数理解析研究所講究録
第 1963 巻 2015 年 150-161

150

STRONG CONVERGENCE THEOREMS FORASYMPTOTICALLY REGULAR NONEXPANSIVE SEMIGROUPSBY SOME ITERATIONSSACHIKO ATSUSHIBAABSTRACT. In this paper, we study Browder’s iterations and Halpern’s iterations for nonexpansive semigroups in Banach spaces. Then, we prove strong
convergence theorems for uniformly asymptotically regular nonexpansive semigroups in Banach spaces.

1. INTRODUCTION

and letand norm
Let $H$ be a real Hilbert space with inner product$H$$CarrowC$Then,
is called
a mapping :be a nonempty closed convex subset of .
nonexpansive if $\Vert Tx-Ty\Vert\leq\Vert x-y\Vert$ for all $y\in C$ . We denote by $F(T)$ the setof fixed points of . Let be an element of and for each with $0<t<1$ , letbe a unique element of satisfying $x_{t}=tx+(1-t)Tx_{t}$ . In 1967, Browder [6]proved the following strong convergence theorem.

$C$$\Vert\cdot\Vert$$\rangle$

$T$

$x,$

$T$

$C$

$x$

$t$

$C$

$x_{t}$

Theorem 1.1. Let $H$ be a Hilbert space, let $C$ be a nonempty bounded closed
convex subset of $H$ and let be a nonexpansive mapping of into itself. Let be
be a unique element of
an element of and for each with $0<t<1$ , letconverges strongly to the element ofsatisfying $x_{t}=tx+(1-t)Tx_{t}$ . Then,$F(T)$ nearest to
as
$C$

$T$

$C$

$t$

$x_{t}$$x$$C$$\{x_{t}\}$$x$$t\downarrow 0.$

Reich [21] and Takahashi and Ueda [35] extended Browder’s result to those ofa Banach space. Using the idea of Shimizu and Takahashi [22, 23] and the notionof sequence of means, Shioji and Takahashi [25] proved the strong convergence ofOn
Browder’s type sequences for nonexpansive semigroups (see also [26, 27, 28
the other hand, Domingues Benavides, Acedo and Xu [11] proved Browder’s typestrong convergence theorems for uniformly asymptotically regular one-parameter2010 Mathematics Subject Classification. Primary $47H09,$ $47H10.$
Key words and phrases. Fixed point, iteration, nonexpansive mapping, nonexpansive semigroup, strong convergence.

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nonexpansive semigroups. Acedo and Suzuki [17] generalized Domingues Benavides, Acedo and Xu’s results concerning the condition of the sequences in real
numbers. Author [1] studied Browder’s type iterations for nonexpansive semigroups
and proved strong convergence theorems for uniformly asymptotically regular nonexpansive semigroups in Hilbert spaces by using the idea of [6, 11, 17, 33, 34].
In this paper, we study Browder’s iterations and Halpern’s iterations for nonexpansive semigroups in Banach spaces. Then, we prove strong convergence theorems
for uniformly asymptotically regular nonexpansive semigroups in Banach spaces using the idea of [1, 6, 11, 17, 33, 34].

2. PRELIMINARIES AND NOTATIONS
Throughout this paper, we denote by and the set of all positive integers andthe set of all real numbers, respectively. We also denote byandthe set of
all nonnegative integers and the set of all nonnegative real numbers, respectively.
Let be a real Banach space with norm . We denote by
the set $\{x\in E$ :. Let
be the dual space of a Banach space $E$ . The value ofat$x\in E$ will be denoted by. Let be a closed subset of a Banach space andlet be a mapping of into itself. We denote by $F(T)$ the set $\{x\in C : x=Tx\}.$We denote by the identity operator on $E$ . The duality mapping from $E$ into
is defined by$\mathbb{R}$$\mathbb{N}$$\mathbb{Z}^{+}$$E$$\Vert x\Vert\leq r\}$$\mathbb{R}^{+}$$B_{r}$$\Vert$$E^{*}$$x^{*}\in E^{*}$$\langle x,$$T$$x^{*}\rangle$

$C$

$C$

$I$

$J$

$2^{E^{*}}$$J(x)=\{y^{*}\in E^{*}:\langle x, y^{*}\rangle=\Vert x\Vert^{2}=\Vert y^{*}\Vert^{2}\}, x\in E.$

From the Hahn-Banach theorem, we see that
Let $E$ be a smooth Banach space. Then,for all $x\in E.$
is said to be weakly sequentially
in $E$ which converges weakly to$J(x)\neq\emptyset$$J$

continuous at zero if for every sequence
converges $weakly^{*}$ to $0\in E^{*}$
Banach space is said to be smooth if$\{x_{n}\}$$0\in E,$ $\{J(x_{n})\}$$E$$\lim_{tarrow 0}\frac{\Vert x+ty\Vert-\Vert x\Vert}{t}$exists for each and in , where $S_{1}=\{u\in E : \Vert u\Vert=1\}$ . The norm of $E$ issaid to be uniformly G\^ateaux diﬀerentiable if for each in , the limit is attained
uniformly for in . We know that if
is smooth, then the duality mapping
is single-valued and norm to weak star continuous and that if the norm of $E$ is
uniformly G\^ateaux diﬀerentiable, then the duality mapping is single-valued andnorm to weak star, uniformly continuous on each bounded subset of $E.$
A closed convex subset $C$ of a Banach space is said to have normal structure iffor each bounded closed convex subset $K$ of which contains at least two points,
there exists an element of $K$ which is not a diametral point of $K$ . It is wellknown that a closed convex subset of a uniformly convex Banach space has normal
structure and a compact convex subset of a Banach space has normal structure (see
$x$

$y$

$S_{1}$$y$$x$$S_{1}$

$E$

$E$

$C$

$S_{1}$

152[34]). We also know that uniformly smooth Banach space has normal structure (see[34]). Every weakly compact convex subset of a Banach space satisfying Opial’s
condition has normal structure (see [14]). We note that closed convex subset of a
Banach space is said to have the fixed point property for nonexpansive mappings
if for every bounded closed convex subset $K$ of , every nonexpansive mapping on$K$ , has a fixed point. We also know that every weakly compact convex subset with
Opial property has fixed point property.
Let be a nonempty closed convex subset of $E$ and let $K$ be a nonempty subset
of . A mapping $P$ of onto $K$ is said to be sunny if $P(Px+t(x-Px))=Px$
is a retraction if $Px=x$
for each $x\in C$ and $t\geq 0$ with $Px+t(x-Px)\in C.$
for each $x\in K$ . We know from [8, Theorem 3] and [20, Lemma 2.7] the following
lemma (see also [34]).
$C$

$E$

$C$

$C$

$C$

$C$

$P$

be a convex subset
Let
be a smooth Banach space, letLemma 2.1 ([8,20$K$$Then_{f}$a retraction of onto $K$ is sunny andof and let be a subset of C.
nonexpansive if and only if$\langle x-Px,$ $J(y-Px)\rangle\leq 0$
for all $x\in C$ and $y\in K.$
$C$

$E$

$P$

$E$

$C$

Hence, there is at most one sunny nonexpansive retraction of$C$onto $K.$

onto $K,$ $K$ is said to be a
If there is a sunny nonexpansive retraction ofsunny nonexpansive retract of . The following theorem related to the existenceof nonexpansive retractions was proved in [9, 10].
$C$

$C$

be a nonempty
Let $E$ be a reflexive Banach space, let
Theorem 2.2 ([9, 10
closed convex subset of $E$ and let be a nonexpansive mapping of into itself with. If has a fixed point in every nonempty bounded closed convex subset$E$of such that leaves invariant, then $F(T)$ is a nonexpansive retract of $C.$
$C$

$C$

$T$

$F(T)\neq\emptyset$

$T$

$T$

Let be a mean on positive integers , i.e., a continuous linear functional on
if and only ifsatisfying $||\mu\Vert=1=\mu(1)$ . We know that is a mean on$\inf\{a_{n} : n\in N\}\leq\mu(f)\leq\sup\{a_{n} : n\in N\}$ for each $f=(a_{1}, a_{2}, \ldots)\in\iota\infty.$
Occasionally, we use
instead of $\mu(f)$ . So, a Banach limit is a mean on
satisfying $\mu_{n}(a_{n})=\mu_{n}(a_{n+1})$ . Let $f=(a_{1}, a_{2}, \ldots)\in l^{\infty}$ and let be a Banach
limit on N. Then,$\mathbb{N}$$\mu$$l^{\infty}$$\mathbb{N}$$\mu$$\mu_{n}(a_{n})$$\mu$$\mathbb{N}$$\mu$$\varliminf_{narrow\infty}a_{n}\leq\mu(f)=\mu_{n}(a_{n})\leq\varlimsup_{narrow\infty}a_{n}.$In particular, if $a_{n}arrow a$ , then $\mu(f)=\mu_{n}(a_{n})=a$ (see [32, 34
lemma was proved in [35] (see also [21,32
To prove our results, we need the following propositions.

The following

Proposition 2.3 $([16, 24 Let a be a real$ number $and let (a_{0}, a_{1}, . . .)$such thatfor all Banach limits and $\varlimsup_{narrow\infty}(a_{n+1}-a_{n})\leq 0$ . Then,$\in\iota\infty$$\mu_{n}(a_{n})\leq b$$\varlimsup_{narrow\infty}a_{n}\leq b.$

$\mu$

153
Lemma 2.4 ([35]). Let be a nonempty closed convex
of a Banach space
G\^ateauxwith a uniformly
be a bounded sequence indiﬀerentiable norm. Let$E$
and let be a Banach limit. Let $z\in C$ . Then,$C$$\mathcal{S}ubset$$\{x_{n}\}$$\mu$$\mu_{n}\Vert x_{n}-z\Vert^{2}=\min_{y\in C}\mu_{n}\Vert x_{n}-y\Vert^{2}$

if and only if
mapping of$\mu_{n}\langle y-z,$$J(x_{n}-z)\rangle\leq 0$

for each $y\in C$ , where$J$

is the duality$E.$

We write $x_{n}arrow x$$( or \lim_{narrow\infty}x_{n}=x)$

to indicate that the sequence$\{x_{n}\}$of vectors

in $H$ converges strongly to . We also write $x_{n}arrow x$ $( or w-\lim_{narrow\infty}x_{n}=x)$ to indicatethat the sequenceof vectors in $H$ converges weakly to . In a Hilbert space,
it is well known that $x_{n}arrow x$ andimply $x_{n}arrow x.$
Let be a semitopological semigroup. A semitopological semigroup is calledright (resp. left) reversible if any two closed left (resp. right) ideals of
have
nonvoid intersection. If is right reversible, $(S, \leq)$ is a directed system when thebinary relation
on is defined by $s\leq t$ if and only if, whereis the closure of . A commutative semigroup is a directed system
when the binary relation is defined by $s\leq t$ if and only if $\{s\}\cup(S+s)\supset\{t\}\cup(S+t)$ .
Let $C$ be a nonempty closed convex subset of a Hilbert space $H$ . A family$S=\{T(t) : t\in S\}$ of mappings ofinto itself is said to be a nonexpansive$C$semigroup on if it satisfies the following conditions:(i) For each $t\in S,$ $T(t)$ is nonexpansive;(ii) $T(ts)=T(t)T(s)$ for each $s\in S.$
We denote by $F(S)$ the set of common fixed points of , i.e., $F(S)= \bigcap_{t\in S}F(T(t))$ .
To prove our main result, we need the following lemma.
$x$$\{x_{n}\}$$x$$\Vert x_{n}\Vertarrow\Vert x\Vert$

$S$

$S$

$S$

$S$

$(\leq$$S$$S$$\{\mathcal{S}\}\cup\overline{Ss}\supset\{t\}\cup\overline{St},$$\overline{A}$$A$$s,$ $t\in$

$S$

$C$

$t,$

$S$

Lemma 2.5 (([11]; See also [36])). Let be a Banach space, Let
be a sequence$\lim_{n}\alpha_{n}=0$
be a sequence of , 1 withof nonnegative real numbers, let
and
. Let
be a sequence of nonnegative real numbers withand letbe a sequence of real numbers with
Suppose that$E$$\{\alpha_{n}\}$$\sum_{n=1}^{\infty}\alpha_{n}=\infty$$\varlimsup_{narrow\infty}\beta_{n}\leq 0$$\{\mathcal{S}_{n}\}$

$[0$

$]$

$\{\beta_{n}\}$$\varlimsup_{narrow\infty}\gamma_{n}\leq 0.$$\{\gamma_{n}\}$$\mathcal{S}_{n+1}\leq(1-\alpha_{n})(s_{n}+\beta_{n})+\alpha_{n}\gamma_{n}$for all$n\in \mathbb{N}$. Then,$\lim_{narrow\infty}s_{n}=0.$

3. LEMMAS
In this section, we give some lemmas which plays an important role in the proofof the main theorems (see also [1,3,31

Lemma 3.1 ([3]). Let be a nonempty closed convex subset of a Banach space $E,$
and let be a commutative semigroup. Let $\mathcal{S}=\{T(t) : t\in S\}$ be a nonexpansive
$C$

$S$

154
semigroup on
such that $F(S)\neq\emptyset$ . Let $m$ be a positive integer and let $t\in S.$
Let be an element of and for each with $0<\alpha<1$ , let $Q(u, \alpha)$ be the uniqueelement of satisfying
$C$

$C$

$u$

$\alpha$$C$$Q(u, \alpha)=\alpha u+(1-\alpha)(T(t))^{m}Q(u, \alpha)$

Assume that.

is smooth. Then, for every $v\in F(S)$ ,$E$$\Vert Q(u, \alpha)-v\Vert^{2}\leq\langle u-v, J(Q(u, \alpha)-v)\rangle$(1)$\langle u-Q(u, \alpha) , J(v-Q(u, \alpha \leq 0$

(2)

and

hold.

Lemma 3.2 ([3]). Let be a nonempty closed convex subset of a Banach space $E,$
and let be a commutative semigroup. Let $S=\{T(t) : t\in S\}$ be a nonexpansive. Let
semigroup on
such thatbe a sequence inand letbe a sequence in such that $0<\alpha_{n}<1$ . Let $u\in C$ , let $t\in S$ , and let $\{Q(u, n)\}$be the sequence defined by
$C$

$S$

$C$

$F(S)\neq\emptyset$$\mathbb{Z}^{+}$$\{m_{n}\}$$\{\alpha_{n}\}$$\mathbb{R}$$Q(u, n)=\alpha_{n}u+(1-\alpha_{n})(T(t))^{m_{n}}Q(u, n)$. Assume that is smooth. Then, the following hold:for each(i) If for every $u\in C,$ $\{Q(u, n)\}$ has a subsequence converging strongly to anelement (say Pu) of $F(S)$ , then is the unique sunny nonexpansive retraction
from onto $F(S)$ .
(ii) If for every $u\in C$ , every subsequence of $\{Q(u, n)\}$ has a subsequence converging strongly to an element of $F(S)$ , then $\{Q(u, n)\}$ converges strongly to anelement (say Pu) of $F(S)$ and is the unique sunny nonexpansive retraction
from onto $F(S)$ .
$E$$n\in \mathbb{N}$

$P$

$C$

$P$

$C$

Lemma 3.3 ([3]). Let $E$ be a Banach space, letbe a locally weakly compact
convex subset of $E$ , and let be a commutative semigroup. Let $S=\{T(t) : t\in S\}$. Let
be a nonexpansive semigroup on
such thatbe a sequence$m_{n}arrowN$in.
such thatorLet
be a sequence
for some. Let $u\in C$ , let $t\in S$ , and letin such that $0<\alpha_{n}<1$ , andbe thesequence defined by
$C$

$S$

$C$

$\mathbb{Z}^{+}$$F(S)\neq\emptyset$$\{m_{n}\}$$N\in \mathbb{Z}^{+}$$m_{n}arrow\infty$$\{\alpha_{n}\}$$\alpha_{n}arrow 0$$\mathbb{R}$$\{x_{n}\}$$x_{n}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{m_{n}}x_{n}$for each. Assume thatis smooth, the normalized duality mappingof$E$ is weakly sequentiallycontinuous at zero and $C$ has the Opial property. Assume
also thatconverges weakly to some $x\in F(S)$ . Then,converges strongly.
$n\in \mathbb{N}$$\{x_{n}\}$

$E$

$J$

$\{x_{n}\}$Lemma 3.4 ([3]). Let $E$ be a Banach space whose norm is uniformly G\^ateaux
diﬀerentiable, letbe a weakly compact convex subset of $E$ , and let be a commutative semigroup. Let $S=\{T(t) : t\in S\}$ be a nonexpansive semigroup on
$C$

$S$

$C$

155
such that $F(S)\neq\emptyset$ . Let. Let
be a sequence insuch that $0<\alpha_{n}<1$ . Let $u\in C$ , let $t\in S$ , and let$\mathbb{Z}^{+}$$\{m_{n}\}$be a sequence inbe the sequence defined$\{\alpha_{n}\}$$\{x_{n}\}$$\mathbb{R}$$by$$x_{n}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{m_{n}}x_{n}$. Assume thathas the fixed point property for nonexpansive
mappings. Let be a Banach limit and let $g(y)=\mu_{n}\Vert y-x_{n}\Vert$ . Putfor each$n\in \mathbb{N}$

$C$

$\mu$

$K= \{z\in C:g(z)=\min_{y\in C}g(y)\}$and assume $K\cap F(S)\neq\emptyset$ . Then, every subsequence of
converging strongly to an element in $K\cap F(S)$ .

$\{x_{n}\}$has a subsequence

4. BROWDER’S TYPE ITERATIONIn this section, we study strong convergence theorems for uniformly asymptotically regular nonexpansive semigroups in Banach spaces. Let
be a nonempty
closed convex subset of a Banach space , letbe a commutative semigroupand let $S=\{T(t) : t\in S\}$ be a nonexpansive semigroup on . We say that anonexpansive semigroup $S=\{T(t):t\in S\}$ is asymptotically regular if
$C$

$E$

$S$

$C$

$\lim_{s\in S}\Vert T(h)T(s)x-T(s)x\Vert=0$

for all $h\in S$ and $x\in C$ (see also [33,34
role in the proof of main theorems.

The following lemma plays an important

Lemma 4.1 ([3]). Let
be a nonempty closed convex subset of a Banach space, and letbe a commutative semigroup. Let $S=\{T(t) : t\in S\}$ be a nonexpansive semigroup on
such that. Assume that $S=\{T(t) : t\in S\}$ isasymptotically regular, that is,
$C$

$E$

$S$

$C$

$F(S)\neq\emptyset$$\lim_{t\in S}\Vert T(h)T(t)x-T(t)x\Vert=0$

for all $h\in S$ and $x\in C$ . Then,$F(T(h))=F(S)$for each $h\in S.$
We say that a nonexpansive semigroup $S=\{T(t) : t\in S\}$ is uniformly asymptotically regular if for every $h\in S$ and every bounded subset $K$ of $C,$$\lim_{s\in S}\sup_{x\in K}\Vert T(h)T(s)x-T(s)x\Vert=0.$

holds.
Several authors proved Browder’s convergence theorems for uniformly asymptotically regular one-parameter nonexpansive semigroups (see [11, 17, 31]).

156
be a locally weakly compact
Theorem 4.2 ([3]). Let
be a Banach space, letconvex subset of , and let be a commutative semigroup. Let $S=\{T(t)$ :such thatbe a uniformly asymptotically regular nonexpansive semigroup on
or $m_{n}arrow N$ for somesuch that. Let
be a sequence in
. Let
. Let
be a sequence in such that $0<\alpha_{n}<1$ , and$u\in C$ , let $t\in S$ , and let
be the sequence defined by
$C$

$E$

$E$

$t\in$

$S$

$C$

$S\}$

$F(S)\neq\emptyset$$\mathbb{Z}^{+}$$\{m_{n}\}$$N\in \mathbb{Z}^{+}$$m_{n}arrow\infty$$\alpha_{n}arrow 0$$\mathbb{R}$$\{\alpha_{n}\}$$\{x_{n}\}$$x_{n}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{m_{n}}x_{n}$. Assume that is smooth, the normalized duality mapping of is
weakly sequentially continuous at zero andhas the Opial property. Then,converges strongly to Pu, whereis the unique sunny nonexpansive retraction$F(S)$.
onto
fromfor each$E$$n\in \mathbb{N}$

$J$

$C$

$E$

$\{x_{n}\}$

$P$

$C$

Theorem 4.3 ([2]). Let $E$ be a Banach space whose norm is uniformly G\^ateaux
diﬀerentiable, letbe a weakly compact convex subset of $E$ and let be a semigroup. Let $\mathcal{S}=\{T(t) : t\in S\}$ be a uniformly ymptotically regular nonexpansive
semigroup on $C$ such that $F(S)\neq\emptyset$ . Let
be a sequence insuch that. Let
or $m_{n}arrow N$ for somebe a sequence in such that
$t\in
S$
$u\in
C$
$0<\alpha_{n}<1$ , and
, let
. Let
, and letbe the sequencebydefined$S$$C$$a\mathcal{S}$$\mathbb{Z}^{+}$$\{m_{n}\}$$N\in \mathbb{Z}^{+}$$m_{n}arrow\infty$$\mathbb{R}$$\{\alpha_{n}\}$$\alpha_{n}arrow 0$$\{x_{n}\}$$x_{n}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{m_{n}}x_{n}$Assume thathas the fixed point property for nonexpansive
Then,converges strongly to Pu, where
mappings.
is the unique sunnynonexpansive retraction fromonto $F(S)$ .

for each$n\in \mathbb{N}$

.

$C$

$P$

$\{x_{n}\}$$C$

Let be a nonempty closed convex subset of $E$ . A family $S=\{T(t) : t\in \mathbb{R}^{+}\}$of mappings of into itself satisfying the following conditions is said to be a oneparameter nonexpansive semigroup on $C$ :$T(t)$ is nonexpansive;(i) For each(ii) $T(t+s)=T(t)T(s)$ for every;(iii) for each $x\in C,$ $t\mapsto T(t)x$ is continuous.
, that is,
In the case whenis a uniformly asymptotically regular
one-parameter nonexpansive semigroup, we have the following strong convergence
theorem for a one-parameter nonexpansive semigroup by Theorem 4.2 (see also[11, 17$C$$C$$t\in \mathbb{R}^{+},$$t,$ $s\in \mathbb{R}^{+}$$S=\mathbb{R}^{+}$$S$

Theorem 4.4. Let $E$ be a Banach space and let
be a locally weakly compact
convex subset of E. Let $S=\{T(t) : t\in \mathbb{R}^{+}\}$ be a uniformly asymptotically regular
one-parameter nonexpansive semigroup on
such that
. Let
be a
$m_{n}arrowN$sequence insuch that
or
. Let
be a
for some$C$$C$$\mathbb{Z}^{+}$$m_{n}arrow\infty$$F(S)\neq\emptyset$$N\in \mathbb{Z}^{+}$$\{m_{n}\}$$\{\alpha_{n}\}$157
sequence in such that $0<\alpha_{n}<1$ , andLet
be the sequence defined by$\mathbb{R}$$\alpha_{n}arrow 0$. Let $u\in C$ and let $t\in(0, \infty)$ .

$\{x_{n}\}$$x_{n}=\alpha_{n}u+(1-\alpha_{n})T(t^{m_{n}})x_{n}$for each. Then,converges strongly to Pu, where
nonexpansive retraction fromonto $F(S)$ .
$n\in \mathbb{N}$$\{x_{n}\}$$P$

is the unique sunny$C$

We also have the following strong convergence theorem for a one-parameternonexpansive semigroup by Theorem 4.3:
Theorem 4.5. Let $E$ be a Banach space whose norm is uniformly G\^ateaux diﬀerentiable and let be a weakly compact convex subset of E. Let $S=\{T(t):t\in \mathbb{R}^{+}\}$
be a uniformly asymptotically regular one-parameter nonexpansive semigroup on
such that. Let
be a sequence insuch thator $m_{n}arrow N$. Let
be a sequence in such that $0<\alpha_{n}<1$ , andfor some
Let $u\in C$ and let $t\in(0, \infty)$ . Let
be the sequence defined by
$C$

$C$

$F(S)\neq\emptyset$$\mathbb{Z}^{+}$$\{m_{n}\}$$N\in \mathbb{Z}^{+}$$\{\alpha_{n}\}$$m_{n}arrow\infty$$\mathbb{R}$$\alpha_{n}arrow 0.$$\{x_{n}\}$$x_{n}=\alpha_{n}u+(1-\alpha_{n})T(t^{m_{n}})x_{n}$for each. Then,converges strongly to Pu, where
nonexpansive retraction fromonto $F(S)$ .
$n\in \mathbb{N}$$\{x_{n}\}$$P$

is the unique sunny$C$

5. HALPERN’S TYPE ITERATIONSIn this section, we study strong convergence theorems for uniformly asymptotically regular nonexpansive semigroups in Banach spaces. Let $C$ be a nonempty
closed convex subset of a Banach space $E$ , let be a commutative semigroup andlet $S=\{T(t) : t\in S\}$ be a nonexpansive semigroup on $C$ . Domingues Benavides,
Acedo and Xu [11] proved Halpern’s convergence theorems for uniformly asymptotically regular one-parameter nonexpansive semigroups (see [11,17,31
The
following lemmas plays an important role in the proof of our main theorem (see[11, 17, 31, 1$S$

Lemma 5.1 ([4]). Let $E$ be a Banach space, letbe a locally weakly compact
convex subset of $E$ , and let be a commutative semigroup. Let $S=\{T(t) : t\in S\}$be a nonexpansive semigroup on
satisfying uniformly asymptotically regularity$F(S)\neq\emptyset$
condition with. Let
be a sequence insuch that] forevery $n=1$ , 2, . . . ,, and. Let $u\in C$ , let $t\in S,$
and letbe the sequence defined by
$C$

$S$

$C$

$\{\alpha_{n}\}$$\alpha_{n}arrow 0,$$\frac{\alpha}{\alpha_{n-1}}arrow 1$$\mathbb{R}$$\alpha_{n}\subset(0,1$$\sum_{n=1}^{\infty}\alpha_{n}=\infty$$\{x_{n}\}$$x_{n+1}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{n}x_{n}$. Assume that is smooth, the normalized duality mapping
is weakly sequentially continuous at zero andhas the Opial property, orfor each$n\in \mathbb{N}$

$E$

$C$

$J$

$E$

of

$E$

is a

158
Banach space whose norm is uniformly G\^ateaux diﬀerentiable. Then,$\lim_{narrow\infty}\Vert x_{n+1}-x_{n}\Vert=0.$

be a locally weakly compact
Lemma 5.2 ([4]). Let
be a Banach space, letconvex subset of , and let be a commutative semigroup. Let $S=\{T(t)$ : $t\in$such thatbe a uniformly asymptotically regular nonexpansive semigroup on] for every $n=1,2,$. Let
be a sequence in such that$t\in
S$ , and let
$u\in
C$
, let
. Let
be the, andsequence defined by
$C$

$E$

$S$

$E$

$C$

$S\}$

$F(S)\neq\emptyset$$\alpha_{n}arrow 0,$$\frac{\alpha}{\alpha_{n-1}}arrow 1$$\alpha_{n}\subset(0,1$$\mathbb{R}$$\{\alpha_{n}\}$$\ldots,$$\{x_{n}\}$$\sum_{n=1}^{\infty}\alpha_{n}=\infty$$x_{n+1}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{n}x_{n}$for each $n\in N$ . Assume thatis smooth, the normalized duality mapping of $E$
is weakly sequentially continuous at zero and $C$ has the Opial property, or $E$ is a
Banach space whose norm is uniformly G\^ateaux diﬀerentiable. Then,Pu, $J(x_{n}-Pu)\rangle\leq 0$
$J$

$E$

$\varlimsup_{narrow\infty}\langle u-$

We prove strong convergence theorems for uniformly asymptotically regular nonexpansive semigroups in Banach spaces by using the idea of [1, 11, 13, 31, 34].
be a locally weakly compact
be a Banach space, let
Theorem 5.3 ([4]). Let$E$ ,convex subset ofand let be a commutative semigroup. Let $S=\{T(t)$ : $t\in$such thatbe a uniformly asymptotically regular nonexpansive semigroup on] for every $n=1,2,$. Let
be a sequence in such that
. Let $u\in C$ , let $t\in S$ , and let, andbe thesequence defined by
$C$

$E$

$S$

$C$

$S\}$

$F(S)\neq\emptyset$$\alpha_{n}arrow 0,$$\{\alpha_{n}\}$$\frac{\alpha}{\alpha_{n}}\mapsto-1arrow 1$$\alpha_{n}\subset(0,1$$\mathbb{R}$$\ldots,$$\{x_{n}\}$$\sum_{n=1}^{\infty}\alpha_{n}=\infty$$x_{n+1}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{n}x_{n}$. Assume that is smooth, the normalized duality mapping of
has the Opial property, oris weakly sequentially continuous at zero andis a Banach space whose norm is uniformly G\^ateaux diﬀerentiable. Then,converges strongly to Pu, whereis the unique sunny nonexpansive retraction$F(S)$.
from onto

for each$J$$E$$n\in \mathbb{N}$

$C$

$E$

$E$

$\{x_{n}\}$

$P$

$C$

, that is,is a uniformly asymptotically regular
In the case when
one-parameter nonexpansive semigroup, we have the following strong convergence
theorem for a one-parameter nonexpansive semigroup by Theorem 5.3 (see also[11, 17$S=\mathbb{R}^{+}$$S$

be as in Theorem 5.3. Let $S=\{T(t):t\in \mathbb{R}^{+}\}$
Theorem 5.4 ([4]). Let andbe a uniformly asymptotically regular one-parameter nonexpansive semigroup on
such that $0<\alpha_{n}\leq 1,$. Let
be a sequence insuch that$t\in(O,\infty)$
$u\in
C$
. Let
be the sequence defined by. Letand let
and
$E$

$C$

$C$

$F(S)\neq\emptyset$$\frac{\alpha}{\alpha_{n-1}}arrow 1$$\{\alpha_{n}\}$$\mathbb{R}$$\{x_{n}\}$$x_{n+1}=\alpha_{n}u+(1-\alpha_{n})T(t^{n})x_{n}$$\alpha_{n}arrow 0$

159
for each. Then,converges strongly to Pu, where
nonexpansive retraction fromonto $F(S)$ .
$n\in \mathbb{N}$$\{x_{n}\}$$P$

is the unique sunny$C$

We know thatsuch that$f$: $Carrow C$ is said to be a contraction on $C$ if there exists $r\in(0,1)$$\Vert f(x)-f(y)\Vert\leq r\Vert x-y\Vert$for each $y\in C$ . Using [30] and Theorem 5.3, we obtain the following strong
convergence theorem by the viscosity approximation methods (see also [18, 4$x,$

Theorem 5.5 ([4]). Let $E,$ $C,$
and $\mathcal{S}=\{T(t):t\in S\}$ be as in Theorem 5.3.
Let be a contraction on C. Let
be a sequence in such that $0<\alpha_{n}\leq 1,$
and. Let $u\in C$ , let $t\in S$ , and let
be the sequence defined
$S$

$f$

$\alpha_{n}arrow 0$$\{\alpha_{n}\}$$\mathbb{R}$$\{x_{n}\}$$\frac{\alpha}{\alpha_{n-1}}arrow 1$$by$$x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})(T(t))^{n}x_{n}$. Then,converges strongly to Pu, where
nonexpansive retraction fromonto $F(S)$ .

for each$n\in \mathbb{N}$$\{x_{n}\}$$P$

is the unique sunny$C$

In the case when

, that is, $S$ is a uniformly asymptotically regular
one-parameter nonexpansive semigroup, we have the following strong convergence
theorem for a one-parameter nonexpansive semigroup by Theorems 5.3 and 5.5 (seealso [4, 11, 17, 18, 30$S=\mathbb{R}^{+}$

Theorem 5.6 ([4]). Let and be as in Theorem 5.3. Let $S=\{T(t):t\in \mathbb{R}^{+}\}$
be a uniformly asymptotically regular one-parameter nonexpansive semigroup on. Let be a contraction on C. Let
such thatbe a sequence insuch that $0<\alpha_{n}\leq 1,$. Let $u\in C$ and let $t\in(0, \infty)$ , andandbe the sequence defined by
let
$E$

$C$

$C$

$F(S)\neq\emptyset$$f$$\{\alpha_{n}\}$$\alpha_{n}arrow 0$$\mathbb{R}$$\frac{\alpha}{\alpha_{n-1}}arrow 1$$\{x_{n}\}$$x_{n+1}=\alpha_{n}f(x_{n})+(1-\alpha_{n})T(t^{n})x_{n}$for each. Then,converges strongly to Pu, where
nonexpansive retraction fromonto $F(S)$ .
$n\in \mathbb{N}$$\{x_{n}\}$$P$

is the unique sunny$C$

ACKNOWLEDGEMENTS
The author is supported by Grant-in-Aid for Scientific Research No. 26400196
from Japan Society for the Promotion of Science.

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[36] H.K. Xu, Another control condition in an iterative method for nonexpansive mappings(S. Atsushiba) DEPARTMENT 0F SCIENCE EDUCATION, GRADUATE SCHOOL OF EDUCATION SclENCE OF TEACHING AND LEARNING, UNIVERSITY OF YAMANASHI, 4-4-37, TAKEDA KOFU, YAMANASHI 400-8510, JAPAN
$E$-mail address: asachiko@yamanashi.ac.jp