Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 965751, 11 pages
doi:10.1155/2012/965751
Research Article
Iterative Algorithm and Δ-Convergence Theorems
for Total Asymptotically Nonexpansive Mappings
in CAT(0) Spaces
J. F. Tang,1 S. S. Chang,2 H. W. Joseph Lee,3 and C. K. Chan3
1
Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming,
Yunnan 650221, China
3
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong
2
Correspondence should be addressed to S. S. Chang, changss@yahoo.cn
Received 9 July 2012; Accepted 9 August 2012
Academic Editor: Yongfu Su
Copyright q 2012 J. F. Tang 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.
The main purpose of this paper is first to introduce the concept of total asymptotically
nonexpansive mappings and to prove a Δ-convergence theorem for finding a common fixed
point of the total asymptotically nonexpansive mappings and the asymptotically nonexpansive
mappings. The demiclosed principle for this kind of mappings in CAT0 space is also proved in
the paper. Our results extend and improve many results in the literature.
1. Introduction
A metric space X is a CAT0 space if it is geodesically connected and if every geodesic
triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane. Fixed point
theory in a CAT0 space was first studied by Kirk 1, 2. He showed that every nonexpansive
mapping defined on a bounded closed convex subset of a complete CAT0 space always has
a fixed point. Since then the fixed point theory for various mappings in CAT0 space has been
developed rapidly and many papers have appeared 3–10. On the other hand, Browder 11
introduced the demiclosed principle which states that if X is a uniformly convex Banach
space, C is a nonempty closed convex subset of X, and if T : C → C is nonexpansive
mapping, then I −T is demiclosed at each y ∈ X, that is, for any sequence {xn } in C conditions
xn → x weakly and I − T xn → y strongly imply that I − T x y where I is the identity
mapping of X. Xu 12 proved the demiclosed principle for asymptotically nonexpansive
mappings in the setting of a uniformly convex Banach space. Nanjaras and Panyanak 13
2
Abstract and Applied Analysis
proved the demiclosed principle for asymptotically nonexpansive mappings in CAT0 space
and obtained a Δ-convergence theorem for the Krasnosel’skii-Mann iteration.
Motivated and inspired by the researches going on in this direction, especially inspired
by Nanjaras and Panyanak, and so forth 13, the purpose of this paper is to introduce
a general mapping, namely, total asymptotically nonexpansive mapping and to prove its
demiclosed principle in CAT0 space. As a consequence, we construct a hierarchical iterative
algorithm to study the fixed point of the total asymptotically nonexpansive mappings and
obtain a Δ-convergence theorem.
2. Preliminaries and Lemmas
Let X, d be a metric space and x, y ∈ X with dx, y l. A geodesic path from x to y is a
isometry c : 0, l → X such that c0 x, cl y. The image of a geodesic path is called
geodesic segment. A space X, d is a uniquely geodesic space if every two points of X are
joined by only one geodesic segment. A geodesic triangle Δx1 , x2 , x3 in a geodesic metric
space X, d consists of three points x1 , x2 , x3 in X the vertices of Δ and a geodesic segment
between each pair of vertices the edges of Δ. A comparison triangle for the geodesic triangle
Δx1 , x2 , x3 in X, d is a triangle Δx1 , x2 , x3 : Δx1 , x2 , x3 in the Euclidean space R2 such
that dR2 xi , xj dxi , xj for i, j ∈ {1, 2, 3}.
A geodesic space is said to be a CAT0 space if for each geodesic triangle Δx1 , x2 , x3 in X and its comparison triangle Δ : Δx1 , x2 , x3 in R2 , the CAT0 inequality
d x, y ≤ dE2 x, y .
2.1
is satisfied for all x, y ∈ Δ and x, y ∈ Δ.
In this paper, we write 1 − tx ⊕ ty for the unique point z in the geodesic segment
joining from x to y such that
dx, z td x, y ,
d y, z 1 − td x, y .
2.2
We also denote by x, y the geodesic segment joining from x to y, that is, x, y {1 − tx ⊕
ty : t ∈ 0, 1}.
A subset C of a CAT0 space X is said to be convex if x, y ⊂ C for all x, y ∈ C.
Lemma 2.1 see 14. A geodesic space X is a CAT(0) space, if and only if the following inequality
2
2
2
d 1 − tx ⊕ ty, z ≤ 1 − tdx, z2 td y, z − t1 − td x, y
2.3
is satisfied for all x, y, z ∈ X and t ∈ 0, 1. In particular, if x, y, z are points in a CAT(0) space and
t ∈ 0, 1, then
d 1 − tx ⊕ ty, z ≤ 1 − tdx, z td y, z .
2.4
Abstract and Applied Analysis
3
Let {xn } be a bounded sequence in a CAT0 space X. For x ∈ X, one sets
rx, {xn } lim sup dx, xn .
n→∞
2.5
The asymptotic radius r{xn } of {xn } is given by
r{xn } inf {rx, {xn }},
x∈X
2.6
the asymptotic radius rC {xn } of {xn } with respect to C ⊂ X is given by
rC {xn } inf {rx, {xn }},
x∈C
2.7
the asymptotic center A{xn } of {xn } is the set
A{xn } {x ∈ X : rx, {xn } r{xn }},
2.8
the asymptotic center AC {xn } of {xn } with respect to C ⊂ X is the set
AC {xn } {x ∈ C : rx, {xn } rC {xn }}.
2.9
Recall that a bounded sequence {xn } in X is said to be regular if r{xn } r{un } for
every subsequence {un } of {xn }.
Proposition 2.2 see 15. If {xn } is a bounded sequence in a complete CAT(0) space X and C is a
closed convex subset of X, then
1 there exists a unique point u ∈ C such that
ru, {xn } inf rx, {xn };
x∈C
2.10
2 A{xn } and AC {xn } are both singleton.
Lemma 2.3 see 16. If C is a closed convex subset of a complete CAT(0) space X and if {xn } is a
bounded sequence in C, then the asymptotic center of {xn } is in C.
Definition 2.4 see 17. A sequence {xn } in a CAT0 space X is said to Δ-converge to x ∈ X
if x is the unique asymptotic center of {un } for every subsequence {un } of {xn }. In this case
one writes Δ − limn → ∞ xn x and call x the Δ-limit of {xn }.
Lemma 2.5 see 17. Every bounded sequence in a complete CAT(0) space always has a Δconvergent subsequence.
4
Abstract and Applied Analysis
Let {xn } be a bounded sequence in a CAT0 space X and let C be a closed convex
subset of X which contains {xn }. We denote the notation
{xn } w
iff Φw inf Φx,
x∈C
2.11
where Φx : lim supn → ∞ dxn , x.
Now one gives a connection between the “” convergence and Δ-convergence.
Proposition 2.6 see 13. Let {xn } be a bounded sequence in a CAT(0) space X and let C be a
closed convex subset of X which contains {xn }. Then
1 Δ − limn → ∞ xn x implies that {xn } x;
2 {xn } x and {xn } is regular imply that Δ − limn → ∞ xn x;
Let C be a closed subset of a metric space X, d. Recall that a mapping T : C → C is
said to be nonexpansive if
d T x, T y ≤ d x, y ,
∀x, y ∈ X.
2.12
T is said to be asymptotically nonexpansive if there is a sequence {kn } ⊂ 1, ∞ with
limn → ∞ kn 1 such that
d T n x, T n y ≤ kn d x, y ,
∀n ≥ 1, x, y ∈ X.
2.13
T is said to be closed if, for any sequence {xn } ⊂ C with dxn , x → 0 and dT xn , y →
0, then T x y.
T is called L-uniformly Lipschitzian, if there exists a constant L > 0 such that
d T n x, T n y ≤ Ld x, y ,
∀x, y ∈ C, n ≥ 1.
2.14
Definition 2.7. Let X, d be a metric space and let C be a closed subset of X. A mapping
T : C → C is said to be {vn }, {μn }, ζ−total asymptotically nonexpansive if there exist nonnegative real sequences {vn }, {μn } with vn → 0, μn → 0 n → ∞ and a strictly increasing
continuous function ζ : 0, ∞ → 0, ∞ with ζ0 0 such that
d T n x, T n y ≤ d x, y vn ζ d x, y μn ,
∀n ≥ 1, x, y ∈ C.
2.15
Remark 2.8. 1 It is obvious that If T is uniformly Lipschitzian, then T is closed.
2 From the definitions, it is to know that, each nonexpansive mapping is a
asymptotically nonexpansive mapping with sequence {kn 1}, and each asymptotically
nonexpansive mapping is a total asymptotically nonexpansive mapping with vn kn − 1,
μn 0, for all n ≥ 1, and ζt t, t ≥ 0.
Abstract and Applied Analysis
5
Lemma 2.9 demiclosed principle for total asymptotically nonexpansive mappings. Let C
be a closed and convex subset of a complete CAT(0) space X and let T : C → C be a L-uniformly
Lipschitzian and {vn }, {μn }, ζ−total asymptotically nonexpansive mapping. Let {xn } be a bounded
sequence in C such that limn → ∞ dxn , T xn 0 and xn w. Then T w w.
Proof. By the definition, xn w if and only if AC {xn } {w}. By Lemma 2.3, we have
A{xn } {w}.
Since limn → ∞ dxn , T xn 0, by induction we can prove that
lim dxn , T m xn 0,
n→∞
∀m ≥ 1.
2.16
In fact, it is obvious that, the conclusion is true for m 1. Suppose the conclusion
holds for m ≥ 1, now we prove that the conclusion is also true for m 1. In fact, since T is a
L-uniformly Lipschitzian mapping, we have
d xn , T m1 xn ≤ d xn , T xn d T xn , T T m xn
≤ d xn , T xn Ld xn , T m xn −→ 0 as n −→ ∞.
2.17
Equation 2.16 is proved. Hence for each x ∈ C and m ≥ 1 from 2.16 we have
Φx : lim sup dxn , x lim sup dT m xn , x.
n→∞
n→∞
2.18
In 2.18 taking x T m w, m ≥ 1, we have
ΦT m w lim sup dT m xn , T m w
n→∞
≤ lim sup dxn , w vm ζdxn , w μm .
2.19
n→∞
Let m → ∞ and taking superior limit on the both sides, it gets that
lim sup ΦT m w ≤ Φw.
m→∞
2.20
Furthermore, for any n, m ≥ 1 it follows from inequality 2.3 with t 1/2 that
d
2
w ⊕ T mw
xn ,
2
≤
1 2
1
1
d xn , w d2 xn , T m w − d2 w, T m w.
2
2
4
2.21
6
Abstract and Applied Analysis
Let n → ∞ and taking superior limit on the both sides of the above inequality, for any m ≥ 1
we get
1
1
w ⊕ T mw 2 1
≤ Φw2 ΦT m w2 − dw, T m w2 .
Φ
2
2
2
4
2.22
Since A{xn } {w}, we have
w ⊕ T mw 2 1
1
1
≤ Φw2 ΦT m w2 − dw, T m w2 ,
Φw2 ≤ Φ
2
2
2
4
∀m ≥ 1,
2.23
which implies that
d2 w, T m w ≤ 2ΦT m w2 − 2Φw2 .
2.24
By 2.20 and 2.24, we have limm → ∞ dw, T m w 0. This implies that limm → ∞ dw,
T m1 w 0. Since T is uniformly Lipschitzian, T is uniformly continuous. Hence we have
T w w. This completes the proof of Lemma 2.9.
The following proposition can be obtained from Lemma 2.9 immediately which is a
generalization of Kirk and Panyanak 17 and Nanjaras and Panyanak 13.
Proposition 2.10. Let C be a closed and convex subset of a complete CAT(0) space X and let T :
C → C be an asymptotically nonexpansive mapping. Let {xn } be a bounded sequence in C such that
limn → ∞ dxn , T xn 0 and Δ − limn → ∞ xn w. Then T w w.
Definition 2.11 see 18. Let X be a CAT0 space then X is uniformly convex, that is, for any
given r > 0, ∈ 0, 2 and λ ∈ 0, 1, there exists a ηr, 2 /8 such that, for all x, y, z ∈ X,
⎫
dx,
z ≤ r ⎬
2
r,
d y, z ≤ r
⇒ d 1 − λx ⊕ λy, z ≤ 1 − 2λ1 − λ
⎭
8
d x, y ≥ r
2.25
where the function η : 0, ∞ × 0, 2 → 0, 1 is called the modulus of uniform convexity of
CAT0.
Lemma 2.12 see 14. If {xn } is a bounded sequence in a complete CAT(0) space with A{xn } {x}, {un } is a subsequence of {xn } with A{un } {u}, and the sequence {dxn , u} converges, then
x u.
Lemma 2.13. Let {an }, {bn }, and {δn } be sequences of nonnegative real numbers satisfying the
inequality
an1 ≤ 1 δn an bn .
∞
If Σ∞
n1 δn < ∞ and Σn1 bn < ∞, then {an } is bounded and limn → ∞ an exists.
2.26
Abstract and Applied Analysis
7
Lemma 2.14 see 13. Let X be a CAT(0) space, x ∈ X be a given point and {tn } be a sequence in
b, c with b, c ∈ 0, 1 and 0 < b1 − c ≤ 1/2. Let {xn } and {yn } be any sequences in X such that
lim sup dxn , x ≤ r,
n→∞
lim sup d yn , x ≤ r,
n→∞
lim d 1 − tn xn ⊕ tn yn , x r,
2.27
n→∞
for some r ≥ 0. Then
lim d xn , yn 0.
n→∞
2.28
3. Main Results
In this section, we will prove our main theorem.
Theorem 3.1. Let C be a nonempty bounded closed and convex subset of a complete CAT(0) space X.
Let S : C → C be a asymptotically nonexpansive mapping with sequence {kn } ⊂ 1, ∞, kn → 1
and T : C → C be a uniformly L-Lipschitzian and {vn }, {μn }, ζ−total asymptotically nonexpansive
mapping such that F FS ∩ FT / ∅. From arbitrary x1 ∈ C, defined the sequence {xn } as follows:
yn αn Sn xn ⊕ 1 − αn xn ,
xn1 βn T n yn ⊕ 1 − βn xn
3.1
for all n ≥ 1, where {βn } is a sequence in (0, 1). If the following conditions are satisfied:
∞
∞
i Σ∞
n1 vn < ∞; Σn1 μn < ∞; Σn1 kn − 1 < ∞;
ii there exists a constant M∗ > 0 such that ζr ≤ M∗ r, r ≥ 0;
iii there exist constants b, c ∈ 0, 1 with 0 < b1 − c ≤ 1/2 such that {αn } ⊂ b, c;
n
iv Σ∞
n1 sup{dz, S z : z ∈ B} < ∞ for each bounded subset B of C.
Then the sequence {xn }Δ-converges to a fixed point of F.
Proof. We divide the proof of Theorem 3.1 into four steps.
I First we prove that for each p ∈ F the following limit exists
lim d xn , p .
n→∞
3.2
8
Abstract and Applied Analysis
In fact, for each p ∈ F, we have
d yn , p d αn Sn xn ⊕ 1 − αn xn , p
≤ αn d Sn xn , p 1 − αn d xn , p
αn d Sn xn , Sn p 1 − αn d xn , p
≤ αn kn d xn , p 1 − αn d xn , p
1 αn kn − 1d xn , p ,
d xn1 , p d βn T n yn ⊕ 1 − βn xn , p
≤ βn d T n yn , p 1 − βn d xn , p
βn d T n yn , T n p 1 − βn d xn , p
≤ βn d yn , p vn ζ d yn , p μn 1 − βn d xn , p
≤ βn d yn , p vn M∗ d yn , p μn 1 − βn d xn , p
≤ d xn , p βn αn kn − 1d xn , p βn vn M∗ 1 αn kn − 1d xn , p μn
≤ 1 kn − 1 vn M∗ 1 αn kn − 1d xn , p μn .
3.3
It follows from Lemma 2.13 that {dxn , p} is bounded and limn → ∞ dxn , p exists. Without
loss of generality, we can assume limn → ∞ dxn , p c ≥ 0.
II Next we prove that
lim dxn , T xn 0.
3.4
d T n yn , p d T n yn , T n p
≤ d yn , p vn ζ d yn , p μn
≤ 1 vn M∗ d yn , p μn
≤ 1 vn M∗ 1 αn kn − 1d xn , p μn
3.5
n→∞
In fact, since
for all n ∈ N and p ∈ F, we have
lim sup d T n yn , p ≤ c.
3.6
lim d βn T n yn ⊕ 1 − βn xn , p lim d xn1 , p c,
3.7
n→∞
On the other hand, since
n→∞
n→∞
Abstract and Applied Analysis
9
by Lemma 2.14, we have
lim d T n yn , xn 0.
3.8
n→∞
From condition iv, we have
d xn , yn dxn , 1 − αn xn ⊕ αn Sn xn ≤ αn dxn , Sn xn −→ 0
3.9
n −→ ∞.
Hence from 3.8 and 3.9 we have that
dxn , T n xn ≤ d xn , T n yn d T n yn , T n xn
≤ d xn , T n yn Ld yn , xn −→ 0
3.10
n −→ ∞.
By 3.9 and 3.10 it gets that
dxn1 , T n xn d 1 − βn xn ⊕ βn T n yn , T n xn
≤ 1 − βn dxn , T n xn βn d T n yn , T n xn
≤ 1 − βn dxn , T n xn βn Ld yn , xn −→ 0 n −→ ∞.
3.11
Hence from 3.10 and 3.11 we have that
dxn , xn1 −→ 0 n −→ ∞.
3.12
Again since T is uniformly L-Lipschitzian, from 3.10 and 3.12 we have that
dxn , T xn ≤ dxn , xn1 d xn1 , T n1 xn1 d T n1 xn1 , T n1 xn d T n1 xn , T xn
≤ L 1dxn , xn1 d xn1 , T n1 xn1 LdT n xn , xn −→ 0
n −→ ∞.
3.13
Equation 3.4 is proved.
III Now we prove that
wω xn :
and wω xn consists exactly of one point.
{un }⊂{xn }
A{un } ⊂ F
3.14
10
Abstract and Applied Analysis
In fact, let u ∈ wω xn , then there exists a subsequence {un } of {xn } such that A{un } {u}. By Lemmas 2.5 and 2.3, there exists a subsequence {νn } of {un } such that Δ−limn → ∞ νn ν ∈ C. By Lemma 2.9, we have ν ∈ FT ⊂ F. By Lemma 2.12, u ν. This shows that
wω xn ⊂ F.
Let {un } be a subsequence of {xn } with A{un } {u} and let A{xn } {x}. Since
u ∈ wω xn ⊂ F and {dxn , u} converges, by Lemma 2.12, we have x u. This shows that
wω xn consists of exactly one point.
IV Finally we prove {xn }Δ-converges to a point of F.
In fact, it follows from 3.2 that {dxn , p} is convergent for each p ∈ F. By 3.4
limn → ∞ dxn , T xn 0. By 3.14 wω xn ⊂ F and wω xn consists of exactly one point. This
shows that {xn }Δ-converges to a point of F.
This completes the proof of Theorem 3.1.
Conflict of Interests
The authors declare that they have no competing interests.
Authors’ Contribution
All the authors contributed equally to the writing of the present article. And they also read
and approved the final paper.
Acknowledgments
The authors would like to express their thanks to the referees for their helpful suggestions and
comments. This study was supported by the Scientific Research Fund of Sichuan Provincial
Education Department 12ZB346.
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