Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 173621, 11 pages
doi:10.1155/2011/173621
Research Article
Strong Convergence of an Implicit Algorithm in
CAT(0) Spaces
Hafiz Fukhar-ud-din,
1
Abdul Aziz Domlo,
2
and Abdul Rahim Khan
3
1
Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan
2
Department of Mathematics, Taibah University, Madinah Munawarah 30002, Saudi Arabia
3
Department 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
Copyright q 2011 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 CAT0 spaces. Our work improves
and extends several recent results from the current literature.
1. Introduction
A metric space X, d is said to be a length space if any two points of X are joined by a rectifiable
in X, d is a triangle Δx
1
,x
2
,x
3
:
Δ
x
1
, x
2
, x
3
in R
2
such that d
R
2
x
i
, x
j
dx
i
,x
j
for i, j ∈{1, 2, 3}. Such a triangle always
exists see 1.
2 Fixed Point Theory and Applications
1
⊕ y
2
/2, then the CAT0 inequality implies
d
2
x,
y
1
⊕ y
2
2
≤
1
2
d
2
x, y
1
1
2
d
2
x, y
1 − α
d
2
z, y
− α
1 − α
d
2
x, y
,
1.3
for any α ∈ 0, 1 and x, y, z ∈ X.
Let us recall that a geodesic metric space is a CAT0 space if and only if it satisfies the (CN)
inequality see 1, page 163. Moreover, if X is a CAT0 metric space and x, y ∈ X, then for
any α ∈ 0, 1, there exists a unique point αx ⊕ 1 − αy ∈ x, y such that
d
z, αx ⊕
1 − α
y ≤ 1 u
n
dx, y for all
x, y ∈ C and n ≥ 1, ii asymptotically quasi-nonexpansive if FT
/
φ and there is a sequence
{u
n
}⊂0, ∞ with lim
n →∞
u
n
0 such that dT
n
x, p ≤ 1 u
n
dx, p for all x ∈ C, p ∈ FT
and n ≥ 1, iii generalized asymptotically quasi-nonexpansive 5 if FT
/
∅ and there exist
two sequences of real numbers {u
n
} and {c
n
} with lim
n →∞
u
n
0 lim
n →∞
N
i1
FT
i
/
φ. We say condition A is
satisfied if there exists a nondecreasing function f : 0, ∞ → 0, ∞ with f00, fr > 0
for all r ∈ 0, ∞ and at least one T ∈{T
i
: i ∈ I} such that dx, Tx ≥ fdx, F for all x ∈ C
where dx, Finf{dx, p : p ∈ F}.
Fixed Point Theory and Applications 3
If in definition iii, c
n
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 {x
n
} be a sequence in a metric space X, d,andletC be a subset of X.Wesay
that {x
n
} is: vi of monotone typeA with respect to C if for each p ∈ C, there exist two
sequences {r
n
} and {s
n
} of nonnegative real numbers such that
< ∞,
∞
n1
s
n
< ∞
and dx
n1
,C ≤ 1 r
n
dx
n
,Cs
n
also see 6.
From the above definitions, it is clear that sequence of monotone typeA is a sequence
of monotone typeB but the converse is not true, in general.
Recently, numerous papers have appeared on the iterative approximation of fixed
points of asymptotically nonexpansive asymptotically quasi-nonexpansive maps through
Mann, Ishikawa, and implicit iterates in uniformly convex Banach spaces, convex metric
spaces and CAT0 spaces see, e.g., 5, 7–16.
Using the concept of convexity in CAT0 spaces, a generalization of Sun’s implicit
algorithm 15 is given by
x
0
∈ C,
x
1
α
.
.
x
N
α
N
x
N−1
⊕
1 − α
N
T
N
x
N
,
x
N1
α
N1
x
N
⊕
1 − α
N1
T
2N
⊕
1 − α
2N1
T
3
1
x
2N1
,
.
.
.,
1.5
where 0 ≤ α
n
≤ 1.
Starting from arbitrary x
0
, the above process in the compact form is written as
x
n
α
n
x
n−1
⊕
T
kn
i
n
x
n
,n≥ 1,
1.7
where n k − 1N i, i in ∈ I and k kn ≥ 1 is a positive integer such that kn →∞
as n →∞.
The algorithms 1.6-1.7 exist as follows.
Let X be a CAT0 space. Then, the following inequality holds:
d
λx ⊕
1 − λ
z, λy ⊕
1 − λ
w
≤ λd
x, y
x
0
⊕1−α
1
T
1
x
for all x ∈ C. The existence of x
1
is guaranteed if S has a fixed point. For any x, y ∈ C,we
have
d
Sx, Sy
≤
1 − α
1
d
T
1
x, T
1
y
≤
Theorem XO see 16, Theorem 2. Let {T
i
: i ∈ I} be nonexpansive selfmaps on a closed convex
subset C of a Hilbert space with F
/
φ,letx
0
∈ C, and let {α
n
} be a sequence in 0, 1 such that
lim
n →∞
α
n
0. Then, the sequence x
n
α
n
x
n−1
1 − α
n
Tx
n
,wheren ≥ 1 and T
n
T
n
mod N,
converges weakly to a point in F.
r
n
∀n ≥ n
0
for some n
0
≥ 1. 1.10
If
∞
n1
s
n
< ∞,thenlim
n →∞
r
n
exists.
2. Convergence in CAT(0) Spaces
We establish some convergence results for the algorithm 1.6 to a common fixed point of a
finite family of uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive
selfmaps in the general class of CAT0 spaces. The following result extends Theorem XO; our
methods of proofs are based on the ideas developed in 15.
Theorem 2.1. Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset of
X.Let{T
i
: i ∈ I} be N uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive
selfmaps of C with {u
in
: i ∈ I} if and only if lim inf
n →∞
dx
n
,F0.
Proof. First, we show that {x
n
} is of monotone type(A) and monotone type(B) with respect to F.Let
p ∈ F. Then, from 1.6,weobtainthat
d
x
n
,p
d
α
n
x
n−1
⊕
1 − α
n
T
kn
i
≤ α
n
d
x
n−1
,p
1 − α
n
d
x
n
,p
u
ikn
d
x
n
,p
c
.
2.1
Since α
n
∈ δ, 1 − δ, the above inequlaity gives that
d
x
n
,p
≤ d
x
n−1
,p
u
ikn
δ
d
x
n
,p
1
δ − u
ikn
c
ikn
.
2.3
Let 1 v
ikn
δ/δ − u
ikn
1 u
ikn
/δ − u
ikn
and γ
ikn
1/δ − 11 v
ikn
c
ikn
.
Since
∞
kn1
u
ikn
< ∞ for all i ∈ I, therefore lim
kn →∞
u
Now, from 2.3,forkn ≥ n
1
/N 1, we get that
d
x
n
,p
≤
1 v
ikn
d
x
n−1
,p
γ
ikn
, 2.4
d
x
n
,F
≤
n →∞
dx
n
,p0. Since 0 ≤ dx
n
,F ≤ dx
n
,p, we have
lim inf
n →∞
dx
n
,F0.
Conversely, suppose that lim inf
n →∞
dx
n
,F0. Applying Lemma 1.1 to 2.5,we
have that lim
n →∞
dx
n
,F exists. Further, by assumption lim inf
n →∞
dx
n
,F0, we conclude
that lim
n →∞
dx
x
n
,p
N
i1
∞
k
n
1
γ
ikn
<Md
x
n
,p
N
i1
∞
k
i1
∞
jn
γ
ij
≤ /4 for all n ≥ n
0
. So, we can find p
∗
∈ F such that
dx
n
0
,p
∗
≤ /4M. Hence, for all n ≥ n
0
and m ≥ 1, we have that
d
x
nm
,x
n
≤ d
x
nm
x
n
0
,p
∗
N
i1
∞
jn
0
γ
ij
2
⎛
⎝
Md
x
n
0
,p
∗
N
n →∞
x
n
z. Since C is closed, therefore
z ∈ C. Next, we show that z ∈ F. Now, the following two inequalities:
d
z, p
≤ d
z, x
n
d
x
n
,p
∀p ∈ F, n ≥ 1,
d
z, x
n
≤ d
z, p
,n≥ 1. 2.9
That is,
|
d
z, F
− d
x
n
,F
|
≤ d
z, x
n
,n≥ 1. 2.10
As lim
n →∞
x
n
z and lim
n →∞
dx
n
,F0, we conclude that z ∈ F.
We deduce some results from Theorem 2.1 as follows.
} converges strongly to a
common fixed point of the maps {T
i
: i ∈ I} if and only if there exists some subsequence {x
n
j
} of {x
n
}
which converges to p ∈ F.
Corollary 2.3. Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset
of X.Let{T
i
: i ∈ I} be N uniformly L-Lipschitzian and asymptotically quasi-nonexpansive selfmaps
of C with {u
in
}⊂0, ∞ such that
∞
n1
u
in
< ∞ for all i ∈ I. Starting from arbitaray x
0
∈ C, define
the sequence {x
n
} by the algorithm 1.6,where{α
n
}⊂δ, 1 − δ for some δ ∈ 0, 1/2. Then, {x
} by the algorithm
1.7,where{α
n
}⊂δ, 1−δ for some δ ∈ 0, 1/2. Then, {x
n
} is of monotone type(A) and monotone
type(B) with respect to F. Moreover, {x
n
} converges strongly to a common fixed point of the maps
{T
i
: i ∈ I} if and only if lim inf
n →∞
dx
n
,F0.
Proof. Take λx ⊕ 1 − λy λx 1 − λy in Corollary 2.3.
The lemma to follow establishes an approximate sequence, and as a consequence of
that, we find another strong convergence theorem for 1.6.
Lemma 2.5. Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset of
X.Let{T
i
: i ∈ I} be N uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive
selfmaps of C with {u
in
}, {c
in
}⊂0, ∞ such that
∞
n →∞
dx
n
,p exists proved in Theorem 2.1. So, there
exists R>0andx
0
∈ X such that x
n
∈ B
R
x
0
{x : dx, x
0
<R} for all n ≥ 1. Denote
dx
n−1
,T
kn
in
x
n
by σ
n
.
We claim that lim
n →∞
σ
n
0.
≤ α
n
d
2
x
n−1
,p
1 − α
n
1 u
ikn
d
x
n
,p
c
ikn
2
− α
n
x
n−1
,x
∗
− d
2
x
n
,x
∗
1 − α
n
1 u
ikn
1 v
ikn
d
x
n−1
,x
1 − α
n
d
2
x
n−1
,p
u
ikn
v
ikn
γ
ikn
c
ikn
M − d
2
x
n
,p
v
ikn
γ
ikn
c
ikn
.
For m ≥ 1, we have that
2δ
3
m
n1
σ
2
n
≤ d
2
x
0
,p
− d
2
x
m
,p
When m →∞, we have that
∞
n1
σ
2
n
< ∞ as
∞
kn1
σ
ikn
< ∞.
Hence,
lim
n →∞
σ
n
0.
2.16
Fixed Point Theory and Applications 9
Further,
d
x
n
,x
n−1
n
2.17
implies that lim
n →∞
dx
n
,x
n−1
0.
For a fixed j ∈ I, we have dx
nj
,x
n
≤ dx
nj
,x
nj−1
··· dx
n
,x
n−1
, and hence
lim
n →∞
d
x
nj
,x
n
T
kn
i
n
x
n
,Tx
n
≤ σ
n
Ld
T
kn−1
i
n
x
n
,x
n
≤ σ
n
L
L
2
d
x
n
,x
n−N
Lσ
n−N
Ld
x
n−N−1
,x
n
,
2.19
which together with 2.16 and 2.18 yields that lim
n →∞
dx
n−1
,Tx
n
0.
Since
d
n
x
n
0.
2.21
Hence, for all l ∈ I,
d
x
n
,T
nl
x
n
≤ d
x
n
,x
nl
d
x
nl
,T
nl
x
x
nl
,
2.22
together with 2.18 and 2.21 implies that
lim
n →∞
d
x
n
,T
nl
x
n
0 ∀l ∈ I.
2.23
Thus, lim
n →∞
dx
n
,T
l
x
n
0 for all l ∈ I.
10 Fixed Point Theory and Applications
Theorem 2.6. Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset of
} by algorithm 1.6. Then, {x
n
} converges strongly to a common fixed point of the maps
in {T
i
: i ∈ I}.
Proof. Without loss of generality, we may assume that T
1
is either semicompact or satisfies
condition A.IfT
1
is semicompact, then there exists a subsequence {x
n
j
} of {x
n
} such that
x
n
j
→ x
∗
∈ C as j →∞.Now,Lemma 2.5 guarantees that lim
n →∞
dx
n
j
,T
l
x
: i ∈ I} be N uniformly L-Lipschizian and asymptotically quasi-nonexpansive selfmaps
of C with {u
in
}⊂0, ∞ such that
∞
n1
u
in
< ∞ for all i ∈ I. Suppose that there exists one member
T in {T
i
: i ∈ I} which is either semicompact or satisfies condition (A). From arbitaray x
0
∈ C, define
the sequence {x
n
} by algorithm 1.6,where{α
n
}⊂δ, 1 − δ for some δ ∈ 0, 1/2. Then, {x
n
}
converges strongly to a common fixed point of the maps in {T
i
: i ∈ I}.
Corollary 2.8. Let X, d be a complete CAT0 space, and let C be a nonempty closed convex subset
of X.Let{T
i
: i ∈ I} be N asymptotically nonexpansive selfmaps of C with {u
in
The author A. R. Khan gratefully acknowledges King Fahd University of Petroleum and
Minerals and SABIC for supporting research project no. SB100012.
References
1 M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, vol. 319 of Grundlehren der
Mathematischen Wissenschaften, Springer, Berlin, Germany, 1999.
Fixed Point Theory and Applications 11
2 M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied
Mathematics, Wiley-Interscience, New York, NY, USA, 2001.
3 F. Bruhat and J. Tits, “Groupes r
´
eductifs sur un corps local,” Institut des Hautes
´
Etudes Scientifiques.
Publications Math
´
ematiques, no. 41, pp. 5–251, 1972.
4 S. Dhompongsa and B. Panyanak, “On Δ-convergence theorems in CAT0 spaces,” Computers &
Mathematics with Applications, vol. 56, no. 10, pp. 2572–2579, 2008.
5 S. Imnang and S. Suantai, “Common fixed points of multistep Noor iterations with errors for a finite
family of generalized asymptotically quasi-nonexpansive mappings,” Abstract and Applied Analysis,
vol. 2009, Article ID 728510, 14 pages, 2009.
6 H. Y. Zhou, G. L. Gao, G. T. Guo, and Y. J. Cho, “Some general convergence principles with
applications,” Bulletin of the Korean Mathematical Society, vol. 40, no. 3, pp. 351–363, 2003.
7 H. Fukhar-ud-din and A. R. Khan, “Approximating common fixed points of asymptotically
nonexpansive maps in uniformly convex Banach spaces,” Computers & Mathematics with Applications,
vol. 53, no. 9, pp. 1349–1360, 2007.
8 H. Fukhar-ud-din and A. R. Khan, “Convergence of implicit iterates with errors for mappings with
unbounded domain in Banach spaces,” International Journal of Mathematics and Mathematical Sciences,
no. 10, pp. 1643–1653, 2005.
9 H. Fukhar-ud-din, A. R. Khan, D. O’Regan, and R. P. Agarwal, “An implicit iteration scheme with
problems and fixed point problems in Banach spaces,” Journal of Computational and Applied
Mathematics, vol. 225, pp. 20–30, 2009.
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