Volume 2011, Article ID 173621,11pages doi:10.1155/2011/173621
Research Article
Strong Convergence of an Implicit Algorithm in
CAT(0) Spaces
Hafiz Fukhar-ud-din,
1Abdul Aziz Domlo,
2and Abdul Rahim Khan
31Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan
2Department of Mathematics, Taibah University, Madinah Munawarah 30002, Saudi Arabia 3Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals,
Dhahran 31261, Saudi Arabia
Correspondence should be addressed to Abdul Rahim Khan,[email protected]
Received 23 November 2010; Accepted 23 December 2010
Academic Editor: Qamrul Hasan Ansari
Copyrightq2011 Hafiz Fukhar-ud-din 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.
We establish strong convergence of an implicit algorithm to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive maps in CAT0spaces. Our work improves and extends several recent results from the current literature.
1. Introduction
A metric spaceX, dis said to be alength spaceif any two points ofXare joined by a rectifiable pathi.e., a path of finite length, and the distance between any two points ofXis taken to be the infimum of the lengths of all rectifiable paths joining them. In this case,dis said to be alength metricotherwise known as aninner metricorintrinsic metric. In case no rectifiable path joins two points of the space, the distance between them is taken to be∞.
Ageodesic pathjoiningx∈Xtoy∈Xor, more briefly, ageodesicfromxtoyis a map
cfrom a closed interval0, l⊂RtoXsuch thatc0 x,cl y, anddct, ct |t−t|
for allt, t∈0, l. In particular,cis an isometry, anddx, y l. The imageαofcis called a geodesicor metricsegmentjoiningxandy. We sayXisiageodesic spaceif any two points ofXare joined by a geodesic andiiuniquely geodesicif there is exactly one geodesic joining
xandyfor eachx, y∈X, which we will denote byx, y, called the segment joiningxtoy. Ageodesic triangleΔx1, x2, x3in a geodesic metric spaceX, dconsists of three points
inXtheverticesofΔand a geodesic segment between each pair of verticestheedgesofΔ. Acomparison trianglefor geodesic triangleΔx1, x2, x3inX, dis a triangleΔx1, x2, x3 :
Δx1, x2, x3inR2 such thatdR2xi, xj dxi, xjfori, j ∈ {1,2,3}. Such a triangle always
A geodesic metric space is said to be a CAT0 space if all geodesic triangles of appropriate size satisfy the following CAT0comparison axiom.
LetΔbe a geodesic triangle inX, and letΔ⊂R2be a comparison triangle forΔ. Then
Δis said to satisfy the CAT0inequalityif for allx, y∈Δand all comparison pointsx, y∈Δ,
dx, y≤dx, y. 1.1
Complete CAT0spaces are often calledHadamard spacessee2. If x, y1, y2are points of
a CAT0 space andy0 is the midpoint of the segment y1, y2, which we will denote by
y1⊕y2/2, then the CAT0inequality implies
d2
x,y1⊕y2
2
≤ 1 2d
2x, y
1
1
2d
2x, y
2
−1
4d
2y
1, y2
. 1.2
The inequality1.2is theCNinequality of Bruhat and Titz3. The above inequality has
been extended in4as
d2z, αx⊕1−αy≤αd2z, x 1−αd2z, y−α1−αd2x, y, 1.3
for anyα∈0,1andx, y, z∈X.
Let us recall thata geodesic metric space is aCAT0space if and only if it satisfies the (CN) inequalitysee1, page 163. Moreover, ifXis a CAT0metric space andx, y∈X, then for anyα∈0,1, there exists a unique pointαx⊕1−αy∈x, ysuch that
dz, αx⊕1−αy≤αdz, x 1−αdz, y, 1.4
for anyz∈Xandx, y {αx⊕1−αy:α∈0,1}.
A subsetCof a CAT0spaceXis convex if for anyx, y∈C, we havex, y⊂C. LetT be a selfmap on a nonempty subsetCofX. Denote the set of fixed points ofT
byFT {x ∈ C : Tx x}. We sayT is: i asymptotically nonexpansive if there is a sequence{un} ⊂ 0,∞with limn→ ∞un 0 such thatdTnx, Tny ≤ 1undx, yfor all
x, y∈Candn≥1,iiasymptotically quasi-nonexpansive ifFT/φand there is a sequence {un} ⊂0,∞with limn→ ∞un 0 such thatdTnx, p≤1undx, pfor allx∈C,p∈FT
andn≥1,iiigeneralized asymptotically quasi-nonexpansive5ifFT/∅and there exist two sequences of real numbers {un} and {cn} with limn→ ∞un 0 limn→ ∞cn such that
dTnx, p ≤ dx, p 1u
ndx, p cn for allx ∈ C,p ∈FTandn ≥ 1,ivuniformly
L-Lipschitzian if for someL >0,dTnx, Tny≤ Ldx, yfor allx, y ∈Candn≥ 1, andv
semicompact if for any bounded sequence{xn}inCwithdxn, Txn → 0 asn → ∞, there is
a convergent subsequence of{xn}.
Denote the indexing set{1,2,3, . . . , N}byI. Let{Ti :i∈I}be the set ofNselfmaps
ofC. Throughout the paper, it is supposed thatF Ni1FTi/φ. We say conditionAis
satisfied if there exists a nondecreasing functionf :0,∞ → 0,∞withf0 0,fr>0 for allr ∈0,∞and at least oneT ∈ {Ti :i∈I}such thatdx, Tx≥fdx, Ffor allx∈C
If in definition iii, cn 0 for all n ≥ 1, then T becomes asymptotically
quasi-nonexpansive, and hence the class of generalized asymptotically quasi-nonexpansive maps includes the class of asymptotically quasi-nonexpansive maps.
Let{xn}be a sequence in a metric space X, d, and letC be a subset ofX. We say
that{xn}is: vi of monotone typeAwith respect to Cif for eachp ∈ C, there exist two
sequences{rn}and{sn}of nonnegative real numbers such that ∞
n1rn < ∞, ∞
n1sn < ∞
anddxn1, p≤1rndxn, p sn,viiof monotone typeBwith respect toCif there exist
sequences{rn}and{sn}of nonnegative real numbers such that∞n1rn < ∞,∞n1sn < ∞
anddxn1, C≤1rndxn, C snalso see6.
From the above definitions, it is clear that sequence of monotone typeAis a sequence of monotone typeBbut the converse is not true, in general.
Recently, numerous papers have appeared on the iterative approximation of fixed points of asymptotically nonexpansiveasymptotically quasi-nonexpansivemaps through Mann, Ishikawa, and implicit iterates in uniformly convex Banach spaces, convex metric spaces and CAT0spacessee, e.g.,5,7–16.
Using the concept of convexity in CAT0spaces, a generalization of Sun’s implicit algorithm15is given by
x0∈C,
x1α1x0⊕1−α1T1x1,
x2α2x1⊕1−α2T2x2,
.. .
xNαNxN−1⊕1−αNTNxN,
xN1αN1xN⊕1−αN1T12xN1,
.. .
x2Nα2Nx2N−1⊕1−α2NTN2x2N,
x2N1α2N1x2N⊕1−α2N1T13x2N1,
.. .,
1.5
where 0≤αn≤1.
Starting from arbitraryx0, the above process in the compact form is written as
xnαnxn−1⊕1−αnTiknnxn, n≥1, 1.6
wheren k−1Ni,iin∈Iandkkn≥1 is a positive integer such thatkn → ∞
In a normed space, algorithm1.6can be written as
x0∈C, xnαnxn−1 1−αnTiknnxn, n≥1, 1.7
wheren k−1Ni,iin∈Iandkkn≥1 is a positive integer such thatkn → ∞
asn → ∞.
The algorithms1.6-1.7exist as follows.
LetXbe a CAT0space. Then, the following inequality holds:
dλx⊕1−λz, λy⊕1−λw≤λdx, y 1−λdz, w, 1.8
for allx, y, z, w∈Xsee17.
Let{Ti :i∈I}be the set ofNuniformlyL-Lipschitzian selfmaps ofC. We show that
1.6exists. Letx0∈Candx1 α1x0⊕1−α1T1x1. DefineS:C → Cby:Sxα1x0⊕1−α1T1x
for allx ∈C. The existence ofx1is guaranteed ifShas a fixed point. For anyx, y ∈ C, we
have
dSx, Sy≤1−α1d
T1x, T1y
≤1−α1Lx−y. 1.9
Now, S is a contraction if 1− α1L < 1 or L < 1/1 −α1. As α1 ∈ 0,1, therefore S
is a contraction even if L > 1. By the Banach contraction principle, S has a unique fixed point. Thus, the existence ofx1 is established. Similarly, we can establish the existence of
x2, x3, x4, . . .. Thus, the implicit algorithm1.6is well defined. Similarly, we can prove that
1.7exists.
For implicit iterates, Xu and Ori16proved the following theorem.
Theorem XOsee16, Theorem 2. Let{Ti :i∈I}be nonexpansive selfmaps on a closed convex subsetCof a Hilbert space withF /φ, letx0 ∈ C, and let{αn} be a sequence in0,1 such that
limn→ ∞αn0. Then, the sequencexnαnxn−1 1−αnTxn, wheren≥1andTnTnmodN, converges weakly to a point inF.
They posed the question: what conditions on the maps {Ti : i ∈ I} and or the
parameters{αn}are sufficient to guarantee strong convergence of the sequence in Theorem
XO?
The aim of this paper is to study strong convergence of iterative algorithm1.6for the class of uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps on a CAT0space. Thus, we provide a positive answer to Xu and Ori’s question for the general class of maps which contains asymptotically quasi-nonexpansive, asymptotically nonexpansive, quasi-nonexpansive, and nonexpansive maps in the setup of CAT0spaces. It is worth mentioning that if an implicit iteration algorithm without an error term converges, then the method of proof generally carries over easily to algorithm with bounded error terms. Thus, our results also hold if we add bounded error terms to the implicit iteration scheme considered. Our results constitute generalizations of several important known results.
Lemma 1.1 see 14, Lemma 1.1. Let {rn} and {sn} be two nonnegative sequences of real numbers, satisfying the following condition:
rn1≤1snrn ∀n≥n0 for somen0≥1. 1.10
If∞n1sn<∞, thenlimn→ ∞rnexists.
2. Convergence in CAT(0) Spaces
We establish some convergence results for the algorithm1.6to a common fixed point of a finite family of uniformlyL-Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps in the general class of CAT0spaces. The following result extends Theorem XO; our methods of proofs are based on the ideas developed in15.
Theorem 2.1. LetX, dbe a complete CAT0space, and letCbe a nonempty closed convex subset of
X. Let{Ti:i∈I}beNuniformlyL-Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps ofCwith{uin},{cin} ⊂ 0,∞such that∞n1uin < ∞and∞n1cin < ∞for alli∈ I. Suppose thatFis closed. Starting from arbitraryx0 ∈C, define the sequence{xn}by the algorithm
1.6, where{αn} ⊂δ,1−δfor someδ∈0,1/2. Then,{xn}is of monotone type(A) and monotone type(B) with respect toF. Moreover,{xn}converges strongly to a common fixed point of the maps
{Ti:i∈I}if and only iflim infn→ ∞dxn, F 0.
Proof. First, we show that{xn}is of monotone type(A) and monotone type(B) with respect toF. Let
p∈F. Then, from1.6, we obtain that
dxn, p
d αnxn−1⊕1−αnTiknnxn, p
≤αnd
xn−1, p 1−αnd Tiknnxn, p
≤αnd
xn−1, p 1−αn
dxn, p
uikndxn, p
cikn
≤αnd
xn−1, p1−αnuikndxn, p
1−αncikn.
2.1
Sinceαn∈δ,1−δ, the above inequlaity gives that
dxn, p
≤dxn−1, p
uikn
δ d
xn, p
1
δ−1
cikn. 2.2
On simplification, we have that
dxn, p
≤ δ
δ−uiknd
xn−1, p
1
δ −1
δ
δ−uikncikn. 2.3
Let 1vikn δ/δ−uikn 1uikn/δ−uiknandγikn 1/δ−11vikncikn.
natural numbern1 such thatuikn < δ/2 forkn≥n1/N1 orn > n1. Then, we have that
∞
kn1vikn<2/δ∞kn1uikn<∞. Similarly,∞kn1γikn<∞.
Now, from2.3, forkn≥n1/N1, we get that
dxn, p
≤1vikndxn−1, pγikn, 2.4
dxn, F≤
1vikndxn−1, F γikn. 2.5
These inequalities, respectively, prove that {xn} is a sequence of monotone typeA and
monotone typeBwith respect toF.
Next, we prove that {xn}converges strongly to a common fixed point of the maps
{Ti:i∈I}if and only if lim infn→ ∞dxn, F 0.
If xn → p ∈ F, then limn→ ∞dxn, p 0. Since 0 ≤ dxn, F ≤ dxn, p, we have
lim infn→ ∞dxn, F 0.
Conversely, suppose that lim infn→ ∞dxn, F 0. Applying Lemma 1.1to2.5, we
have that limn→ ∞dxn, Fexists. Further, by assumption lim infn→ ∞dxn, F 0, we conclude
that limn→ ∞dxn, F 0. Next, we show that{xn}is a Cauchy sequence.
Sincex≤expx−1forx≥1, therefore from2.4, we have
dxnm, p
≤exp ⎛ ⎝N
i1 ∞
kn1
vikn
⎞
⎠dxn, pN i1
∞
kn1
γikn
< Mdxn, p
N
i1 ∞
kn1
γikn,
2.6
for the natural numbers m, n, where M exp{Ni1
∞
kn1vikn} 1 < ∞. Since
limn→ ∞dxn, F 0, therefore for any > 0, there exists a natural numbern0 such that
dxn, F < /4Mand N
i1
∞
jnγij ≤ /4 for alln ≥ n0. So, we can find p∗ ∈ F such that
dxn0, p∗≤/4M. Hence, for alln≥n0andm≥1, we have that
dxnm, xn≤d
xnm, p∗
dxn, p∗
< Mdxn0, p∗
N
i1 ∞
jn0
γijMd
xn0, p∗
N
i1 ∞
jn0
γij
2 ⎛ ⎝Mdxn
0, p∗
N
i1 ∞
jn0
γijMd
xn0, p∗
⎞⎠ ≤2 M 4M 4 . 2.7
This proves that{xn}is a Cauchy sequence. Let limn→ ∞xn z. SinceCis closed, therefore
z∈C. Next, we show thatz∈F. Now, the following two inequalities:
dz, p≤dz, xn d
xn, p
∀p∈F, n≥1,
dz, xn≤d
z, pdxn, p
give that
−dz, xn≤dz, F−dxn, F≤dz, xn, n≥1. 2.9
That is,
|dz, F−dxn, F| ≤dz, xn, n≥1. 2.10
As limn→ ∞xnzand limn→ ∞dxn, F 0, we conclude thatz∈F.
We deduce some results fromTheorem 2.1as follows.
Corollary 2.2. Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset ofX. Let{Ti : i ∈ I}beN uniformlyL-Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps ofCwith{uin},{cin} ⊂0,∞such that∞n1uin < ∞and∞n1cin < ∞ for alli∈I. Suppose thatFis closed. Starting from arbitarayx0∈C, define the sequence{xn}by the algorithm1.6, where{αn} ⊂δ,1−δfor someδ ∈0,1/2. Then,{xn}converges strongly to a common fixed point of the maps{Ti:i∈I}if and only if there exists some subsequence{xnj}of{xn} which converges top∈F.
Corollary 2.3. LetX, dbe a complete CAT0space, and letCbe a nonempty closed convex subset ofX. Let{Ti:i∈I}beNuniformlyL-Lipschitzian and asymptotically quasi-nonexpansive selfmaps ofCwith{uin} ⊂0,∞such that
∞
n1uin<∞for alli∈I. Starting from arbitarayx0∈C, define
the sequence{xn}by the algorithm1.6, where{αn} ⊂δ,1−δfor someδ∈0,1/2. Then,{xn} is of monotone type(A) and monotone type(B) with respect toF. Moreover,{xn}converges strongly to a common fixed point of the maps{Ti:i∈I}if and only iflim infn→ ∞dxn, F 0.
Proof. Follows fromTheorem 2.1withcin0 for alln≥1.
Corollary 2.4. Let X be a Banach space, and let Cbe a nonempty closed convex subset ofX. Let {Ti :i ∈I}beNasymptotically quasi-nonexpansive self-maps ofCwith{uin} ⊂ 0,∞such that ∞
n1uin<∞for alli∈I. Starting from arbitarayx0∈C, define the sequence{xn}by the algorithm
1.7, where{αn} ⊂δ,1−δfor someδ∈0,1/2. Then,{xn}is of monotone type(A) and monotone type(B) with respect toF. Moreover,{xn}converges strongly to a common fixed point of the maps
{Ti:i∈I}if and only iflim infn→ ∞dxn, F 0.
Proof. Takeλx⊕1−λyλx 1−λyinCorollary 2.3.
The lemma to follow establishes an approximate sequence, and as a consequence of that, we find another strong convergence theorem for1.6.
Lemma 2.5. LetX, dbe a complete CAT0space, and letCbe a nonempty closed convex subset of
X. Let{Ti:i∈I}beNuniformlyL-Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps ofCwith{uin},{cin} ⊂ 0,∞such that
∞
n1uin < ∞and ∞
n1cin < ∞for alli∈ I. Suppose thatFis closed. Let{αn} ⊂δ,1−δfor someδ∈0,1/2. From arbitarayx0 ∈C, define
Proof. Note that{xn}is bounded as limn→ ∞dxn, pexistsproved inTheorem 2.1. So, there
existsR > 0 andx0 ∈ X such thatxn ∈ BRx0 {x : dx, x0 < R}for alln ≥ 1. Denote
dxn−1, Tiknnxnbyσn.
We claim that limn→ ∞σn0.
For anyp∈F, apply1.3to1.6and get
d2xn, p
d2 αnxn−1⊕1−αnTiknnxn, p
≤αnd2
xn−1, p
1−αn
1uikn
dxn, p
cikn 2
−αn1−αnd2 Tiknnxn, xn−1
2.11
further, using2.4, we obtain
2δ3σn2≤αnd2xn−1, x∗−d2xn, x∗
1−αn
1uikn1vikndxn−1, x∗ 1uiknγikncikn2,
2.12
which implies that
2δ3σn2≤αnd2
xn−1, p 1−αnd2
xn−1, p
uiknviknγikncikn
M−d2xn, p
, 2.13
for some consantM >0. This gives that
2δ3σn2≤d2
xn−1, p−d2xn, p
σiknM, 2.14
whereσiknuiknviknγikncikn.
Form≥1, we have that
2δ3
m
n1
σn2≤d2
x0, p
−d2xm, p
m
kn1
σiknM
≤d2x0, p
m
kn1
σiknM.
2.15
Whenm → ∞, we have that∞n1σn2<∞as ∞
kn1σikn<∞.
Hence,
lim
Further,
dxn, xn−1≤1−αnd Tiknnxn, xn−1
1−αnσn≤1−δσn
2.17
implies that limn→ ∞dxn, xn−1 0.
For a fixedj∈I, we havedxnj, xn≤dxnj, xnj−1 · · ·dxn, xn−1, and hence
lim
n→ ∞d
xnj, xn
0 ∀j∈I. 2.18
Forn > N,n n−NmodN. Also,n kn−1Nin. Hence,n−N kn−1−
1Nin kn−NNin−N.
That is,kn−N kn−1 andin−N in.
Therefore, we have
dxn−1, Tnxn≤d xn−1, Tiknnxn
d Tiknnxn, Txn
≤σnLd Tiknn−1xn, xn
≤σnL2dxn, xn−N Ld Tiknn−−NNxn−N, xn−N−1
Ldxn−N−1, xn
σnL2dxn, xn−N Lσn−NLd
xn−N−1, xn
,
2.19
which together with2.16and2.18yields that limn→ ∞dxn−1, Txn 0.
Since
dxn, Txn≤dxn, xn−1 dxn−1, Tnxn, 2.20
we have
lim
n→ ∞dxn, Tnxn 0. 2.21
Hence, for alll∈I,
dxn, Tnlxn≤dxn, xnl dxnl, Tnlxnl dTnlxnl, Tnlxn
≤1Ldxn, xnl dxnl, Tnlxnl,
2.22
together with2.18and2.21implies that
lim
n→ ∞dxn, Tnlxn 0 ∀l∈I. 2.23
Theorem 2.6. LetX, dbe a complete CAT0space, and letCbe a nonempty closed convex subset of
X. Let{Ti:i∈I}beN-uniformlyL-Lipschitzian and generalized asymptotically quasi-nonexpansive selfmaps ofCwith{uin},{cin} ⊂ 0,∞such that
∞
n1uin < ∞and ∞
n1cin < ∞for alli∈ I. Suppose thatFis closed, and there exists one memberTin{Ti:i∈I}which is either semicompact or satisfies condition (A). Let{αn} ⊂δ,1−δfor someδ∈0,1/2. From arbitarayx0∈C, define the
sequence{xn}by algorithm1.6. Then,{xn}converges strongly to a common fixed point of the maps in{Ti:i∈I}.
Proof. Without loss of generality, we may assume thatT1 is either semicompact or satisfies
conditionA. IfT1 is semicompact, then there exists a subsequence{xnj}of{xn}such that
xnj → x∗∈Casj → ∞. Now,Lemma 2.5guarantees that limn→ ∞dxnj, Tlxnj 0 for alll∈I and sodx∗, Tlx∗ 0 for alll∈I. This implies thatx∗∈F. Therefore, lim infn→ ∞dxn, F 0.
IfT1 satisfies conditionA, then we also have lim infn→ ∞dxn, F 0. Now,Theorem 2.1
gaurantees that{xn}converges strongly to a point inF.
Finally, we state two corollaries to the above theorem.
Corollary 2.7. LetX, dbe a complete CAT0space and letCbe a nonempty closed convex subset ofX. Let{Ti:i∈I}beNuniformlyL-Lipschizian and asymptotically quasi-nonexpansive selfmaps ofCwith{uin} ⊂0,∞such that∞n1uin <∞for alli∈I. Suppose that there exists one member
Tin{Ti:i∈I}which is either semicompact or satisfies condition (A). From arbitarayx0∈C, define
the sequence{xn}by algorithm1.6, where{αn} ⊂ δ,1−δfor some δ ∈ 0,1/2. Then,{xn} converges strongly to a common fixed point of the maps in{Ti:i∈I}.
Corollary 2.8. LetX, dbe a complete CAT0space, and letCbe a nonempty closed convex subset ofX. Let{Ti:i∈I}beNasymptotically nonexpansive selfmaps ofCwith{uin} ⊂0,∞such that ∞
n1uin < ∞for alli∈ I. Suppose that there exists one memberT in{Ti : i ∈I}which is either semicompact or satisfies condition (A). From arbitraryx0∈C, define the sequence{xn}by algorithm
1.6, where{αn} ⊂δ,1−δfor someδ∈0,1. Then,{xn}converges strongly to a common fixed point of the maps in{Ti:i∈I}.
Remark 2.9. The corresponding approximation results for a finite family of asymptotically quasi-nonexpansive maps on: i uniformly convex Banach spaces 5, 14, 15,ii convex metric spaces13,iiiCAT0spaces12are immediate consequences of our results. Remark 2.10. Various algorithms and their strong convergence play an important role in finding a common element of the set of fixedcommon fixedpoint for different classes of mappingsand the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces; for details we refer to18–20.
Acknowledgments
The author A. R. Khan gratefully acknowledges King Fahd University of Petroleum and Minerals and SABIC for supporting research project no. SB100012.
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