ON WEAK CONVERGENCE TO FIXED POINTS OF
NONEXPANSNE
MAPPINGS IN BANACH SPACES筑波大学・大学院経営システム科学 鈴木智成(Tomonari Sllzllki)
東京工業大学大学院情報理工学研究科 高橋 渉(Wataru Takahaslii)
ABSTRACT. In this paper, we prove the following weak convergence theorem: Let
$C$ be a nonempty closed convex subset of a uniformly convex Banach space $E$
which satisfies Opial’s conditionor whosenorm is Fr\’echet differentiable. Let $T$ be
anonexpansivemappingfrom$C$intoitself with a fixed point. Suppose that $\{x_{n}\}$ is
givenby $x_{1}\in C$ and $x_{n+1}=\alpha_{n}T\beta_{n}Tx_{n}+(1-\beta_{n}\rangle$$x_{\hslash}$]$+(1-\alpha_{n})x_{n}$ for all$n\geq 1$, where $\{\alpha_{n}\}$ and $\{\beta_{n}\}$ are sequences in $[0,1]$ such that $\sum_{n=1}^{\infty}\alpha_{n}($1- $\alpha_{n})=\infty$ and
$\lim_{narrow}\sup_{\infty}\rho \mathfrak{n}<1$, or $\sum_{\mathfrak{n}=\iota^{\alpha}}^{\infty}n\beta_{n}=\infty$ and lin$\sup\beta_{n}<1$. Then $\{x_{n}\}$ converges
weaidy to afixed point of$T$. This is a$\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}1\mathrm{i}\mathrm{z}\mathrm{a}\mathrm{t}\mathrm{i}narrow\infty \mathrm{o}\mathrm{n}$
ofthe results ofTan andXu,
and Takahashi and Kim.
1.
INTRODUCTION
Let $E$ bea real Banach space and let $C$ be a nonempty closed convexsubset of$E$.
Then a napping $T$ bom $C$ into itselfis $\mathrm{c}\mathrm{a}\mathrm{U}\mathrm{e}\mathrm{d}$nonexpansive if
$||Tx-\tau y||\leq||x-y|\{$
for $\mathrm{a}\mathrm{U}x,y\in C$
.
For a mapping $T$ bom $C$ into itself, we denote by $F(T)$ the set offixed points of $T$
.
Now, we considerthe $\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{W}\mathrm{i}\mathrm{n}\mathrm{g}$ iteration scheme: $x_{1}\in C$ and(1) $X_{n+\iota}=\alpha nT[\beta_{n}TX_{n}+(1-\beta_{n})x_{n}|+(1-\alpha_{n})x_{\hslash}$ ffir all$n\geq 1$,
where $\{\alpha_{\pi}\}$ and $\{\beta_{n}\}$ are sequencesin [$0,1\iota$
.
Such an iteration scheme was introducedby Ishikawa [3]; $\mathrm{s}\infty$ also Mann [4]. Recently Tan and Xu [8] proved the $\mathrm{f}\mathrm{o}\mathrm{u}_{\mathrm{o}\mathrm{W}}\mathrm{i}\mathrm{n}\mathrm{g}$
interesting result (Corouary 1): Let $C$ be a nonempty closed convex subset of a
uniformly convex Banach space $E$ which satisfies Opial’s condition or whose norm is
Fr\’echet differentiable and let $T$ be a nonexpansive mapping from $C$ into $\mathrm{i}\dot{\mathrm{t}}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}$
with a
fixedpoint. Then for any initial data $x_{1}$ in $C$, the iterates $\{x_{n}\}$ defined by (1), where
$\{\alpha_{n}\}$ and $\{\beta_{n}\}$ are chosen so that $\Sigma_{n=1}^{\infty}\alpha n(1-\alpha_{n})=\infty,$ $\Sigma_{n=1}^{\infty}\beta n(1-\alpha n)<\infty$ and
$\lim\sup\beta_{n}<1$,
converge
weakly to a fixed point of $T$. On the other $\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{d}_{\mathit{7}}$Takahashi$\mathfrak{n}arrow\infty$
and Kim [$\overline{(}|$ proved the $\mathrm{f}\mathrm{o}\mathrm{u}_{\mathrm{o}\mathrm{W}}\mathrm{i}\mathrm{n}\mathrm{g}$ (Corollaxy 2): Let $C,$ $E$ and $T$ be as above and
suppose $\alpha_{n}\in[a, b]$ and $\beta_{n}\in[0, b]$, or $\alpha_{n}\in[a, 1]$ and $\beta_{n}\in[a, b.]$ for some $a,$ $b$ with
1991 Mathematics Subject Classification. Primary $47\mathrm{H}10$, Secondary $47\mathrm{H}09,47\mathrm{H}17$.
Key $wo\mathrm{r}d_{\mathit{8}}$and
$ph_{\Gamma qs}es$. Fixed$\mathrm{p}\mathrm{o}i\mathrm{n}\mathrm{t}$, Nonexpansive napping, Ishikawa iteration.
$0<a\leq b<1$
.
Then for any initial data $x_{1}$ in $C$, the iterates $\{x_{\pi}\}$ defined by (1)converge weakly to a fixed point of $T$
.
Note that Tan and Xu’s result is applicableto the case of $\alpha_{n}=1-1/n$ and $\beta_{n}=1/n$ for all $n\geq 1$, while Takahashi and Kim’s result is applicable to the case of $\alpha_{n}=\beta_{n}=1/2$ for all $n\geq 1$.
In this paper, notivated by these two results, we prove the following weak conver-gence theorem: Let $C,$ $E$ and $T$ be as above and suppose $\sum_{n=1}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty$ and
$\lim_{narrow}\sup_{\infty}\beta_{n}<1$, or $\Sigma_{n=1}^{\infty}\alpha_{n}\beta_{n}=\infty$ and $\lim_{narrow}\sup_{\infty}\beta_{n}<1$
.
Then for any initial data $x_{1}$in $C$, theiterates $\{x_{n}\}$ defined by (1) converge weaklyto a fixed point of$T$
.
Comparethis with Tan and Xu’s result [8] and Takahashi and Kim’s result [7].
2. PRELIMINARIES
Let $E$ be a Banach space. For each $\epsilon$ with $0\leq\epsilon\leq 2$,
we
define the modulus$\delta(\epsilon)$ of convexity of $E$ by
$\delta(\epsilon)=\inf\{1-\frac{||X+y||}{2}$ : $||x||\leq 1,$ $||y||\leq 1,$$||_{X}-y||\geq\epsilon\}$ .
Note that $\delta$ is nondecreasing
and
$|| \lambda_{X}+(1-\lambda)y||\leq\max\{||_{X}||, ||y||\}[1-2\lambda(1-\lambda)\cdot\delta(\frac{||X-y||}{\max\{||X||,||y||\}})]$
forevery $x,$$y\in E\backslash \{0\}$ and$\lambda\in[0,1]$; see [2]. $E$ is called uniformly convex if$\delta(\epsilon)>0$
for all $\epsilon>0$
.
Thenorm
of $E$ is called Fr\’echet differentiable if for each $x\in E$ with$||x||=1,$ $\mathrm{I}\mathrm{i}\mathrm{m}_{0^{\frac{||X+ty||-||x||}{t}}}tarrow$
exists and is attained uniformly in $y\in E$ with $||y||=1$;
see [2]. $E$ is said to satisfy Opial’s condition [5] if for any sequence $\{x_{n}\}$ in $E$ such
that $\{x_{n}\}$
converges
weakly to $z\in E,$$\lim_{narrow}\inf_{\infty}||x_{n}-z||<\mathrm{I}\mathrm{i}\mathrm{m}\inf_{narrow\infty}||x_{n}-y||$ for all $y\in E$
with $y\neq z$. All Hilbert spaces and $l^{p}(1<p<\infty)$ satisfy Opial’s condition, while $L^{\mathrm{p}}$
with $1<p<\infty$ and $p\neq 2$ do not. The following lemma w\’as proved by Reich [6];
see also [7].
Lemma
1. Let $C$ be a nonempty closed convex subsetof
a uniformly convex Banach space $E$ whose norm is Fr\’echetdifferentiable
and let$\{T_{1}, T_{2,3}T, \cdots\}$ be a sequence
of
nonexpansive mappingsfrom
$C$ intoitself
such that $\bigcap_{n=1}\infty F(\tau_{n})$ is nonempty. Let$x\in C$ and $S_{n}=T_{n}T_{n-}1\ldots T_{1}$
for
all $n\geq 1$.
Then the set $( \bigcap_{n=1}^{\infty}\overline{c\mathit{0}}\{s_{m^{X}}$ : $m\geq$$n \})\cap(\bigcap_{n=1}^{\infty}F(\tau n))$ consists
of
at most one point, where $\overline{co}\{s_{m}X : m\geq n\}$ isthe closure
of
the convex hullof
$\{S_{m}x:m\geq n\}$.3. WEAK CONVERGENCE THEOREM
In this section,
we
prove thefollowing theorem which generalizesthe results ofTanTheorem. Let $C$ be a nonempty dosed
convex
subsetof
a uniformlyconvex
Banachspace $E$ which
satisfies
Opial’s condition or whose norm $i_{\mathit{8}}$ Fr\’echetdiffe
rentiable. Let$T$ be a nonexpansive mapping
from
$C$ intoitself
with afixed
point. Suppose that $\{x_{n}\}$ is given by $x_{1}\in C$ and $x_{n+1}=\alpha_{n}T[\beta_{nn}T_{X}+(1-\beta_{n})x_{n}]+(1-\alpha_{n})_{X_{n}}$for
all $n\geq 1$,
where $\{\alpha_{n}\}$ and $\{\beta_{n}\}$ are sequences in $[0,1]$ such that $\Sigma_{n=1}^{\infty}\alpha n(1-\alpha_{n})=\infty$ and
$\lim\sup\beta_{n}<1_{f}$
or
$\Sigma_{n=1}^{\infty}\alpha_{n}\beta n=\infty$ and$\lim_{narrow}\sup_{\infty}\beta_{n}<1$
.
Then $\{x_{n}\}$ converges weakly $to^{narrow\infty}afixed$ pointof
$T$.
Before $\mathrm{p}\mathrm{r}\mathrm{o}\dot{\mathrm{v}}$ling it, we
need
some
definitions and lemmas. We denote by $\mathrm{N}$ theset of positive integers. Let $I$ be an infinite subset of N. If $\{\lambda_{n}\}$ is a sequence of
nonnegative numbers, then we denote by $\{\lambda_{i} : i\in I\}$ the subsequence of $\{\lambda_{n}\}$
.
Lemma 2. Let $\{\lambda_{n}\}$ and $\{\mu_{n}\}$ be sequences
of
nonnegative numbers such that $\sum_{n=1}^{\infty}$$\lambda_{n}=\infty$ and $\sum_{n=1}^{\infty}\lambda_{n}\mu_{n}<\infty$
.
Thenfor
$\epsilon>0$, there exists aninfinite
subset Iof
$\mathrm{N}$such that $\sum\{\lambda_{j} : j\in \mathrm{N}\backslash I\}\leq\epsilon$ and the subsequence $\{\mu: : i\in I\}$
of
$\{\mu_{n}\}$ convergesto $0$
.
Proof.
For each $\epsilon>0$, first take $p_{0}\in \mathrm{N}$ with $\Sigma_{n=_{\mathrm{P}+}}^{\infty}01\lambda n\mu n\leq\epsilon/2$.
From $\Sigma_{n=1}^{\infty}\lambda_{n}=$ $\infty$ and $\Sigma_{n=1}^{\infty}\lambda_{n}\mu_{n}<\infty$,
we have $\lim\inf\mu_{n}narrow\infty=0$.
So, thereexists
$p_{1}\in \mathrm{N}$ such that$p_{1}>p\mathrm{o},$ $\mu p_{1}<1$ and
$\sum\{\lambda_{j\mu_{j}} : j>p_{1}\}\leq\frac{\epsilon}{2\cdot 2^{2}}$
.
Similarly we can take $p_{2},p_{3},$ $\cdots\in \mathrm{N}$ such that $p_{k}>p_{k-1},$ $\mu_{\mathrm{p}k}<1/k$ and $\sum\{\lambda_{\mathrm{j}\mu_{j}:j}>p_{k}\}\leq\frac{\epsilon}{(k+1)\cdot 2k+1}$
for all $k=2,3,$$\cdots$
.
Define$I= \{1,2, \cdots,p_{0}\}\cup(_{k=}\bigcup_{1}^{\infty}\{n$ : $p_{k}-1<n \leq pk,\mu_{n}<\frac{1}{k}\})$
.
$\mathrm{T}\dot{\mathrm{h}}\mathrm{e}\mathrm{n},$
$\{\mu_{i} : i\in I\}$.is a subsequence of $\{\mu_{n}\}$ such that $\mu_{i}arrow 0$.
We
also have$\sum\{\lambda_{j} : j\in \mathrm{N}\backslash I\}=\sum k=1\infty\sum\{\lambda_{n}$: $pk-1<n \leq pk,\mu n\geq\frac{1}{k}\}$
.
Putting $S_{k}=\{n:pk-1<n\leq p_{k}, \mu_{n}\geq 1/k\}$, we have
$\frac{1}{k}\sum\{\lambda_{n} : n\in S_{k}\}\leq\sum\{\lambda_{n}\mu_{n} : n\in s_{k}\}\leq\sum\{\lambda_{j}\mu j : j>pk-1\}$
$\leq\frac{\epsilon}{k\cdot 2^{k}}$ and
hen.
ceThis completes the proof. $\square$
Lemma 3. Let$\{\lambda_{n}\}$ and $\{\mu_{n}\}$ be sequences
of
$n$onnegative numbers such fhat $\lambda_{n+1}\leq$$\lambda_{n}+\mu_{n}$
for
all $n\in$ N. Suppose there exists a subsequence $\{\mu: : i\in I\}$of
$\{\mu_{n}\}$ suchthat $\mu_{i}arrow 0,$ $\lambda_{i}arrow\alpha$ and $\sum\{\mu_{j} : j\in \mathrm{N}\backslash I\}<\infty$
.
Then $\lambda_{n}arrow\alpha$.Proof.
Fix $\epsilon>0$ and take $n_{0}\in I$ such that [$\lambda_{i}-\alpha|\leq\epsilon$ and $\mu_{i}\leq\epsilon$for all $i\geq n_{\mathrm{O}}$ and$\sum\{\mu_{j} : j>n_{\mathrm{O}},j\in \mathrm{N}\backslash I\}\leq$
. $\epsilon$
.
For $n\in \mathrm{N}\backslash I$ with $n>n_{0}$, putting $k= \max\{i\in I$ :$i<n\}$ and $\ell=\min\{i\in I:\iota>n\}$, we have
$\lambda_{n}\leq\lambda_{n-1}+\mu_{n}-1\leq\cdots\leq\lambda_{k}+\sum_{\mathrm{j}=k}^{-1}\mu j\leq\lambda k+\mu_{k}+\epsilon\leq\alpha+3n\epsilon$
and
$\lambda_{n}\geq\lambda_{n+\iota}-\mu n\geq\cdots\geq\lambda_{\ell-}\sum_{nj=}^{1}\ell-\mu j\geq\lambda\ell^{-}\epsilon\geq\alpha-2\epsilon>\alpha-3\epsilon$
.
So, we obtain the desired result. $\square$
Lemma 4. Let $C$ be a closed convex subset
of
a $unifo\mathit{7}mly$convex
Banach space $E$and let $T$ be a $nonexpan\mathit{8}ive$ mapping
from
$C$ into $it\mathit{8}elf$with afixed
point. Supposethat $\{x_{n}\}$ is given by $x_{1}\in C$ and $x_{n+1}=\alpha_{n}T[\beta_{n}Tx_{n}+(1-\beta_{n})x_{n}]+(1-\alpha_{n})x_{n}$
for
all $n\in \mathrm{N}_{J}$ where $\alpha_{n},\beta_{n}\in[0,1]$
.
Then the following hold:(i)
If
$\sum_{n=1}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty$ and$\lim_{narrow}\sup_{\infty}\beta\hslash<1$, then $\lim_{narrow\infty}||Tx_{n}-X_{n}||=0_{j}$ (ii)
if
$\Sigma_{n=1}^{\infty}\alpha_{n}\beta n=\infty$ and$\lim_{narrow}\sup_{\infty}\beta_{n}<1$, then $narrow\infty \mathrm{b}||Tx-nX_{n}||=0$
.
Pmof.
We may assume that there exists $b\in(0,1)$ such that $\beta_{n}\leq b$ for all $n\in$ N.Fix $w\in F(T)$ and put $y_{n}=\beta_{n}Tx_{n}+(1-\beta_{n})_{X}n$ for all $n\in \mathrm{N}$
.
Then bythe definitionof $\{x_{n}\}$, we have $||_{X_{n+1}}-w||=||\alpha Tyn+(n1-\alpha n)x_{n}-w||$ $\leq\alpha_{n}||Tyn-w||+(1-\alpha n)||x-nw||$ $\leq\alpha_{n}||yn-w||+(1-\alpha n)||x-nw||$ $=\alpha_{n}||\beta_{nn}\tau_{x}+(1-\beta n)x_{n}-w\mathrm{f}|+(1-\alpha n)||x_{n}-w||$ $\leq\alpha_{n}(\beta_{n}||\tau X_{n}-w||+(1-\beta_{n})||X_{n}-w||)+(1-\alpha n)||xn-w||$ $\leq||_{X_{n}}-w||$
and hence the limit of$\{||x_{n}-w\mathrm{I}|\}$exists. Put $c= \lim_{narrow\infty}||x_{n}-w||$. If$c=0$, then (i) and
for all.$n\in \mathrm{N}$, we obtain $||x_{n+1^{-w}}||=||\alpha_{n}(Ty_{n}-w)+(1-\alpha n)(X_{n}-w)||$ $\leq||_{X_{n}-}w||[1-2\alpha_{n}(1-\alpha_{n})\cdot\delta(\frac{||\tau_{y_{n}-}Xn||}{||x_{n}-w||})]$
.
Since $||x_{n}-w||-||_{X}n+1^{-w||}$ $\geq 2||x_{n}-w|[\cdot\alpha_{n}(1-\alpha_{n})\cdot\delta(\frac{||Ty_{n}-x_{n}||}{||x_{n}-w||})$ $\geq 2c\cdot\alpha_{n}(1-\alpha_{n})\cdot\delta(\frac{||\tau_{y_{\mathfrak{n}}-}x_{n}||}{||x_{n}-w||})$for all $n\in \mathrm{N}$,
we
have$\sum_{n=1}^{\infty}\alpha_{n}(1-\alpha_{n})\cdot\delta.(\frac{||Ty_{n}-xn||}{||x_{n}-w||})<\infty$
.
By Lemna 2, there exists an infinite subset $I_{1}$ of $\mathrm{N}$ such that
(2) $\sum\{\alpha_{j}(1-\alpha j) : j\in \mathrm{N}\backslash I_{1}\}<\infty$
and $\{S(_{||}^{\ovalbox{\tt\small REJECT}-}Ty.:x)x.-w1|$ : $i\in I_{1}\}$ converges to $0$. Since
$c= \lim_{narrow\infty}||x_{n}-w||>0$, we obtain
$\{||\tau_{y_{i^{-x}}}.|| : i\in I_{1}\}$ converges to $0$. From
$||T_{X_{i}}-x_{i}||\leq||\tau X.\cdot-\tau y*\cdot||+||Ty_{i}-x_{i}||$ $\leq||_{X_{i^{-y}}}i||+||\tau y_{i}-X_{i}||$
$=\beta_{i}||TX_{i}-X_{i}||+||Ty_{i}-X.\cdot||$
$\leq b||TX_{i}-xi||+||\tau yi-X:||$,
we
obtain$\lim_{arrow}.\cdot\sup_{\infty}||Tx_{i}-x_{i}||\leq\lim_{iarrow}\sup_{\infty}\frac{1}{(1-b)}||\tau y_{i^{-}}x_{i}||=0$
.
Hence we have
Since
$||Tx+1-nx+n\iota||$
$\leq||T_{X_{n+}}1^{-}\tau(\alpha_{nn}T_{X}+(1-\alpha_{n})x_{n})||+||T(\alpha_{n}TX_{n}+(1-\alpha_{n})Xn)-Tx_{n}||$
$+||Tx_{n}-(\alpha_{n}T_{X_{n}}+(1-\alpha_{n})x_{n})||+||\alpha_{n}T_{X_{n}}+(1-\alpha_{n})_{X_{n}}-Xn+1||$ $\leq 2||\alpha_{n}\tau x_{\hslash}+(1-\alpha_{n})xn-Xn+1||+||\alpha_{nn}T_{X}+(1-\alpha_{n})xn-x_{n}||$
$+(1-\alpha_{n})||\tau Xn-xn||$ $=2\alpha_{n}||Txn-Ty_{n}||+||\tau_{x-}nXn||$ $\leq 2\alpha_{n}||X_{n}-y_{n}||+||\tau x_{n}-Xn||$ $=(1+2\alpha_{n}\beta n)||\tau x_{n}-x_{n}||$
and
$||\tau_{X_{n+1^{-}}}xn+1||$ $\leq||\tau X_{n+}1^{-T(y_{n}}\alpha_{n}\tau+(1-\alpha_{n})y_{n})||+||T(\alpha_{n}Ty_{n}+(1-\alpha_{n})y_{\mathrm{n}})-Ty_{n}||$$+||Tyn-(\alpha_{n}\tau y_{n}+(1-\alpha n)yn)||+||\alpha nTyn+(1-\alpha n)y_{n}-xn+1||$
$\leq 2||\alpha_{n}Ty_{n}+(1-\alpha_{n})y_{n}-X_{n+}1||+||\alpha_{n}\tau_{yn}+(1-\alpha_{n})yn-yn||$ $+(1-\alpha_{n})||\tau y_{n}-y_{n}||$ $=2(1-\alpha_{n})||xn-y_{n}||+||Tyn-y_{n}||$ $\leq 2(1-\alpha_{n})||x_{n}-y_{n}||+||\tau_{y_{n}-}TXn||+||Tx_{n}-y_{n}||$ $\leq 2(1-\alpha_{n})||x_{n}-y_{n}||+||y_{n}-X_{n}||+|!^{Tx_{n}-}y_{n}||$ $=(1+2(1-\alpha n)\beta_{n})||\tau x-nnX||$
for all $n\in \mathrm{N}$, we obtain
(4) $||T_{X}n+1-Xn+1||\leq(1+4\alpha_{n}(1-\alpha n)\beta_{n})||\tau xn-xn||$.
Since$\{||Tx_{n}-X_{n}||\}$ isbounded,from Lemma3, (2), (3) and(4), weobtain$\lim_{narrow\infty}||Tx_{n}-$
$x_{n}||=0$
.
We next prove (ii). From $||Tx_{n}-w||\leq||x_{n}-w||$ for all $n\in \mathrm{N}$, we obtain$||x_{n+1^{-w||}}\leq\alpha_{n}||y_{n}-w||+(1-\alpha_{n})||X_{n}-w||$
$=\alpha_{n}||\beta_{n}(\tau x_{n}-w)+(1-\beta n)(x_{n}-w)||+(1-\alpha n)||xn-w||$
$\leq\alpha_{n}||x_{n}-w||[1-2\beta_{n}(1-\beta_{n})\cdot\delta(\frac{||\tau x_{n}-X_{n}||}{||x_{n}-w||})]$
From
$||x_{n}-w||-||_{X}n+1^{-w||}$
$\geq 2||_{X_{\hslash}}-w||\cdot\alpha_{n}\beta_{n}(1-\beta n)\cdot\delta(\frac{||Tx_{nn}-x||}{||x_{n}-w||})$
$\geq 2_{C}\cdot\alpha_{n}\beta n(1-b)\cdot\delta(\frac{||Txn-X_{n}||}{||x_{n}-w||})$
for all $n\in \mathrm{N}$, we have
$\sum_{n=1}^{\infty}\alpha_{n}\beta n.\delta(\frac{||Tx_{n}-X_{n}||}{||x_{n}-w||})<\infty$
.
By Lemma 2, there exists an infinite subset $I_{2}$ of$\mathrm{N}$ such that (5) $\sum\{\alpha_{j}\beta_{i} : j\in \mathrm{N}\backslash I_{2}\}<\infty$
and $\{\delta(\frac{\mathrm{I}|Tx.-x}{||x:-w||})$ : $i\in I_{2}\}$ converges to $0$
.
Since $c= \lim_{narrow\infty}||x_{n}-w||>0$, we obtain(6) $\lim_{iarrow\infty}||Tx_{i}-X_{i}||=0$
.
Since $\{||\tau_{x_{n}}-x_{n}||\}$ is bounded, from Lemma 3, (4), (5) and (6), we
obtain
$\lim_{narrow\infty}|1\tau_{x_{n}-}$$x_{n}||=^{0}$
.
Proof
of
Theorem. Notethat byLemma4and Browder [1],aweak subsequential limitof the sequence $\{x_{n}\}$ is a fixed point of $T$
.
Since $E$ is reflexive and $\{x_{n}\}$ is bounded,to complete the proof, we prove that $\{x_{n}\}$ has at most one weak subsequential limit.
In the case that $E$ satisfies Opial’s condition, we assume that
$z_{1}$ and $z_{2}$ are two
distinct weak sequential limit of the subsequence $\{x_{i} : i\in I\}$ and $\{x_{j} : j\in J\}$ of
$\{x_{n}\}$ respectively. We obtain
$\lim_{narrow\infty}||x-nZ1||=\lim||xi-Z_{1}||<\lim||_{X_{i}}-Z2|iarrow\infty iarrow\infty|=\lim_{\infty narrow}||_{X}n-Z_{2}||$
$= \lim_{jarrow\infty}||x_{j}-z_{2}||<\lim_{jarrow\infty}||x_{j}-z_{1}||=\lim_{narrow\infty}||x_{n}-Z_{1}||$
.
This is a contradiction. In the case that the norm of $E$ is Re’chet differentiable, for
each $n\in \mathrm{N}$, we define a nonexpansive mapping $T_{n}$ from $C$ into itself by
$T_{n}(x)=\alpha_{n}T[\beta_{n}T_{X}+(1-\beta n)x]+(1-\alpha_{n})_{X}$
.
Then $\{x_{n}\}$ can be written as $x_{n+1}=\tau_{n}\tau_{n-1}\ldots\tau_{11}X$ and $F(T)\subset F(T_{n})$ for all $n\in$ N.
Let $z$ be a subsequential limit of $\{x_{n}\}$ and put $S_{n}=T_{nn-11}T\cdots T$ for all$n\in \mathrm{N}$
.
Then$z \in(\bigcap_{n=1^{\overline{C}}}\infty \mathit{0}\{s_{m}x : m\geq n\})\cap(\bigcap_{n}^{\infty}=1F(\tau_{n}))$
.
So, by Lemma 1, $\{x_{n}\}$ has at mostone weak subsequential limit. This completes the proof.
Corollary 1 (Tan and Xu [8]). Let $C$ be a nonempty closed convex subset
of
auniformly convex Banach space $E$ which $satisfie\mathit{8}$ Opial’s condition or whose
norm
isFr\’echet
differentiable.
Let$T$ be a nonexpansive mappingfrom
$C$ intoitself
with afixed
point. $Suppo\mathit{8}e$ that $\{x_{n}\}$ is given by$x_{1}\in C$ and $x_{n+1}=\alpha_{\pi}T[\beta_{nn}Tx+(1-\beta_{n})x_{n}]+$
$(1-\alpha_{n})x_{n}$
for
all $n\in \mathrm{N}_{f}$ where $\alpha_{n},\beta_{n}\in[0,1]$ such that $\sum_{n=1}^{\infty}\alpha n(1-\alpha_{n})=\infty$,$\sum_{n=1}^{\infty}\beta_{n}(1-\alpha_{n})<\infty$ and
$\lim_{narrow}\sup_{\infty}\beta_{n}<1$, Then $\{x_{n}\}$ converges weakly to a
fixed
point
of
$T$.Corollary 2 (Takahashi and Kim [7]). Let$C$ be a nonemptyclosed convex subset
of
a uniformlyconvexBanach space $E$ which$\mathit{8}atisfieS$ Opial’s condition or whose normis Fr\’echet
differentiable.
Let $T$ be a nonexpansive mappingfrom
$C$ intoitself
with afixed
point. Suppose that $\{x_{n}\}$ is given by $x_{1}\in C$ and$x_{n+1}=\alpha_{n}T[\beta_{n}\tau x_{\mathfrak{n}}+(1-\beta_{n})$$x_{n}]+(1-\alpha_{n})x_{n}$
for
all$n\in \mathrm{N}$,
where $\alpha_{n},\beta_{n}\in[0,1]$ such that$\alpha_{n}\in[a, b]$ and$\beta_{n}\in[0, b]$or $\alpha_{n}\in[a, 1]$ and $\beta_{n}\in[a,b]$
for
some a, $b$ with $0<a\leq b<1$.
Then $\{x_{n}\}$ convergesweakly to a
fixed
pointof
$T$.
Proof.
It is obvious that $\lim\sup\beta_{n}\leq b<1$.
In the caseof$\alpha_{n}\in[a, b]$ and $\beta_{n}\in[0, b]$,we obtain $\sum_{n=1}^{\infty}\alpha(n1-\alpha_{n}^{narrow\infty})\geq\sum_{n=\iota^{a(-b)}}^{\infty}1=\infty$
.
In the case of $\alpha_{n}\in[a, 1]$ and
$\beta_{n}\in[a, b]$ we obtain $\sum_{n=1}^{\infty}\alpha_{n}\beta n\geq\Sigma_{n=1}^{\infty}a^{2}=\infty$
.
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(T. Suzuki) GRADUATE SCIIOOL OF SYSTEMS MANAGEMENT, THE UNIVERSITY OF TSUKUBA,
3-29-1 OTSUKA BUNKYO-KU, TOKYO 112, JAPAN
$E$-mail address, T. Suzuki: [email protected]
(W. Takahashi) DEPARTMENT OF MATHEMATICAL AND COMPUTING SCIENCES, TOKYO
INSTI-TUTE OF TECHNOLOGY, OHOKAYAMA, MEGURO-KU, TOKYO 152, JAPAN
