Volume 2009, Article ID 819036,12pages doi:10.1155/2009/819036
Research Article
Strong Convergence Theorems for
Common Fixed Points of Multistep Iterations
with Errors in Banach Spaces
Feng Gu
1and Qiuping Fu
21Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University,
Hangzhou, Zhejiang 310036, China
2Mathematics group, West Lake High Middle School, Hangzhou, Zhejiang 310012, China
Correspondence should be addressed to Feng Gu,[email protected]
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 Wangkeeree2006, and many others.
Copyrightq2009 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 onCif there exists a sequence{rn}in0,∞with limn→ ∞rn0
such that for eachx, y∈C,
Tnx−Tny≤1rnx−y, ∀n≥1. 1.1
If rn ≡ 0, then T is known as a nonexpansive mapping. T is called asymptotically nonexpansive in the intermediate sense1providedT is uniformly continuous and
lim sup
n→ ∞
sup
x,y∈C
From the above definitions, it follows that asymptotically nonexpansive mapping must be asymptotically nonexpansive in the intermediate sense.
LetCbe a nonempty subset of normed spaceX, and LetTi :C → Cbemmappings. For a givenx1∈Cand a fixedm∈NNdenotes the set of all positive integers, compute the iterative sequencesxn1, . . . , xnmdefined by
xn1αn1Tikxnβ
1
n xnγn1un1,
xn2αn2Tikx 1
n βn2xnγn2un2,
xn3αn3Tikx
2
n βn3xnγn3un3,
.. .
xnm−1αnm−1Tikx m−2
n βnm−1xnγnm−1unm−1, xn1 xnmαnmTikx
m−1
n βnmxnγnmunm, ∀n≥1,
1.3
where n k −1m i, {un1},{un2}, . . . ,{unm} are bounded sequences in C and {αni}, {βi
n },{γni}, are appropriate real sequences in0,1such thatαniβniγni 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 Wangkeeree2, and many others.
2. Preliminaries
Definition 2.1see1. A Banach spaceX is said to be a uniformly convex if the modulus of convexity ofXis
δX inf
1− xy
2 : x y1, x−y
>0, ∀∈0,2. 2.1
Lemma 2.2see3. Let{an},{bn}, and{γn}be three nonnegative real sequences satisfying the following condition:
an1 ≤
1γnanbn, ∀n≥1, 2.2
where∞n1γn<∞and∞n1bn<∞. Then
1limn→ ∞anexists;
Lemma 2.3see4. LetX be a uniformly convex Banach space and0 < α ≤ tn ≤ β < 1for all
n≥1. Suppose that{xn}and{yn}are two sequences ofXsuch that
lim sup
n→ ∞ xn ≤a,
lim sup
n→ ∞
yn≤a,
lim
n→ ∞tnxn 1−tnyna,
2.3
for somea≥0. Then
lim
n→ ∞xn−yn0. 2.4
3. Main Results
Lemma 3.1. LetXbe a uniformly convex Banach space,{xn},{yn}are two sequences ofX,α, β ∈ 0,1and{αn}be a real sequence. If there existsn0∈Nsuch that
i0< α≤αn≤β <1for alln≥n0; iilim supn→ ∞ xn ≤a;
iiilim supn→ ∞ yn ≤a;
ivlimn→ ∞ αnxn 1−αnyn a,
thenlimn→ ∞ xn−yn 0.
Proof. The proof is clear byLemma 2.3.
Lemma 3.2. LetXbe a uniformly convex Banach space, letCbe a nonempty closed bounded convex subset ofX, and letTi : C → Cbemasymptotically nonexpansive mappings in the intermediate sense such thatF mi1FTi/∅. Put
Gik sup
x,y∈C
Tk
i x−Tiky−x−y∨0, ∀k≥1, 3.1
so that∞k1Gik<∞. Let{αni},{βni}, and{γni}be real sequences in0,1satisfying the following
condition:
iαniβniγni1for alli∈ {1,2, . . . , m}andn≥1;
ii∞n1γni <∞for alli∈ {1,2, . . . , m}.
If {xn} is the iterative sequence defined by 1.3, then, for each p ∈ F mi1FTi, the limit
Proof. For eachq∈F, we note that
xn1−qαn1Tikxnβ
1
n xnγn1un1−q
≤αn1Tikxn−qβ
1
n xn−qγn1un1−q
≤αn1xn−qαn1Gikβn1xn−qγn1un1−q
αn1βn1
xn−qαn1Gikγn1un1−q
≤xn−qdn1,
3.2
wheredn1αn1Gikγn1 un1−q . Since
∞
n1
Gik i∈I
∞
k1
Gik<∞, 3.3
we see that
∞
n1
dn1<∞. 3.4
It follows from3.2that
xn2−q≤αn2xn1−qαn2Gikβn2xn−qγn2un2−q
≤αn2 xn−qdn1
αn2Gikβn2xn−qγn2un2−q
αn2βn2
xn−qαn2dn1αn2Gikγn2un2−q
≤xn−qdn2,
3.5
wheredn2αn2dn1αn2Gikγn2 un2−q . Since
∞
n1
Gik <∞,
∞
n1
dn1<∞, 3.6
we see that
∞
n1
It follows from3.5that
xn3−q≤αn3xn2−qαn3Gikβn3xn−qγn3un3−q
≤αn3 xn−qdn1
αn3Gikβn3xn−qγn3un3−q
αn3βn3
xn−qαn3dn2αn3Gikγn3un3−q
≤xn−qdn3,
3.8
wheredn3αn3dn2αn3Gikγn3 un3−q , and so
∞
n1
dn3<∞. 3.9
By continuing the above method, there are nonnegative real sequences{dnk}such that
∞
n1
dnk<∞,
xnk−q≤xn−qdnk, ∀k∈ {1,2, . . . , m}.
3.10
This together withLemma 2.2gives that limn→ ∞ xn−q exists. This completes the proof.
Lemma 3.3. LetXbe a uniformly convex Banach space, letCbe a nonempty closed bounded convex subset ofX, and letTi : C → Cbemasymptotically nonexpansive mappings in the intermediate sense such thatF mi1FTi/∅. Put
Gik sup
x,y∈C
Tikx−Tiky−x−y∨0, ∀k≥1, 3.11
so that∞k1Gik<∞. Let the sequence{xn}be defined by1.3whenever{αni},{βni},{γni}satisfy
the same assumptions as inLemma 3.2for eachi∈ {1,2, . . . , m}and the additional assumption that there existsn0∈Nsuch that0< α≤αnm−1, αnm≤β <1for alln≥n0. Then we have the following:
1limn→ ∞ Tikx
m−1
n −xn 0;
2limn→ ∞ Tikx
m−2
n −xn 0.
Proof. 1Taking eachq∈F, it follows fromLemma 3.2that limn→ ∞ xn−q exists. Let
lim
for somea≥0. We note that
xnm−1−q≤xn−qdnm−1, ∀n≥1, 3.13
where{dnm−1}is a nonnegative real sequence such that
∞
n1
dnm−1<∞. 3.14
It follows that
lim sup
n→ ∞
xnm−1−q≤lim sup n→ ∞
xn−q
lim
n→ ∞xn−q
a,
3.15
which implies that
lim sup
n→ ∞
Tikxnm−1−q≤lim sup n→ ∞
xnm−1−qGik
lim
n→ ∞
xnm−1−q
≤a.
3.16
Next, we observe that
Tikxnm−1−qγnm unm−xn≤Tikx
m−1
n −qγnm unm−xn. 3.17
Thus we have
lim sup
n→ ∞
Tikx
m−1
n −qγnm unm−xn≤a. 3.18
Also,
xn−qγnm unm−xn≤xn−qγnmunm−xn 3.19
gives that
lim sup
n→ ∞
Note that
a lim
n→ ∞
xnm−q
lim
n→ ∞
αnmTikx
m−1
n βnmxnγnmunm−q
lim
n→ ∞
αnmTikx
m−1
n 1−αnm
xn−γnmxn
γnmunm− 1−αnm
q−αnmq
lim
n→ ∞
αnmTikx m−1
n −αnmqαnmγnmunm−αnmγnmxn
1−αnm
q−γnmxnγnmunm−αnmγnmunmαnmγnmxn
lim
n→ ∞
αnm Tikx
m−1
n −qγnm unm−xn
1−αnm xn−qγnm unm−xn.
3.21
This together with3.18,3.20, andLemma 3.1, gives
lim
n→ ∞
Tikx
m−1
n −xn0. 3.22
This completes the proof of1.
2For eachn≥1,
xn−qxn−Tk
i x m−1
n Tikx m−1
n −q
≤xn−Tk i x
m−1
n xnm−1−qGik.
3.23
Since
lim
n→ ∞
xn−Tikxnm−10nlim→ ∞Gik, 3.24
we obtain
a lim
n→ ∞xn−q≤lim infn→ ∞
It follows that
a≤lim inf
n→ ∞
xnm−1−q
≤lim sup
n→ ∞
xnm−1−q
≤a,
3.26
which implies that
lim
n→ ∞
xnm−1−qa. 3.27
On the other hand, we note that
xnm−2−q≤xn−qdnm−2, ∀n≥1, 3.28
where{dnm−2}is a nonnegative real sequence such that
∞
n1
dnm−2<∞. 3.29
Thus we have
lim sup
n→ ∞
xnm−2−q≤lim sup n→ ∞
xn−q
a,
3.30
and hence
lim sup
n→ ∞
Tikxnm−2−q≤lim sup n→ ∞
xnm−2−qGik
≤a.
3.31
Next, we observe that
Tikxnm−2−qγnm−1 unm−1−xn≤Tikx
m−2
n −qγnm−1unm−1−xn. 3.32
Thus we have
lim sup
n→ ∞
Also,
xn−qγnm−1 unm−1−xn≤xn−qγnm−1unm−1−xn 3.34
gives that
lim sup
n→ ∞
xn−qγnm−1 unm−1−xn≤a. 3.35
Note that
a lim
n→ ∞
xnm−1−q
lim
n→ ∞
αnm−1Tikxnβ
m−1
n xnγnm−1unm−1−q
lim
n→ ∞
αnm−1 Tikx
m−2
n −qγnm−1 unm−1−xn
1−αnm−1 xn−qγnm−1 unm−1−xn.
3.36
Therefore, it follows from3.33,3.35, andLemma 3.1that
lim
n→ ∞
Tikxnm−2−xn0. 3.37
This completes the proof.
Theorem 3.4. LetX be a uniformly convex Banach space and letCbe a nonempty closed bounded convex subset ofX. LetTi :C → Cbemasymptotically nonexpansive mappings in the intermediate sense such that F mi1FTi/∅ and there exists one memberT in {Ti}mi1which is completely continuous. Put
Gik sup
x,y∈C
Tikx−Tiky−x−y∨0, ∀k≥1, 3.38
so that∞k1Gik<∞. Let the sequence{xn}be defined by1.3whenever{αni},{βni},{γni}satisfy
the same assumptions as inLemma 3.2for eachi∈ {1,2, . . . , m}and the additional assumption that there existsn0 ∈Nsuch that0 < α ≤ αnm−1, αnm ≤ β < 1for alln ≥ n0. Then{xnk}converges
strongly to a common fixed point of the mappings{Ti}mi1.
Proof. FromLemma 3.3, it follows that
lim
n→ ∞
Tikxnm−1−xn0nlim→ ∞Tikx
m−2
which implies that
xn1−xn xnm−xn
≤αnmTikx m−1
n −xnγnm−1unm−1−xn−→0, n−→ ∞,
3.40
and so
xnl−xn −→0, n−→ ∞. 3.41
It follows from3.22,3.37that
Tnkxn−xn≤Tikxn−Tikx
m−1
n Tikx
m−1 n −xn
≤xn−xnm−1GikTikx
m−1 n −xn
≤αnm−1Tikx
m−2
n −xnGikγnm−1unm−1−xn
Tikxnm−1−xn−→0, n−→ ∞.
3.42
Letσn Tk
i xn−xn for alln > n0. Then we have
xn−Tnxn ≤xn−TnkxnTnkxn−Tnxn
≤xn−TikxnLTnk−1xn−xn
≤σnLTnk−1xn−Tnk−−m1xn−mTnk−−m1xn−m−xn−m xn−m−xn
.
3.43
Notice thatn≡n−mmodm. ThusTnTn−mand the above inequality becomes
xn−Tnxn ≤σnL2 xn−xn−m Lσn−m xn−m−xn , 3.44
and so
lim
n→ ∞ xn−Tnxn 0. 3.45
Since
xn−Tnlxn ≤ xn−xnl xnl−Tnlxnl Tnlxnl−Tnlxn
≤1L xn−xnl xnl−Tnlxnl , ∀l∈ {1,2, . . . , m},
we have
lim
n→ ∞ xn−Tnlxn 0, ∀l∈ {1,2, . . . , m}, 3.47
and so
lim
n→ ∞ xn−Tlxn 0, ∀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{T1xnk}of
{T1xn}such thatT1xnk → q∈Cask → ∞. Moreover, by3.48, we have
lim
n→ ∞ xnk −T1xnk 0, 3.49
which implies thatxnk → qask → ∞. By3.48again, we have
q−Tlq lim
n→ ∞ xnk−Tlxnk 0, ∀l∈ {1,2, . . . , m}. 3.50
It follows thatq∈F. Since limn→ ∞ xn−q exists, we have
lim
n→ ∞xn−q0, 3.51
that is,
lim
n→ ∞x
m n lim
n→ ∞xnq. 3.52
Moreover, we observe that
xnk−q≤xn−qdnk, 3.53
for allk1,2, . . . , m−1 and
lim
n→ ∞d
k
n 0. 3.54
Therefore,
lim
n→ ∞x
k
n q, 3.55
Remark 3.5. Theorem 3.4improves and extends the corresponding results of Plubtieng and Wangkeeree2in the following ways.
1The iterative process{xn}defined by1.3in2is replaced by the new iterative process{xn}defined by1.3in 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. Ifm 3 andT1 T2 T3 T in Theorem 3.4, we obtain strong convergence theorem for Noor iteration scheme with error for asymptotically nonexpansive mappingT 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.
