert as
follows:
zn1 αn u 1 − αn T zn ,
n ≥ 0,
1.7
where u, z0 ∈ C are arbitrary but fixed and {αn } is a sequence in 0, 1. Lions 4 proved the
following.
Theorem 1.2. In a Hilbert space, if {αn } satisfies the following conditions:
K1 limn → ∞ αn 0;
K2 ∞
n0 αn ∞;
K3 limn → ∞ |αn − αn−1 |/α2n1 0.
Then {zn } converges strongly to the nearest point projection of u onto FixT .
The Banach space versions of Theorems 1.1 and 1.2 were obtained by Reich 5. He
proved the following.
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3
Theorem 1.3. In a uniformly smooth Banach space X, both zt defined in 1.6 and {zn } defined in
1.7 converge strongly to a same fixed point of T. If one defines Q : C → FixT by
Qu limzt ,
1.8
t→0
then Q is the sunny nonexpansive retraction from C onto FixT . Namely, Q satisfies the property:
Qx − Qy2 ≤ x − y, J Qx − Qy ,
x, y ∈ C,
1.9
where J is the duality mapping of X.
Moudafi 6 introduced a viscosity approximation method and proved the strong
convergence of both the implicit and explicit methods in Hilbert spaces. Xu 7 extended
Moudafi’s results in Hilbert spaces. Given a real number t ∈ 0, 1 and a contraction f ∈ ΠC ,
f
define a contraction Tt : C → C by
f
Tt x tfx 1 − tT x,
x ∈ C.
1.10
f
Let xt ∈ C be the unique fixed point of Tt . Thus,
xt tfxt 1 − tT xt .
1.11
Corresponding explicit iterative process is defined by
xn1 αn fxn 1 − αn T xn ,
1.12
where x0 ∈ C is arbitrary but fixed and {αn } is a sequence in 0, 1. It was proved by Xu
7 that under certain appropriate conditions on {αn }, both xt defined in 1.11 and {xn }
defined in 1.12 converged strongly to x∗ ∈ C, which is the unique solution of the variational
inequality:
I − f x∗ , x − x∗ ≥ 0,
x ∈ FixT .
1.13
Xu 7 also extended Moudafi’s results to the setting of Banach spaces and proved the strong
convergence of both the implicit method 1.11 and explicit method 1.12 in uniformly
smooth Banach spaces.
In order to deal with some problems involving the common fixed points of infinite
family of nonexpansive mappings, W-mapping is often used, see 12–20. Let {Tk }∞
k1 : C →
be
a
real
number
sequence
C be an infinite family of nonexpansive mappings and let {ξk }∞
k1
4
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such that 0 < ξk < 1 for every k ∈ N. For any n ∈ N, we define a mapping Wn of C into itself
as follows:
Un,n1 I,
Un,n ξn Tn Un,n1 1 − ξn I,
Un,n−1 ξn−1 Tn−1 Un,n 1 − ξn−1 I,
..
.
Un,k ξk Tk Un,k1 1 − ξk I,
1.14
Un,k−1 ξk−1 Tk−1 Un,k 1 − ξk−1 I,
..
.
Un,2 ξ2 T2 Un,3 1 − ξ2 I,
Wn Un,1 ξ1 T1 Un,2 1 − ξ1 I.
∞
Such Wn is called the W-mapping generated by {Tk }∞
k1 and {ξk }k1 , see 12, 13.
Yao et al. 10 introduced the following iterative algorithm for infinite family of nonexpansive mappings. Let X be a uniformly convex Banach space with a uniformly Gâteaux
differentiable norm and C a nonempty closed convex subset of X. Sequence {xn } is defined
by
xn1 αn u 1 − αn Wn xn ,
n ≥ 0,
1.15
where u, x0 ∈ C are arbitrary but fixed and {αn } ⊂ 0, 1. It was proved that under certain
appropriate conditions on {αn }, the sequence {xn } generated by 1.15 converges strongly to
a common fixed point of {Tk }∞
k1 13.
Since W-mapping contains many composite operations of {Tk }, it is complicated and
needs large computational work. In this paper, we introduce a new iterative algorithm for
solving the common fixed point problem of infinite family of nonexpansive mappings. Let X
be a real uniformly smooth Banach space, C a nonempty closed convex subset of X, {Tk }∞
k1 :
C → C an infinite family of nonexpansive mappings with the nonempty set of common fixed
points ∞
k1 FixTk , and f ∈ ΠC . Given any x0 ∈ C, define a sequence {xn } by
xn1 αn fxn 1 − αn Ln xn ,
n ≥ 0,
1.16
where {αn } ⊂ 0, 1, Ln nk1 ωk /sn Tk , Sn nk1 ωk and wk > 0 with ∞
k1 ωk 1.
Under certain appropriate conditions on {αn }, we prove that {xn } converges strongly to x∗ ∈
∞
k1 FixTk , which solves the following variational inequality:
x∗ − fx∗ , J x∗ − p ≤ 0,
p∈
∞
k1
FixTk ,
1.17
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5
where J is the duality mapping of X. Because Ln doesn’t contain many composite operations
of {Tk }, this algorithm is brief and needs less computational wok.
We will use M to denote a constant, which may be different in different places.
2. Preliminaries
Let B {x ∈ X : x 1} denotes the unit sphere of X. A Banach space X is said to be
strictly convex, if x y/2 < 1 holds for all x, y ∈ B, x / y. A Banach space X is said to be
uniformly convex if for each ε ∈ 0, 2, there exists a constant δ > 0 such that for any x, y ∈ B,
x − y ≥ ε implies x y/2 ≤ 1 − δ. It is known that a uniformly convex Banach space is
reflexive and strictly convex, see 21.
The norm of X is said to be Gâteaux differentiable if
lim
t→0
x ty − x
2.1
t
exists for each x, y ∈ B and in this case X is said to be smooth. The norm of X is said to be
uniformly Gâteaux differentiable if for each y ∈ B, the limit 2.1 is attained uniformly for
x ∈ B. The norm of X is said to be Frêchet differentiable, if for each x ∈ B, the limit 2.1 is
attained uniformly for y ∈ B. The norm of X is said to be uniformly Frêchet differentiable,
if the limit 2.1 is attained uniformly for x, y ∈ B and in this case X is said to be uniformly
smooth.
Let D be a nonempty subset of C. A mapping Q : C → D is said to be sunny 22 if
Qx tx − Qx Qx,
∀x ∈ C, t ≥ 0,
2.2
whenever x tx − Qx ∈ C. A mapping Q : C → D is called a retraction if Qx x for all
x ∈ D. Furthermore, Q is sunny nonexpansive retraction from C onto D if Q is a retraction
from C onto D which is also sunny and nonexpansive.
A subset D of C is called a sunny nonexpansive retraction of C if there exits a sunny
nonexpansive retraction from C onto D.
Lemma 2.1 see 22. Let C be a closed convex subset of a smooth Banach space X. Let D be a
nonempty subset of C and Q : C → D be a retraction. Then the following are equivalent.
a Q is sunny and nonexpansive.
b Qx − Qy2 ≤ x − y, JQx − Qy, for all x, y ∈ C.
c x − Qx, Jy − Qx ≤ 0, for all x ∈ C, y ∈ D.
Lemma 2.2 see 23. Let {sn } be a sequence of nonnegative real numbers satisfying
sn1 ≤ 1 − γn sn γn βn δn ,
n ≥ 0,
2.3
6
Journal of Applied Mathematics
where {γn } ⊂ 0, 1, {βn } and {δn } satisfy the following conditions:
A1
∞
n0 γn
∞;
A2 lim supn → ∞ βn ≤ 0;
A3 δn ≥ 0 n ≥ 0,
∞
n1
δn < ∞.
Then limn → ∞ sn 0.
Lemma 2.3 see 24. In a Banach space X, the following inequality holds:
x y2 ≤ x2 2 y, j x y ,
∀x, y ∈ X,
2.4
where jx y ∈ Jx y.
Lemma 2.4 see 25. Let C be a closed convex subset of a strictly convex Banach space X. Let
{Tn : n ∈ N} be a sequence of nonexpansive mappings on C. Suppose ∞
n1 FixTn is nonempty.
∞
Let {λn } be a sequence of positive numbers with n1 λn 1. Then a mapping S on C defined by
∞
Sx ∞
n1 FixTn holds.
n1 λn Tn x for x ∈ C is well defined, nonexpansive and FixS Lemma 2.5 see 7. Let X be a uniformly smooth Banach space, C a closed convex subset of X,
T : C → C a nonexpansive mapping with FixT /
∅, and f ∈ ΠC . Then {xt } defined by
xt tfxt 1 − tT xt
2.5
converges strongly to a point in FixT . If we define a mapping Q : ΠC → FixT by
Q f : limxt ,
t→0
f ∈ ΠC ,
2.6
then Qf solves the variational inequality:
I − f Q f , J Q f − p ≤ 0,
f ∈ ΠC , p ∈ FixT .
2.7
Lemma 2.6. Let X be a Banach space, {xk } a bounded sequence of X, and {ωk } a sequence of positive
∞
numbers with ∞
k1 ωk 1. Then
k1 ωk xk is convergent in X.
Lemma 2.7. Let X be Banach space, {Tk : k ∈ N} a sequence of nonexpansive mappings on X with
∞
∞
∅, and {ωk } a sequence of positive numbers with ∞
k1 ωk 1. Let T k1 ωk Tk ,
k1 FixT
m k /
m
Lm k1 ωk /Sm Tk , and Sm k1 ωk . Then Lm uniformly converges to T in each bounded
subset S of X.
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7
Proof. For all x ∈ S, we observe that
m
∞
ωk
Tk x − ωk Tk x
Lm x − T x k1 Sm
k1
m
∞
ωk − ωk Sm
Tk x −
ωk Tk x
k1
Sm
km1
m
∞
1 − Sm
≤
ωk Tk x ωk Tk x
k1 Sm
km1
≤
m
∞
1 − Sm ωk Tk x ωk Tk x
Sm k1
km1
≤
∞
1 − Sm
MM
ωk ,
Sm
km1
2.8
where M supx∈S,k≥1 Tk x < ∞. Taking m → ∞ in above last inequality, we have that
lim Lm x − T x 0
2.9
m→∞
holds uniformly for x ∈ S and this completes the proof.
3. Main Results
Theorem 3.1. Let C be a nonempty closed convex subset of a real uniformly smooth Banach space
of nonexpansive mappings with ∞
∅, and {ωk }
X, {Tk }∞
k1 : C → C an infinite family
k1 FixTk /
n
∞
a sequence of positive numbers with k1 ωk 1. Let Ln k1 ωk /Sn Tk , Sn nk1 ωk , and
f ∈ ΠC with coefficient α ∈ 0, 1. Given any x0 ∈ C, let {xn } be a sequence generated by
xn1 αn fxn 1 − αn Ln xn ,
n ≥ 0,
3.1
where {αn } ⊂ 0, 1 satisfies the following conditions:
A1 limn → ∞ αn 0;
A2 ∞
n0 αn ∞;
A3 either ∞
n0 |αn1 − αn | < ∞ or limn → ∞ αn1 /αn 1.
Then {xn } converges strongly to x∗ ∈
∞
k1
FixTk , which solves the following variational inequality:
x∗ − fx∗ , J x∗ − p ≤ 0,
p∈
∞
k1
FixTk .
3.2
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Proof.
Step 1. We show that {xn } is bounded.
Noticing nonexpansiveness of Ln , take a p ∈ ∞
k1 FixTk to derive that
xn1 − p αn fxn 1 − αn Ln xn − p
≤ αn fxn − p 1 − αn Ln xn − p
αn fxn − f p f p − p 1 − αn Ln xn − p
≤ αn fxn − f p αn f p − p 1 − αn Ln xn − p
≤ ααn xn − p αn f p − p 1 − αn xn − p
1 − 1 − ααn xn − p αn f p − p 3.3
f p −p 1 − 1 − ααn xn − p 1 − ααn
1−α
f p −p , xn − p .
≤ max
1−α
By induction, we obtain
f p −p xn1 − p ≤ max
, x0 − p ,
1−α
n ≥ 0,
3.4
and {xn } is bounded, so are {Tk xn }, {Ln xn }, and {fxn }.
Step 2. We prove that limn → ∞ xn1 − xn 0.
By 3.1, We have
xn1 − xn αn fxn 1 − αn Ln xn − αn−1 fxn−1 − 1 − αn−1 Ln−1 xn−1 αn fxn − αn fxn−1 αn fxn−1 − αn−1 fxn−1 1 − αn Ln xn − 1 − αn Ln−1 xn−1 1 − αn Ln−1 xn−1 − 1 − αn−1 Ln−1 xn−1 ≤ ααn xn − xn−1 |αn − αn−1 |fxn−1 1 − αn Ln xn − Ln−1 xn−1 |αn − αn−1 |Ln−1 xn−1 ααn xn − xn−1 |αn − αn−1 | fxn−1 Ln−1 xn−1 1 − αn Ln xn − Ln−1 xn−1 ≤ ααn xn − xn−1 |αn − αn−1 | fxn−1 Ln−1 xn−1 1 − αn Ln xn − Ln xn−1 1 − αn Ln xn−1 − Ln−1 xn−1 Journal of Applied Mathematics
9
≤ 1 − 1 − ααn xn − xn−1 1 − αn Ln xn−1 − Ln−1 xn−1 |αn − αn−1 | fxn−1 Ln−1 xn−1 ≤ 1 − 1 − ααn xn − xn−1 1 − αn Ln xn−1 − Ln−1 xn−1 |αn − αn−1 |M,
3.5
where M supn≥1 fxn−1 Ln−1 xn−1 . At the same time, we observe that
∞ n
n−1
ωk
ωk
Tk xn−1 −
Tk xn−1 Ln xn−1 − Ln−1 xn−1 S
S
n1
n1 k1 n
k1 n−1
∞ n−1
−ωn ωk
ωn
Tk xn−1 Tn xn−1 S
S
S
n
n
n−1
n1
k1
∞
∞
n−1
ωn
ωn ωk
≤
T
x
T
x
n
n−1
k
n−1
S
S S
n
n1
k1 n n−1
∞
ωn
n−1
∞ ωn ωk
≤
Tn xn−1 Tk xn−1 S
S
S
n
n1
n1 k1 n n−1
≤
∞
ωn
n1
Sn
M
∞
2M
n1
Sn
∞
ωn
n1
Sn
3.6
M
ωn ,
where M supk≥1,n≥1 Tk xn−1 . Applying Lemma 2.6 and compatibility test of series, we have
∞
Ln xn−1 − Ln−1 xn−1 < ∞.
n1
3.7
Put
γn 1 − ααn ,
βn |αn − αn−1 |M
,
1 − ααn
3.8
δn 1 − αn Ln xn−1 − Ln−1 xn−1 .
It follows that
xn1 − xn ≤ 1 − γn xn − xn−1 γn βn δn .
3.9
10
Journal of Applied Mathematics
It is easily seen from A2, A3, and 3.7 that
∞
γn ∞,
∞
δn < ∞.
lim supβn ≤ 0,
n→∞
n1
n1
3.10
Applying Lemma 2.2 to 3.9, we obtain
lim xn1 − xn 0.
3.11
n→∞
Step 3. We show that limn → ∞ xn − T xn 0.
Indeed we observe that
xn − T xn ≤ xn − xn1 xn1 − T xn xn − xn1 αn fxn 1 − αn Ln xn − T xn ≤ xn − xn1 αn fxn − T xn 1 − αn Ln xn − T xn .
3.12
Hence, by 3.11, A1, and Lemma 2.7, we have
lim xn − T xn 0.
3.13
lim sup fx∗ − x∗ , Jxn − x∗ ≤ 0,
3.14
n→∞
Step 4. We prove that
n→∞
where x∗ limt → 0 xt with xt being the fixed point of the contraction
x −→ tfx 1 − tT x.
3.15
From Lemma 2.5, we have x∗ ∈ FixT and
By Lemma 2.4, we have x∗ ∈
I − f x∗ , J x∗ − p ≤ 0,
∞
k1
p ∈ FixT .
3.16
FixTk and
I − f x∗ , J x∗ − p ≤ 0,
p∈
∞
FixTk .
k1
3.17
By xt tfxt 1 − tT xt , we have
xt − xn t fxt − xn 1 − tT xt − xn .
3.18
Journal of Applied Mathematics
11
It follows from Lemma 2.3 that
2
xt − xn 2 tfxt − xn 1 − tT xt − xn ≤ 1 − t2 T xt − xn 2 2tfxt − xn , Jxt − xn ≤ 1 − t2 T xt − T xn T xn − xn 2 2tfxt − xt , Jxt − xn 2txt − xn , Jxt − xn ≤ 1 − t2 xt − xn T xn − xn 2 2tfxt − xt , Jxt − xn 3.19
2txt − xn , Jxt − xn ≤ 1 − t2 xt − xn 2 bn t 2tfxt − xt , Jxt − xn 2txt − xn 2 ,
where bn t T xn − xn 2xt − xn T xn − xn → 0 n → ∞. It follows from above last
inequality that
xt − fxt , Jxt − xn ≤
1
t
xt − xn 2 bn t.
2
2t
3.20
Taking n → ∞ in 3.20 yields
lim supxt − fxt , Jxt − xn ≤
n→∞
t
M,
2
3.21
where M ≥ xt − xn 2 for all n ≥ 1 and t ∈ 0, 1. Taking t → 0 in 3.21, we have
lim sup lim supxt − fxt , Jxt − xn ≤ 0.
t→0
n→∞
3.22
Noticing the fact that two limits are interchangeable due to the fact the duality mapping J is
norm-to-norm uniformly continuous on bounded sets, it follows from 3.22, we have
lim sup fx∗ − x∗ , Jxn − x∗ lim sup lim supxt − fxt , Jxt − xn n→∞
n→∞
t→0
lim sup lim supxt − fxt , Jxt − xn t→0
≤ 0.
Hence 3.14 holds.
Step 5. Finally, we prove that xn → x∗ n → ∞.
n→∞
3.23
12
Journal of Applied Mathematics
Indeed we observe that
2
xn1 − x∗ 2 αn fxn 1 − αn Ln xn − x∗ 2
αn fxn − fx∗ 1 − αn Ln xn − x∗ αn fx∗ − x∗ 2
≤ αn fxn − fx∗ 1 − αn Ln xn − x∗ 2αn fx∗ − x∗ , Jxn1 − x∗ 2
≤ αn fxn − fx∗ 1 − αn Ln xn − x∗ 2αn fx∗ − x∗ , Jxn1 − x∗ 3.24
≤ ααn xn − x∗ 1 − αn xn − x∗ 2
2αn fx∗ − x∗ , Jxn1 − x∗ ≤ 1 − 1 − ααn 2 xn − x∗ 2 2αn fx∗ − x∗ , Jxn1 − x∗ ≤ 1 − 1 − ααn xn − x∗ 2 2αn fx∗ − x∗ , Jxn1 − x∗ .
By view of 3.14 and condition A2, it follows from Lemma 2.2 that xn → x∗ n → ∞. This
completes the proof.
Corollary 3.2. Let C be a nonempty closed convex subset of a real uniformly smooth Banach space
of nonexpansive mappings with ∞
X, {Tk }∞
/ ∅, {ωk } a
k1 : C → C an infinite family
k1 FixTk
n
∞
sequence of positive numbers with k1 ωk 1. Let Ln k1 ωk /Sn Tk , Sn nk1 ωk , and
u ∈ C. Given any x0 ∈ C, let {xn } be a sequence generated by
xn1 αn u 1 − αn Ln xn ,
n ≥ 0,
3.25
where {αn } ⊂ 0, 1 satisfies the following conditions:
A1 limn → ∞ αn 0;
A2 ∞
n0 αn ∞;
A3 either ∞
n0 |αn1 − αn | < ∞ or limn → ∞ αn1 /αn 1.
Then {xn } converges strongly to x∗ ∈ ∞
k1 FixTk , which solves the following variational inequality:
x∗ − u, J x∗ − p ≤ 0,
p∈
∞
k1
FixTk .
3.26
Acknowledgment
This paper is supported by the Fundamental Research Funds for the Central Universities
ZXH2011D005.
Journal of Applied Mathematics
13
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