Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 695183, 20 pages
doi:10.1155/2012/695183
Research Article
The General Iterative Methods for Asymptotically
Nonexpansive Semigroups in Banach Spaces
Rabian Wangkeeree and Pakkapon Preechasilp
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand
Correspondence should be addressed to Rabian Wangkeeree, rabianw@nu.ac.th
Received 5 August 2012; Accepted 12 December 2012
Academic Editor: Abdelaziz Rhandi
Copyright q 2012 R. Wangkeeree and P. Preechasilp. 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 introduce the general iterative methods for finding a common fixed point of asymptotically
nonexpansive semigroups which is a unique solution of some variational inequalities. We prove
the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits
a weakly continuous duality mapping. The main result extends various results existing in the
current literature.
1. Introduction
Let E be a normed linear space. Let T be a self-mapping on E. Then T is said to be
asymptotically nonexpansive if there exists a sequence {kn } ⊂ 1, ∞ with limn → ∞ kn 1 such
that for each x, y ∈ E,
n
T x − T n y ≤ kn x − y,
∀n ≥ 1.
1.1
The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk 1 as an
important generalization of the class of nonexpansive maps i.e., mappings T : E → E such
that T x − T y ≤ x − y, for all x, y ∈ E. We use FT to denote the set of fixed points of
T , that is, FT {x ∈ E : T x x}. A self-mapping f : E → E is a contraction on E if there
exists a constant α ∈ 0, 1 such that
fx − f y ≤ αx − y,
∀x, y ∈ E.
1.2
2
Abstract and Applied Analysis
We use ΠE to denote the collection of all contractions on E. That is, ΠE {f : f is a contraction
on E}.
A family S {T s : 0 ≤ s < ∞} of mappings of E into itself is called a strongly continuous semigroup of Lipschitzian mappings on E if it satisfies the following conditions:
i T 0x x for all x ∈ E;
ii T s t T sT t for all s, t ≥ 0;
iii for each t > 0, there exists a bounded measurable function Lt : 0, ∞ → 0, ∞
such that T tx − T ty ≤ Lt x − y, for all x, y ∈ E;
iv for all x ∈ E, the mapping t → T tx is continuous.
A strongly continuous semigroup of Lipchitszian mappings S is called strongly continuous
semigroup of nonexpansive mappings if Lt 1 for all t > 0 and strongly continuous semigroup of
asymptotically nonexpansive if lim supt → ∞ Lt ≤ 1. Note that for asymptotically nonexpansive
semigroup S, we can always assume that the Lipchitszian constant {Lt }t>0 is such that Lt ≥ 1
for each t > 0, Lt is nonincreasing in t, and limn → ∞ Lt 1; otherwise we replace Lt , for each
t > 0, with Lt : max{sups≥t Ls , 1}. We denote by FS the set of all common fixed points of S,
that is,
FS : {x ∈ E : T tx x, 0 ≤ t < ∞} FT t.
t≥0
1.3
S is called uniformly asymptotically regular on C 2, 3 if for all h ≥ 0 and any bounded subset
B of C,
lim supT hT tx − T tx 0,
t → ∞ x∈B
1.4
and almost uniformly asymtotically regular on C 4 if
1 t
1 t
T sx ds −
T sx ds 0.
lim supT h
t → ∞ x∈B t 0
t 0
1.5
Let u ∈ C. Then, for each t ∈ 0, 1 and for a nonexpansive map T , there exists a unique
point xt ∈ C satisfying the following condition:
xt 1 − tT xt tu,
1.6
since the mapping Gt x 1 − tT x tu is a contraction. When H is a Hilbert space and T
is a self-map, Browder 5 showed that {xt } converges strongly to an element of FT which
is nearest to u as t → 0 . This result was extended to more various general Banach space by
Morales and Jung 6, Takahashi and Ueda 7, Reich 8, and a host of other authors.
Many authors see, e.g., 9, 10 have also shown the convergence of the path xn 1 − αn T n xn αn u, in Banach spaces for asymptotically nonexpansive mapping self-map T
under some conditions on αn .
Abstract and Applied Analysis
3
It is an interesting problem to extend the above results to a strongly continuous semigroup of nonexpansive mappings and a strongly continuous semigroup of asymptotically
nonexpansive mappings.
Let S be a strongly continuous semigroup of nonexpansive self-mappings. In 1998
Shioji and Takahashi 11 introduced, in Hilbert space, the implicit iteration
1
un 1 − αn tn
tn
T sun ds αn u,
1.7
u ∈ C, n ≥ 0,
0
where {αn } is a sequence in 0, 1, {tn } is a sequence of positive real numbers divergent
to ∞. Under certain restrictions to the sequence {αn }, Shioji and Takahashi proved strong
convergence of 1.7 to a member of FS. Recently, Zegeye et al. 4 introduced the
implicit 1.7 and the following explicit iteration process for a semigroup of asymptotically
nonexpansive mappings:
un1 1 − αn 1
tn
tn
T sun ds αn u,
u ∈ C, n ≥ 0,
1.8
0
where tn ∈ R and αn ∈ 0, 1 in a reflexive strictly convex Banach space with a uniformly
Gâteaux differentiable norm. Suppose, in addition, that S is almost uniformly asymptotically
regular. Then the implicit sequence 1.7 and explicit sequence 1.8 converge strongly to a
point of FS.
On the other hand, by a gauge function ϕ we mean a continuous strictly increasing
function ϕ : 0, ∞ → 0, ∞ such that ϕ0 0 and ϕt → ∞ as t → ∞. Let E∗ be the dual
∗
space of E. The duality mapping Jϕ : E → 2E associated to a gauge function ϕ is defined by
Jϕ x f ∗ ∈ E∗ : x, f ∗ xϕx, f ∗ ϕx ,
∀x ∈ E.
1.9
In particular, the duality mapping with the gauge function ϕt t, denoted by J, is
referred to as the normalized duality mapping. Clearly, there holds the relation Jϕ x ϕx/xJx for all x / 0 see 12. Browder 12 initiated the study of certain classes
of nonlinear operators by means of the duality mapping Jϕ . Following Browder 12, we say
that a Banach space E has a weakly continuous duality mapping if there exists a gauge ϕ for
which the duality mapping Jϕ x is single valued and continuous from the weak topology
to the weak∗ topology; that is, for any {xn } with xn x, the sequence {Jϕ xn } converges
weakly∗ to Jϕ x. It is known that lp has a weakly continuous duality mapping with a gauge
function ϕt tp−1 for all 1 < p < ∞. Set
t
ϕτdτ,
∀t ≥ 0,
1.10
Jϕ x ∂Φx,
∀x ∈ E,
1.11
Φt 0
then
where ∂ denotes the subdifferential in the sense of convex analysis.
4
Abstract and Applied Analysis
In a Banach space E having a weakly continuous duality mapping Jϕ with a gauge
function ϕ, an operator A is said to be strongly positive 13 if there exists a constant γ > 0
with the property
Ax, Jϕ x ≥ γxϕx,
αI − βA sup αI − βA x, Jϕ x , α ∈ 0, 1, β ∈ −1, 1,
x≤1
1.12
1.13
where I is the identity mapping. If E : H is a real Hilbert space, then the inequality 1.12
reduces to
Ax, x ≥ γx2
1.14
∀x ∈ H.
A typical problem is to minimize a quadratic function over the set of the fixed points of a
nonexpansive mapping on a real Hilbert space H:
1
min Ax, x − x, b,
x∈C 2
1.15
where C is the fixed point set of a nonexpansive mapping T on H and b is a given point
in H. In 2009, motivated and inspired by Marino and Xu 14, Li et al. 15 introduced the
following general iterative procedures for the approximation of common fixed points of a
one-parameter nonexpansive semigroup {T s : s ≥ 0} on a nonempty closed convex subset
C in a Hilbert space:
1
yn αn γf yn I − αn A
tn
xn1
1
αn γfxn I − αn A
tn
tn
T syn ds,
n ≥ 0,
0
tn
1.16
T sxn ds,
n ≥ 0,
0
where {αn } and {tn } are sequences in 0, 1 and 0, ∞, respectively, A is a strongly positive
bounded linear operator on H, and f is a contraction on H. And their convergence
theorems can be proved under some appropriate control conditions on parameter {αn } and
{tn }. Furthermore, by using these results, they obtained two mean ergodic theorems for
nonexpansive mappings in a Hilbert space.
All of the above brings us to the following conjectures.
Question 1. Could we obtain strong convergence theorems for the general class of strongly
continuous semigroup of asymptotically nonexpansive mappings in more general Banach
spaces? such as a reflexive Banach space which admits a weakly continuous duality mapping
Jϕ , where ϕ : 0, ∞ → 0, ∞ is a gauge function.
In this paper, inspired and motivated by Shioji and Takahashi 11, Zegeye et al. 4,
Marino and Xu 14, Li et al. 15, and Wangkeeree et al. 13, we prove the strong convergence theorems of the iterative approximation methods 1.16 for the general class of
Abstract and Applied Analysis
5
the strongly continuous semigroup of asymptotically nonexpansive mappings in a reflexive
Banach space which admits a weakly continuous duality mapping Jϕ , where ϕ : 0, ∞ →
0, ∞ is a gauge function and A is a strongly positive bounded linear operator on a Banach
space E. The results in this paper generalize and improve some well-known results in Shioji
and Takahashi 11, Li et al. 15, and many others.
2. Preliminaries
Throughout this paper, let E be a real Banach space and E∗ be its dual space. We write xn x
resp., xn ∗ x to indicate that the sequence {xn } weakly resp., weak∗ converges to x; as
usual xn → x will symbolize strong convergence. Let U {x ∈ E : x 1}. A Banach space
E is said to uniformly convex if, for any ∈ 0, 2, there exists δ > 0 such that, for any x, y ∈ U,
x − y ≥ implies x y/2 ≤ 1 − δ. It is known that a uniformly convex Banach space is
reflexive and strictly convex see also 16. A Banach space E is said to be smooth if the limit
limt → 0 x ty − x/t exists for all x, y ∈ U. It is also said to be uniformly smooth if the
limit is attained uniformly for x, y ∈ U.
Now we collect some useful lemmas for proving the convergence result of this paper.
The first part of the next lemma is an immediate consequence of the subdifferential
inequality and the proof of the second part can be found in 10.
Lemma 2.1 see 10. Assume that a Banach space E has a weakly continuous duality mapping Jϕ
with gauge ϕ.
i For all x, y ∈ E, the following inequality holds:
Φ x y ≤ Φx y, Jϕ x y .
2.1
In particular, for all x, y ∈ E,
x y2 ≤ x2 2 y, J x y .
2.2
ii Assume that a sequence {xn } in E converges weakly to a point x ∈ E.
Then the following identity holds:
lim sup Φ xn − y lim sup Φxn − x Φ y − x ,
n→∞
n→∞
∀x, y ∈ E.
2.3
The next valuable lemma is proved for applying our main results.
Lemma 2.2 see 13, Lemma 3.1. Assume that a Banach space E has a weakly continuous duality
mapping Jϕ with gauge ϕ. Let A be a strong positive linear bounded operator on E with coefficient
γ > 0 and 0 < ρ ≤ ϕ1A−1 . Then I − ρA ≤ ϕ11 − ργ.
In the following, we also need the following lemma that can be found in the existing
literature 17, 18.
6
Abstract and Applied Analysis
Lemma 2.3 see 18, Lemma 2.1. Let {an } be a sequence of a nonnegative real number satisfying
the property
an1 ≤ 1 − γn an γn βn ,
where {γn } ⊆ 0, 1 and {βn } ⊆ R such that
converges to zero, as n → ∞.
∞
n ≥ 0,
2.4
∞ and lim supn → ∞ βn ≤ 0. Then {an }
n0 γn
3. Main Theorem
Theorem 3.1. Let E be a reflexive Banach space which admits a weakly continuous duality mapping
Jϕ with gauge ϕ such that ϕ is invariant on 0, 1. Let S {T s : s ≥ 0} be a strongly continuous
semigroup of asymptotically nonexpansive mappings on E with a sequence {Lt } ⊂ 1, ∞ and
FS / ∅. Let f ∈ ΠE with coefficient α ∈ 0, 1, let A be a strongly positive bounded linear operator
with coefficient γ > 0 and 0 < γ < ϕ1γ/α, and let {αn } and {tn } be sequences of real numbers such
that 0 < αn < 1, tn > 0. Then the following holds.
t
i If 1/tn 0n Ls ds − 1/αn < ϕ1γ − γα, for all n ≥ 0, then there exists a sequence {yn } ⊂
E defined by
yn αn γf yn
1
I − αn A
tn
tn
T syn ds,
n ≥ 0.
3.1
0
ii Suppose, in addition, that S is almost uniformly asymptotically regular and the real
sequences {αn } and {tn } satisfy the following conditions:
B1 limn → ∞ tn ∞;
B2 limn → ∞ αn 0;
t
B3 limn → ∞ 1/tn 0n Ls ds − 1/αn 0.
Then {yn } converges strongly as n → ∞ to a common fixed point x in FS which solves the
variational inequality:
A − γf x,
Jϕ x − z ≤ 0,
z ∈ FS.
3.2
Proof. We first show the uniqueness of a solution of the variational inequality 3.2. Suppose
both x ∈ FS and x∗ ∈ FS are solutions to 3.2, then
A − γf x,
Jϕ x − x∗ ≤ 0,
≤ 0.
A − γf x∗ , Jϕ x∗ − x
3.3
Adding 3.3, we obtain
A − γf x − A − γf x∗ , Jϕ x − x∗ ≤ 0.
3.4
Abstract and Applied Analysis
7
Noticing that for any x, y ∈ E,
A − γf x − A − γf y, Jϕ x − y
A x − y , Jϕ x − y − γ fx − f y , Jϕ x − y
≥ γ x − yϕ x − y − γ fx − f y Jϕ x − y ≥ γΦ x − y − γαΦ x − y
γ − γα Φ x − y
≥ γϕ1 − γα Φ x − y ≥ 0.
3.5
Therefore x x∗ and the uniqueness is proved. Below, we use x to denote the unique solution
of 3.2.
Since limn → ∞ αn 0, we may assume, without the loss of generality, that αn <
ϕ1A−1 .
For each integer n ≥ 0, define a mapping Gn : E → E by
1
Gn y αn γf y I − αn A
tn
tn
∀y ∈ E.
T sy ds,
3.6
0
We show that Gn is a contraction mapping. For any x, y ∈ E,
Gn x − Gn y 1 tn
1 tn
T sx ds − αn γf y − I − αn A
T sy ds
αn γfx I − αn A
tn 0
tn 0
t
1 n
1 tn
≤ αn γ fx − f y
I − αn A
T sx ds −
T sy ds tn 0
tn 0
1 tn
≤ αn γαx − y ϕ1 1 − αn γ
Ls ds x − y
tn 0
1 tn
1 tn
αn γα ϕ1
Ls ds − ϕ1αn γ
Ls ds x − y
tn 0
tn 0
t
1 n
1 tn
x − y.
≤
Ls ds − αn ϕ1γ
Ls ds − γα
tn 0
tn 0
Since 0 < 1/tn tn
0
0<
3.7
Ls ds − 1/αn < ϕ1γ − γα, we have
1/tn tn
0
Ls ds − 1
αn
1
< ϕ1γ − γα ≤ ϕ1γ
tn
tn
0
Ls ds − γα.
3.8
8
Abstract and Applied Analysis
t
t
It then follows that 0 < 1/tn 0n Ls ds − αn ϕ1γ1/tn 0n Ls ds − γα < 1. We have Gn is
t
t
a contraction map with coefficient 1/tn 0n Ls ds − αn ϕ1γ1/tn 0n Ls ds − γα. Then, for
each n ≥ 0, there exists a unique yn ∈ E such that Gn yn yn , that is,
1
yn αn γf yn I − αn A
tn
tn
T syn ds,
n ≥ 0.
3.9
0
Hence i is proved.
ii We first show that {yn } is bounded. Letting p ∈ FS and using Lemma 2.2, we
can calculate the following:
tn
1
yn − p αn γf yn I − αn A
T syn ds − p
tn 0
αn γf yn − αn γf p αn γf p
1 tn
I − αn A
T syn ds − I − αn Ap − αn Ap
tn 0
≤ αn γ f yn − f p αn γf p − A p t
1 tn
1 n
ϕ1 1 − αn γ T syn ds −
T sp ds
tn 0
tn 0
1 tn
≤ αn γα yn − p αn γf p − A p ϕ1 1 − αn γ
Ls dsyn − p
tn 0
tn
1
≤ αn γαyn − p αn γf p − A p ϕ1
Ls dsyn − p
tn 0
1 tn
− ϕ1αn γ
Ls dsyn − p
tn 0
1 tn
≤ αn γαyn − p αn γf p − A p Ls dsyn − p
tn 0
1 tn
− ϕ1αn γ
Ls dsyn − p.
tn 0
3.10
Thus, we get that
yn − p ≤
αn γf p − A p .
t
t
1 − αn γα − 1/tn 0n Ls ϕ1αn γ1/tn 0n Ls ds
3.11
Abstract and Applied Analysis
9
Calculating the right-hand side of the above inequality, we have
αn γf p − A p t
Ls ϕ1αn γ1/tn 0n Ls ds ϕ1αn γ
αn γf p − A p t
t
1 − αn γα − 1/tn 0n Ls ϕ1αn γ1/tn 0n Ls ds ϕ1αn γ − ϕ1αn γ
αn γf p − A p t
t
ϕ1αn γ − αn αγ 1 − 1/tn 0n Ls ds ϕ1αn γ1/tn 0n Ls ds − ϕ1αn γ
αn γf p − A p t
t
αn ϕ1γ − αγ − 1/tn 0n Ls ds − 1 ϕ1αn γ 1/tn 0n Ls ds − 1
1 − αn γα − 1/tn tn
0
3.12
αn γf p − A p .
t
αn ϕ1γ − αγ − 1 − ϕ1αn γ 1/tn 0n Ls ds − 1
Thus, we get that
γf p − A p ,
ϕ1γ − αγ − 1 − αn γ dn
yn − p ≤ 3.13
t
where dn 1/tn 0n Ls ds − 1/αn . Thus, there exists N > 0 such that yn − p ≤
γfp − Ap/ϕ1γ − γα, for all n ≥ N. Therefore, {yn } is bounded and hence {fyn }
t
and {1/tn 0n T syn ds} are also bounded.
t
Let δtn yn 1/tn 0n T syn ds. Then, from 3.1, we get
yn − δt yn αn γf yn − Aδt yn −→ 0
n
n
as n −→ ∞.
3.14
Since S is almost uniformly asymptotically regular and 3.14, we have
yn − T hyn ≤ yn − δtn yn δtn yn − T hδtn yn T hδtn yn − T hyn ≤ yn − δtn yn δtn yn − T hδtn yn Ls δtn yn − yn −→ 0.
3.15
It follows from the reflexivity of E and the boundedness of sequence {yn } that there exists
{ynj } which is a subsequence of {yn } converging weakly to w ∈ E as j → ∞. Since Jϕ is
weakly sequentially continuous, we have by Lemma 2.1 that
lim sup Φ ynj − y lim sup Φ ynj − w Φ y − w ,
j →∞
j →∞
∀x ∈ E.
3.16
10
Abstract and Applied Analysis
Let
Hx lim sup Φ ynj − y ,
∀y ∈ E.
3.17
H y Hw Φ y − w ,
∀y ∈ E.
3.18
j →∞
It follows that
For h ≥ 0, from 3.15 we obtain
HT hw lim sup Φ ynj − T hw lim sup Φ T hynj − T hw
j →∞
j →∞
3.19
≤ lim sup Φ ynj − w Hw.
j →∞
On the other hand, however,
HT hw Hw ΦT hw − w.
3.20
It follows from 3.19 and 3.20 that
ΦT hw − w HT hw − Hw ≤ 0.
3.21
This implies that T hw w for all h ≥ 0, and so w ∈ FS. Next, we show that ynj → w
t
as j → ∞. In fact, since Φt 0 ϕτdτ, for all t ≥ 0, and ϕ : 0, ∞ → 0, ∞ is a gauge
function, then for 1 ≥ k ≥ 0, ϕkx ≤ ϕx and
Φkt kt
0
ϕτdτ k
t
0
ϕkxdx ≤ k
t
0
ϕxdx kΦt.
3.22
Abstract and Applied Analysis
11
Following Lemma 2.1, we have
1 tn
Φ yn − w Φ I − αn A
T syn ds − I − αn Aw
tn 0
αn γf yn − γfw γfw − Aw 1 tn
≤ Φ I − αn A
T syn ds − I − αn Aw αn γ f yn − fw tn 0
αn γfw − Aw, Jϕ yn − w
t
1 tn
1 n
≤ Φ ϕ1 1 − αn γ T syn ds −
T sw ds αn γαyn − w
tn 0
tn 0
αn γ fw − fw, Jϕ yn − w
1 tn
≤ Φ ϕ1 1 − αn γ
Ls ds
yn − w αn γα yn − w
tn 0
αn γfw − Aw, Jϕ yn − w
1 tn
≤ Φ ϕ1 1 − αn γ
Ls ds αn γα yn − w
tn 0
αn γfw − Aw, Jϕ yn − w
1 tn
≤ ϕ1 1 − αn γ
Ls ds αn γα Φ yn − w
tn 0
αn γfw − Aw, Jϕ yn − w .
3.23
This implies that
Φ yn − w ≤
1 − ϕ1 1 − αn γ
1
1/tn tn
0
αn γfw − Aw, Jϕ yn − w ,
Ls ds αn γα
3.24
also
Φ yn − w ≤ 1
γfw − Aw, Jϕ yn − w ,
ϕ1γ − αγ − 1 − αn γ dn
3.25
12
Abstract and Applied Analysis
t
where dn 1/tn 0n Ls ds − 1/αn . Now observing that ynj w implies Jϕ ynj − w ∗ 0,
we conclude from the above inequality that
Φ ynj − w −→ 0
as j −→ ∞.
3.26
Hence ynj → w as j → ∞. Next, we prove that w solves the variational inequality 3.2. For
any z ∈ FS, we observe that
1
yn −
tn
tn
T syn ds
1
z−
tn
−
0
yn − z, Jϕ yn − z −
1
tn
tn
0
tn
T sz ds , Jϕ yn − z
0
1
T syn ds −
tn
tn
T sz ds, Jϕ yn − z
0
tn
tn
1
1
≥ Φ yn − z − T syn ds −
T szdsJϕ yn − z tn 0
tn 0
1
≥ Φ yn − z −
tn
tn
3.27
Ls dsyn − zJϕ yn − z 0
tn
1
Φ yn − z −
Ls dsΦ yn − z
tn 0
1 tn
1−
Ls ds Φ yn − z .
tn 0
Since
yn αn γf yn
tn
1
I − αn A
tn
3.28
T syn ds,
0
we can derive that
A − γf
yn
1
−
αn
1
yn −
tn
tn
0
T syn ds
A yn − A
1
tn
tn
T syn ds
.
0
3.29
Abstract and Applied Analysis
13
Since Φ is strictly increasing and yn − p ≤ M for some M > 0, we have Φyn − p ≤ ΦM.
Thus
A − γf yn , Jϕ yn − z
1 tn
1 tn
1
yn −
T syn ds − z −
T sz ds , Jϕ yn − z
−
αn
tn 0
tn 0
t
1 n
A yn − A
T syn ds , Jϕ yn − z
tn 0
t
t
1/tn 0n Ls ds − 1 1 n
≤
Φ yn − z A yn − A
T syn ds , Jϕ yn − z
αn
tn 0
t
1/tn 0n Ls ds − 1
1 tn
≤
ΦM A yn −
T syn ds , Jϕ yn − z .
αn
tn 0
3.30
Notice that
ynj −
1
tnj
tn
0
j
T synj ds −→ w −
1
tnj
tn
j
T sw ds w − w 0.
3.31
0
Now using B3 and replacing n with nj in 3.30 and letting j → ∞, we have
A − γf w, Jϕ w − z ≤ 0.
3.32
So, w ∈ FS is a solution of the variational inequality 3.2, and hence w x by the
uniqueness. In a summary, we have shown that each cluster point of {yn } at n → ∞ equals
x.
Therefore, yn → x as n → ∞. This completes the proof.
If A ≡ I, the identity mapping on E, and γ 1, then Theorem 3.1 reduces to the
following corollary.
Corollary 3.2. Let E be a reflexive Banach space which admits a weakly continuous duality mapping
Jϕ with gauge ϕ such that ϕ is invariant on 0, 1. Let S {T s : s ≥ 0} be a strongly continuous
semigroup of asymptotically nonexpansive mappings on E with a sequence {Lt } ⊂ 1, ∞ and
FS /
∅. Let f ∈ ΠE with coefficient α ∈ 0, 1 and let {αn } and {tn } be sequences of real numbers
such that 0 < αn < 1 and tn > 0. Then the following holds.
i If 1/tn by
tn
0
Ls ds − 1 < αn 1 − α, for all n ∈ N, then there exists a sequence {yn } defined
yn αn f yn
1
1 − αn tn
tn
0
T syn ds,
n ≥ 0.
3.33
14
Abstract and Applied Analysis
ii Suppose, in addition, that S is almost uniformly asymptotically regular and the real
sequences {αn } and {tn } satisfy the following:
B1 limn → ∞ tn ∞;
B2 limn → ∞ αn 0;
t
B3 limn → ∞ 1/tn 0n Ls ds − 1/αn 0.
Then {yn } converges strongly as n → ∞ to a common fixed point x in FS which solves the
variational inequality:
I − f x,
Jϕ x − z ≤ 0,
z ∈ FS.
3.34
If E : H is a Hilbert space and S {T s : s ≥ 0} is a strongly continuous semigroup
of nonexpansive mappings on H, then we have Lt ≡ 1 and Theorem 3.1 reduces to the
following corollary.
Corollary 3.3 see 15, Theorem 3.1. Let H be a real Hilbert space. Suppose that f : H → H is
a contraction with coefficient α ∈ 0, 1 and S {T s : s ≥ 0} a strongly continuous semigroup of
nonexpansive mappings on H such that FS /
∅. Let A be a strongly positive bounded linear operator
with coefficient γ > 0 and let {αn } and {tn } be sequences of real numbers such that 0 < αn < 1, tn > 0
such that limn → ∞ tn ∞ and limn → ∞ αn 0, then for any 0 < γ < γ/α, there is a unique {yn } in
H such that
yn αn γf yn
1
I − αn A
tn
tn
T syn ds,
n≥0
3.35
0
and the iterative sequence {yn } converges strongly as n → ∞ to a common fixed point x in FS
which solves the variational inequality:
A − γf x,
x − z ≤ 0, z ∈ FS.
3.36
Theorem 3.4. Let E be a reflexive strictly convex Banach space which admits a weakly continuous
duality mapping Jϕ with gauge ϕ such that ϕ is invariant on 0, 1. Let S {T s : s ≥ 0} be
a strongly continuous semigroup of asymptotically nonexpansive mappings on E with a sequence
∅. Let f ∈ ΠE with coefficient α ∈ 0, 1; let A be a strongly positive
{Lt } ⊂ 1, ∞ and FS /
bounded linear operator with coefficient γ > 0 and 0 < γ < ϕ1γ/α. For any x0 ∈ C, let the sequence
{xn } be defined by
xn1
1
αn γfxn I − αn A
tn
tn
T sxn ds,
n ≥ 0.
3.37
0
Suppose, in addition, that S is almost uniformly asymptotically regular. Let {αn } and {tn } be
sequences of real numbers such that 0 < αn < 1, tn > 0,
C1 limn → ∞ tn ∞;
C2 limn → ∞ αn 0,
C3 limn → ∞ 1/tn ∞
tn
0
n0
αn ∞;
Ls ds − 1/αn 0.
Abstract and Applied Analysis
15
Then {xn } converges strongly as n → ∞ to a common fixed point x in FS which solves the
variational inequality 3.2.
Proof. First we show that {xn } is bounded. By condition C3 and given 0 < ε < ϕ1γ − αγ
t
there exists N > 0 such that 1/tn 0n Ls ds − 1/αn < ε for all n ≥ N. Thus
1 − ϕ1γαn
1
tn
tn
Ls ds − 1
≤
0
1
tn
tn
Ls ds − 1 < εαn ,
3.38
0
for all n ≥ N. Since limn → ∞ αn 0, we may assume, without the loss of generality, that
αn < ϕ1A−1 .
Claim that xn − p ≤ M, n ≥ 0, where M : max{x0 − p, . . . , xN − p, fp −
p/ϕ1γ − αγ − ε}.
Let p ∈ FS. Then from 3.56 we get that
tn
1
xn1 − p αn γfxn I − αn A
T sxn ds − p
tn 0
1 tn
≤ αn γfxn I − αn A
T sxn ds − αn A p − I − αn Ap
tn 0
≤ αn γfxn − αn γf p αn γf p − αn A p t
1 n
1 tn
I − αn A
T sxn ds −
T sp ds tn 0
tn 0
≤ αn γfxn − αn γf p αn γf p − αn A p 1 tn
1 tn
I − αn A
T sxn ds −
T sp ds
tn 0
tn 0
≤ αn γf p − Ap αn αγ xn − p ϕ1 1 − αn γ
αn γf p − Ap ≤ αn γf p − Ap αn αγ ϕ1 1 − αn γ
1
αn αγ tn
tn
0
1
tn
1
tn
tn
Ls ds xn − p
0
tn
Ls ds
xn − p
0
1
Ls ds − ϕ1αn γ
tn
tn
Ls ds xn − p
0
≤ αn γf p − Ap
1 tn
1 1 − ϕ1αn γ
Ls ds − 1 − αn ϕ1γ − αγ xn − p
tn 0
16
Abstract and Applied Analysis
≤ αn f p − p 1 − αn ϕ1γ − αγ εαn xn − p
αn f p − p 1 − αn ϕ1γ − αγ − ε xn − p
f p − p
≤ max , xn − p .
ϕ1γ − αγ − ε
3.39
By induction,
f p − p
xn − p ≤ max , xN − p ,
ϕ1γ − αγ − ε
∀n ≥ N,
3.40
t
and hence {xn } is bounded, so are {fxn } and {1/tn 0n T sxn ds}. Let δtn xn :
tn
1/tn 0 T sxn ds. Then, since αn → 0 as n → ∞, we obtain that
xn1 − δtn xn αn γfxn − Aδtn xn −→ 0 as n −→ ∞.
3.41
For any h > 0, we have
T hxn1 − xn1 ≤ T hxn1 − T hδtn xn T hδtn xn − δtn xn δtn xn − xn1 ≤ Lh xn1 − δtn xn T hδtn xn − δtn xn δtn xn − xn1 ,
3.42
it follows from 3.41 and S is almost uniformly asymptotically regular that
T hxn1 − xn1 −→ 0
as n −→ ∞.
3.43
lim sup γfx
≤ 0.
− Ax,
Jϕ xn − x
3.44
Next, we prove that
n→∞
Let {xnk } be a subsequence of {xn } such that
lim γfx
lim sup γfx
.
− Ax,
Jϕ xnk − x
− Ax,
Jϕ xn − x
k→∞
n→∞
3.45
It follows from the reflexivity of E and the boundedness of sequence {xnk } that there exists
{xnki } which is a subsequence of {xnk } converging weakly to w ∈ E as i → ∞. Since Jϕ is
weakly continuous, we have by Lemma 2.1 that
lim sup Φ xnki − x lim sup Φ xnki − w Φx − w,
i→∞
i→∞
∀x ∈ E.
3.46
Abstract and Applied Analysis
17
Let
Hx lim sup Φ xnki − x ,
i→∞
∀x ∈ E.
3.47
It follows that
Hx Hw Φx − w,
∀x ∈ E.
3.48
From 3.43, for each h > 0, we obtain
HT hw lim sup Φ xnki − T hw lim sup Φ T hxnki − T hw
i→∞
i→∞
≤ lim sup Φ xnki − w Hw.
3.49
i→∞
On the other hand, however,
HT hw Hw ΦT hw − w.
3.50
It follows from 3.49 and 3.50 that
ΦT hw − w HT hw − Hw ≤ 0.
3.51
This implies that T hw w for all h > 0, and so w ∈ FS. Since the duality map Jϕ is single
valued and weakly continuous, we get that
lim γfx
lim sup γfx
− Ax,
Jϕ xn − x
− Ax,
Jϕ xnk − x
n→∞
k→∞
− Ax,
Jϕ xnki − x
lim γfx
i→∞
3.52
A − γf x,
Jϕ x − w ≤ 0
as required. Finally, we show that xn → x as n → ∞. It follows from Lemma 2.1i that
1 tn
Φxn1 − x
Φ I − αn A
T sxn ds − I − αn Ax
tn 0
αn γfxn − γfx
γfx
− Ax
18
Abstract and Applied Analysis
1 tn
≤ Φ I − αn A
T sxn ds − I − αn Ax αn γ fxn − fx
tn 0
αn γfx
− Ax,
Jϕ yn − x
t
1 tn
1 n
≤ Φ ϕ1 1 − αn γ T sxn ds −
T sxds
αn γαxn − x
tn 0
tn 0
αn γ fx
− fx,
Jϕ yn − x
1 tn
Ls ds xn − x
αn γαxn − x
≤ Φ ϕ1 1 − αn γ
tn 0
αn γfx
− Ax,
Jϕ xn − x
1 tn
≤ Φ ϕ1 1 − αn γ
Ls ds αn γα xn − x
tn 0
αn γfx
− Ax,
Jϕ xn − x
1 tn
≤ ϕ1 1 − αn γ
Ls ds αn γα Φxn − x
tn 0
αn γfx
− Ax,
Jϕ xn − x
1 tn
≤ 1 − ϕ1αn γ
Ls ds − 1 1 − αn ϕ1γ − γα Φxn − x
tn 0
αn γfx
− Ax,
Jϕ xn − x
≤ 1 − αn ϕ1γ − γα Φxn X − x
1 − ϕ1αn γ
1
tn
tn
Ls ds − 1 M
0
αn γfx
,
− Ax,
Jϕ xn − x
3.53
where M > 0 such that Φxn − x
≤ M. Put
sn αn ϕ1γ − γα ,
t
1/tn 0n Ls ds − 1
1 − ϕ1αn γ
1
− Ax,
Jϕ xn − x
M .
tn γfx
αn
ϕ1γ − γα
ϕ1γ − γα
3.54
Then 3.53 is reduced to
Φxn1 − x
≤ 1 − sn Φxn − x
sn t n .
3.55
Abstract and Applied Analysis
19
→ 0 as n → ∞; that is,
Applying Lemma 2.3 to 3.55, we conclude that Φxn1 − x
xn → x as n → ∞. This completes the proof.
If A ≡ I, the identity mapping on E, and γ 1, then Theorem 3.4 reduces to the
following corollary.
Corollary 3.5. Let E be a reflexive strictly convex Banach space which admits a weakly continuous
duality mapping Jϕ with gauge ϕ such that ϕ is invariant on 0, 1. Let S {T s : s ≥ 0} be
a strongly continuous semigroup of asymptotically nonexpansive mappings from C into C with a
∅. Let f ∈ ΠE with coefficient α ∈ 0, 1 and the sequence {xn } be
sequence {Lt } ⊂ 1, ∞, FS /
defined by x0 ∈ C,
xn1 αn fxn 1 − αn 1
tn
tn
T sxn ds,
n ≥ 0.
3.56
0
Suppose, in addition, that S is almost uniformly asymptotically regular. Let {αn } and {tn } be
sequences of real numbers such that 0 < αn < 1, tn > 0,
C1 limn → ∞ tn ∞;
C2 limn → ∞ αn 0,
C3 limn → ∞ 1/tn ∞
tn
0
n0
αn ∞;
Ls ds − 1/αn 0.
Then {xn } converges strongly as n → ∞ to a common fixed point x in FS which solves the
variational inequality:
I − f x,
Jϕ x − z ≤ 0,
z ∈ FS.
3.57
If E : H is a Hilbert space and S {T s : s ≥ 0} is a strongly continuous semigroup
of nonexpansive mappings on H, then we have Lt ≡ 1 and Theorem 3.4 reduces to the
following corollary.
Corollary 3.6 see 15, Theorem 3.2. Let H be a real Hilbert space. Suppose that f : H → H is
a contraction with coefficient α ∈ 0, 1 and S {T s : s ≥ 0} a strongly continuous semigroup of
nonexpansive mappings on H such that FS /
∅. Let A be a strongly positive bounded linear operator
with coefficient γ > 0 and 0 < γ < γ/α and let the sequence {xn } be defined by x0 ∈ C,
xn1 αn γfxn I − αn A
1
tn
tn
T sxn ds,
n ≥ 0.
3.58
0
Let {αn } and {tn } be sequences of real numbers such that 0 < αn < 1, tn > 0,
C1 limn → ∞ tn ∞;
C2 limn → ∞ αn 0,
∞
n0
αn ∞.
Then {xn } converges strongly as n → ∞ to a common fixed point x in FS which solves the
variational inequality:
A − γf x,
x − z ≤ 0,
z ∈ FS.
3.59
20
Abstract and Applied Analysis
Acknowledgment
R. Wangkeeree is supported by Naresuan University.
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