Volume 2010, Article ID 719631,13pages doi:10.1155/2010/719631
Research Article
Weak and Strong Convergence of an Implicit
Iteration Process for an Asymptotically
Quasi-
I
-Nonexpansive Mapping in Banach Space
Farrukh Mukhamedov and Mansoor Saburov
Department of Computational & Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, P.O. Box 141, 25710 Kuantan, Malaysia
Correspondence should be addressed to Farrukh Mukhamedov,[email protected]
Received 31 August 2009; Accepted 6 December 2009
Academic Editor: Mohamed A. Khamsi
Copyrightq2010 F. Mukhamedov and M. Saburov. 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.
We prove the weak and strong convergence of the implicit iterative process to a common fixed point of an asymptotically quasi-I-nonexpansive mappingT and an asymptotically quasi-nonexpansive mappingI, defined on a nonempty closed convex subset of a Banach space.
1. Introduction
LetKbe a nonempty subset of a real normed linear spaceXand letT :K → Kbe a mapping. Denote byFTthe set of fixed points ofT, that is,FT {x∈K:Txx}. Throughout this paper, we always assume thatFT/∅. Now let us recall some known definitions.
Definition 1.1. A mappingT :K → Kis said to be
inonexpansive, ifTx−Ty ≤ x−yfor allx, y∈K;
iiasymptotically nonexpansive, if there exists a sequence {λn} ⊂ 1,∞ with
limn→ ∞λn1 such thatTnx−Tny ≤λnx−yfor allx, y∈Kandn∈N;
iiiquasi-nonexpansive, ifTx−p ≤ x−pfor allx∈K, p∈FT;
ivasymptotically quasi-nonexpansive, if there exists a sequence {μn} ⊂ 1,∞with
Note that from the above definitions, it follows that a nonexpansive mapping must be asymptotically nonexpansive, and an asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive, but the converse does not holdsee1.
IfKis a closed nonempty subset of a Banach space andT :K → Kis nonexpansive, then it is known thatTmay not have a fixed pointunlike the case ifTis a strict contraction, and even when it has, the sequence{xn}defined byxn1Txnthe so-calledPicard sequence
may fail to converge to such a fixed point.
In2,3Browder studied the iterative construction for fixed points of nonexpansive mappings on closed and convex subsets of a Hilbert space. Note that for the past 30 years or so, the studies of the iterative processes for the approximation of fixed points of nonexpansive mappings and fixed points of some of their generalizations have been flourishing areas of research for many mathematicianssee for more details1,4.
In 5 Diaz and Metcalf studied quasi-nonexpansive mappings in Banach spaces. Ghosh and Debnath6established a necessary and sufficient condition for convergence of the Ishikawa iterates of a quasi-nonexpansive mapping on a closed convex subset of a Banach space. The iterative approximation problems for nonexpansive mapping, asymptotically nonexpansive mapping and asymptotically quasi-nonexpansive mapping were studied extensively by Goebel and Kirk7, Liu8, Wittmann9, Reich10, Gornicki11, Schu
12Shioji and Takahashi 13, and Tan and Xu14 in the settings of Hilbert spaces and uniformly convex Banach spaces.
There are many methods for approximating fixed points of a nonexpansive mapping. Xu and Ori15introduced implicit iteration process to approximate a common fixed point of a finite family of nonexpansive mappings in a Hilbert space. Recently, Sun16has extended an implicit iteration process for a finite family of nonexpansive mappings, due to Xu and Ori, to the case of asymptotically quasi-nonexpansive mappings in a setting of Banach spaces. In17 it has been studied the weak and strong convergence of implicit iteration process with errors to a common fixed point for a finite family of nonexpansive mappings in Banach spaces, which extends and improves the mentioned paperssee also18,19for applications and other methods of implicit iteration processes.
There are many concepts which generalize a notion of nonexpansive mapping. One of such concepts isI-nonexpansivity of a mappingT20. Let us recall some notions.
Definition 1.2. LetT :K → K,I:K → Kbe two mappings of a nonempty subsetKof a real
normed linear spaceX. ThenTis said to be
iI-nonexpansive, ifTx−Ty ≤ Ix−Iyfor allx, y∈K;
iiasymptotically I-nonexpansive, if there exists a sequence {λn} ⊂ 1,∞ with
limn→ ∞λn1 such thatTnx−Tny ≤λnInx−Inyfor allx, y∈Kandn≥1; iiiasymptotically quasi I-nonexpansive mapping, if there exists a sequence{μn} ⊂
1,∞with limn→ ∞μn 1 such thatTnx−p ≤ μnInx−pfor all x ∈ K, p ∈
FT∩FIandn≥1.
Remark 1.3. IfFT∩FI/∅then an asymptoticallyI-nonexpansive mapping is
asymptot-ically quasi-I-nonexpansive. But, there exists a nonlinear continuous asymptotically quasi
I-nonexpansive mappings which is asymptoticallyI-nonexpansive.
I-nonexpansive mappings were investigated in20. In23the weak convergence of three-step Noor iterative scheme for an I-nonexpansive mapping in a Banach space has been established. Recently, in24the weak and strong convergence of implicit iteration process to a common fixed point of a finite family ofI-asymptotically nonexpansive mappings were studied. Assume that the family consists of oneI-asymptotically nonexpansive mappingT. Now let us consider an iteration method used in24, forT, which is defined by
x1∈K,
xn1 1−αnxnαnInyn,
yn
1−βn
xnβnTnxn.
n≥1, 1.1
where{αn}and{βn}are two sequences in0,1. From this formula one can easily see that the
employed method, indeed, is not implicit iterative processes. The used process is some kind of modified Ishikawa iteration.
Therefore, in this paper we will extend of the implicit iterative process, defined in16, to I-asymptotically quasi-nonexpansive mapping defined on a uniformly convex Banach space. Namely, letKbe a nonempty convex subset of a real Banach spaceXandT :K → K
be an asymptotically quasiI-nonexpansive mapping, and letI:K → Kbe an asymptotically quasi-nonexpansive mapping. Then for given two sequences{αn}and{βn}in0,1we will
consider the following iteration scheme:
x0∈K,
xn 1−αnxn−1αnTnyn,
yn
1−βn
xnβnInxn.
n≥1, 1.2
In this paper we will prove the weak and strong convergences of the implicit iterative process1.2to a common fixed point ofT andI. All results presented here generalize and extend the corresponding main results of15–17in a case of one mapping.
2. Preliminaries
Throughout this paper, we always assume thatXis a real Banach space. We denote byFT
and DTthe set of fixed points and the domain of a mapping T, respectively. Recall that a Banach space X is said to satisfy Opial condition 25, if for each sequence{xn} in X, xn
converging weakly toximplies that
lim inf
n→ ∞ xn−x<lim infn→ ∞ xn−y. 2.1
for ally∈Xwithy /x.It is well known thatsee26inequality2.1is equivalent to
lim sup
n→ ∞ xn−x<lim supn→ ∞
Definition 2.1. LetK be a closed subset of a real Banach space X and letT : K → K be a mapping.
iA mapping T is said to be semicloseddemiclosed at zero, if for each bounded sequence {xn} in K, the conditions xn converges weakly to x ∈ K and Txn
converges strongly to 0 implyTx0.
iiA mappingT is said to be semicompact, if for any bounded sequence{xn} inK
such thatxn−Txn → 0, n → ∞,then there exists a subsequence{xnk} ⊂ {xn}
such thatxnk → x∗∈Kstrongly.
iiiTis called a uniformlyL-Lipschitzian mapping, if there exists a constantL >0 such thatTnx−Tny ≤Lx−yfor allx, y∈Kandn≥1.
The following lemmas play an important role in proving our main results.
Lemma 2.2see12. LetXbe a uniformly convex Banach space and letb, cbe two constants with
0 < b < c <1.Suppose that{tn}is a sequence inb, cand{xn}and{yn}are two sequences inX
such that
lim
n→ ∞tnxn 1−tnynd, lim supn→ ∞ xn ≤d, lim supn→ ∞ yn≤d, 2.3
holds somed≤0.Thenlimn→ ∞xn−yn0.
Lemma 2.3 see 14. Let {an} and {bn} be two sequences of nonnegative real numbers with ∞
n1bn<∞.If one of the following conditions is satisfied:
ian1≤anbn, n≥1, iian1≤1bnan, n≥1,
then the limitlimn→ ∞anexists.
3. Main Results
In this section we will prove our main results. To formulate one, we need some auxiliary results.
Lemma 3.1. LetX be a real Banach space and letKbe a nonempty closed convex subset ofX.Let
T :K → Kbe an asymptotically quasiI-nonexpansive mapping with a sequence{λn} ⊂1,∞and
I :K → Kbe an asymptotically quasi-nonexpansive mapping with a sequence{μn} ⊂1,∞such
thatFFT∩FI/∅.SupposeA∗supnαn, Λ supnλn≥1, Msupnμn≥1and{αn}and {βn}are two sequences in0,1which satisfy the following conditions:
i∞n1λnμn−1αn<∞, iiA∗<1/Λ2M2.
If{xn}is the implicit iterative sequence defined by1.2, then for eachp∈FFT∩FIthe limit
Proof. SinceFFT∩FI/∅,for any givenp∈F,it follows from1.2that
xn−p1−αn
xn−1−p
αn
Tnyn−p
≤1−αnxn−1−pαnTnyn−p ≤1−αnxn−1−pαnλnInyn−p ≤1−αnxn−1−pαnλnμnyn−p.
3.1
Again from1.2we derive that
yn−p1−βn
xn−p
βn
Inxn−p
≤1−βnxn−pβnμnxn−p ≤1−βn
μnxn−pβnμnInxn−p
≤μnxn−p,
3.2
which means
yn−p≤μnxn−p≤λnμnxn−p. 3.3
Then from3.3one finds
xn−p≤1−αnxn−1−pαnλ2nμ2nxn−p, 3.4
and so
1−αnλ2nμ2n
xn−p≤1−αnxn−1−p. 3.5
By conditioniiwe haveαnλ2nμn2 ≤A∗Λ2M2<1,and therefore
1−αnλ2nμ2n≥1−A∗Λ2M2>0. 3.6
Hence from3.5we obtain
xn−p≤ 1−αn
1−αnλ2nμ2n
xn−1−p
1
λ2 nμ2n−1
αn
1−αnλ2nμ2n
xn−1−p
≤
1
λ2 nμ2n−1
αn
1−A∗Λ2M2 xn−1−p.
By puttingbn λ2nμ2n−1αn/1−A∗Λ2M2the last inequality can be rewritten as follows: xn−p≤1bnxn−1−p. 3.8
From conditioniwe find
∞
n1
bn
1 1−A∗Λ2M2
∞
n1
λ2nμ2n−1
αn
1
1−A∗Λ2M2
∞
n1
λnμn−1
λnμn1
αn
≤ ΛM1
1−A∗Λ2M2
∞
n1
λnμn−1
αn<∞.
3.9
Denotinganxn−1−pin3.8one gets
an1≤1bnan, 3.10
andLemma 2.3implies the existence of the limit limn→ ∞an. This means the limit
lim
n→ ∞xn−pd 3.11
exists, whered≥0 is a constant. This completes the proof.
Now we prove the following result.
Theorem 3.2. LetXbe a real Banach space and letKbe a nonempty closed convex subset ofX.Let
T : K → K be a uniformlyL1-Lipschitzian asymptotically quasi-I-nonexpansive mapping with a
sequence{λn} ⊂ 1,∞and let I : K → K be a uniformlyL2-Lipschitzian asymptotically
quasi-nonexpansive mapping with a sequence {μn} ⊂ 1,∞such that F FT∩FI/∅. Suppose
A∗ supnαn, Λ supnλn ≥1, M supnμn ≥1,and{αn}and{βn}are two sequences in0,1
which satisfy the following conditions:
i∞n1λnμn−1αn<∞, iiA∗<1/Λ2M2.
Then the implicitly iterative sequence {xn}defined by1.2converges strongly to a common fixed
point inF FT∩FI/∅if and only if
lim inf
n→ ∞ dxn, F 0. 3.12
Proof. The necessity of condition 3.12 is obvious. Let us proof the sufficiency part of
theorem.
For any givenp∈F,we havesee3.8
xn−p≤1bnxn−1−p, 3.13
here as beforebn λ2nμ2n−1αn/1−A∗Λ2M2with ∞
n1bn<∞.Hence, one finds
dxn, F≤1bndxn−1, F. 3.14
From 3.14 due to Lemma 2.3 we obtain the existence of the limit limn→ ∞dxn, F. By
condition3.12, one gets
lim
n→ ∞dxn, F lim infn→ ∞ dxn, F 0. 3.15
Let us prove that the sequence{xn}converges to a common fixed point ofTandI.In
fact, due to 1t≤exptfor allt >0,and from3.13, we obtain
xn−p≤expbnxn−1−p. 3.16
Hence, for any positive integersm, n,from3.16with∞n1bn<∞we find xnm−p≤expbnmxnm−1−p
≤expbnmbnm−1xnm−2−p
≤ · · ·
≤exp
nm
in1
bi xn−p
≤exp
∞
i1
bi xn−p,
3.17
which means that
xnm−p≤Wxn−p 3.18
for allp∈F, whereW exp∞i1bi<∞.
Since limn→ ∞dxn, F 0, then for any givenε > 0, there exists a positive integer
numbern0such that
dxn0, F<
ε
W. 3.19
Therefore there existsp1∈Fsuch that
xn0−p1< ε
Consequently, for alln≥n0from3.18we derive
xn−p1≤Wxn
0−p1
< W· ε W
ε,
3.21
which means that the strong convergence of the sequence{xn}is a common fixed pointp1of
TandI.This proves the required assertion.
We need one more auxiliary result.
Proposition 3.3. LetX be a real uniformly convex Banach space and letK be a nonempty closed
convex subset of X. Let T : K → K be a uniformly L1-Lipschitzian asymptotically quasi-I
-nonexpansive mapping with a sequence {λn} ⊂ 1,∞ and let I : K → K be a uniformly L2
-Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence{μn} ⊂ 1,∞such that
FFT∩FI/∅.SupposeA∗infnαn, A∗supnαn, Λ supnλn≥1, Msupnμn≥1and {αn}and{βn}are two sequences in0,1which satisfy the following conditions:
i∞n1λnμn−1αn<∞, ii0< A∗≤A∗<1/Λ2M2,
iii0< B∗infnβn≤supnβnB∗<1.
Then the implicitly iterative sequence{xn}defined by1.2satisfies the following:
lim
n→ ∞xn−Txn0, nlim→ ∞xn−Ixn0. 3.22
Proof. First, we will prove that
lim
n→ ∞xn−T nx
n0, lim n→ ∞xn−I
nx
n0. 3.23
According toLemma 3.1for anyp∈FFT∩FIwe have limn→ ∞xn−pd. It
follows from1.2that
xn−p1−αn
xn−1−p
αn
Tnyn−p−→d, n−→ ∞. 3.24
By means of asymptotically quasi-I-nonexpansivity of T and asymptotically quasi-nonexpansivity ofIfrom3.3we get
lim sup
n→ ∞
Tny
n−p≤lim sup n→ ∞ λnμn
yn−p≤lim sup n→ ∞ λ
2
nμ2nxn−pd. 3.25
Now using
lim sup
n→ ∞
with3.25and applyingLemma 2.2to3.24one finds
lim
n→ ∞xn−1−T ny
n0. 3.27
Now from1.2and3.27we infer that
lim
n→ ∞xn−xn−1nlim→ ∞αn
Tnyn−xn−10. 3.28
On the other hand, we have
xn−1−p≤xn−1−Tny
nTnyn−p
≤xn−1−Tnynλnμnyn−p,
3.29
which implies
xn−1−p−xn−1−Tny
n≤λnμnyn−p. 3.30
The last inequality with3.3yields that
xn−1−p−xn−1−Tny
n≤λnμnyn−p≤λ2nμ2xn−p. 3.31
Then3.27and3.24with the Squeeze theorem imply that
lim
n→ ∞yn−pd. 3.32
Again from1.2we can see that
yn−p1−βn
xn−p
βn
Inxn−p−→d, n−→ ∞. 3.33
From3.11one finds
lim sup
n→ ∞ Inx
n−p≤lim sup n→ ∞
μnxn−pd. 3.34
Now applyingLemma 2.2to3.33we obtain
lim
n→ ∞xn−I
nx
Consider
xn−Tnxn ≤ xn−xn−1xn−1−TnynTnyn−Tnxn ≤ xn−xn−1xn−1−TnynL1yn−xn xn−xn−1xn−1−TnynL1βnInxn−xn xn−xn−1xn−1−TnynL1βnInxn−xn.
3.36
Then from3.27,3.28, and3.35we get
lim
n→ ∞xn−T
nx
n0. 3.37
Finally, from
xn−Txn ≤ xn−TnxnTnxn−Txn
≤ xn−TnxnL1Tn−1xn−xn
≤ xn−TnxnL1Tn−1xn−Tn−1xn−1
Tn−1xn−1−xn−1xn−1−xn
≤ xn−TnxnL1
L1xn−xn−1
Tn−1xn−1−xn−1xn−1−xn
≤ xn−TnxnL1L11xn−xn−1L1Tn−1xn−1−xn−1
3.38
with3.28and3.37we obtain
lim
n→ ∞xn−Txn0. 3.39
Analogously, one has
xn−Ixn ≤ xn−InxnL2L21xn−xn−1L2In−1xn−1−xn−1, 3.40
which with3.28and3.35implies
lim
n→ ∞xn−Ixn0. 3.41
Theorem 3.4. Let X be a real uniformly convex Banach space satisfying Opial condition and let
K be a nonempty closed convex subset of X. Let E : X → X be an identity mapping, let
T : K → K be a uniformlyL1-Lipschitzian asymptotically quasi-I-nonexpansive mapping with a
sequence{λn} ⊂ 1,∞, and,I : K → K be a uniformlyL2-Lipschitzian asymptotically
quasi-nonexpansive mapping with a sequence {μn} ⊂ 1,∞such that F FT∩FI/∅. Suppose
A∗ infnαn, A∗ supnαn, Λ supnλn ≥ 1, M supnμn ≥ 1,and {αn}and {βn}are two
sequences in0,1satisfying the following conditions:
i∞n1λnμn−1αn<∞, ii0< A∗≤A∗<1/Λ2M2.
iii0< B∗infnβn≤supnβnB∗<1.
If the mappingsE−T andE−I are semiclosed at zero, then the implicitly iterative sequence{xn}
defined by1.2converges weakly to a common fixed point ofT andI.
Proof. Let p ∈ F, then according to Lemma 3.1 the sequence {xn −p} converges. This
provides that{xn}is a bounded sequence. SinceXis uniformly convex, then every bounded
subset ofX is weakly compact. Since{xn}is a bounded sequence in K,then there exists a
subsequence{xnk} ⊂ {xn}such that{xnk}converges weakly toq∈K.Hence from3.39and 3.41it follows that
lim
nk→ ∞xnk−Txnk0, nklim→ ∞xnk−Ixnk0. 3.42
Since the mappingsE−T and E−I are semiclosed at zero, therefore, we find Tq qand
Iqq,which meansq∈FFT∩FI.
Finally, let us prove that{xn}converges weakly toq.In fact, suppose the contrary, that
is, there exists some subsequence{xnj} ⊂ {xn}such that{xnj}converges weakly toq1∈Kand
q1/q. Then by the same method as given above, we can also prove thatq1∈FFT∩FI.
Takingpqandpq1and using the same argument given in the proof of3.11, we
can prove that the limits limn→ ∞xn−qand limn→ ∞xn−q1exist, and we have
lim
n→ ∞xn−qd, nlim→ ∞xn−q1d1, 3.43
where dand d1 are two nonnegative numbers. By virtue of the Opial condition ofX, one
finds
dlim sup
nk→ ∞
xnk−q<lim sup
nk→ ∞
xnk−q1d1
lim sup
nj→ ∞
xnj−q1<lim sup nj→ ∞
xnj−qd.
3.44
This is a contradiction. Hence q1 q.This implies that {xn}converges weakly to q.This
completes the proof ofTheorem 3.4.
Theorem 3.5. Let X be a real uniformly convex Banach space and let K be a nonempty closed
convex subset of X. Let T : K → K be a uniformly L1-Lipschitzian asymptotically quasi-I
-nonexpansive mapping with a sequence {λn} ⊂ 1,∞ and I : K → K be a uniformly L2
-Lipschitzian asymptotically quasi-nonexpansive mapping with a sequence{μn} ⊂ 1,∞such that
FFT∩FI/∅.SupposeA∗infnαn, A∗supnαn, Λ supnλn≥1, Msupnμn≥1and {αn}and{βn}are two sequences in0,1satisfying the following conditions:
i∞n1λnμn−1αn<∞, ii0< A∗≤A∗<1/Λ2M2.
iii0< B∗infnβn≤supnβnB∗<1
If at least one mapping of the mappingsT andIis semicompact, then the implicitly iterative sequence
{xn}defined by1.2converges strongly to a common fixed point ofTandI.
Proof. Without any loss of generality, we may assume thatTis semicompact. This with3.39
means that there exists a subsequence{xnk} ⊂ {xn}such thatxnk → x∗strongly andx∗ ∈K.
SinceT, Iare continuous, then from3.39and3.41we find
x∗−Tx∗ lim
nk→ ∞xnk −Txnk0, x
∗−Ix∗ lim
nk→ ∞xnk−Ixnk0. 3.45
This shows thatx∗ ∈ F FT∩FI.According toLemma 3.1the limit limn→ ∞xn−x∗
exists. Then
lim
n→ ∞xn−x
∗ lim
nk→ ∞xnk−x
∗0,
3.46
which means that{xn}converges tox∗∈F.This completes the proof.
Note that all results presented here generalize and extend the corresponding main results of15–17in a case of one mapping.
Acknowledgment
The authors acknowledge the MOSTI Grant 01-01-08-SF0079.
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