数理解析研究所講究録
第 1906 巻 2014 年 24-33

24

STRONG CONVERGENCE THEOREMS FOR FAMILIES OFNONEXPANSIVE MAPPINGS BY BROWDER’S TYPEITERATIONSSACHIKO ATSUSHIBAABSTRACT. In this paper, we study Browder’s type iterations for nonexpansive semigroups in Banach spaces. Then, we study strong convergence theorems for uniformly
asymptotically regular nonexpansive semigroups in Banach spaces. We also givestrong convergence theorems for the nonexpansive semigroups in Banach spaces by
the viscosity approximation method.

1. INTRODUCTION
and letand norm
Let $H$ be a real Hilbert space with inner product$H$$CarrowC$is called
mapping:Then,abe a nonempty closed convex subset of .
nonexpansive if $\Vert Tx-Ty\Vert\leq\Vert x-y\Vert$ for all $y\in C$. We denote by $F(T)$ the set of
fixed points of . Let be an element of and for each with $0<t<1$ , let be aunique element of satisfying $x_{t}=tx+(1-t)Tx_{t}$ . In 1967, Browder [5] proved thefollowing strong convergence theorem.
$\Vert$$\rangle$$C$$\Vert$

$T$

$x,$

$T$

$C$

$x$

$t$

$x_{t}$$C$, let $C$ be a nonempty bounded closed convex
Theorem 1.1. Let $H$ be a Hilbert
subset of $H$ and let be a nonexpansive mapping of into itself. Let be an element
of $C$ and for each with $0<t<1$ , let be a unique element of $C$ satisfying$\mathcal{S}pace$

$C$

$T$

$t$

$x$

$x_{t}$$x_{t}=tx+(1-t)Tx_{t}.$Then,$\{x_{t}\}$converges strongly to the element of $F(T)$ nearest to

$x$

as

$t\downarrow 0.$

Reich [18] and Takahashi and Ueda [31] extended Browder’s result to those of a
Banach space. Using the idea of Shimizu and Takahashi [19, 20] and the notion ofsequence of means, Shioji and Takahashi [21] proved the strong convergence of Browder’s type sequences for nonexpansive semigroups (see also [22, 23, 24 On the other
hand, Domingues Benavides, Acedo and Xu [10] proved Browder’s type strong convergence theorems for uniformly asymptotically regular one-parameter nonexpansive
semigroups. Acedo and Suzuki [14] 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 [5, 10, 14, 29, 30]. The author [1] also proved strong
convergence theorems for the nonexpansive semigroups by the viscosity approximation2010 Mathematics Subject Classification. Primary $47H09,$ $47H10.$
Key words and phrases. Fixed point, iteration, nonexpansive mapping, nonexpansive semigroup,
strong convergence.

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method. Furthermore, author [2] proved strong convergence theorems for uniformly
asymptotically regular nonexpansive semigroups in a smooth Banach space which satisfies Opial’s condition.
In this paper, we study Browder’s type iterations for nonexpansive semigroups in Banach spaces. Then, we study strong convergence theorems for uniformly asymptoticallyregular nonexpansive semigroups in Banach spaces by using the idea of [1, 5, 10, 14,
29, 30]. Furthermore, we also give strong convergence theorems for the nonexpansive
semigroups in Banach spaces by the viscosity approximation method.
2. PRELIMINARIES

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 allnonnegative integers and the set of all nonnegative real numbers, respectively.
Let $E$ 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$C$$x\in E$ will be denoted by. Let be a closed subset of a Banach space and let
be a mapping of into itself. We denote by $F(T)$ the set $\{x\in C:x=Tx\}.$We denote by the identity operator on $E$ . The duality mapping from $E$ into
is defirled by$\mathbb{N}$$\mathbb{R}$$\mathbb{R}^{+}$$\mathbb{Z}^{+}$$B_{r}$$\Vert$$\Vert x\Vert\leq r\}$$E^{*}$$x^{*}\in E^{*}$$\langle x,$$T$$x^{*}\rangle$

$C$

$I$

$J$

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

From the Hahn-Banach theorem, we see thatfor all $x\in E.$$E$
Let be a smooth Banach space. Then, is said to be weakly sequentially contin$J(x)\neq\emptyset$$J$uous at zero if for every sequence
in $E$ which converges weakly to $0\in E,$ $\{J(x_{n})\}$
converges $weakly^{*}$ to $0\in E^{*}.$
We say that a Banach space $E$ satisfies Opial’s condition [16] if for each sequence
in $E$ which converges weakly to$\{x_{n}\}$$\{x_{n}\}$$x,$$\varliminf_{narrow\infty}\Vert x_{n}-x\Vert<\varliminf_{narrow\infty}\Vert x_{n}-y\Vert$(1)for each $y\in E$ with $y\neq x$ . If $E$ is reflexive Banach space with weakly sequentiallycontinuous duality mapping, then $E$ satisfies Opial’s condition. Each Hilbert space andthe sequence spaces with $1<p<\infty$ satisfy Opial’s condition (see [16]). Though an-space with $p\neq 2$ does not usually satisfy Opial’s condition, each separable Banach
space can be equivalently renormed so that it satisfies Opial’s condition (see [11,16
In a reflexive Banach space, this condition is equivalent to the analogous condition fora bounded net which has been introduced in [13]. It is well known that this conditionis equivalent to the analogous condition of(see [4]).
$\ell^{p}$$I\nearrow$$\varlimsup$

Proposition 2.1. Let $H$ be a Hilbert space. Let
weakly to $x\in H$ . Then,$\{x_{n}\}$be a sequence in $H$ converging$\varliminf_{narrow\infty}\Vert x_{n}-x\Vert<\varliminf_{narrow\infty}\Vert x_{n}-y\Vert$for each $y\in E$ with

$y\neq x.$

(2)

26
Banach space$E$is said to be smooth if$\lim_{tarrow 0}\frac{\Vert x+ty\Vert-\Vert x\Vert}{t}$exists for each and in , where $S_{1}=\{u\in E : \Vert u\Vert=1\}$ . The norm of $E$ issaid to be ur iformly G\^ateaux diﬀerentiable if for each in , the limit is attained
uniformly for in . We know that if $E$ is smooth, then the duality mapping issingle-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 and norm to weak
star, uniformly continuous on each bounded subset of $E.$
of a Banach space $E$ is said to have normal structure
A closed convex subset
if for each bounded closed convex subset $K$ of $C$ which contains at least two points,
there exists an element of $K$ which is not a diametral point of $K$ . It is well-known
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 [30]). We
also know that uniformly smooth Banach space has normal structure (see [30]). Every
weakly compact convex subset of a Banach space satisfying Opial’s condition has normal structure (see [12]). We note that closed convex subset $C$ of a Banach space $E$ issaid to have the fixed poirlt property for nonexparlsive mappings if for every boundedclosed 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 $C$ be a nonempty closed convex subset of $E$ and let $K$ be a nonempty subset of$C$. A mapping $P$ of $C$ onto $K$ is said to be sunny if $P(Px+t(x-Px))=Px$ for each$x\in C$ and $t\geq 0$ with $Px+t(x-Px)\in C.$ $P$ is a retraction if $Px=x$ for each $x\in K.$
We know from [7, Theorem 3] and [17, Lemma 2.7] the following lemma (see also [30]).
$x$

$y$

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

be a smooth Banach space, let $C$ be a convex
Lemma 2.2 ([7, 17 Let$E$ and let $K$ be a subsetof C. Then, a retraction $P$ of $C$ onto $K$ is sunny andof$none\varphi$ ansive
if and only if$sub_{\mathcal{S}}et$$E$$\langle x-Px,$ $J(y-Px)\rangle\leq 0$

for all $x\in C$

and$y\in K.$

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

If there is a sunny nonexpansive retraction of $C$ onto $K,$ $K$ is said to be a sunny nonexpansive retract of $C$ . The following theorem related to the existence of nonexpansive
retractions was proved in [8, 9].

Theorem 2.3 ([8,9 Let be a reflexive Banach space, let be a nonempty closed
convex subset of $E$ and let be a nonexpansive mapping of $C$ into itself with
nonempty bounded closed convex subset of $E$ such that
If has a fixed point inleaves invariant, then $F(T)$ is a nonexpansive retract of $C.$
$C$

$E$

$F(T)\neq\emptyset.$

$T$

$T$

$ever^{v}y$$T$

Let be a mean on positive integers , i.e., a continuous linear functional on
satisfying $\Vert\mu||=1=\mu(1)$ . We know that is a mean on if and only if$\mathbb{N}$$\mu$$\mathbb{N}$$\mu$$\inf\{a_{n}:n\in \mathbb{N}\}\leq\mu(f)\leq\sup\{a_{n}:n\in \mathbb{N}\}$$l^{\infty}$

27
for each $f=(a_{1}, a_{2}, \ldots)\in l^{\infty}$ . Occasionally, we use
Banach limit is a mean on satisfying$\mu_{n}(a_{n})$instead of $\mu(f)$ . So,$a$$\mathbb{N}$

$\mu$

.
be a Banach limit on$\mu_{n}(a_{n})=\mu_{n}(a_{n+1})$Let

and let$f=(a_{1}, a_{2}, \ldots)\in l^{\infty}$$\mu$$\mathbb{N}$. Then,$\varliminf_{narrow\infty}a_{n}\leq\mu(f)=\mu_{n}(a_{n})\leq\varlimsup_{narrow\infty}a_{n}.$The following lemma

In particular, if $a_{n}arrow a$ , then $\mu(f)=\mu_{n}(a_{n})=a$ (see [28, 30
was proved in [31] (see also [18,28

Lemma 2.4 ([31]). Let $C$ be a nonempty closed convex subset of a Banach space witha uniformly G\^ateaux diﬀerentiable norm. Let
be a bounded sequence in and let
be a Banach limit. Let $z\in C$ . Then,$E$$\{x_{n}\}$$\mu$$\mu_{n}\Vert x_{n}-z\Vert^{2}=\min_{y\in C}\mu_{n}\Vert x_{n}-y\Vert^{2}$

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

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

is the duality mapping$E.$of vectors in
We write $x_{n}arrow x$
to indicate that the sequence$H$ converges strongly toto indicate that
. We also write $x_{n}arrow x$
the sequenceof vectors in $H$ converges weakly to . In a Hilbert space, it is well
imply
known thatandis calledLet
be a semitopological semigroup. A semitopological semigroupright (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 the binary relation$t\in S$ , whereis
on is defined by $s\leq t$ if and only if
the closure of . A commutative semigroup is a directed system when the binary
relation is defined by $s\leq t$ if and only if $\{\mathcal{S}\}\cup(S+s)\supset\{t\}\cup(S+t)$ .
Let
be a nonempty closed convex subset of a Hilbert space $H$ . A family $S=$$\{T(t) : t\in S\}$ of mappings ofinto itself is said to be a nonexpansive semigroup on
if it satisfies the following conditions:(i) For each $t\in S,$ $T(t)$ is nonexpansive;(ii) $T(ts)=T(t)T(s)$ for each $s\in S.$
We denote by $F(S)$ the set of common fixed points of , i.e., $F(S)= \bigcap_{t\in S}F(T(t))$ .
$\{x_{n}\}$$( or \lim_{narrow\infty}x_{n}=x)$$x$$( or w-\lim_{narrow\infty}x_{n}=x)$$\{x_{n}\}$$x_{n}arrow x$$x$$\Vert x_{n}\Vertarrow\Vert x\Vert$$x_{n}arrow x.$

$S$

$S$

$S$

$S$

$”\leq$$S$$\{s\}\cup\overline{Ss}\supset\{t\}\cup\overline{St},$$A$$\overline{A}$

$s,$

$S$

$C$

$C$

$C$

$t,$

$S$

3. LEMMAIn this section, we give some lemmas which plays an important role in the proof ofour main results (see also [1, 2, 27

Lemma 3.1 ([2]). Let
be a nonempty closed convex
of a Banach space $E,$
and let be a commutative semigroup. Let $S=\{T(t) : t\in S\}$ be a nonexpansive. Let $m$ be a positive integer and let $t\in S$ . Let
semigroup on
such that$C$
be an element of and for each with $0<\alpha<1$ , let $Q(u, \alpha)$ be the unique elementof satisfying$C$$\mathcal{S}ubset$

$S$

$C$

$F(S)\neq\emptyset$$u$$\alpha$$C$$Q(u, \alpha)=\alpha u+(1-\alpha)(T(t))^{m}Q(u, \alpha)$.

28
Assume thatis 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$(3)$\langle u-Q(u, \alpha) , J(v-Q(u, \alpha \leq 0$

(4)

and

hold.
Lemma 3.2 ([2]). Let $C$ 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 nonexpansivebe. Let
and letbe a sequence insemigroup on
such that
$t\in
S$
$u\in
C$
$0<\alpha_{n}<1$$\{Q(u,
n)\}$
,
be
the
,
let
.
let
and
Let
a sequence in such thatsequence defined by
$S$

$C$

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

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 an element(say Pu) of $F(S)$ and $P$ is the unique sunny nonexpansive retractionfrom $C$ onto
$F(S)$ .

Lemma 3.3 ([2]). Let $E$ be a Banach space, let $C$ be a locally weakly compact convex
of $E$ , and let be a commutative semigroup. Let $S=\{T(t) : t\in S\}$ be a
such
. Let
be a sequence innonexpansive semigroup on $C$ such that. Let
be a sequence in such thator $m_{n}arrow N$ for somethat
$0<\alpha_{n}<1$ , andbe the sequence defined by. Let $u\in C$ , let $t\in S$ , and let$S$$\mathcal{S}ubset$$F(S)\neq\emptyset$$N\in \mathbb{Z}^{+}$$m_{n}arrow\infty$$\mathbb{Z}^{+}$$\{m_{n}\}$$\mathbb{R}$$\{\alpha_{n}\}$$\{x_{n}\}$$\alpha_{n}arrow 0$$x_{n}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{m_{n}}x_{n}$. Assume thatis $smooth_{f}$ the normalized duality mapping of $E$ isweakly sequentially continuous at zero and has the Opial property. Assume also that
. Then,converges strongly.
converges weakly to somefor each$n\in \mathbb{N}$

$J$

$E$

$C$

$\{x_{n}\}$$x\in F(\mathcal{S})$$\{x_{n}\}$G\^ateaux diﬀerLemma 3.4 ([3]). Let $E$ be a Banach space whose norm is$E$$C$
entiable, letbe a locally weakly compact convex subset of , and let be a commutative emigroup. Let $S=\{T(t) : t\in S\}$ be a nonexpansive semigroup on $C$ such
that
. Let
be a sequence in. Let
be a sequence inthat$0<\alpha_{n}<1$ . Let $u\in C$ , let $t\in S$ , and let
be the sequence defined by$unifor\tau r\iota ly$$S$$\mathcal{S}$$F(\mathcal{S})\neq\emptyset$$\mathbb{Z}^{+}$$\{m_{n}\}$$\mathbb{R}\mathcal{S}uch$$\{\alpha_{n}\}$$\{x_{n}\}$$x_{n}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{m_{\mathfrak{n}}}x_{n}$

Let

$\mu$

. Assume thathas the fixed point property for nonexpansive mappings.
be a Banach limit and let $g(y)=\mu_{n}\Vert y-x_{n}$ Put

for each$n\in \mathbb{N}$$C$$K= \{z\in C:g(z)=\min_{y\in C}g(y)\}$and assume $K\cap F(S)\neq\emptyset$ . Then, every subsequence ofverging strongly to an element in $K\cap F(S)$ .

$\{x_{n}\}$has a subsequence con-29
4. STRONG CONVERGENCE THEOREMSIn this section, we study strong convergence theorems for uniformly asymptoticallyregular nonexpansive semigroups in Banach spaces.
Let be a nonempty closed convex subset of a Banach space $E$ , let be a commutative semigroup and let $\mathcal{S}=\{T(t) : t\in S\}$ be a nonexpansive semigroup on . We
say that a nonexpansive semigroup $S=\{T(t) : t\in S\}$ is asymptotically regular if
$C$

$S$

$C$

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

for all $h\in S$ and $x\in C$ (see also [29, 30 The following lemma plays an importantrole in the proof of our main results (see [1, 14
Lemma 4.1 ([2]). Let $C$ be a nonempty closed convex
of a Banachand let be a commutative semigroup. Let $S=\{T(t) : t\in S\}$ be a $nonexpan\mathcal{S}ive$
semigroup on $C$ such that
. Assume that $S=\{T(t) : t\in S\}$ is ymptoticallyregular, that is,$\mathcal{S}ubset$$\mathcal{S}paceE,$$S$$F(\mathcal{S})\neq\emptyset$$a\mathcal{S}$$\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 for every bounded subset $K$ of $C,$$\lim_{s\in S_{x}}\sup_{\in K}\Vert T(h)T(s)x-T(\mathcal{S})x\Vert=0.$

holds.
Several authors proved strong convergence theorems for uniformly asymptoticallyregular one-parameter nonexpansive semigroups by Browder’s type iterations (see also
[6, 10, 14, 27
We study strong convergence theorems for uniformly asymptoticallyregular nonexpansive semigroups in Banach spaces by using the idea of [1, 5, 10, 27,
29, 30].
Theorem 4.2 ([2]). Let $E$ be a Banach $space_{f}$ let be a locally weakly compact convex
subset of $E$ , and let
be a commutative semigroup. Let $S=\{T(t) : t\in S\}$ be a
uniformly asymptotically regular nonexpansive semigroup on $C$ such that $F(S)\neq\emptyset.$
Let
. Let
be a sequence inor $m_{n}arrow N$ for somesuch that. Let $u\in C$ , letbe a sequence in such that $0<\alpha_{n}<1$ , andandbe the sequence defined by
let
$C$

$S$

$\mathbb{Z}^{+}$$\{m_{n}\}$$\mathbb{R}$$\{\alpha_{n}\}$$N\in \mathbb{Z}^{+}$$m_{n}arrow\infty$$\alpha_{n}arrow 0$$t\in S_{Z}$$\{x_{n}\}$$x_{n}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{m_{n}}x_{n}$that $E$ is smooth, the normalized duality mappingof $E$
the Opial property. Then,is weakly sequentially continuous at zero andconverges strongly to Pu, where $P$ is the unique sunny nonexpansive retraction fromfor each$n\in \mathbb{N}.$$J$$A_{\mathcal{S}\mathcal{S}}ume$$Cha\mathcal{S}$$C$

onto $F(S)$ .

$\{x_{n}\}$30
Theorem 4.3 ([3]). Let $E$ be a Banach space whose norm is uniformly G\^ateaux differentiable, let be a locally weakly compact convex subset of $E$ , and let be a commutative semigroup. Let $\mathcal{S}=\{T(t) : t\in S\}$ be a uniformly asymptotically regular. Let
be a sequence innonexpansive semigroup on such that$m_{n}arrowN$. Let
be a sequence in such thator
that
for some$0<\alpha_{n}<1$ , andbe the sequence defined by. Let $u\in C$ , let $t\in S$ , and let
$S$

$C$

$C$

$F(S)\neq\emptyset$$N\in \mathbb{Z}^{+}$$m_{n}arrow\infty$$\mathbb{Z}^{+}such$$\{m_{n}\}$$\mathbb{R}$$\{\alpha_{n}\}$$\{x_{n}\}$$\alpha_{n}arrow 0$$x_{n}=\alpha_{n}u+(1-\alpha_{n})(T(t))^{m_{n}}x_{n}$has the fixed point property for nonexpansive mapstrongly
to Pu, where $P$ is the unique sunny nonexpansive
converges
Then,pings.
$C$
retraction from onto $F(S)$ .

for each$n\in \mathbb{N}$. Assume that$C$$\{x_{n}\}$5. DEDUCED THEOREMS
We can deduce some strong convergence theorems from our main results.
We know that : $Carrow C$ is said to be a contraction on if there exists $r\in(O, 1)$such that$C$$f$$\Vert f(x)-f(y)\Vert\leq r\Vert x-y\Vert$for each $y\in C$ . Using [26] and Theorem 4.2, we obtain the following strong convergence theorem by the viscosity approximation methods (see also [1, 2, 3, 15Theorem 5.1 ([2]). Let $E$ be a Banach space, let be a locally weakly compact convex
subset of $E$ , and let be a commutative semigroup. Let $S=\{T(t) : t\in S\}$ be a
uniformly asymptotically regular nonexpansive semigroup on $C$ such thatorsuch thatbe a sequence inLet be a contraction on C. Let$0<\alpha_{n}<1$,and$m_{n}arrow N$
thatequence
such
in
be
.
Let
for some
be the sequence defined by. Let $t\in S$ and let
$x,$

$C$

$S$

$F(\mathcal{S})\neq\emptyset.$$\mathbb{Z}^{+}$$\{m_{n}\}$$f$$N\in \mathbb{Z}^{+}$$\{\alpha_{n}\}$$m_{n}arrow\infty$$\mathbb{R}$$a\mathcal{S}$$\{x_{n}\}$$\alpha_{n}arrow 0$$x_{n}=\alpha_{n}f(x_{n})+(1-\alpha_{n})(T(t))^{rn_{n}}x_{n}$is smooth, the normalized duality mapping of $E$
at zero and $C$ has the Opial property. Then,is weakly sequentiallyconverges strongly to Pu, where $P$ is the unique sunny nonexpansive retraction from.
onto
Using [26] and Theorem 4.3, we also obtain the following strong convergence theoremby the viscosity approximation methods (see also [1, 2, 3, 15Theorem 5.2 ([3]). Let $E$ be a Banach space whose norm is uniformly G\^ateaux differentiable, let $C$ be a locally weakly compact convex subset of $E$ , and let be a commutative semigroup. Let $S=\{T(t) : t\in S\}$ be a uniformly asymptotically regularnonexpansive emigroup on $C$ such that $F(S)\neq\emptyset$ . Let be a contraction on C. Let. Let
or $m_{n}arrow N$ for somesuch thatbe a sequence in$t\inS$$0<\alpha_{n}<1$
be the. Let
and let, andbe a sequence in such thatsequence defined byfor each$n\in \mathbb{N}$. Assume that

$J$

$E$

$\{x_{n}\}$$continuou\mathcal{S}$$C$$F(\mathcal{S})$

$S$

$f$

$\mathcal{S}$$\mathbb{Z}^{+}$$\{m_{n}\}$$N\in \mathbb{Z}^{+}$$m_{n}arrow\infty$$\alpha_{n}arrow 0$$\mathbb{R}$$\{\alpha_{n}\}$$\{x_{n}\}$$x_{n}=\alpha_{n}f(x_{n})+(1-\alpha_{n})(T(t))^{m_{\mathfrak{n}}}x_{n}$has the fixed point property for nonexpansive mapto Pu, where $P$ is the unique sunny nonexpansive
strongly
converges
Then,pings.
$C$
retraction from onto $F(S)$ .

for each$n\in \mathbb{N}$. Assume that$\{x_{n}\}$

$C$

31
Let
be a nonempty closed convex subset of $E$ . A familyof mappings ofinto itself satisfying the following conditions is said to be a oneparameter nonexpansive semigroup on $C$ :$T(t)$ is nonexpansive;(i) For each(ii) $T(t+\mathcal{S})=T(t)T(s)$ for every;$t\mapsto
T(t)x$$x\inC,$(iii) for eachis continuous.
In the case when
, that is, $S$ is a uniformly asymptotically regular oneparameter nonexpansive semigroup, we have the following strong convergence theorem
for the semigroup by Theorem 4.2 (see also [10, 14$C$$\mathcal{S}=\{T(t) :t\in \mathbb{R}^{+}\}$$C$$t\in \mathbb{R}^{+},$$t,$$\mathcal{S}\in \mathbb{R}^{+}$$S=\mathbb{R}^{+}$

Theorem 5.3 ([2]). Let $E$ be a Banach space, let $C$ be a locally weakly compact convex
subset of E. Let $S=\{T(t) : t\in \mathbb{R}^{+}\}$ be a uniformly ymptotically regular oneparameter nonexpansive semigroup on such that. Let
be a sequence$m_{n}arrowN$insuch that. Letorbe a sequence infor somesuch that $0<\alpha_{n}<1$ , and. Let $u\in C$ and let $t\in(0, \infty)$ . Let
be thesequence defined by$a\mathcal{S}$$C$$\mathbb{Z}^{+}$$F(\mathcal{S})\neq\emptyset$$N\in \mathbb{Z}^{+}$$m_{n}arrow\infty$$\mathbb{R}$$\{m_{n}\}$$\{\alpha_{n}\}$$\alpha_{n}arrow 0$$\{x_{n}\}$$x_{n}=\alpha_{n}u+(1-\alpha_{n})T(t^{m_{n}})x_{n}$for each. Assume thatsmooth, the normalized duality mappingof $E$
is weakly sequentiallyat zero andthe Opial property. Then,converges strongly to Pu, where $P$ is the unique sunny nonexpansive retraction from$C$
onto $F(S)$ .
$n\in \mathbb{N}$$J$$Ei\mathcal{S}$$continuou\mathcal{S}$$Cha\mathcal{S}$$\{x_{n}\}$We also have the following strong convergence theorems for a one-parameter nonexpansive semigroup by Theorem 4.3 (see also [10, 14
Theorem 5.4 ([3]). Let $E$ be a Banach space whose norm is uniformly G\^ateaux diﬀerentiable, let be a locally weakly compact convex subset of E. Let
be a uniformly asymptotically regular one-parameter nonexpansive semigroup on suchthat $F(S)\neq\emptyset$ . Letor $m_{n}arrow N$ for
be a sequence insuch that
. Let
some
. Let
be a sequence in such that $0<\alpha_{n}<1$ , and$u\in C$ and let $t\in(0, \infty)$ . Let
sequenceby
be thdefined$C$$\mathcal{S}=\{T(t) :t\in \mathbb{R}^{+}\}$$C$$\mathbb{Z}^{+}$$\{m_{n}\}$$N\in \mathbb{Z}^{+}$$m_{n}arrow\infty$$\alpha_{n}arrow 0$$\mathbb{R}$$\{\alpha_{n}\}$$\{x_{n}\}$$e$$x_{n}=\alpha_{n}u+(1-\alpha_{n})T(t^{m_{n}})x_{n}$for each. Assume thathas the fixed point property for nonexpansive mappings. Then,converges strongly to Pu, where $P$ is the unique sunny nonexpansive
retraction from $C$ onto $F(S)$ .
$n\in \mathbb{N}$$C$$\{x_{n}\}$Using [26] and Theorem 5.3, we obtain the following strong convergence theorem by
the viscosity approximation methods (see also [1, 2, 10, 14, 15,26

Theorem 5.5 ([2]). Let $E$ be a Banach space, let be a locally weakly compact convex
subset of E. Let
be a uniformly asymptotically regular oneparameter nonexpansive semigroup on. Let be a contractionthat$m_{n}arrow
N$on C. Letbe equence insuch thatorfor someLetbe a sequence in such that $0<\alpha_{n}<1$ , and. Let $t\in(O, \infty)$ and let
be the sequence defined by$C$$\mathcal{S}=\{T(t) :t\in \mathbb{R}^{+}\}$$C\mathcal{S}uch$$\{m_{n}\}$$\{\alpha_{n}\}$$\mathbb{Z}^{+}$$a\mathcal{S}$$\mathbb{R}$$F(S)\neq\emptyset$$f$$N\in \mathbb{Z}^{+}.$$m_{n}arrow\infty$$\alpha_{n}arrow 0$$\{x_{n}\}$$x_{n}=\alpha_{n}f(x_{n})+(1-\alpha_{n})T(t^{m_{n}})x_{n}$32is smooth, the normalized duality mapping of $E$
is weakly sequentially continuous at zero and $C$ has the Opial property. Then,converges strongly to Pu, where is the unique sunny nonexpansive retraction from$C$
onto $F(S)$ .

for each$n\in \mathbb{N}$. Assume that

$J$

$E$

$\{x_{n}\}$$P$

Using [26] and Theorem 5.4, we obtain the following strong convergence theorem by
the viscosity approximation methods (see also [1, 2, 10, 14, 15,26

Theorem 5.6 ([3]). Let $E$ be a Banach space whose norm is uniformly G\^ateaux diﬀerentiable, let be a locally weakly compact convex subset of E. Let $S=\{T(t):t\in \mathbb{R}^{+}\}$
be a $unif_{07}mly$ asymptotically regular one-parameter nonexpansive emigroup on such
such thatbe a sequence in. Let be a contraction on C. Let
thatsuch that$m_{n}arrow\infty$ or $m_{n}arrow N$. Let
be a sequence infor some$t\in(O,\infty)$$0<\alpha_{n}<1$ , andsequence
be
the
and
let
. Let
defined by
$C$

$C$

$\mathcal{S}$$F(S)\neq\emptyset$$\{m_{n}\}$$f$$N\in \mathbb{Z}^{+}$$\alpha_{n}arrow 0$$\{\alpha_{n}\}$$\mathbb{Z}^{+}$$\mathbb{R}$$\{x_{\mathfrak{n}}\}$$x_{n}=\alpha_{n}f(x_{n})+(1-\alpha_{n})T(t^{m_{n}})x_{n}$that $C$ has the fixed point property for nonexpansive mapconverges strongly to Pu, where $P$ is the unique sunny nonexpansive
pings. Then,
retraction from $C$ onto $F(S)$ .

for each$n\in \mathbb{N}.$$As\mathcal{S}ume$$\{x_{n}\}$ACKNOWLEDGEMENTS
The author is supported by Grant-in-Aid for Scientific Research No. 22540120 fromJapan Society for the Promotion of Science.

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DEPARTMENT OF SCIENCE EDUCATION, GRADUATE SCHOOL OF EDUCATION SCIENCE OF TEACHING AND LEARNING, UNIVERSITY OF YAMANASHI, 4-4-37, TAKEDA KOFU, YAMANASH1400-8510,

JAPAN-mail address: asachiko@yamanashi.ac.jp$E$