Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 327434, 17 pages
doi:10.1155/2012/327434
Research Article
Common Fixed Points for Asymptotic
Pointwise Nonexpansive Mappings in Metric and
Banach Spaces
P. Pasom1 and B. Panyanak1, 2
1
2
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to B. Panyanak, banchap@chiangmai.ac.th
Received 26 October 2011; Accepted 8 December 2011
Academic Editor: Rudong Chen
Copyright q 2012 P. Pasom and B. Panyanak. 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.
Let C be a nonempty bounded closed convex subset of a complete CAT0 space X. We prove
that the common fixed point set of any commuting family of asymptotic pointwise nonexpansive
mappings on C is nonempty closed and convex. We also show that, under some suitable
nk
conditions, the sequence {xk }∞
k1 defined by xk1 1 − tmk xk ⊕ tmk Tm ym−1k , ym−1k 1 −
nk
nk
tm−1k xk ⊕ tm−1k Tm−1 ym−2k , ym−2k 1 − tm−2k xk ⊕ tm−2k Tm−2 ym−3k , . . . , y2k 1 − t2k xk ⊕
t2k T2nk y1k , y1k 1 − t1k xk ⊕ t1k T1nk y0k , y0k xk , k ∈ N, converges to a common fixed point
of T1 , T2 , . . . , Tm where they are asymptotic pointwise nonexpansive mappings on C, {tik }∞
k1 are
sequences in 0, 1 for all i 1, 2, . . . , m, and {nk } is an increasing sequence of natural numbers.
The related results for uniformly convex Banach spaces are also included.
1. Introduction
A mapping T on a subset C of a Banach space X is said to be asymptotic pointwise
nonexpansive if there exists a sequence of mappings αn : C → 0, ∞ such that
n
T x − T n y ≤ αn xx − y,
1.1
where lim supn → ∞ αn x ≤ 1, for all x, y ∈ C. This class of mappings was introduced by Kirk
and Xu 1, where it was shown that if C is a bounded closed convex subset of a uniformly
convex Banach space X, then every asymptotic pointwise nonexpansive mapping T : C → C
always has a fixed point. In 2009, Hussain and Khamsi 2 extended Kirk-Xu’s result to the
case of metric spaces, specifically to the so-called CAT0 spaces. Recently, Kozlowski 3
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defined an iterative sequence for an asymptotic pointwise nonexpansive mapping T : C → C
by x1 ∈ C and
xk1 1 − tk xk tk T nk yk ,
yk 1 − sk xk sk T nk xk ,
k ∈ N,
1.2
where {tk } and {sk } are sequences in 0, 1 and {nk } is an increasing sequence of natural
numbers. He proved, under some suitable assumptions, that the sequence {xk } defined by
1.2 converges weakly to a fixed point of T where X is a uniformly convex Banach space
which satisfies the Opial condition and {xk } converges strongly to a fixed point of T provided
T r is a compact mapping for some r ∈ N. On the other hand, Khan et al. 4 studied the
iterative process defined by
n
ym−1n ,
xn1 1 − αmn xn αmn Tm
n
ym−1n 1 − αm−1n xn αm−1n Tm−1
ym−2n ,
n
ym−2n 1 − αm−2n xn αm−2n Tm−2 ym−3n ,
..
.
1.3
y2n 1 − α2n xn α2n T2n y1n ,
y1n 1 − α1n xn α1n T1n y0n ,
y0n xn ,
n ∈ N,
where T1 , . . . , Tm are asymptotically quasi-nonexpansive mappings on C and {αin }∞
n1 are
sequences in 0, 1 for all i 1, 2, . . . , m.
In this paper, motivated by the results mentioned above, we ensure the existence of
common fixed points for a family of asymptotic pointwise nonexpansive mappings in a
CAT0 space. Furthermore, we obtain and strong convergence theorems of a sequence
defined by
nk
ym−1k ,
xk1 1 − tmk xk ⊕ tmk Tm
nk
ym−2k ,
ym−1k 1 − tm−1k xk ⊕ tm−1k Tm−1
nk
ym−2k 1 − tm−2k xk ⊕ tm−2k Tm−2 ym−3k ,
..
.
y2k 1 − t2k xk ⊕
y1k 1 − t1k xk ⊕
y0k xk ,
1.4
t2k T2nk y1k ,
t1k T1nk y0k ,
k ∈ N,
where T1 , . . . , Tm are asymptotic pointwise nonexpansive mappings on a subset C of a
complete CAT0 space and {tik }∞
k1 are sequences in 0, 1 for all i 1, 2, . . . , m, and {nk }
is an increasing sequence of natural numbers. We also note that our method can be used to
prove the analogous results for uniformly convex Banach spaces.
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3
2. Preliminaries
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. It is
well-known that any complete, simply connected Riemannian manifold having nonpositive
sectional curvature is a CAT0 space. Other examples include Pre-Hilbert spaces see 5, Rtrees see 6, Euclidean buildings see 7, and the complex Hilbert ball with a hyperbolic
metric see 8. For a thorough discussion of these spaces and of the fundamental role they
play in geometry, we refer the reader to Bridson and Haefliger 5.
Fixed point theory in CAT0 spaces was first studied by Kirk see 9, 10. He showed
that every nonexpansive single-valued 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 single-valued and multivalued mappings in CAT0 spaces has been rapidly developed,
and many papers have appeared see, e.g., 2, 11–22 and the references therein. It is worth
mentioning that fixed point theorems in CAT0 spaces specially in R-trees can be applied
to graph theory, biology, and computer science see, e.g., 6, 23–26.
Let X, d be a metric space. A geodesic path joining x ∈ X to y ∈ X or, more briefly, a
geodesic from x to y is a map c from a closed interval 0, l ⊂ R to X such that c0 x, cl y, and dct, ct |t − t | for all t, t ∈ 0, l. In particular, c is an isometry and dx, y l.
The image α of c is called a geodesic or metric segment joining x and y. When it is unique,
this geodesic is denoted by x, y. The space X, d is said to be a geodesic space if every two
points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one
geodesic joining x and y for each x, y ∈ X. A subset Y ⊂ X is said to be convex if Y includes
every geodesic segment joining any two of its points.
A geodesic triangle x1 , x2 , x3 in a geodesic space X, d consists of three points x1 ,
x2 , x3 in X the vertices of and a geodesic segment between each pair of vertices the
edges of . A comparison triangle for geodesic triangle x1 , x2 , x3 in X, d is a triangle
x1 , x2 , x3 : x1 , x2 , x3 in the Euclidean plane E2 such that dE2 xi , xj dxi , xj for
i, j ∈ {1, 2, 3}.
A geodesic space is said to be a CAT0 space if all geodesic triangles satisfy the
following comparison axiom.
CAT0: Let be a geodesic triangle in X, and let be a comparison triangle for .
Then, is said to satisfy the CAT0 inequality if for all x, y ∈ and all comparison points
x, y ∈ ,
d x, y ≤ dE2 x, y .
2.1
Let x, y ∈ X, by Lemma 2.1iv of 14 for each t ∈ 0, 1, there exists a unique point
z ∈ x, y such that
dx, z td x, y ,
d y, z 1 − td x, y .
2.2
We will use the notation 1 − tx ⊕ ty for the unique point z satisfying 2.2. We now collect
some elementary facts about CAT0 spaces.
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Journal of Applied Mathematics
Lemma 2.1. Let X be a complete CAT(0) space.
i [5, Proposition 2.4] If C is a nonempty closed convex subset of X, then, for every x ∈ X,
there exists a unique point P x ∈ C such that dx, P x inf{dx, y : y ∈ C}.
Moreover, the map x → P x is a nonexpansive retract from X onto C.
ii [14, Lemma 2.4] For x, y, z ∈ X and t ∈ 0, 1, we have
d 1 − tx ⊕ ty, z ≤ 1 − tdx, z td y, z .
2.3
iii [14, Lemma 2.5] For x, y, z ∈ X and t ∈ 0, 1, we have
2
2
2
d 1 − tx ⊕ ty, z ≤ 1 − tdx, z2 td y, z − t1 − td x, y .
2.4
We now give the concept of Δ-convergence and collect some of its basic properties. Let
{xn } be a bounded sequence in a CAT0 space X. For x ∈ X, we set
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
and the asymptotic center A{xn } of {xn } is the set
A{xn } {x ∈ X : rx, {xn } r{xn }}.
2.7
It is known from Proposition 7 of 27 that, in a CAT0 space, A{xn } consists of
exactly one point.
Definition 2.2 see 28, 29. 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, we write Δ-limx x and call x the Δ-limit of {xn }.
n
n
Lemma 2.3. Let X be a complete CAT(0) space.
i [28, page 3690] Every bounded sequence in X has a Δ-convergent subsequence.
ii [30, Proposition 2.1] If C is a closed convex subset of a complete CAT(0) space and if {xn }
is a bounded sequence in C, then the asymptotic center of {xn } is in C.
iii [14, Lemma 2.8] If {xn } is a bounded sequence in a complete CAT(0) space with A{xn } {x} and {un } is a subsequence of {xn } with A{un } {u} and the sequence {dxn , u}
converges, then x u.
Recall that a mapping T : X → X is said to be nonexpansive 31 if
d T x, T y ≤ d x, y ,
∀x, y ∈ X,
2.8
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5
where T is called asymptotically nonexpansive 32 if there is a sequence {kn } of positive
numbers with the property limn → ∞ kn 1 and such that
d T n x, T n y ≤ kn d x, y ,
∀n ≥ 1, x, y ∈ X,
2.9
where T is called an asymptotic pointwise nonexpansive mapping 1 if there exists a sequence of
functions αn : X → 0, ∞ such that
d T n x, T n y ≤ αn xd x, y ,
∀n ≥ 1, x, y ∈ X,
2.10
where lim supn → ∞ αn x ≤ 1. The following implications hold.
T is nonexpansive ⇒ T is asymptotically nonexpansive
⇒ T is asymptotic pointwise nonexpansive.
2.11
A point x ∈ X is called a fixed point of T if x T x. We shall denote by FT the set of fixed
points of T . The existence of fixed points for asymptotic pointwise nonexpansive mappings
in CAT0 spaces was proved by Hussain and Khamsi 2 as the following result.
Theorem 2.4. Let C be a nonempty bounded closed convex subset of a complete CAT(0) space X.
Suppose that T : C → C is an asymptotic pointwise nonexpansive mapping. Then, FT is nonempty
closed and convex.
3. Existence Theorems
Let M be a metric space and F a family of subsets of M. Then, we say that F defines a convexity structure on M if it contains the closed balls and is stable by intersection.
Definition 3.1 see 2. Let F be a convexity structure on M. We will say that F is compact if
∅ for
any family {Aα }α∈Γ of elements of F has a nonempty intersection provided α∈F Aα /
any finite subset F ⊂ Γ.
Let X be a complete CAT0 space. We denote by CX the family of all closed convex
subsets of X. Then, CX is a compact convexity structure on X see, e.g., 2.
The following theorem is an extension of Theorem 4.3 in 33. For an analog of this
result in uniformly convex Banach spaces, see 34.
Theorem 3.2. Let C be a nonempty bounded closed and convex subset of a complete CAT(0) space
X. Then, for any commuting family S of asymptotic pointwise nonexpansive mappings on C, the set
FS of common fixed points of S is nonempty closed and convex.
Proof. Let T be the family of all finite intersections of the fixed point sets of mappings in
the commutative family S. We first show that T has the finite intersection property. Let
T1 , T2 , . . . , Tn ∈ S. By Theorem 2.4, FT1 is a nonempty closed and convex subset of C. We
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Journal of Applied Mathematics
assume that A : k−1
j1 FTj is nonempty closed and convex for some k ∈ N with 1 < k ≤ n.
For x ∈ A and j ∈ N with 1 ≤ j < k, we have
Tk x Tk ◦ Tj x Tj ◦ Tk x.
3.1
Thus, Tk x is a fixed point of Tj , which implies that Tk x ∈ A; therefore, A is invariant under
Tk . Again, by Theorem 2.4, Tk has a fixed point in A, that is,
k
F Tj FTk A /
∅.
3.2
j1
By induction,
compact,
n
j1
FTj / ∅. Hence, T has the finite intersection property. Since CX is
FS T ∈T
T/
∅.
3.3
Obviously, the set is closed and convex.
As a consequence of Lemma 2.1i and Theorem 3.2, we obtain an analog of Bruck’s
theorem 35.
Corollary 3.3. Let C be a nonempty bounded closed and convex subset of a complete CAT(0) space
X. Then, for any commuting family S of nonexpansive mappings on C, the set FS of common fixed
points of S is a nonempty nonexpansive retract of C.
4. Convergence Theorems
Throughout this section, X stands for a complete CAT0 space. Let C be a closed convex
subset of X. We shall denote by TC the class of all asymptotic pointwise nonexpansive
mappings from C into C. Let T1 , . . . , Tm ∈ TC, without loss of generality, we can assume that
there exists a sequence of mappings αn : C → 0, ∞ such that for all x, y ∈ C, i 1, . . . , m,
and n ∈ N, we have
d Tin x, Tin y ≤ αn xd x, y ,
lim sup αn x ≤ 1.
n→∞
4.1
Let an x max{αn x, 1}. Again, without loss of generality, we can assume that
d Tin x, Tin y ≤ an xd x, y ,
lim an x 1,
n→∞
an x ≥ 1,
4.2
for all x, y ∈ C, i 1, . . . , m, and n ∈ N. We define bn x an x − 1, then, for each x ∈ C, we
have limn → ∞ bn x 0.
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7
The following definition is a mild modification of 3, Definition 2.3.
Definition 4.1. Define Tr C as a class of all T ∈ TC such that
∞
sup bn x < ∞,
4.3
n1 x∈C
an is a bounded function for every n ∈ N.
Let T1 , . . . , Tm ∈ Tr C, and let {tik }∞
k1 ⊂ 0, 1 be bounded away from 0 and 1 for all
i 1, 2, . . . , m, and {nk } an increasing sequence of natural numbers. Let x1 ∈ C, and define a
sequence {xk } in C as
nk
xk1 1 − tmk xk ⊕ tmk Tm
ym−1k ,
nk
ym−2k ,
ym−1k 1 − tm−1k xk ⊕ tm−1k Tm−1
nk
ym−2k 1 − tm−2k xk ⊕ tm−2k Tm−2 ym−3k ,
..
.
y2k 1 − t2k xk ⊕
y1k 1 − t1k xk ⊕
y0k xk ,
4.4
t2k T2nk y1k ,
t1k T1nk y0k ,
k ∈ N.
We say that the sequence {xk } in 4.4 is well defined if lim supk → ∞ ank xk 1. As in 3, we
observe that limk → ∞ ak x 1 for every x ∈ C. Hence, we can always choose a subsequence
{ank } which makes {xk } well defined.
Lemma 4.2 see 36, Lemma 2.2. Let {an } and {un } be sequences of nonnegative real numbers
satisfying
an1 ≤ 1 un an ,
∀n ∈ N,
∞
un < ∞.
4.5
n1
Then, (i) limn an exists, (ii) if lim infn an 0, then limn an 0.
Lemma 4.3 see 37, 38. Suppose {tn } is a sequence in b, c for some b, c ∈ 0, 1 and {un }, {vn }
are sequences in X such that lim supn dun , w ≤ r, lim supn dvn , w ≤ r, and limn d1 − tn un ⊕
tn vn , w r for some r ≥ 0. Then,
lim dun , vn 0.
n→∞
4.6
Lemma 4.4. Let C be a nonempty closed convex subset of X and T1 , . . . , Tm ∈ Tr C. Let
⊂ a, b ⊂ 0, 1 and {nk } ⊂ N be such that {xk } in 4.4 is well defined. Assume that
{tik }∞
k1
∅. Then,
F : m
i1 FTi /
a there exists a sequence {vk } in 0, ∞ such that ∞
k1 vk < ∞ and dxk1 , p ≤
1 vk m dxk , p, for all p ∈ F and all k ∈ N,
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Journal of Applied Mathematics
b there exists a constant M > 0 such that dxkl , p ≤ Mdxk , p, for all p ∈ F and
k, l ∈ N.
Proof. a Let p ∈ F and vk supx∈C bnk x for all k ∈ N. Since
have ∞
k1 vk < ∞. Now,
∞
k1
supx∈C bnk x < ∞, we
d y1k , p ≤ 1 − t1k d xk , p t1k d T1nk xk , p
≤ 1 − t1k d xk , p t1k 1 bnk p d xk , p
1 t1k bnk p d xk , p
≤ 1 vk d xk , p .
4.7
Suppose that dyjk , p ≤ 1 vk j dxk , p holds for some 1 ≤ j ≤ m − 2. Then,
nk
d yj1k , p ≤ 1 − tj1k d xk , p tj1k d Tj1
yjk , p
≤ 1 − tj1k d xk , p tj1k 1 bnk p d yjk , p
≤ 1 − tj1k d xk , p tj1k 1 vk j1 d xk , p
j1 j 1 j ··· j 2 − r r
1 − tj1k tj1k 1 d xk , p
vk
r!
r1
j1
j 1 j ··· j 2−r r 1 tj1k
vk d xk , p
r!
r1
j1 ≤ 1 vk d xk , p .
4.8
By induction, we have
d yik , p ≤ 1 vk i d xk , p ,
∀i 1, 2, . . . , m − 1.
4.9
This implies
nk
d xk1 , p ≤ 1 − tmk d xk , p tmk d Tm
ym−1k , p
≤ 1 − tmk d xk , p tmk 1 bnk p d ym−1k , p
≤ 1 − tmk d xk , p tmk 1 vk 1 vk m−1 d xk , p
≤ 1 − tmk d xk , p tmk 1 vk m d xk , p
m
mm − 1 · · · m − r 1 r
1 − tmk tmk 1 d xk , p
vk
r!
r1
m
mm − 1 · · · m − r 1
r
1 tmk
vk d xk , p
r!
r1
m ≤ 1 vk d xk , p .
This completes the proof of a.
4.10
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9
b We observe that 1 αn ≤ enα holds for all n ∈ N and α ≥ 0. Thus, by a, for
k, l ∈ N, we have
d xkl , p ≤ 1 vkl−1 m d xkl−1 , p
≤ exp{mvkl−1 }d xkl−1 , p ≤ · · · ≤ exp m
≤ exp m
∞
kl−1
vi d xk , p
i1
4.11
vi d xk , p .
i1
The proof is complete by setting M exp{m
∞
i1
vi }.
Theorem 4.5. Let C be a nonempty closed convex subset of X and T1 , . . . , Tm ∈ Tr C. Let {tik }∞
k1 ⊂
∅.
Then,
{xk }
a, b ⊂ 0, 1 and {nk } ⊂ N be such that {xk } in 4.4 is well defined. Assume that F /
converges to some point in F if and only if lim infk → ∞ dxk , F 0, where dx, F infp∈F dx, p.
Proof. The necessity is obvious. Now, we prove the sufficiency. From Lemma 4.4a, we have
d xk1 , p ≤ 1 vk m d xk , p ,
∀p ∈ F, ∀k ∈ N.
4.12
This implies
m
dxk1 , F ≤ 1 vk dxk , F 1
m
mm − 1 · · · m − r 1
r!
r1
vkr
dxk , F.
4.13
∞ m
r
Since ∞
r1 mm − 1 · · · m − r 1/r!vk < ∞. By Lemma 4.2ii, we
k1 vk < ∞, then
k1
get limk → ∞ dxk , F 0. Next, we show that {xk } is a Cauchy sequence. From Lemma 4.4b,
there exists M > 0 such that
d xkl , p ≤ Md xk , p ,
∀p ∈ F, k, l ∈ N.
4.14
Since limk → ∞ dxk , F 0, for each ε > 0, there exists k1 ∈ N such that
dxk , F <
ε
,
2M
∀k ≥ k1 .
4.15
Hence, there exists z1 ∈ F such that
dxk1 , z1 <
ε
.
2M
4.16
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By 4.14 and 4.16, for k ≥ k1 , we have
dxkl , xk ≤ dxkl , z1 dxk , z1 ≤ Mdxk1 , z1 Mdxk1 , z1 ε < 2M
2M
ε.
4.17
This shows that {xk } is a Cauchy sequence and so converges to some q ∈ C. We next show
that q ∈ F. Let L sup{a1 x : x ∈ C}. Then, for each > 0, there exists k2 ∈ N such that
d xk , q <
,
21 L
∀k ≥ k2 .
4.18
Since limk → ∞ dxk , F 0, there exists k3 ≥ k2 such that
dxk , F <
,
21 L
∀k ≥ k3 .
4.19
Thus, there exists z2 ∈ F such that
dxk3 , z2 <
.
21 L
4.20
By 4.18 and 4.20, for each i 1, 2, . . . , m, we have
d Ti q, q ≤ d Ti q, Ti xk3 dTi xk3 , z2 dz2 , xk3 d xk3 , q
≤ Ld xk3 , q Ldxk3 , z2 dxk3 , z2 d xk3 , q
≤ 1 Ld xk3 , q 1 Ldxk3 , z2 < 1 L
1 L
21 L
21 L
.
4.21
Since is arbitrary, we have Ti q q for all i 1, 2, . . . , m. Hence, q ∈ F.
As an immediate consequence of Theorem 4.5, we obtain the following.
Corollary 4.6. Let C be a nonempty closed convex subset of X and T1 , . . . , Tm ∈ Tr C. Let {tik }∞
k1 ⊂
∅. Then, {xk }
a, b ⊂ 0, 1 and {nk } ⊂ N be such that {xk } in 4.4 is well defined. Assume that F /
converges to a point p ∈ F if and only if there exists a subsequence {xkj } of {xk } which converges to
p.
Definition 4.7. A strictly increasing sequence {nk } ⊂ N is called quasiperiodic 39 if the
sequence {nk1 − nk } is bounded or equivalently if there exists a number p ∈ N such that
any block of p consecutive natural numbers must contain a term of the sequence {nk }. The
smallest of such numbers p will be called a quasiperiod of {nk }.
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11
Lemma 4.8. Let C be a nonempty closed convex subset of X and T1 , . . . , Tm ∈ Tr C. Let {tik }∞
k1 ⊂
δ, 1 − δ for some δ ∈ 0, 1/2 and {nk } ⊂ N be such that {xk } in 4.4 is well defined. Then,
i limk → ∞ dxk , p exists for all p ∈ F,
ii limk → ∞ dxk , Tjnk yj−1k 0, for all j 1, 2, . . . , m,
iii if the set J {k ∈ N : nk1 1 nk } is quasiperiodic, then limk → ∞ dxk , Tj xk 0, for
all j 1, 2, . . . , m.
Proof. i Follows from Lemmas 4.2i and 4.4a.
ii Let p ∈ F, then, by i, we have limk → ∞ dxk , p exists. Let
lim d xk , p c.
4.22
k→∞
By 4.9 and 4.22, we get that
lim sup d yjk , p ≤ c,
k→∞
for 1 ≤ j ≤ m − 1.
4.23
Note that
nk
d xk1 , p ≤ 1 − tmk d xk , p tmk d Tm
ym−1k , p
≤ 1 − tmk d xk , p tmk 1 vk d ym−1k , p
..
.
≤ 1 − tmk tm−1k · · · tj1k 1 vk m−j d xk , p
tmk tm−1k · · · tj1k 1 vk m−j d yjk , p .
4.24
Thus,
d xk , p ≤
d xk , p
δm−j
−
d xk1 , p
δm−j 1 vk m−j
d yjk , p ,
4.25
so that
c ≤ lim inf d yjk , p ,
k→∞
for 1 ≤ j ≤ m − 1.
4.26
From 4.23 and 4.26, we have
lim d yjk , p c,
k→∞
for each j 1, 2, . . . , m − 1.
4.27
That is
lim d
k→∞
for each j 1, 2, . . . , m − 1.
1 − tjk xk ⊕ tjk Tjnk yj−1k , p c,
4.28
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We also obtain from 4.23 that
lim sup d Tjnk yj−1k , p ≤ c,
k→∞
for each j 1, 2, . . . , m − 1.
4.29
By Lemma 4.3, we get that
lim d Tjnk yj−1k , xk 0,
k→∞
for each j 1, 2, . . . , m − 1.
4.30
For the case j m, by 4.1, we have
nk
d Tm
ym−1k , p ≤ 1 bnk p d ym−1k , p ≤ 1 bnk p 1 vnk m−1 d xk , p .
4.31
But since limk → ∞ dxk , p c, then
nk
lim sup d Tm
ym−1k , p ≤ c.
4.32
nk
lim d 1 − tmk xk ⊕ tmk Tm
ym−1k , p lim d xk1 , p c.
4.33
k→∞
Moreover,
k→∞
k→∞
Again, by Lemma 4.3, we get that
nk
ym−1k , xk 0.
lim d Tm
k→∞
4.34
Thus, 4.30 and 4.34 imply that
lim d Tjnk yj−1k , xk 0,
k→∞
for each j 1, 2, . . . , m.
4.35
iii For j 1, from ii, we have
lim d T1nk xk , xk 0.
4.36
d Tjnk xk , xk ≤ d Tjnk xk , Tjnk yj−1k d Tjnk yj−1k , xk
≤ ank xk d xk , yj−1k d Tjnk yj−1k , xk
nk
≤ ank xk tj−1k d xk , Tj−1
yj−2k d Tjnk yj−1k , xk .
4.37
k→∞
If j 2, 3, . . . , m, then we have
Journal of Applied Mathematics
13
By ii and lim supk → ∞ ank xk 1, we get
lim sup d Tjnk xk , xk 0,
k→∞
for j 2, 3, . . . , m.
4.38
∀j 1, 2, . . . , m.
4.39
By 4.36 and 4.38, we have
lim d Tjnk xk , xk 0,
k→∞
By the construction of the sequence {xk }, we have from 4.35 that
lim dxk1 , xk 0.
4.40
k→∞
Next, we show that
lim d Tj xk , xk 0,
k→∞
∀j 1, 2, . . . , m.
4.41
It is enough to prove that dTj xk , xk → 0 as k → ∞ though J. Indeed, let p be a quasiperiod
of J, and let ε > 0 be given. Then, there exists N1 ∈ N such that
ε
lim d Tj xk , xk < ,
k→∞
3
∀k ∈ J such that k ≥ N1 .
4.42
By the quasiperiodicity of J, for each l ∈ N, there exists il ∈ J such that |l − il | ≤ p. Without
loss of generality, we can assume that l ≤ il ≤ l p the proof for the other case is identical.
Let M sup{a1 x : x ∈ C}. Then, M ≥ 1. Since liml → ∞ dxl1 , xl 0 by 4.40, there exists
N2 ∈ N such that
dxl1 , xl <
ε
,
3pM
∀l ≥ N2 .
4.43
This implies that
ε
dxil , xl ≤ dxil , xil −1 · · · dxl1 , xl ≤ p
3pM
ε
.
3M
4.44
By the definition of T , we have
d Tj xil , Tj xl ≤ Mdxil , xl ≤ M
ε
3M
ε
.
3
4.45
Let N max{N1 , N2 }. Then, for l ≥ N, we have from 4.42, 4.44, and 4.45 that
ε ε
ε
≤ ε.
d xl , Tj xl ≤ dxl , xil d xil , Tj xil d Tj xil , Tj xl <
3M 3 3
4.46
14
Journal of Applied Mathematics
To prove that dTj xk , xk → 0 as k → ∞ though J. Since J {k ∈ N : nk1 nk 1} is
quasiperiodic, for each k ∈ J, we have
d xk , Tj xk ≤ dxk , xk1 d xk1 , Tjnk1 xk1 d Tjnk1 xk1 , Tjnk1 xk d Tjnk 1 xk , Tj xk
≤ dxk , xk1 d xk1 , Tjnk1 xk1 ank1 xk1 dxk1 , xk a1 xk d Tjnk xk , xk .
4.47
From this, together with 4.39 and 4.40, we can obtain that dTj xk , xk → 0 as k → ∞
through J.
The following lemmas can be found in 3 see also 2.
Lemma 4.9. Let C be a nonempty closed convex subset of X, and let T
limn → ∞ dxn , T xn 0, then limn → ∞ dxn , T l xn 0 for every l ∈ N.
∈
Tr C. If
Lemma 4.10. Let C be a nonempty closed convex subset of X, and let T ∈ Tr C. Suppose {xn } is a
bounded sequence in C such that limn dxn , T xn 0 and Δ-limn xn w. Then, T w w.
By using Lemmas 2.3 and 4.10, we can obtain the following result. We omit the proof
because it is similar to the one given in 38.
Lemma 4.11. Let C be a closed convex subset of X, and let T : C → C be an asymptotic pointwise
nonexpansive mapping. Suppose {xn } is a bounded sequence in C such that limn dxn , T xn 0
and dxn , v converges for each v ∈ FT , then ωw xn ⊂ FT . Here, ωw xn A{un } where
the union is taken over all subsequences {un } of {xn }. Moreover, ωw xn consists of exactly one point.
Now, we are ready to prove our Δ-convergence theorem.
Theorem 4.12. Let C be a nonempty closed convex subset of X and T1 , . . . , Tm ∈ Tr C. Let
for some δ ∈ 0, 1/2 and {nk } ⊂ N be such that {xk } in 4.4 is well defined.
{tik }∞
k1 ⊂ δ, 1 − δ
∅ and the set J {k ∈ N : nk1 1 nk } is quasiperiodic. Then,
Suppose that F : m
i1 FTi /
{xk }Δ-converges to a common fixed point of the family {Ti : i 1, 2, . . . , m}.
Proof. Let p ∈ F, by Lemma 4.8, limk → ∞ dxk , p existsm and hence {xk } is bounded. Since
limk → ∞ dxk , Tj xk 0 for all j 1, 2, . . . , m, then by Lemma 4.11 ωw xk ⊂ FTj for all
j 1, 2, . . . , m, and hence ωw xk ⊂ m
j1 FTj F. Since ωw xn consists of exactly one
point, then {xk }Δ-converges to an element of F.
Before proving our strong convergence theorem, we recall that a mapping T : C → C
is said to be semicompact if C is closed and, for any bounded sequence {xn } in C with
limn → ∞ dxn , T xn 0, there exists a subsequence {xnj } of {xn } and x ∈ C such that
limk → ∞ xnk x.
Theorem 4.13. Let C be a nonempty closed convex subset of X and T1 , . . . , Tm ∈ Tr C such that Til
is semicompact for some i ∈ {1, . . . , m} and l ∈ N. Let {tik }∞
δ for some δ ∈ 0, 1/2
k1 ⊂ δ, 1 − ∅ and the set
and {nk } ⊂ N be such that {xk } in 4.4 is well defined. Suppose that F : m
i1 FTi /
J {k ∈ N : nk1 1 nk } is quasiperiodic. Then, {xk } converges to a common fixed point of the
family {Ti : i 1, 2, . . . , m}.
Journal of Applied Mathematics
15
Proof. By Lemma 4.8, we have
lim dxk , Ti xk 0,
k→∞
for i 1, . . . , m.
4.48
Let i ∈ {1, . . . , m} be such that Til is semicompact. Thus, by Lemma 4.9,
lim d xk , Til xk 0.
4.49
k→∞
We can also find a subsequence {xnj } of {xk } such that limj → ∞ xkj q ∈ C. Hence, from
4.48, we have
d q, Ti q lim d xkj , Ti xkj 0,
j →∞
∀i 1, . . . , m.
4.50
Thus, q ∈ F, and, by Corollary 4.6, {xk } converges to q. This completes the proof.
5. Concluding Remarks
One may observe that our method can be used to obtain the analogous results for uniformly
convex Banach spaces. Let C be a nonempty closed convex subset of a Banach space X and
fix x1 ∈ C. Define a sequence {xk } in C as
nk
ym−1k ,
xk1 1 − tmk xk tmk Tm
nk
ym−2k ,
ym−1k 1 − tm−1k xk tm−1k Tm−1
nk
ym−2k 1 − tm−2k xk tm−2k Tm−2 ym−3k ,
..
.
5.1
y2k 1 − t2k xk t2k T2nk y1k ,
y1k 1 − t1k xk t1k T1nk y0k ,
y0k xk ,
k ∈ N,
where T1 , . . . , Tm ∈ Tr C, {tik }∞
k1 are sequences in 0, 1 for all i 1, 2, . . . , m, and {nk } is an
increasing sequence of natural numbers.
Theorem 5.1. Let X be a uniformly convex Banach space with the Opial property, and let C be a
for some δ ∈
nonempty closed convex subset of X. Let T1 , . . . , Tm ∈ Tr C, {tik }∞
k1 ⊂ δ, 1 − δ
∅
0, 1/2, and let {nk } ⊂ N be such that {xk } in 5.1 is well defined. Suppose that F : m
i1 FTi /
and the set J {k ∈ N : nk1 1 nk } is quasiperiodic. Then, {xk } converges weakly to a common
fixed point of the family {Ti : i 1, 2, . . . , m}.
Theorem 5.2. Let C be a nonempty closed convex subset of a uniformly convex Banach space X and
T1 , . . . , Tm ∈ Tr C such that Til is semicompact for some i ∈ {1, . . . , m} and l ∈ N. Let {tik }∞
k1 ⊂
δ, 1 − δ for some δ ∈ 0, 1/2, and let {nk } ⊂ N be such that {xk } in 5.1 is well defined. Suppose
∅ and the set J {k ∈ N : nk1 1 nk } is quasiperiodic. Then, {xk }
that F : m
i1 FTi /
converges strongly to a common fixed point of the family {Ti : i 1, 2, . . . , m}.
16
Journal of Applied Mathematics
Acknowledgments
This research is partially supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The first author also thanks the Graduate School of
Chiang Mai University, Thailand.
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