Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2008, Article ID 350483, 10 pages
doi:10.1155/2008/350483
Research Article
An Implicit Iterative Scheme for an Infinite
Countable Family of Asymptotically Nonexpansive
Mappings in Banach Spaces
Shenghua Wang, Lanxiang Yu, and Baohua Guo
School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
Correspondence should be addressed to Shenghua Wang, sheng-huawang@hotmail.com
Received 6 May 2008; Accepted 24 August 2008
Recommended by William Kirk
Let K be a nonempty closed convex subset of a reflexive Banach space E with a weakly continuous
dual mapping, and let {Ti }∞
i1 be an infinite countable family of asymptotically nonexpansive
mappings with the sequence {kin } satisfying kin ≥ 1 for each i 1, 2, . . ., n 1, 2, . . ., and
limn→∞ kin 1 for each i 1, 2, . . .. In this paper, we introduce a new implicit iterative scheme
generated by {Ti }∞
i1 and prove that the scheme converges strongly to a common fixed point of
,
which
solves
some certain variational inequality.
{Ti }∞
i1
Copyright q 2008 Shenghua Wang 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.
1. Introduction and preliminaries
Let E be a Banach space and let K be a nonempty closed convex subset of E. Let T : K→K be
a mapping. Then T is called nonexpansive if
T x − T y ≤ x − y
1.1
for all x, y ∈ K. T is called asymptotically nonexpansive if there exists a sequence {kn } ⊂
1, ∞ that converges to 1 as n→∞ such that
n
T x − T n y ≤ kn x − y
1.2
for all x, y ∈ K and all n ≥ 1. Obviously, a nonexpansive mapping is asymptotically
nonexpansive. In 1, Goebel and Kirk originally introduced the concept of asymptotically
nonexpansive mappings and proved that if E is a uniformly convex Banach space and K is
a nonempty closed convex bounded subset of E, then every asymptotically nonexpansive
2
Fixed Point Theory and Applications
self-mapping on K has a fixed point. After that, many authors began to study the convergence
of the iterative scheme generated by asymptotically nonexpansive mappings 2–12.
In 8, the authors introduced an iterative scheme generated by a finite family of
asymptotically nonexpansive mappings:
xn αn xn−1 1 − αn Trlnn 1 xn ,
n ≥ 1,
1.3
where {αn } is a sequence in 0, 1, {Ti }N
i1 : K→K are N asymptotically nonexpansive
mappings, where K is a nonempty closed convex subset of a uniformly convex Banach
space satisfying Opial’s condition 13, and where n ln N rn for some integers ln ≥ 0
φ, then {xn } generated by 1.3
and 1 ≤ rn ≤ N. Then the authors proved that if ∩N
i1 FTi /
strongly converges to a common fixed point of {Ti }N
.
i1
Let K be a nonempty closed convex subset of a uniformly convex Banach space E. Let
S : K→K be a nonexpansive mapping and let T : K→K be an asymptotically nonexpansive
mapping. In 10, the authors introduced the following modified Ishikawa iteration sequence
with errors with respect to S and T :
yn a
n Sxn bn
T n xn cn
vn ,
xn1 an Sxn bn T n yn cn un ,
∀n ≥ 1,
1.4
where {a
n }, {bn
}, {cn
} are three real numbers sequences in 0, 1 satisfying a
n bn
cn
1, {an }, {bn }, {cn } are also three real numbers sequences in 0, 1 satisfying an bn cn 1,
and {un } and {vn } are given bounded sequences in K. Then the authors proved that the
sequence {xn } generated by 1.4 strongly converges to a common fixed point of S and T if
some certain conditions are satisfied.
Let K be a nonempty closed convex subset of a Banach space E and let f : K→K be a
contraction with efficient λ 0 < λ < 1 such that
fx − fy ≤ λx − y
1.5
for all x, y ∈ K. Shahzad and Udomene 9 studied the following implicit and explicit
iterative schemes for an asymptotically nonexpansive mapping T with the sequence {kn }
in a uniformly smooth Banach space:
tn
f xn kn
tn
f xn 1−
kn
xn xn1
1−
tn n
T xn ,
kn
tn n
T xn ,
kn
1.6
where {tn } is a sequence in 0, 1. They proved that the sequence {xn } converges strongly to
the unique solution of some variational inequality if the sequence {tn } satisfies some certain
conditions and the mapping T satisfies T xn − xn →0 as n→∞.
Quite recently, Ceng et al. 12 introduced the following two implicit and explicit
iterative schemes generated by a finite family of asymptotically nonexpansive mappings
Shenghua Wang et al.
3
{Ti }N
i1 with the same sequence {kn } in a reflexive Banach space with a weakly continuous
duality map:
1
1 − tn f xn xn xn 1 −
kn
kn
1
1 − tn xn1 1 −
f xn xn kn
kn
tn n
T xn ,
kn rn
tn n
T xn ,
kn rn
1.7
where rn n mod N and {tn } is a sequence in 0, 1. Then they proved that if the control
sequence {tn } satisfies some certain condition and Ti xn −xn →0 as n→∞ for each i 1, 2, . . . , N,
then both schemes 1.7 strongly converge a common fixed point x∗ of {Ti }N
i1 which solves
the variational inequality
I − fx∗ , J p − x∗ ≥ 0,
p∈
N
F Ti ,
1.8
i1
where FTi denotes the set of fixed points of the mapping Ti for each i 1, 2, . . . , N.
Let E be a Banach space and let E∗ be the dual space of E. Given a continuous strictly
increasing function ϕ : R →R such that ϕ0 0 and limt→∞ ϕt ∞, we associate a
∗
possibly multivalued generalized duality map Jϕ : E→2E , defined as
Jϕ x x∗ ∈ E∗ : x∗ x xϕ x , x∗ ϕ x
1.9
for every x ∈ E. We call the function ϕ a gauge. If ϕt t for all t ≥ 0, then we call Jϕ a
normalized duality mapping and write it as J.
A Banach space E is said to have a weakly continuous generalized duality map if there
exists a continuous strictly increasing function ϕ : R →R such that ϕ0 0, limt→∞ ϕt ∞,
and Jϕ is single valued and sequentially continuous from E with the weak topology to E∗
with the weak∗ topology. For instance, every lp -space 1 < p < ∞ has a weakly continuous
generalized duality map for ϕt tp−1 .
t
For each t ≥ 0, let Φt 0 ϕxdx. The following property may be seen in many
literatures.
Property 1.1. Let E be a real Banach space and let Jϕ be the duality map associated with the
gauge ϕ. Then for all x, y ∈ E and jx y ∈ Jϕ x y one holds
Φ x y ≤ Φ x y, jx y .
1.10
x y2 ≤ x2 2 y, jx y
1.11
One also holds
for all x, y ∈ E and jx y ∈ Jx y.
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Fixed Point Theory and Applications
Lemma 1.2 see 14. Let E be a Banach space satisfying a weakly continuous duality map and let K
be a nonempty closed convex subset of E. Let T : K→K be an asymptotically nonexpansive mapping
with fixed point. Then I − T is demiclosed at zero.
2. Strong convergence results
In this section, let E be a reflexive Banach space with a weakly continuous duality map Jϕ ,
where ϕ is a gauge and let K be a nonempty closed convex subset of E. Let {Ti }∞
i1 : K→K be
an infinite countable family of asymptotically nonexpansive mappings such that
n
T x − T n y ≤ kin x − y
i
i
2.1
for all x, y ∈ K, where the sequence {kin } ⊂ 1, ∞ and limn→∞ kin 1 for each i 1, 2, . . . .
For each n 1, 2, . . . , let bn
sup{kin | i 1, 2, . . .} and assume
sup bn
| n 1, 2, . . . < ∞,
lim bn
b < ∞.
2.2
n→∞
Taking bn max{bn
, b} for each n 1, 2, . . . , obviously, we have
lim bn b ≥ 1,
b sup bn | n 1, 2, . . . < ∞.
n→∞
2.3
Moreover, the following inequality
n
T x − T n y ≤ bn x − y
i
i
2.4
holds for all x, y ∈ K and each i 1, 2 . . . .
Take an integer r > 1 arbitrarily. For each n ≥ 1, define the mapping Sni : K→K by
Sni Tn−1ri
2.5
S11 T1 , . . . , S1r Tr , S21 Tr1 , . . . , S2r T2r , . . . .
2.6
for each i 1, 2, . . . , r, that is,
Shenghua Wang et al.
5
For each i 1, 2, . . . , r, let {αni } ⊂ 0, 1 be a sequence real numbers. For each n ≥ 1,
define the mapping Wn of K into itself by
Wn Unr αnr Snnr Unr−1 1 − αnr I,
2.7
where
Un1 αn1 Snn1 1 − αn1 I,
Un2 αn2 Snn2 Un1 1 − αn2 I,
2.8
..
.
Unr−1 αnr−1 Snnr−1 Unr−2 1 − αnr−1 I.
We call Wn a W-mapping generated by Sn1 , Sn2 , . . . , Snr and αn1 , αn2 , . . . , αnr.
Let f : K→K be a λ-contraction with 0 < λ < 1/b
r . Take a sequence of real
numbers{tn } ⊂ 0, b such that
lim tn 0,
n→∞
tn <
b1 − bnr λ
,
1 − λbnr
n ≥ 1.
2.9
Note that since λ < 1/b
r , one has 0 < b1 − bnr λ/1 − λbnr ≤ b. Therefore, the sequence {tn }
can be taken easily to satisfy the condition 2.9, for example, tn 1/nb1 − bnr λ/1 − λbnr .
Then, we introduce an implicit iterative scheme
xn 1−
b − tn tn
xn r1 f Wn xn r1 Wn xn ,
r1
bn
bn
bn
b
n ≥ 1.
2.10
By using the following lemmas, we will prove that the implicit scheme 2.10 is well defined.
Lemma 2.1. Let {Ti }∞
i1 : K→K be an infinite countable family of asymptotically nonexpansive
mappings with the sequences {kin } and let Wn be a W-mapping generated by 2.7 for each n φ, then ∩∞
1, 2, . . . . If ∩∞
i1 FTi /
i1 FTi ⊂ FWn for each n 1, 2, . . . .
Proof. The conclusion is obtained directly from the definition of Wn .
Lemma 2.2. Let {Ti }∞
i1 : K→K with the sequences {kin } and let Wn be the W-mapping generated
by 2.7 for each n 1, 2, . . . . Then one holds
Wn x − Wn y ≤ bnr x − y
for all n ≥ 1 and all x, y ∈ K.
2.11
6
Fixed Point Theory and Applications
Proof. For any x, y ∈ K all n ≥ 1, we first see noting that bn ≥ 1
Un1 x − Un1 y αn1 Sn 1 − αn1 I x − αn1 Sn 1 − αn1 I y
n1
n1
≤ αn1 Snn1 x − Snn1 y 1 − αn1 x − y
n
n
αn1 Tn−1r1
x − Tn−1r1
y 1 − αn1 x − y
≤ αn1 kn−1r1n x − y 1 − αn1 x − y
≤ αn1 bn x − y 1 − αn1 x − y
≤ αn1 bn x − y 1 − αn1 bn x − y
bn x − y,
Un2 x − Un2 y αn2 Sn Un1 1 − αn2 I x − αn2 Sn Un1 1 − αn2 I y
n2
n2
≤ αn2 Snn2 Un1 x − Snn2 Un1 y 1 − αn2 x − y
n
n
αn2 Tn−1r2
Un1 x − Tn−1r2
Un1 y 1 − αn2 x − y
≤ αn2 kn−1r2n Un1 x − Un1 y 1 − αn2 x − y
≤ αn2 bn Un1 x − Un1 y 1 − αn2 x − y
≤ αn2 bn2 x − y 1 − αn1 bn2 x − y
2.12
bn2 x − y.
Similarly, for each i 3, . . . , r − 1, we have
Uni x − Uni y ≤ bni x − y.
2.13
Hence,
Wn x − Wn y αnr Snnr Un r−1 1 − αnr I x − αnr Snnr Un r−1 1 − αnr I y
≤ αnr Snnr Un r−1 x − Snnr Un r−1 y 1 − αnr x − y
≤
bnr x
2.14
− y.
This completes the proof.
Now we prove that the implicit scheme 2.10 is well defined. Since 0 < tn < b1 −
bnr λ/1 − λbnr , we obtain
b
b − tn
tn
λ
< 1.
bn
bn
2.15
tn
b − tn x r1 f Wn x r1 Wn x
r1
bn
bn
bn
2.16
0<1−
bnr1
Hence, the mapping
x → T x :
1−
b
Shenghua Wang et al.
7
is a contraction on K. In fact, to see this, taking any x, y ∈ K, by Lemma 2.2 we have
b
tn b − tn 1
−
T x − T y x
−
f
W
y
x
−
W
y
x
−
y
fW
W
n
n
n
n
bnr1
bnr1
bnr1
b − tn λbnr
b
tn
≤ 1 − r1 x − y x − y r1 bnr x − y
r1
bn
bn
bn
2.17
b − tn
tn
b
x − y
λ
1 − r1 b
b
bn
n
n
≤ x − y,
which implies that the implicit scheme 2.10 is well defined.
For the implicit scheme 2.10, we have strong convergence as follows.
Theorem 2.3. Assume 2.9, FT ∩∞
φ and limn→∞ xn −Ti xn 0 for each i 1, 2, . . . .
i1 FTi /
Then {xn } converges strongly to a common fixed point x ∈ FT , where x solves the variational
inequality
I − fx, Jp − x ≥ 0,
p ∈ FT .
2.18
Proof. First, we prove that {xn } is bounded. By using Property 1.1, Lemmas 2.1, 2.2, for any
z ∈ FT , we have noting 0 < 1 − b/bnr1 b − tn /bn λ tn /bn < 1
b 1
−
xn − z xn − z r1
bn
2
b − tn r1 fz − z bn
b 1
−
≤
xn − z r1
bn
2
b − tn tn f Wn xn − fz r1 Wn xn − z
r1
bn
bn
2
b − tn tn f Wn xn − fz r1 Wn xn − z bnr1
bn
2b − tn fz − z, jxn − z
r1
bn
b − tn 2
tn b ≤
1 − r1 xn − z r1 f Wn xn − f Wn z r1 Wn xn − Wn z
bn
bn
bn
2b − tn fz − z, j xn − z
r1
bn
b
b − tn λ tn 2 xn − z2 2b − tn fz − z, j xn − z
≤ 1 − r1 r1
bn
bn
bn
bn
b
b − tn λ tn 2b − tn 2
≤ 1 − r1 xn − z fz − z, j xn − z
r1
b
b
bn
bn
n
n
2 2b − tn 1 − ηn xn − z fz − z, j xn − z ,
r1
bn
2.19
8
Fixed Point Theory and Applications
where
ηn b
bnr1
−
b − tn
tn
λ−
> 0.
bn
bn
2.20
It follows from 2.19 that
xn − z2 ≤ 2b − tn fz − z, jxn − z .
ηn bnr1
2.21
Since limn→∞ bn b, limn→∞ tn 0, we have
lim
b − tn
n→∞ η br1
n n
1
.
1 − λbr
2.22
Hence, {xn } is bounded.
Now we prove that {xn } strongly converges to a common fixed point x ∈ FT . To see
this, we assume that x is a weak limit point of {xn } and a subsequence {xnj } of {xn } converges
weakly to x. Then by the assumption of the theorem and Lemma 1.2, we have x ∈ FTi for
every i 1, 2, . . . . In 2.21, replacing xn with xnj and z with x, respectively, and then taking
the limit as j→∞, we obtain by the weak continuity of the duality map J
lim xnj − x 0.
2.23
j→∞
Therefore, xnj →x. We further show that x solves the variational inequality
I − fx, Jp − x ≥ 0,
p ∈ FT .
2.24
To see this result, taking any p ∈ FT , then by using Property 1.1, Lemmas 2.1 and 2.2 we
compute
Φ xn − p
b − tn b − tn b tn 1
−
Φ x
x
W
f
W
−
x
−
p
−
p
x
−
p
x
n
n
n
n
n
n
n
bnr1
bnr1
bnr1
bnr1
tn b − tn tn ≤Φ 1 − r1 xn − p r1 Wn xn − p r1 f Wn xn − xn , Jϕ xn − p
bn
bn
bn
b − tn tn
≤ 1 − r1 tn Φ xn − p r1 fWn xn − xn , Jϕ xn − p ,
bn
bn
2.25
Shenghua Wang et al.
9
which implies that
br1 − 1tn Φ xn − p .
xn − f Wn xn , Jϕ xn − p ≤ n
b − tn
2.26
Now in 2.26, replacing xn with xnj and noting limn→∞ bn b and limn→∞ tn 0, we obtain
x − fx, Jϕ x − p lim xnj − f Wnj xnj , Jϕ xnj − p
j→∞
≤ lim sup
bnr1
− 1tnj
j
j→∞
b − tnj
Φ xnj − p 0,
2.27
which implies that x is a solution to 2.24.
Finally, we prove that the sequence {xn } strongly converges to x. It suffices to prove
that the variational inequality 2.24 can have only one solution. To see this, assuming that
both u ∈ FT and v ∈ FT are solutions to 2.24, we have
I − fu, Ju − v ≤ 0,
I − fv, Jv − u ≤ 0.
2.28
I − fu − I − fv, Ju − v ≤ 0.
2.29
Adding them yields
However, since f is a λ-contraction, we have that
1 − λu − v2 ≤ I − fu − I − fv, Ju − v ,
2.30
which implies that u v. This completes the proof.
Remark 2.4. In Theorem 2.3, the condition that limn→∞ Ti xn − xn 0 for each i 1, 2, . . .
is necessary see 9, 12. This theorem shows that if for each n 1, 2, . . ., the supremum
of the sequence {kin }, that is, sup{kin | i 1, 2, . . .}, is finite and the limit of the sequence
sup {kin | i 1, 2, . . .}∞
n1 exists, then by choosing the contraction constant λ and the control
sequence {tn } we can obtain the common fixed point of {Ti }∞
i1 .
Corollary 2.5. Let {Ti }N
i1 K→K be a finite family of asymptotically nonexpansive mappings with the
sequences {kin } and let Wn be a W-mapping generated by T1 , T2 , . . . , TN and αn1 , αn2 , . . . , αnN for
each n 1, 2, . . . . Let the sequence {tn } ⊂ 0, 1 and satisfy tn < 1−knN λ/1−λknN and tn →0, where
kn max{k1n , k2n , . . . , kNn } for each n 1, 2, . . . . Assume that k sup{kn | n 1, 2, . . .} < ∞. Let
f be a contraction with λ0 < λ < 1/kN . Consider the implicit iterative scheme
xn 1−
1
knN1
xn 1 − tn tn
f Wn xn N1 Wn xn .
knN1
kn
2.31
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Fixed Point Theory and Applications
N
φ and Ti xn − xn →0 as n→∞ for each i 1, 2, . . . , N,
If {Ti }N
i1 satisfy the condition ∩i1 FTi /
then {xn } converges strongly to a common fixed point x ∈ ∩N
i1 FTi , where x solves the variational
inequality
I − fx, Jp − x ≥ 0,
p∈
N
F Ti .
2.32
i1
Proof. In Theorem 2.3, take bn kn , b limn→∞ kn 1, b
k, and r N. Then, this corollary
can obtained directly from Theorem 2.3.
Acknowledgment
The work was supported by Youth Foundation of North China Electric Power University.
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