数理解析研究所講究録
第 1570 巻 2007 年 123-136

123

Two New Nonexpansive Mappings andGeometry of Banach Spaces
東京工業大学・大学院情報理工学研究科
高橋渉 (Wataru Takahashi)
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology

Abstract. Our purpose in this article is to discuss new nonlinear operators in a Banach
space which are related to nonexpansive mappings and to obtain convergence theorems forthe operators. We first deal with a nonlinear operator called a relatively nonexpansive mapping which generalizes a nonexpansive mapping in a Hilbert space. Using this operator, we
prove a strong convergence theorem which generalizes Nakajo and Takahashi [29]. We also
obtain another theorem for relatively nonexpansive mappings which is $co$ nnected with Reich’s
theorem [33]. Next, we define another nonlinear operator in a Banach space called a generalized nonexpansive mapping. This mapping also generalizes a nonexpansive mapping in a
Hilbert space. Using this mapping, we also get a strong convergence theorem which is related
to Nakajo and Takahashi [29] and is different from the theorem above. Further, we obtain aweak convergence theorem of Reich’s type. Finally, we prove a strong convergence theorem fornonexpansive mappings in a Banach space which is closedly related to Nakajo and Takahashi
[29].

1

lntroduction

Let $H$ be a real Hilbert space with inner product(
and normand let be a nonempty
closed convex subset of $H$ . Then, a mappingof
into itself is called nonexpansive if$||Tx-Ty\Vert\leq\Vert x-y\Vert$ for all$y\in C$ . We denote by $F(T)$ the set of fixed points of.
Mann [22] introduced the following iterative sequence to approximate a fixed point of anonexpansive mapping: $x_{1}=x$ and$\cdot\rangle$$C$$\Vert\cdot\Vert$$\cdot,$

$C$

$T$

$T$

$x,$

$x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})Tx_{n}$,$n=1,2,$$\ldots$,

whereis a sequence in $[0,1]$ . Reich [33] proved the following weak convergence theorem
for such a sequence. For the proof, see Takahashi [46].
$\{\alpha_{n}\}$Theorem 1.1 (Reich [33]). Let be a closed convex subset of a Hilbert space $H$ and let
be a nonexpansive mapping of into itself such that $F(T)$ is nonempty. Let be the metricprvjection of $H$ onto $F(T)$ . Let $x\in C$ and let
be a sequence defined by $x_{1}=x$ and
$C$

$T$

$C$

$P$

$\{x_{n}\}$$x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})Tx_{n}$,$n=1,2,$$\ldots$

,

124

where

$\{\alpha_{n}\}\subset[0,1]$

satisfies$0\leq\alpha_{n}<1$Then,$\{x_{n}\}$and$\sum_{n=1}^{\infty}\alpha_{n}(1-\alpha_{n})=\infty$.

converges weakly to $z\in F(T)$ , where $z= \lim_{narrow\infty}Px_{n}$ .

Reich [33] proved really such a theorem in a uniformly convex Banach space whose norm
is a Fr\’echet differentiable. On the other hand, we know many problems in nonlinear analysis
and optimization which are formulated as follows: Find$u\in H$such that $O\in Au$ ,

(1.1)

$A$
is a maximal monotone operator from $H$ to $H$ . Such $u\in H$ is called a zero point(or a zero) of $A$ . A well-known method for solving (1.1) in a Hilbert space $H$ is the proximal

wherepoint algorithm: $x_{1}\in H$ and$x_{n+1}=J_{r}.x_{n}$,$n=1,2,$$\ldots$

,

(1.2)

where
and $J_{r}=(I+rA)^{-1}$ for all $r>0$ . This algorithm was first introduced, thenby Martinet [23]. In [39], Rockafellar proved that if $\lim\inf_{narrow\infty}r_{n}>0$ andthe sequence
defined by (1.2) converges weakly to a solution of (1.1). Motivated byRockafellar’s result, Kamimura and Takahashi [16] proved the following convergence theorem.
$\{r_{n}\}\subset(0,\infty)$$A^{-1}0\neq\emptyset$$\{x_{n}\}$Theorem 1.2 (Kamimura and Ihlahashi [16]). Let $H$ be a Hilbert space and let$HxH$ be a maximal monotone operator. Let $J_{r}=(I+rA)^{-1}$ for all $r>0$ and let
be asequence defined as follows: $x_{1}=x\in H$ and$A\subset$$\{x_{n}\}$$x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})J_{r_{n}}x_{n}$where$\{\alpha_{n}\}\subset[0,1]$

and$\{r_{n}\}\subset(0, \infty)$,$n=1,2,$$\ldots$,satisfy$\lim_{narrow}\sup_{\infty}\alpha_{n}<1$

and$\lim_{narrow}\inf_{\infty}r_{n}>0$.

, then the sequence
converges weakly to an element of $A^{-1}0$ , where $v=$
and is the metric projection of $H$ onto $A^{-1}0$ .
Solodov and Svaiter [41] also proved the following strong convergence $th\infty rem$ by the hybrid
method in mathematical programming.
Theorem 1.3 (Solodov and Svaiter [41]). Let $H$ be a Hilbert space and let $A\subset HxH$
be a sequence defined by
be a maximal monotone operator. Let $x\in H$ and let

If$A^{-1}0\neq\emptyset$$\lim_{narrow\infty}Px_{\mathfrak{n}}$$\{x_{n}\}$

$v$

$P$

$\{x_{n}\}$$\{\begin{array}{l}x_{1}=x\in H0=v_{n}+\frac{1}{r_{n}}(y_{n}-x_{n}),v_{n}\in Ay_{n}H_{n}=\{z\in H : \langle z-y_{n},v_{n}\rangle\leq 0\}W_{\mathfrak{n}}=\{z\in H:\langle z-x_{n},x_{1}-x_{n}\rangle\leq 0\}x_{n+1}=P_{H_{\hslash}\cap W_{n}}x_{1},n=1,2,\ldots\end{array}$is a sequence of positive numbers. If $A^{-1}0\neq\phi$ and $\lim\inf_{narrow\infty}r_{n}>0,$ $then\{x_{n}\}-$.
converges strongly towhere$\{r_{n}\}$$P_{A^{-1}0}x_{1}$125

Motivated by Solodov and Svaiter [41], Nakajo and Takahashi [29] proved the following
strong convergence teorem by using the hybrid method for nonexpansive mappings in a Hilbert
space.

Theorem 1.4 (Nakajo and Takahashi [29]). Let be a dosed convex subset of a Hilbert
space $H$ and let be a $none\varphi ansive$ mapping of into itself such that $F(T)$ is nonempty.
Let be the metric projection of $H$ onto $F(T)$ . Let $x_{1}=x\in C$ and
$C$

$T$

$C$

$P$

$\{\begin{array}{ll}y_{n}=\alpha_{n}x_{n}+(1-\alpha_{n}) x_{n},C_{n}=\{z\in C : \Vert y_{n}-z \leq\Vert x_{n}-z\Vert\},Q_{n}=\{z\in C:\langle x_{n}-z x_{1}-x_{n}\rangle\geq 0\},x_{n+1}=P_{C_{n}\cap Q_{n}}(x_{1}), n=1,2, \ldots,\end{array}$

where
onto

$\{\alpha_{n}\}\subset[0,1]$$C_{n}\cap Q_{n}$satisfies. Then,$\{x_{n}\}$$\lim\sup_{narrow\infty}\alpha_{n}<1$

and$P_{C_{n}\cap Q_{n}}$converges strongly to $Px_{1}\in F(T)$ .

is the metric projection of $H$

After Nakajo and Takahashi [29], many reseachers have studied such theorems by hybrid
methods in a Hilbert space; see, for instance, [14, 24, 42, 55]. However, we can not find atheorem for nonexpansive mappings in a Banach space which generalizes Nakajo and Takahashi[29].

Our purpose in this article is to consider new nonlinear operators in a Banach space forextending Nakajo and Takahashi’s result [29] in a Hilbert space to that in a Banach space.
In Section 3, we deal with a nonlinear operator in a Banach space called a relatively nonexpansive mapping which generalizes a nonexpansive mapping in a Hilbert space. We knowthat a relatively nonexpansive mapping in a Banach space is completely different from a nonexpansive mapping in a Banach space. In this section, we state a strong convergence theorem
for relatively nonexpansive mappings which generalizes Nakajo and Takahashi [29]. We also
obtain another $th\infty rem$ for relatively nonexpansive mappings which is connected with Reich’s
theorem [33].
In Section 4, we define another nonlinear operator in a Banach space which generalizes anonexpansive mapping in a Hilbert space. We call such a nonlinear operator a generalized
nonexpansive mapping. In this section, we obtain a strong convergence $th\infty rem$ which isrelated to Nakajo and Takahashi [29] and is different from the result in Section 3. Further, we
obtain a weak convergence theorem of Reich’s type. Finally, in Section 5, we prove a strong

convergence theorem for nonexpansive mappings in a Banach space which is closedly related
to Nakajo and Takahashi [29].

2Preliminaries

Let $E$ be a real Banach space with norm
and let $E$ “ denote the dual of $E$ . We denotethe value ofat $x\in E$ by. Whenis a sequence in $E$ , we denote the strong
convergence ofto $x\in E$ by $x_{n}arrow x$ and the weak convergence by $x_{n}arrow x$ . The modulus
of convexity of $E$ is defined by$||\cdot\Vert$$y^{*}\in E^{*}$$\{x_{n}\}$$\langle x, y^{*}\rangle$$\{x_{n}\}$$\delta$$\delta(\epsilon)=\inf\{1-\frac{||x+y\Vert}{2}$:$||x\Vert\leq 1,$ $\Vert y\Vert\leq 1,$ $||x-y\Vert\geq\epsilon\}$126

for every
every$\epsilon$$\epsilon>0$

with

. If

$0\leq\epsilon\leq 2$$E$. A Banach spaceis uniformly convex, then$\delta$

is said to be uniformly convex ifsatisfies that
and

$E$

$\delta(\epsilon)>0$

for$\delta(\epsilon/r)>0$$\Vert\frac{x+y}{2}\Vert\leq r(1-\delta(\frac{\epsilon}{r}))$. Let be a nonempty closed convex
andfor every $y\in E$ withsubset of a uniformly convex Banach space . Then we know that for any $x\in E$ , there exists
for all $y\in C$ . Putting $z=P_{C}(x)$ , we calla unique element $z\in C$ such thatis defined by
the metric projection of onto . The duality mapping from into$\Vert x\Vert\leq r,$

$x,$

$C$

$\Vert x-y\Vert\geq\epsilon$$\Vert y\Vert\leq r$$E$$\Vert x-z\Vert\leq\Vert x-y\Vert$
$C$

$E$

$P_{C}$$Jx=\{x‘ \in E" :

$E$

$J$

$2^{E^{*}}$\langle x, x^{*}\rangle=\Vert x\Vert^{2}=||x^{*}\Vert^{2}\}$for every $x\in E$ . Let $U=\{x\in E : \Vert x\Vert=1\}$ . The norm of $E$ is said to be G\^ateauxdifferentiable if for each $x,$ $y\in U$ , the limit(2.1)$\lim_{tarrow 0}\frac{||x+ty\Vert-\Vert x\Vert}{t}$is called smooth. The norm ofis said to be uniformly G\^ateaux
exists. In the case,differentiable if for each $y\in U$ , the limit (2.1) is attained uniformly for $x\in U$ . It is also
said to be Fr\’echet differentiable if for each $x\in U$ , the limit (2.1) is attained uniformly for$y\in U$ . A Banach spaceis called uniformly smooth if the limit (2.1) is attained uniformly
for $y\in U$ . It is known that if the norm of is uniformly G\^ateaux differentiable, then the
duality mapping is single valued and uniformly norm to weak’ continuous on each bounded
be a smooth Banach space. Let
be asubset of . We know the following result: Let
nonempty closed convex subset of and $x_{1}\in E$ . Then, $x_{0}=P_{C}(x_{1})$ if and only if.
$E$

$E$

$E$

$E$

$x,$

$J$

$E$

$E$

$C$

$E$

$\langle x_{0}-y, J(x_{1}-x_{0})\rangle\geq 0$$foral1y\in C,$ whereJ is the duality maPping of E.

A Banach space $E$ is said to satisfy Opial’s condition [31] if for any sequence$x_{n}arrow y$ implies$\{x_{n}\}\subset E$,$\lim_{narrow}\inf_{\infty}\Vert x_{n}-y\Vert<\lim_{narrow}\inf_{\infty}\Vert x_{n}-z\Vert$

for all $z\in E$ with $z\neq y$ . A Hilbert space satisfies Opial’s condition.
Letbe a closed convex subset of $E$ . A mapping $T:Carrow E$ is said to be nonexpansive if
for all $y\in C$ . We denote the set of all fixed points of by $F(T)$ . Let
1 Tx-Ty$D$ be a subset ofand let be a mapping of into $D$ . Then is said to be sunny if$C$$\Vert\leq\Vert x-y\Vert$

$T$

$x,$

$C$

$P$

$C$

$P$

$P(Px+t(x-Px))=Px$
mapping $P$ of $C$ into $C$ is said to be awhenever $Px+t(x-Px)\in C$ for $x\in C$ and $t\geq 0.$$P^{2}=P$.
of the convex hull of $D$ by. We denote the closure
retraction if“$D(A)=\{z\inE:
Az\neq\emptyset\}$and range
A multi-valued operator $A:Earrow E$ with domain$R(A)=\cup\{Az:z\in D(A)\}$ is said to be monotone if $\langle x_{1}-x_{2}, y_{1}-y_{2}\rangle\geq 0foreachx_{1}\in D(A)$and $y_{i}\in Ax_{i},$ $i=1,2$ A monotone operator is said to be maximal if its graph $G(A)=$$\{(x, y) : y\in Ax\}$ is not properly contained in the graph of any other monotone operator. The
following theorems are well known; for instance, see [46].
$A$$\overline{c}7D$.

$A$

Theorem 2.1. Let $E$ be a reflexive, stri ctly convex and smooth Banach space and let $A:Earrow$be a monotone operator. Then is maximal if and only if $R(J+rA)=E^{*}for$ all $r>0$ .
$2^{E}$$A$

127

Theorem 2.2. Let$\langle x-y, Jx-Jy\rangle=0$$E$

be a stntctly convex and smooth Banach space and let $x,$ $y\in E.$, then $x=y$ .

If

A duality mapping of a smooth Banach space is said to be weakly sequentially continuous
if $x_{n}arrow x$ implies that $Jx_{n}arrow*Jx,$ $wherearrow*means$ the weak’ convergence.
$J$

3 Relatively nonexpansive mappings
In this section, we first deal with a strong convergence theorem in a Banach space which
generalizes Nakajo and Takahashi’s thmrem $(Th\infty rem1.4)$ in a Hilbert space.
Let $E$ be a reflexive, strictly convex and smooth Banach space. The function $\phi:ExEarrow$is defined by$(-\infty, \infty)$$\phi(x, y)=\Vert x\Vert^{2}-2(x,$ $Jy\rangle$$+\Vert y\Vert^{2}$

for $y\in E$ , where is the duality mapping of $E$ ; see [1] and [18]. Let
be a nonempty
closed convex subset of $E$ and let $x\in E$ . Then there exists a unique element $x_{0}\in C$ such that
$J$

$x,$

$C$

$\phi(x_{0}, x)=\inf\{\phi(z, x) :z\in C\}$

.

(3.1)

Now, we define the mappingof $E$ onto by $Q_{C}x=x_{0}$ , whereis defined by (3.1). Suchis called the generalized projection of $E$ onto . It is easy to see that in a Hilbert space,
the mapping
is coincident with the metric projection.
$C$$Q_{C}$$x_{0}$$Q_{C}$$C$$Q_{C}$

Lemma 3.1. Let $E$ be a smooth Banach space, let be a nonempty closed convex subset of$E$ ,let $x\in E$ and let $x_{0}\in C$ . Then, the following (1) and (2) are equivalent:(1) $\phi(x_{0},x)=\min_{y\in C}\phi(y,x)$ ;(2) ( $x_{0}-y$ , Jx–Jx
for all $y\in C$ .
$C$

$0\rangle$$\geq 0$

Let $E$ be a smooth Banach space. Let
be a closed convex subset of $E$ , and let be a$C$mapping from into itself. We denote by $F(T)$ the set of fixed points of . A point in issaid to be an asymptotic fixed point of $T[36]$ if $C$ contains a sequencewhich converges
weakly to and the strong $\lim_{narrow\infty}(x_{n}-Tx_{n})=0$ . The set of asymptotic fixed points ofwill be denoted by. A mapping from into itself is called relatively nonexpansive if$\hat{F}(T)=F(T)$ and $\phi(p, Tx)\leq\phi(p,x)$ for all $x\in C$ and $p\in F(T)$ .
The following is a strong convergence theorem for relatively nonexpansive mappings in a
Banach space which generalizes Nakajo and Takahashi’s theorem [29] in a Hilbert space.
$C$

$T$

$T$

$C$

$p$

$\{x_{n}\}$

$T$

$p$

$\hat{F}(T)$

$T$

$C$

Theorem 3.2 (Matsushita and $Ih1_{B}hash\ddagger[26]$ ). Let $E$ be a unifomly convex and unifornly smooth Banach space, letbe a nonempty closed convex subset of $E$ , let
be arelatively $none\varphi ansive$ mapping ftom into itself with $F(T)\neq\phi$ and let
be a sequence. Suppose thatis given byof real numbers such that $0\leq\alpha_{n}<1$ and
$C$

$T$

$C$

$\{\alpha_{n}\}$$\lim\sup_{narrow\infty}\alpha_{n}<1$$\{x_{n}\}$$\{\begin{array}{l}x_{1}=x\in Cy_{n}=J^{-1}(\alpha_{n}Jx_{n}+(1-\alpha_{n})JTx_{n})H_{n}=\{z\in C : \phi(z,y_{n})\leq\phi(z,x_{n})\}W_{n}=\{z\in C;\langle x_{n}-z, Jx-Jx_{n})\geq 0\}x_{n+1}=Q_{H_{n}\cap W_{n}^{X}}\end{array}$for all $n=1,2,$$Q_{F(T)}x$, where$\ldots$, where$Q_{F(T)}$is the duality mapping on E. Thenis the generalized projection ffom $C$ onto $F(T)$ .
$J$$\{x_{n}\}$converges strvngly to128

Using Theorem 3.2, we can prove Nakajo and Takahashi’s theorem (Theorem 1.4) as follows:
To show Nakajo and Takahashi’s theorem, it is sufficient to prove that if is nonexpansive,$T$then is relatively nonexpansive. It is obvious that
. If $u\in\hat{F}(T)$ , then there$x_{n}-Tx_{n}arrow0$
that
existssuch
and. Since is nonexpansive, isdemiclosed. So, we have $u=Tu$ . This implies $F(T)=\hat{F}(T)$ . Further, in a Hilbert space $H$ ,we know that
$F(T)\subset\hat{F}(T)$$T$$\{x_{n}\}\subset C$$T$$x_{n}-\Delta u$$T$$\phi(x, y)=\Vert x-y\Vert^{2}$for every $x,y\in H$ . So, $\Vert Tx-Ty\Vert\leq\Vert x-y||$ is equivalent to $\phi(Tx, Ty)\leq\phi(x, y)$ . Therefore,
is relatively nonexpansive. Using $Th\infty rem3.2$ , we obtain the desired result.
Using Theorem 3.2, we can prove a strong convergence threorem for maximal monotone
operators in a Banach space. Before stating the theorem, we define the following resolventsfor maximal monotone operators in a Banach space. Let
be a reflexive, strictly convex
and smooth Banach space, and let be a maximal monotone operator from $E$ to $E$ “. Using$Th\infty rem2.1$ and the strict convexity of, we obtain that for every $r>0$ and $x\in E$ , there$x_{r}\inD(A)$exists a uniquesuch that
$T$

$E$

$A$

$E$

$Jx\in Jx_{r}+rAx_{r}$

.

(3.2)

If $Q_{r}x=x_{r}$ , then we can define a single valued mapping: $Earrow D(A)$ by $Q_{r}=(J+rA)^{-1}J$
and such
is called the relative resolvent of . We know that $A^{-1}0=F(Q_{r})$ for all $r>0$ ;see $[45, 46]$ for more details.
$Q_{r}$$A$$Q_{r}$

Theorem 3.3. Let $E$ be a uniformly convex and uniformly smooth Banach space, let be a
be the relative resolvent of , where $r>0$ .
maximal monotone operator Jbvm $E$ to $E$ “, let$A^{-1}0$$none\varphi
ansive$ mapping on $E$ .
is a relatively
is nonempty, then
If
$A$

$A$

$Q_{r}$$Q_{r}$

Using this result and Theorem 3.2, we prove a strong convergence $th\infty rem$ for relative
resolvents of maximal monotone operators in a Banach space.
Theorem 3.4. Let $E$ be a uniformly convex and uniformly smooth Banach space, let be a
, let
be the relative resolvent of , where $r>0$
maximal monotone operator from $E$ to.
and letbe a sequence of red numbers such that $0\leq\alpha_{n}<1$ andSuppose thatis given by.
$A$$E^{*}$$A$$Q_{r}$$\lim\sup_{narrow\infty}\alpha_{n}<1$$\{\alpha_{n}\}$$\{x_{n}\}$$\{\begin{array}{l}x_{1}=x\in Ey_{n}=J^{-1}(\alpha_{n}Jx_{n}+(1-\alpha_{n})JQ_{r}x_{n})H_{n}=\{z\in E : \phi(z,y_{n})\leq\phi(z,x_{n})\}W_{n}=\{z\in E:\langle x_{n}-z, Jx-Jx_{n}\rangle\geq 0\}x_{n+1}=Q_{H_{n}\cap W_{n^{X}}}\end{array}$, whereis the duality mapping on E. If $A^{-1}0$ is nonempty, thenconverges strongly to $Q_{A^{-1}0}x$ , whereis the genera lized projection from $E$ onto $A^{-1}0$ .

for all $n=1,2,$$\ldots$$J$$\{x_{n}\}$$Q_{A^{-1}0}$

Next, we obtain a weak convergence $th\infty rem$ for relatively nonexpansive mappings in a
Banach space which is connected with Reich [33], Browder and Petryshyn’s theorem [6] andRockafellar’s $th\infty rem[39]$ . Before proving it, we need the following proposition.

Proposition 3.5 (Matsushita and Thkahashi [25]). Let $E$ be a uniformly convex andbe a nonempty closed convex subset of $E$ , and let
uniformly smooth Banach space, let. Let
be a relatively nonexpansive mapping frvm
into itself such that
be a
$C$

$T$

$C$

$F(T)\neq\emptyset$$\{\alpha_{n}\}$129

sequence of real numbers such that $0\leq\alpha_{n}\leq 1$ . Let $x_{1}\in C$ and let$\{x_{n}\}$defined by

be the sequence$x_{n+1}=Q_{C}J^{-1}(\alpha_{n}Jx_{n}+(1-\alpha_{n})JTx_{n})$. Then $\{Q_{F(T)}x_{n}\}$ converyes strongly to a fixed point of , where $Q_{F(T)}$ is thegeneralized projection ftom $C$ onto $F(T)$ .

for $n=1,2,$$T$$\ldots.$

Using Proposition 3.5, we can prove the following weak convergence $th\infty rem$ .

Theorem 3.6 (Matsushita and Takahashi [25]). Let $E$ be a uniformly convex and uniformly smooth Banach space, let be a nonempty closed convex subset of $E$ , and let be a
relatively nonexpansive mapping ffom into itself such that. Let
be a sequence
real
numbers
such
that
of
$C$

$T$

$C$

$0\leq\alpha_{n}\leq 1$

Let $x_{1}\in C$ and let$\{x_{n}\}$$F(T)\neq\emptyset$

and$\lim_{narrow}\inf_{\infty}\alpha_{n}(1-\alpha_{n})>0$$\{\alpha_{n}\}$.

be the sequence defined by$x_{n+1}=Q_{C}J^{-1}(\alpha_{n}Jx_{n}+(1-\alpha_{n})JTx_{n})$for $n=1,2,$. If is weakly sequentially continuous, thenconverges weakly towhere $u= \lim_{narrow\infty}Q_{F(T)}x_{n}$ and $Q_{F(T)}$ is the generdized projection flom onto $F(T)$$\ldots$$J$$\{x_{n}\}$

$u$

,

$C$

Using Theorem 3.6, we can prove the following two weak convergence theorems.

Theorem 3.7 ([6]). Let be a nonempty closed convex subset of a Hilbert space $H,$
be a nonexpansive mapping fivm into itself such thatand let be a real number
such that $0<\lambda<1$ . Let $x_{1}\in C$ and let
be the sequence defined by$C$$\cdot let$$C$

$F(T)\neq\emptyset$$T$$\lambda$$\{x_{n}\}$$x_{n+1}=\lambda x_{n}+(1-\lambda)Tx_{n}$for $n=1,2,$ . Thenthe metric projection fivm$\{x_{n}\}$$\ldots$$C$

converges weakly to , where $u= \lim_{narrow\infty}P_{F(T)}x_{n}$ and $P_{F(T)}$ isonto $F(T)$$u$

Theorem 3.8. Let $E$ be a uniformly convex and uniformly smooth Banach space, let be a
maximal monotone operator fiom $E$ to, letsuch thatbe the relative resolvent
,
A
$r>0$
where
sequence
and
let
be
a
of
of real numbers such that$A$

$E^{*}$$A^{-1}0\neq\emptyset$$Q_{r}$$\{\alpha_{n}\}$$0\leq\alpha_{n}\leq 1$

Let $x_{1}\in E$ and let$\{x_{n}\}$and$\lim_{narrow}\inf_{\infty}\alpha_{n}(1-\alpha_{n})>0$.

be the sequence defined by$x_{n+1}=J^{-1}(\alpha_{n}Jx_{n}+(1-\alpha_{n})JQ_{r}x_{n})$for $n=1,2,$ . If is weakly sequentially continuous, thenconverges weakly toin $A^{-1}0$ , where $u= \lim_{narrow\infty}Q_{A^{-1}0}x_{n}$ andis the generalized projection fivm $E$ onto$\ldots$$A^{-1}0$.

$J$$\{x_{n}\}$$u$$Q_{A^{-1}0}$

Kamimura and Takahashi [18] extended Solodov and Svaiter’s result [41] to the following
by using Lemma 3.1 and the resolvents defined by (3.2).

$th\infty rem$

130

Theorem 3.9 ([18]). Let $E$ be a uniformly convex and uniformly smooth Banach space and let
be a maximal monotone operator from $E$ into $E^{*}such$ that $A^{-1}0\neq\phi$ . Let $Q_{r}=(J+rA)^{-1}J$
be a sequence generated byfor all $r>0$ and let$A$$\{x_{n}\}$$\{\begin{array}{l}x_{1}\in Ey_{n}=Q_{r_{n}}x_{n}H_{n}=\{z\in E:\langle z-y_{n}, Jx_{n}-Jy_{n}\rangle\leq 0\}W_{n}=\{z\in E:\langle z-x_{n}, Jx_{1}-Jx_{n}\rangle\leq 0\}x_{n+1}=Q_{H_{n}\cap W_{n}}x_{1},n=1,2,\ldots\end{array}$is a sequence of positive numbers such that
. Then,, whereis the generalized projection of $E$ onto $A^{-1}0$ .
convefges strongly towhere$\lim\inf_{narrow\infty}r_{\mathfrak{n}}>0$$\{r_{n}\}$$Q_{A^{-1}0}x_{1}$$\{x_{n}\}$$Q_{A^{-1}0}$

Kamimura, Kohsaka and Takahashi [15] also proved a weak convergence theorem of Mann’s
type for maximal monotone operators in a Banach space. Before stating the theorem, we need
the following strong convergence thmrem.

Theorem 3.10 ([15]). Let $E$ be a smooth and uniformly convex Banach space. Let$ExE^{*}$ be a maximal monotone operator such that $A^{-1}0$ is nonempty, let $Q_{r}=(J+rA)^{-1}J$
be the generalized projection of $E$ onto $A^{-1}0$ . Let
be afor all $r>0$ and letsequence defined as follows: $x_{1}=x\in E$ and$A\subset$$Q_{A^{-1}0}$$\{x_{n}\}$$x_{n+1}=J^{-1}(\alpha_{n}J(x_{n})+(1-\alpha_{n})J(Q_{r_{n}}x_{n}))$,$n=1,2,$$\ldots$. Then, the sequence
where
and
to an element of $A^{-1}0$ , which is a unique element $v\in A^{-1}0$ such that$\{r_{n}\}\subset(0, \infty)$$\{\alpha_{n}\}\subset[0,1]$$\{Q_{A^{-1}0}(x_{n})\}$,
converges strongly.

$\lim_{narrow\infty}\phi(v, x_{n})=\underline{\min_{1\nu\in A0}}\lim_{narrow\infty}\phi(y, x_{n})$Using Theorem 3.10, we can prove the following theorem in a Banach space which generalizes
the results of Rockafellar [39] and Kamimura and Takahashi [16] in a Hilbert space.
Theorem 3.11 ([15]). Let $E$ be a smooth and unifomly convex Banach space whose duality
mapping
is weakly sequentially continuous. Let $A\subset ExE^{*}$ be a maximal monotone
operator such that $A^{-1}0$ is nonempty, let $Q_{r}=(J+rA)^{-1}J$ for all $r>0$ and let
be the$A^{-1}0$$E$$x_{1}=x\inE$. Let
be a sequence defined as follows:
generalized projection of ontoand
$x_{n+1}=J^{-1}(\alpha_{n}J(x_{n})+(1-\alpha_{n})J(Q_{r_{n}}x_{n}))$ ,,$n=1,2,$$J$$Q_{A^{-1}0}$$\{x_{n}\}$$\ldots$

where$\{\alpha_{n}\}\subset[0,1]$

and$\{r_{n}\}\subset(0, \infty)$

satisfy$\lim_{narrow}\sup_{\infty}\alpha_{n}<1$Then,$\{x_{n}\}$converges weakly to an element

and
$v$

of

$\lim_{narrow}\inf_{\infty}r_{n}>0$$A^{-1}0$.

, where $v= \lim_{narrow\infty}Q_{A^{-1}0}(x_{n})$ .

As a direct consequence of $Th\infty rem3.11$ , we obtain the following:
Theorem 3.12. Let $E$ be a smooth and uniformly convex Banach space whose duality mapping
is weakly sequentially continuous. Let $A\subset ExE^{*}$ be a maximal monotone operator such
$J$

that $A^{-1}0$ is nonempty, let $Q_{r}=(J+rA)^{-1}J$ for all $r>0$ and let
be the generalized
be a sequence defined as follows: $x_{1}=x\in E$ andprojection of $E$ onto $A^{-1}0$ . Let$Q_{A^{-1}0}$$\{x_{n}\}$$x_{n+1}=Q_{r_{n}}x_{n}$,$n=1,2,$$\ldots$

,

131

where
to an element$\{r_{n}\}\subset(0, \infty)$$v$

ofsatisfies$A^{-1}0$$\lim\inf_{narrow\infty}r_{n}>0$. Then, the sequence$\{x_{n}\}$, where $v= \lim_{narrow\infty}Q_{A0}-1(x_{n})$ .

converges weakly

Problem. If andare uniformly convex Banach spaces, does Theorem 3.12 hold without
assumming that is weakly sequentially continuous ?
$E$$E^{*}$$J$

4 Generalized nonexpansive mappings
Let $E$ be a smooth Banach space and let $D$ be a nonempty closed convex subset of $E$ . A
mapping $R:Darrow D$ is called generalized nonexpansive if
and$F(R)\neq\emptyset$$\phi(Rx, y)\leq\phi(x, y)$,$\forall x\in D,\forall y\in F(R)$,where $F(R)$ is the set of fixed points of . A point inis said to be a generalized
asymptotic fixed point of $T[13]$ if contains a sequence
Jp and thesuch that$\lim_{\mathfrak{n}arrow\infty}(Jx_{n}-JTx_{n})=0$
strong. The set of generalized asymptotic fixed points of will be.
denoted by
Let $E$ be a reflexive and smooth Banach space and let B C
be a maximal monotone
operator. For each
and $x\in E$ , consider the set
$R$

$C$

$p$

$C$

$Jx_{n}arrow*$$\{x_{n}\}$$T$$\check{F}(T)$$E^{*}\cross E$$\lambda>0$$R_{\lambda}x:=\{z\in E:x\in z+\lambda BJ(z)\}$.

consists of one point. We also denote the domain and the range ofby
and $R(R_{\lambda})=D(BJ)$ , respectively. Suchis called the generalized resolvent of$B$
and is denoted by

Then$R_{\lambda}x$$R_{\lambda}$$R(I+\lambda BJ)$

$D(R_{\lambda})=$$R_{\lambda}$$R_{\lambda}=(I+\lambda BJ)^{-1}$We get some properties of$R_{\lambda}$.

and $(BJ)^{-1}0$ .

Proposition 4.1 ([12]). Let $E$ be a reflexive and strictly convex Banach space with a }. Thendifferentiable nom and let B C $E^{*}xE$ be a macimal monotone operator withthe following hold:$I$$\cdot\acute{e}chet$$B^{-1}0\neq\emptyset$

1. $D(R_{\lambda})=E$ for each;
2. $(BJ)^{-1}0=F(R_{\lambda})$ for each
, whereS. $(BJ)^{-1}0$ is closed;
is generalized $none\varphi ansive$ for each4.
$\lambda>0$$\lambda>0$$R_{\lambda}$$F(R_{\lambda})$$\lambda>0$

is the set of fixed points of$R_{\lambda}$;.

Proposition 4.2 ([13]). Let $E$ be a smooth and uniformly convex Banach space, let$E^{*}xE$ be a maximal monotone operator with, and letbe the generalized resolvent. Then.
of for$B\subset$$B^{-1}0\neq\emptyset$$B$$\lambda>0$$R_{\lambda}$$\check{F}(R_{\lambda})=F(R_{\lambda})$Next, we get the following result for generalized nonexpansive mappings.

space E. Let
if and only if$R_{C}$. Let

be a nonempty closed subset of a smooth and strictly convex Banachbe a retraction of onto C. Thenis sunny and generalized noneapansive$Prop_{O8}ition4.3$

$C$

$E$

$R_{C}$$\langle x-R_{C}x, J(R_{C}x)-J(y)\rangle\geq 0$

for each $x\in E$ and $y\in C$ .

132

Let $E$ be a smooth and strictly convex Banach space and let $C$ be a nonempty closed subset
of $E$ . Then, a sunny generalized nonexpansive retraction of $E$ onto $C$ is unique. In fact, let$R,$be two sunny generalized nonexpansive retractions of $E$ onto . Then, by Proposition
4.3, for each $x\in E$ , we have
$C$

$S$

\langle x--Rx, $J(Rx)-J(y)\rangle$$\geq 0$, (x–Sx, $J(Sx)-J(y)\rangle$$\geq 0,$$\forall y\in C$.

From $Rx,$ $Sx\in C$ , we get\langle x--Rx, $J(Rx)-J(Sx)\rangle$$\geq 0$, \langle x--Sx, $J(Sx)-J(Rx)\rangle$$\geq 0$.

From these inequalities, we have\langle Sx--Rx, $J(Rx)-J(Sx)\rangle$$\geq 0$.

Since is strictly convex, we get $Sx=Rx$ .
Before showing an example of sunny generalized nonexpansive retractions, we recall the$E$$f_{0}nowing$

theorem.

be a mamimal monotone
Theorem 4.4 ([34]). Let $E$ be a Banach space and let. If is stnctly convex and has a Fre’chet differentiable norm. Then,
operator withexists and belongs to $A^{-1}0$ .
for each $x\in E,$$A\subset E\cross E^{*}$$A^{-1}0\neq\emptyset$$E^{*}$$\lim_{\lambdaarrow\infty}(J+\lambda A)^{-1}J(x)$Using Theorem 4.4, we get the following result.
Theorem 4.5 ([12]). Let $E$ be a unifomly convex Banach space with a P\dagger \’echet differentiable. Then the following
nom and let B C $E^{*}xE$ be a maximal monotone operator withhold:$\cdot$$B^{-1}0\neq\emptyset$

1. For each $x\in E,$
2. If $Rx$$\lim_{\lambdaarrow\infty}R_{\lambda}xe$$:= \lim_{\lambdaarrow\infty}R_{\lambda}xfor$

retraction of

$E$

onto

ntsts and belongs to $(BJ)^{-1}0$ ;each $x\in E$ , then$(BJ)^{-1}0$.

$R$

is a sunny generalized $none_{W^{nsive}}$

Next, we discuss proximal point algorithms for generalized resolvents of a maximal monotone
operator B C $E^{*}\cross E$ . We start with the following lemma. Compare this lemma with theresults in Kamimura and Takahashi [18], and Kohsaka and Takahashi [20].

Lemma 4.6. Let $E$ be a reflexive, strictly convex, and smooth Banach space, let B C $E^{*}xE$, and $R_{r}=(I+rBJ)^{-1}$ for all $r>0$ . Thenbe a maximal monotone operator with$B^{-1}0\neq\emptyset$$\phi(x, R_{r}x)+\phi(R_{r}x, u)\leq\phi(x,u)$

for all $r>0,$ $u\in(BJ)^{-1}0$ , and $x\in E$ .
The following is a strong convergence theorem for generalized nonexpansive mappinga in a
Banach space which is related to Nakajo and Talrahashi’s theorem [29] in a Hilbert space.
be a unifomly convex and uniTheorem 4.7 (Ibaraki and $Ih]ahashi[13]$ ). Let
be a generalized nonexpansive mapping fhom $E$ into itfomly smooth Banach space, letbe a sequence of real numbers such that $0\leq\alpha_{n}<1$ andself with $F(T)\neq\phi$ and let
is given by. Suppose that
$E$

$T$

$\{\alpha_{n}\}$$\lim_{8}up_{narrow\infty}\alpha_{n}<1$$\{x_{n}\}$$\{\begin{array}{l}x_{\sim}=x\in Ey_{n}=\alpha_{n}x_{n}+(1-\alpha_{\mathfrak{n}})Tx_{n})H_{n}=\{z\in E : \phi(z, y_{n})\leq\phi(z,x_{n})\}W_{n}=\{z\in E;\langle x_{n}-z, Jx-Jx_{n}\rangle\geq 0\}x_{n+1}=R_{H_{n}\cap W_{n}^{X}}\end{array}$

133

for all $n=1,2,$, whereis the duality mapping on E. If $\check{F}(T)=F(T)$ , thenconverges strongly to $R_{F(T)}x$ , where $R_{F(T)}$ is the sunny generalized nonexpansive retraction
from $C$ onto $F(T)$ .
$\ldots$$J$$\{x_{n}\}$We can also prove the following weak convergence theorem, which is a generalization of
Kamimura and Takahashi’s weak convergence theorem (Theorem 1.2).

Theorem 4.8. Let $E$ be a smooth and uniformly convex Banach space whose duality mapping
is weakly sequentially continuous. Let $B\subset E^{*}\cross E$ be a maximal monotone operator, let$R_{r}=(I+rBJ)^{-1}$ for all $r>0$ and let
be a sequence defined as follows: $x_{1}=x\in E$ and$J$$\{x_{n}\}$$x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})R_{r_{n}}x_{n}$

where

and

$\{\alpha_{n}\}\subset[0,1]$$\{r_{n}\}\subset(0, \infty)$$B^{-1}0\neq\emptyset$, then the sequence$\{x_{n}\}$$n=1,2,$$\ldots$,satisfy$\lim_{narrow}\sup_{\infty}\alpha_{n}<1$

If

,

and

$\lim_{narrow}\inf_{\infty}r_{n}>0$.

converg es weakly to an element of $(BJ)^{-1}0$ .

5 Concluding remarks
Recently, Matsushita and Takahashi [27] proved the following strong convergence theorem
for nonexpansive mappings in a Banach space which is related to Nakajo and Takahashi’s
theorem [29].

Theorem 5.1 (Matsushita and $Ib ahash\ddagger[27]$ ). Let $E$ be a unifomly convex and smooth
Banach space, let be a nonempty bounded closed convex subset of and let be a nonexpansive mapping fiom
into itself. Let
be a sequence in defined by
$C$

$E$

$C$

$\{x_{n}\}$

$T$

$C$

$\{\begin{array}{l}x_{1}=x\in CC_{n}=\overline{co}\{z\in C : \Vert z-y_{n}\Vert\leq\Vert z-x_{n}||\}D_{n}=\{z\in C:(x_{n}-z, Jx-Jx_{n}\rangle\leq 0\}x_{n+1}=P_{C_{n}\cap D_{n}}x\end{array}$for all $n=1,2,$, whereis the metric projection $fmmE$ ontoandisa sequence in $(0,1)$ with. Thenconverges strongly to the element $P_{F(T)}x$ , where$P_{F(T)}$ is the the metnc projection
ftom $E$ onto $F(T)$ .
For the proof of Theorem 5.1, Matsushita and Takahashi [27] used essentially the following
Bruck’s theorem [7]:$\ldots$$C_{n}\cap D_{n}$$P_{C_{n}\cap D_{n}}$$t_{n}arrow 0$$\{t_{n}\}$$\{x_{n}\}$Theorem 5.2 (Bruck [7]). Let be a closed convex subset of a unifomly convex Banach
space E. Then for each $r>0$ , there exists a strictly increasing convex continuous function$C$$\lambda:[0, \infty)arrow[0, \infty)$

such that $\lambda(0)=0$ and$\lambda(\Vert\tau(\sum_{j=0}^{n}\lambda_{j}x_{j})-\sum_{j=0}^{n}\lambda_{j}Tx_{j}\Vert)\leq 0\leq j<k\leq n\max(||x_{j}-x_{k}\Vert-||Tx_{j}-Tx_{k}||)$

for all $n\in N,$;
and $T\in Lip(C, 1)$ , whereand}, $B_{r}=\{z\in E : ||z\Vert\leq r\}$ and Lip $(C, 1)$ is the set of all
nonezp ansive mappings of $C$ into $E$ .

$0\leq\lambda_{j}$$\{\lambda_{j}\}\in\Delta_{f}^{n}\{x_{j}\}\subset C\cap B_{r}$$\sum_{j=0}^{n}\lambda_{j}=1$$\Delta^{n}=\{\{\lambda_{0}, \lambda_{1}, \ldots, \lambda_{n}\}$134

Problem. Can we prove Thrrem 5.1 under assuming thatof $E$ and $T:Carrow C$ is a nonexpansive mapping with$C$

$F(T)\neq\emptyset$

is a closed and convex subset?

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