Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 819036, 12 pages
doi:10.1155/2009/819036
Research Article
Strong Convergence Theorems for
Common Fixed Points of Multistep Iterations
with Errors in Banach Spaces
Feng Gu1 and Qiuping Fu2
1
Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University,
Hangzhou, Zhejiang 310036, China
2
Mathematics group, West Lake High Middle School, Hangzhou, Zhejiang 310012, China
Correspondence should be addressed to Feng Gu, gufeng99@sohu.com
Received 19 November 2008; Revised 11 January 2009; Accepted 9 April 2009
Recommended by Yeol Je Cho
We establish strong convergence theorem for multi-step iterative scheme with errors for
asymptotically nonexpansive mappings in the intermediate sense in Banach spaces. Our results
extend and improve the recent ones announced by Plubtieng and Wangkeeree 2006, and many
others.
Copyright q 2009 F. Gu and Q. Fu. 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
Let C be a subset of real normal linear space X. A mapping T : C → C is said to be
asymptotically nonexpansive on C if there exists a sequence {rn } in 0, ∞ with limn → ∞ rn 0
such that for each x, y ∈ C,
n
T x − T n y ≤ 1 rn x − y,
∀n ≥ 1.
1.1
If rn ≡ 0, then T is known as a nonexpansive mapping. T is called asymptotically
nonexpansive in the intermediate sense 1 provided T is uniformly continuous and
lim sup sup T n x − T n y − x − y ≤ 0.
n → ∞ x,y∈C
1.2
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Journal of Inequalities and Applications
From the above definitions, it follows that asymptotically nonexpansive mapping
must be asymptotically nonexpansive in the intermediate sense.
Let C be a nonempty subset of normed space X, and Let Ti : C → C be m mappings.
For a given x1 ∈ C and a fixed m ∈ N N denotes the set of all positive integers, compute the
1
m
iterative sequences xn , . . . , xn defined by
1
1
1
1 1
xn αn Tik xn βn xn γn un ,
2
2
1
2
2 2
3
3
2
3
3 3
xn αn Tik xn βn xn γn un ,
xn αn Tik xn βn xn γn un ,
1.3
..
.
m−1
xn
m−1
αn
m
xn1 xn
m−2
Tik xn
m
m−1
αn Tik xn
1
2
m−1
βn
m
m−1 m−1
un
,
xn γn
m m
βn xn γn un ,
∀n ≥ 1,
m
i
where n k − 1m i, {un }, {un }, . . . , {un } are bounded sequences in C and {αn },
i
i
i
i
i
{βn }, {γn }, are appropriate real sequences in 0, 1 such that αn βn γn 1 for each
i ∈ {1, 2, . . . , m}.
The purpose of this paper is to establish a strong convergence theorem for common
fixed points of the multistep iterative scheme with errors for asymptotically nonexpansive
mappings in the intermediate sense in a uniformly convex Banach space. The results
presented in this paper extend and improve the corresponding ones announced by Plubtieng
and Wangkeeree 2, and many others.
2. Preliminaries
Definition 2.1 see 1. A Banach space X is said to be a uniformly convex if the modulus of
convexity of X is
x y
: x y 1, x − y > 0,
δX inf 1 −
2
∀ ∈ 0, 2.
2.1
Lemma 2.2 see 3. Let {an }, {bn }, and {γn } be three nonnegative real sequences satisfying the
following condition:
an1 ≤ 1 γn an bn ,
where
∞
n1 γn
< ∞ and
∞
n1 bn
< ∞. Then
1 limn → ∞ an exists;
2 If lim infn → ∞ an 0, then limn → ∞ an 0.
∀n ≥ 1,
2.2
Journal of Inequalities and Applications
3
Lemma 2.3 see 4. Let X be a uniformly convex Banach space and 0 < α ≤ tn ≤ β < 1 for all
n ≥ 1. Suppose that {xn } and {yn } are two sequences of X such that
lim supxn ≤ a,
n→∞
lim supyn ≤ a,
2.3
n→∞
lim tn xn 1 − tn yn a,
n→∞
for some a ≥ 0. Then
lim xn − yn 0.
2.4
n→∞
3. Main Results
Lemma 3.1. Let X be a uniformly convex Banach space, {xn }, {yn } are two sequences of X, α, β ∈
0, 1 and {αn } be a real sequence. If there exists n0 ∈ N such that
i 0 < α ≤ αn ≤ β < 1 for all n ≥ n0 ;
ii lim supn → ∞ xn ≤ a;
iii lim supn → ∞ yn ≤ a;
iv limn → ∞ αn xn 1 − αn yn a,
then limn → ∞ xn − yn 0.
Proof. The proof is clear by Lemma 2.3.
Lemma 3.2. Let X be a uniformly convex Banach space, let C be a nonempty closed bounded convex
subset of X, and let Ti : C → C be m asymptotically nonexpansive mappings in the intermediate
∅. Put
sense such that F m
i1 FTi /
Gik sup Tik x − Tik y − x − y ∨ 0,
∀k ≥ 1,
3.1
x,y∈C
i
i
i
so that ∞
k1 Gik < ∞. Let {αn }, {βn }, and {γn } be real sequences in 0, 1 satisfying the following
condition:
i
i
i
i αn βn γn 1 for all i ∈ {1, 2, . . . , m} and n ≥ 1;
i
ii ∞
n1 γn < ∞ for all i ∈ {1, 2, . . . , m}.
If {xn } is the iterative sequence defined by 1.3, then, for each p ∈ F limn → ∞ xn − p exists.
m
i1 FTi ,
the limit
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Journal of Inequalities and Applications
Proof. For each q ∈ F, we note that
1
1
1
1 1
xn − q αn Tik xn βn xn γn un − q
1 1 1 1
≤ αn Tik xn − q βn xn − q γn un − q
1 1
1 1 1
≤ αn xn − q αn Gik βn xn − q γn un − q
1
1 1
1 1
αn βn xn − q αn Gik γn un − q
3.2
1
≤ xn − q dn ,
1
1
1
1
where dn αn Gik γn un − q. Since
∞
Gik ∞
Gik < ∞,
n1
3.3
i∈I k1
we see that
∞
1
dn < ∞.
3.4
n1
It follows from 3.2 that
2
2 1
2
2 2 2
xn − q ≤ αn xn − q αn Gik βn xn − q γn un − q
2 1
2
2 2 2
≤ αn xn − q dn αn Gik βn xn − q γn un − q
2
2 2 1
2
2 2
αn βn xn − q αn dn αn Gik γn un − q
3.5
2
≤ xn − q dn ,
2
2 1
2
2
2
where dn αn dn αn Gik γn un − q. Since
∞
Gik < ∞,
∞
1
dn < ∞,
n1
n1
3.6
we see that
∞
2
dn < ∞.
n1
3.7
Journal of Inequalities and Applications
5
It follows from 3.5 that
3
3 2
3
3 3 3
xn − q ≤ αn xn − q αn Gik βn xn − q γn un − q
3 1
3
3 3 3
≤ αn xn − q dn αn Gik βn xn − q γn un − q
3
3 3 2
3
3 3
αn βn xn − q αn dn αn Gik γn un − q
3.8
3
≤ xn − q dn ,
3
3 2
3
3
3
where dn αn dn αn Gik γn un − q, and so
∞
3
dn < ∞.
3.9
n1
k
By continuing the above method, there are nonnegative real sequences {dn } such that
∞
k
dn < ∞,
3.10
n1
k
k
xn − q ≤ xn − q dn ,
∀k ∈ {1, 2, . . . , m}.
This together with Lemma 2.2 gives that limn → ∞ xn − q exists. This completes the proof.
Lemma 3.3. Let X be a uniformly convex Banach space, let C be a nonempty closed bounded convex
subset of X, and let Ti : C → C be m asymptotically nonexpansive mappings in the intermediate
sense such that F m
/ ∅. Put
i1 FTi Gik sup Tik x − Tik y − x − y ∨ 0,
∀k ≥ 1,
3.11
x,y∈C
i
i
i
so that ∞
k1 Gik < ∞. Let the sequence {xn } be defined by 1.3 whenever {αn }, {βn }, {γn } satisfy
the same assumptions as in Lemma 3.2 for each i ∈ {1, 2, . . . , m} and the additional assumption that
m−1
m
, αn ≤ β < 1 for all n ≥ n0 . Then we have the following:
there exists n0 ∈ N such that 0 < α ≤ αn
m−1
− xn 0;
m−2
− xn 0.
1 limn → ∞ Tik xn
2 limn → ∞ Tik xn
Proof. 1 Taking each q ∈ F, it follows from Lemma 3.2 that limn → ∞ xn − q exists. Let
lim xn − q a,
n→∞
3.12
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Journal of Inequalities and Applications
for some a ≥ 0. We note that
m−1
m−1
− q ≤ xn − q dn
,
xn
∀n ≥ 1,
3.13
where {dm−1
} is a nonnegative real sequence such that
n
∞
m−1
dn
< ∞.
3.14
n1
It follows that
m−1
lim supxn
− q ≤ lim supxn − q
n→∞
n→∞
lim xn − q
n→∞
3.15
a,
which implies that
m−1
m−1
lim supTik xn
− q ≤ lim sup xn
− q Gik
n→∞
n→∞
m−1
lim xn
− q
n→∞
3.16
≤ a.
Next, we observe that
k m−1
m
m
m−1
m m
− q γn un − xn ≤ Tik xn
− q γn un − xn .
Ti xn
3.17
Thus we have
m−1
m
m
lim supTik xn
− q γn un − xn ≤ a.
3.18
m
m
m m
xn − q γn un − xn ≤ xn − q γn un − xn 3.19
m
m
lim supxn − q γn un − xn ≤ a.
3.20
n→∞
Also,
gives that
n→∞
Journal of Inequalities and Applications
7
Note that
m
a lim xn − q
n→∞
m
m−1
m
m m
lim αn Tik xn
βn xn γn un − q
n→∞
m
m−1
m
m
1 − αn xn − γn xn
lim αn Tik xn
n→∞
m m
γn un
m
m − 1 − αn q − αn q
m
m−1
m
m m m
m m
lim αn Tik xn
− αn q αn γn un − αn γn xn
3.21
n→∞
m
m
m m
m m m
m m
1 − αn q − γn xn γn un − αn γn un αn γn xn m
m−1
m
m
lim αn Tik xn
− q γn un − xn
n→∞
m
m
m
xn − q γn un − xn .
1 − αn
This together with 3.18, 3.20, and Lemma 3.1, gives
m−1
lim Tik xn
− xn 0.
n→∞
3.22
This completes the proof of 1.
2 For each n ≥ 1,
m−1 m−1
xn − q − q
xn − Tik xn
Tik xn
m−1
m−1 ≤ xn − Tik xn
− q Gik .
xn
3.23
Since
m−1 lim xn − Tik xn
0 lim Gik ,
3.24
m−1
a lim xn − q ≤ lim infxn
− q.
3.25
n→∞
n→∞
we obtain
n→∞
n→∞
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Journal of Inequalities and Applications
It follows that
m−1
a ≤ lim infxn
− q
n→∞
m−1
≤ lim supxn
− q
3.26
n→∞
≤ a,
which implies that
m−1
lim xn
− q a.
3.27
n→∞
On the other hand, we note that
m−2
m−2
− q ≤ xn − q dn
,
xn
m−2
where {dn
∀n ≥ 1,
3.28
} is a nonnegative real sequence such that
∞
m−2
dn
< ∞.
3.29
n1
Thus we have
m−2
lim supxn
− q ≤ lim supxn − q
n→∞
n→∞
3.30
a,
and hence
m−2
m−2
lim supTik xn
− q ≤ lim sup xn
− q Gik
n→∞
n→∞
3.31
≤ a.
Next, we observe that
k m−2
m−1
m−1
m−2
m−1 m−1
un
− q γn
− xn ≤ Tik xn
− q γn
− xn .
un
Ti xn
3.32
Thus we have
m−2
m−1
m−1
un
lim supTik xn
− q γn
− xn ≤ a.
n→∞
3.33
Journal of Inequalities and Applications
9
Also,
m−1
m−1
m−1 m−1
un
− xn ≤ xn − q γn
− xn un
xn − q γn
3.34
m−1
m−1
lim supxn − q γn
un
− xn ≤ a.
3.35
gives that
n→∞
Note that
m−1
a lim xn
− q
n→∞
m−1 k
m−1
m−1 m−1
lim αn
Ti xn βn
xn γn
un
− q
n→∞
m−1 k m−2
m−1
m−1
Ti xn
un
− q γn
− xn
lim αn
3.36
n→∞
m−1
m−1
m−1
xn − q γn
un
1 − αn
− xn .
Therefore, it follows from 3.33, 3.35, and Lemma 3.1 that
m−2
lim Tik xn
− xn 0.
3.37
n→∞
This completes the proof.
Theorem 3.4. Let X be a uniformly convex Banach space and let C be a nonempty closed bounded
convex subset of X. Let Ti : C → C be m asymptotically nonexpansive mappings in the intermediate
∅ and there exists one member T in {Ti }m
sense such that F m
i1 which is completely
i1 FTi /
continuous. Put
Gik sup Tik x − Tik y − x − y ∨ 0,
∀k ≥ 1,
3.38
x,y∈C
i
i
i
so that ∞
k1 Gik < ∞. Let the sequence {xn } be defined by 1.3 whenever {αn }, {βn }, {γn } satisfy
the same assumptions as in Lemma 3.2 for each i ∈ {1, 2, . . . , m} and the additional assumption that
m−1
m
k
, αn ≤ β < 1 for all n ≥ n0 . Then {xn } converges
there exists n0 ∈ N such that 0 < α ≤ αn
m
strongly to a common fixed point of the mappings {Ti }i1 .
Proof. From Lemma 3.3, it follows that
m−1
m−2
lim Tik xn
− xn 0 lim Tik xn
− xn ,
n→∞
n→∞
3.39
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Journal of Inequalities and Applications
which implies that
m
xn1 − xn xn − xn m m−1
m−1 m−1
≤ αn Tik xn
− xn γn
− xn −→ 0,
un
3.40
n −→ ∞,
and so
xnl − xn −→ 0,
n −→ ∞.
3.41
It follows from 3.22, 3.37 that
k
m−1 m−1
− xn Tn xn − xn ≤ Tik xn − Tik xn
Tik xn
m−1 m−1
≤ xn − xn
− xn Gik Tik xn
m−1 m−1
≤
− xn Gik γn
− xn un
m−1
Tik xn
− xn −→ 0, n −→ ∞.
m−1 k m−2
αn
Ti xn
3.42
Let σn Tik xn − xn for all n > n0 . Then we have
xn − Tn xn ≤ xn − Tnk xn Tnk xn − Tn xn ≤ xn − Tik xn LTnk−1 xn − xn k−1
k−1
≤ σn L Tnk−1 xn − Tn−m
xn−m Tn−m
xn−m − xn−m xn−m − xn .
3.43
Notice that n ≡ n − mmodm. Thus Tn Tn−m and the above inequality becomes
xn − Tn xn ≤ σn L2 xn − xn−m Lσn−m xn−m − xn ,
3.44
lim xn − Tn xn 0.
3.45
and so
n→∞
Since
xn − Tnl xn ≤ xn − xnl xnl − Tnl xnl Tnl xnl − Tnl xn ≤ 1 Lxn − xnl xnl − Tnl xnl ,
∀l ∈ {1, 2, . . . , m},
3.46
Journal of Inequalities and Applications
11
we have
lim xn − Tnl xn 0,
n→∞
∀l ∈ {1, 2, . . . , m},
3.47
and so
lim xn − Tl xn 0,
n→∞
∀l ∈ {1, 2, . . . , m}.
3.48
Since {xn } is bounded and one of Ti is completely continuous, we may assume that T1 is
completely continuous, without loss of generality. Then there exists a subsequence {T1 xnk } of
{T1 xn } such that T1 xnk → q ∈ C as k → ∞. Moreover, by 3.48, we have
lim xnk − T1 xnk 0,
n→∞
3.49
which implies that xnk → q as k → ∞. By 3.48 again, we have
q − Tl q lim xn − Tl xn 0,
k
k
n→∞
∀l ∈ {1, 2, . . . , m}.
3.50
It follows that q ∈ F. Since limn → ∞ xn − q exists, we have
lim xn − q 0,
n→∞
3.51
that is,
m
lim xn
n→∞
lim xn q.
n→∞
3.52
Moreover, we observe that
k
k
xn − q ≤ xn − q dn ,
3.53
for all k 1, 2, . . . , m − 1 and
lim dn 0.
k
3.54
k
3.55
n→∞
Therefore,
lim xn q,
n→∞
for all k 1, 2, . . . , m − 1. This completes the proof.
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Journal of Inequalities and Applications
Remark 3.5. Theorem 3.4 improves and extends the corresponding results of Plubtieng and
Wangkeeree 2 in the following ways.
1 The iterative process {xn } defined by 1.3 in 2 is replaced by the new iterative
process {xn } defined by 1.3 in this paper.
2 Theorem 3.4 generalizes Theorem 3.4 of Plubtieng and Wangkeeree 2 from
a asymptotically nonexpansive mappings in the intermediate sense to a finite family of
asymptotically nonexpansive mappings in the intermediate sense.
Remark 3.6. If m 3 and T1 T2 T3 T in Theorem 3.4, we obtain strong convergence
theorem for Noor iteration scheme with error for asymptotically nonexpansive mapping T in
the intermediate sense in Banach space, we omit it here.
References
1 K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”
Proceedings of the American Mathematical Society, vol. 35, no. 1, pp. 171–174, 1972.
2 S. Plubtieng and R. Wangkeeree, “Strong convergence theorems for multi-step Noor iterations with
errors in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 321, no. 1, pp. 10–23,
2006.
3 Q. Liu, “Iterative sequences for asymptotically quasi-nonexpansive mappings with error member,”
Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 18–24, 2001.
4 J. Schu, “Iterative construction of fixed points of strictly pseudocontractive mappings,” Applicable
Analysis, vol. 40, no. 2-3, pp. 67–72, 1991.