Banach m- (Shigeru Iemoto), (Watalu Takahashi) (Department of Mathematical and Computing Sciences, Tokyo Institute of Technology) 1

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1 Banach m- (Shigeru Iemoto) (Watalu Takahashi) (Department of Mathematical and Computing Sciences Tokyo Institute of Technology) 1 $H$ Hilbert $gg_{1}g_{2}$ $g_{m}$ : $Harrow R$ $C=\{x\in H : \mathit{9}:(x)\leq 0(i=12 \ldots m)\}$ $g(u)=1\mathrm{n}\mathrm{i}\mathrm{n}g(x)x\in C$ $u\in C$ $C$ $f(x)=\{$ $\mathit{9}(x)$ $\infty$ $(x\in C)$ $(x\not\in C)$ : $f$ $Harrow(-\infty \infty]$ $x\in H$ $\partial f(x)=\{z\in H : f(x)+\langle y-x z\rangle\leq f(y)(^{\forall}y\in H)\}$ $H$ $H$ $f$ $u\in H$ $\partial f$ $f(u)= \min_{x\epsilon H}f(x)$ $(xy)$ $(sb)\in\partial f$ $\partial f$ $(x-s y-t)\geq 0$ Rockafellar [25] $\partial f$ $\partial f$ $f(u)=1\mathrm{n}\mathrm{i}\mathrm{n}_{x\in H}f(x)$ $\mathrm{o}\in\partial f(u)$ $A\subset H\mathrm{x}H$ $0\in Au$

2 32 $u\in H$ $A\subset H\mathrm{x}H$ $\lambda>0$ $x\in H$ $J_{\lambda}(x)=\{z\in H : x\in z+\lambda A_{\tilde{\wedge}}\}$ $J_{\lambda}$ $H$ $H$ (cf [29 30]) $A$ \mbox{\boldmath $\lambda$} $x$ $y\in H$ $ J_{\lambda^{X-}}J_{\lambda y} \leq x-y $ $0\in Au$ Hilbert $u=j_{\lambda u}$ $A\subset H\mathrm{x}H$ $\mathrm{o}\in Au$ $u\in H$ Rockafellar [26] (Proximal Point Algorithm) $x_{0}=x\in H$ $\{x_{n}\}$ $\{\lambda_{n}\}\subset(0 \infty)$ $\lambda_{n}a)^{-1}$ 1976 $x_{n+1}=j_{\lambda}x_{n}$ $n=012$ $\ldots$ (1) $\lambda_{n}>0$ \mbox{\boldmath $\lambda$} $=(I+$ Rpckafellar [26] liln $\inf_{narrow\infty}\lambda_{n}>0$ $A^{-1}0\neq\emptyset$ (1) $\{x_{n}\}$ $A^{-1}0$ Br\ ezis-lions [1] Lions [13] Passty [16] G\"uler [4] Solodov-Svaiter [28] Hilbert [8] $x0=x\in H$ $\{x_{n}\}$ $x_{\iota+1}=\alpha_{n}x+(1-\alpha_{n})j_{\lambda_{\iota}}x_{n)}$ $n=012$ $\cdots$ (2) $x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})j_{\lambda_{\iota}}x_{n}$ $n=012$ (3) $\cdots$ $x_{0}=x\in H$ $\{\alpha_{n}\}$ $[01]$ $\{\lambda_{n}\}$ $(0 \infty)$ (2) $\{x_{n}\}$ $A^{-1}0$ (3) $\{x_{n}\}$ $A^{-1}0$ - [9] Hilbert Banach Hilbert $A\subset H\mathrm{x}H$

3 33 Banach Banach $E$ $A\subset E\mathrm{x}E$ $(x y)$ $(s b)\in A$ $j\in J(x-s)$ $\langle y-tj\rangle\geq 0$ $B\subset ExE^{*}$ $(x x^{*}))(y y^{*})\in B$ $\langle x-y x^{*}-y^{*}\rangle\geq 0$ $J$ $E$ $E^{*}\text{}$ - $[8 9]$ $x_{0}=x\in E$ $\{x_{n}\}$ $x_{n+1}=a_{n}x+\beta_{n}x_{n}+\gamma_{n^{j_{\lambda}}}$ $x_{n}+e_{n}$ $n=012$ $\cdots$ (4) $\{\lambda_{n}\}\subset(0 \infty)$ $\{\alpha_{n}\}$ $\{\beta_{n}\}$ $\{\gamma_{n}\}$ $\alpha_{n}+\beta_{n}+\gamma_{n}=1$ $A^{-1}0$ {en} $E$ $[01]$ Hilbert Banach Hilbert 4 Barxach $\mathrm{m}$- 5 2 $R$ $H$ Hilbert Banach $E$ $J$ $x\in E$ $E$ $N$ $J(x)$ $=\{x^{*}\in E^{*} : \langle xx^{*}\rangle= x ^{2}= x^{*} ^{2}\}$ $E$ $E^{*}$ $J(\mathrm{O})=\{0\}$ Hahn-Banach $x\in E$ $J(x)\neq\emptyset$ $E=H$ $J$ $I$ $A\subset ExE$ $(xy)$ $(st)\in A$ $j\in J(x-s)$ $(y-tj)\geq 0$ m- A\subset ExE $R(I+\lambda A)=E$ $\mathrm{v}_{\lambda>0}$ $D(A)=\{x\in E : Ax\neq\emptyset\}$ $A$ $R(A)=\cup\{Ax:x\in D(A)\}$ $A$ $A\subset E\mathrm{x}E$ m- $\lambda>0$ $x\in E$ $J_{\lambda}(x)=\{\approx\in E:x\in z+\lambda Az\}$

4 $E$ 34 $\text{_{}\lambda}$ $E$ $A$ $J_{\lambda}=(I+\lambda A)^{-1}$ $\mathrm{o}\in Au$ $u=j_{\lambda}u$ $A$ $A_{\lambda}= \frac{1}{\lambda}(i-j_{\lambda})$ $x\in E$ $(J_{\lambda^{X}} A_{\lambda^{X}})\in A$ $f$ : $Earrow(-\infty \infty]$ $f(a)\in R$ $a\in E$ $f$ $r\in R$ $\{x\in E:f(x)\leq r\}$ $E$ $f$ $x$ $y\in E$ $\alpha\in(01)$ $f(\alpha x+(1-\alpha)y)\leq\alpha f(x)+(1-a)f(y)$ $f$ : $Earrow(-\infty \infty]$ $x\in E$ $\partial f(x)=\{x^{*}\in E^{*} : f(x)+\langle y-xx^{\alpha})\leq f(y) \forall_{y}\in E\}$ $E$ $E^{*}$ $\partial f$ $f$ $E$ Banach $\epsilon\in[02]$ $\delta(\epsilon)=\inf\{1-\frac{ x+y }{2}$ : $ x \leq 1$ $ y \leq 1$ $ x-y \geq\epsilon\}$ modulus $\mathrm{a}\mathrm{l}$ $\delta$ $[02]$ $[01]$ $E$ $E$ $E$ $\{x_{n}\}$ {y $ x_{n} = y_{n} =1$ $narrow\infty 1\mathrm{i}\ln x_{n}+y_{n} =2$ $1\mathrm{i}_{\mathrm{l}}\mathrm{n}_{narrow\infty} x_{n}-y_{n} =0$ $\epsilon>0$ $\epsilon>0$ $E$ $\delta(\epsilon)>0$ $E$ $ x \leq r$ $ y \leq r$ $ x-y \geq\epsilon>0$ $ \frac{ x+y }{2}\leq^{J}r\{1-\delta(\frac{\epsilon}{r})\}$ $\delta(\frac{\epsilon}{r})>0$ Banach [29] Banadh $S(E)=\{x\in E: x =$ $1\}$ $x$ $y\in S(E)$ $\lim_{tarrow 0}\frac{ x+ty - x }{t}$ $(*)$ Banach $E$ G\^ateaux $S(E)$ $x$ $y$ $(*)$ $E$

5 35 $E$ G\^ateaux $y\in S(E)$ $(*)$ Banach $E$ Rr\ echet $x\in S(E)$ $x\in S(E)$ $(*)$ $y\in S(E)$ Banach $E$ $\mathrm{r}\cdot\text{\ {e}} \mathrm{c}\mathrm{h}\mathrm{e}\mathrm{t}$ $(*)$ $S(E)$ $x$ $y$ $E$ $E$ G\^ateaux $E$ $E$ $E^{*}$ $T$ * $C$ Banach $E$ } $T:Carrow C$ $F(T)$ $C$ $D$ $\approx\in D$ $C$ 2 $\sup_{y\in D} z-y <\sup_{xy\in D} x-y =\delta(d)$ Banach Banach $C$ $C$ Hilbert $C$ $C$ Banach C Banach E D C $D$ $P$ S sunny $x\in C$ $t\geq 0$ $C$ $Px+t(x-Px)\in C$ $P(Px+t(x-Px))=Px$ $C$ $D$ $P$ $x\in C$ $P^{2}x=Px$ $D$ $C$ sunny retract $C$ $D$ sunny sunny [29] 21 $C$ Banach $E$ $P$ $E$ $C$ $x\in E$ $P\mathrm{B}\sim unny$ $y\in C$ $\langle$x Px $J(y-Px))\leq 0$ $E$ $E^{*}$

6 36 3 Hilbert Hilbert Rockafellar [26] 31 (Rockafellar [26]) $H$ bert $A\subset H\mathrm{x}H$ $x_{0}=x\in H$ $\{x_{n}\}$ $x_{n+1}=j_{\lambdax_{n}}$ $n\in N$ $\{\lambda_{n}\}\subset(0 \infty)$ $\lim\inf_{narrow\infty}\lambda_{n}>0$ $A^{-1}0\neq_{-}\emptyset$ $\{x_{n}\}$ $A^{-1}0$ $u$ Br\ ezis-lions [1] Lions [13] Passty [16] Guler [4] Solodov-Svaiter [28] Hilbert - [8] 32 ( - [8]) $H$ Hilbe $A\subset H\mathrm{x}H$ $x_{0}=x\in H$ $\{x_{n}\}$ $y_{n}\approx J_{\lambda_{\iota}}x_{n}$ $x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})y_{n}$ $n\in N$ $ y_{n}-j_{\lambda_{n}x_{n}} \leq\delta_{n}$ $ l\in N$ $\{a_{n}\}\subset[01]$ $\{\lambda_{n}\}\subset$ $(0 \infty)$ $\{\delta_{n}\}\subset[0 \infty)$ $\lim_{narrow}\inf_{\infty}\alpha_{n}<1$ $\lim_{narrow}\inf_{\infty}\lambda_{n}>0\sum_{n=0}^{\infty}\delta_{n}<\infty$ $A^{-1}0\neq\emptyset$ $v= \lim_{narrow\infty}px_{n}$ $P$ $\{x_{n}\}$ $H$ $A^{-1}0$ $u$ $A^{-1}0$ $\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{k}\mathrm{a}\mathrm{f}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{a}\iota\cdot[26]$ Theorem 31 $\alpha_{n}\equiv 0$ 33 ( - [8]) $H$ Hilbert $A\subset H\mathrm{x}H$ $x_{0}=x\in H$ $\{x_{n}\}$ $y_{n}\approx J_{\lambda_{n}}x_{n}$ $x_{n+1}=0_{n} X+(1-a_{n})y_{n}$ $n\in N$

7 37 $ y_{n}-\text{_{}\lambda_{l}}x_{n} \leq\delta_{n}$ $(0\cdot\infty)$ $\{\delta_{n}\}\subset[0 \infty)$ $n\in N$ $\{\alpha_{n}\}\subset[01]$ $\{\lambda_{n}\}\subset$ $n arrow\infty 1\mathrm{i}\iota \mathrm{n}a_{n}=0\sum_{n=0}^{\infty}\alpha_{n}=\infty$ $n arrow\infty 1\mathrm{i}\ln\lambda_{n}=\infty\sum_{n=0}^{\infty}\delta_{n}<\infty$ $A^{-1}0\neq\emptyset$ $\{x_{n}\}$ $A^{-1}0$ $u$ $Px=u$ $P$ $H$ $A^{-1}0$ - 34 ( - [6]) $H$ $H$ ilbe $x0=x\in H$ $A\subset HxH$ $A^{-1}0\neq\emptyset$ $\{x_{n}\}$ $y_{n}\approx J_{\lambda_{n}X_{n}}$ $x_{n+1}=\alpha_{n}x+\beta_{n}x_{n}+\gamma_{n}y_{n}$ $fx\in N$ $ y_{n}-j_{\lambda}x_{n} \leq\delta_{n}$ $\dagger 1\in N$ $\{\alpha_{n}\}$ $\{\beta_{n}\}$ $\{\gamma_{n}\}\subset$ $[01]$ $\{\lambda_{n}\}\subset(0 \infty)$ $\{\delta_{n}\}\subset[0 \infty)$ $a_{n}+\beta_{n}+\gamma_{n}=1$ $\Sigma_{n=0}^{\infty}\lambda_{n}<\infty$ (1) (2) (1) $\{\alpha_{n}\}$ $\{\beta_{n}\}$ $\{\gamma_{n}\}\subset[01]$ $n arrow\infty 1\mathrm{i}\ln\alpha_{n}^{l}=0\sum_{n=0}^{\infty}\alpha_{n}=\infty$ $narrow\infty 1\mathrm{i}\mathrm{m}\beta_{n}=0$ $narrow\infty 1\mathrm{i}\mathrm{m}\lambda n=$ $\{X_{\hslash}\}$ $A^{-1}0$ $u$ $Px=u$ $P$ $H$ $A^{-1}0$ (2) $\{\alpha_{\hslash}\}$ $\{\beta_{n}\}$ $\{\gamma_{n}\}\subset[01]$ $\sum_{n=0}^{\infty}\alpha_{n}<\infty$ $\lim_{narrow}\sup_{\infty}\beta_{n}<1$ $\lim_{narrow}\inf_{\infty}\lambda_{n}>0$ $\mathrm{t}_{x_{n\}}}$ $v=1\mathrm{i}m_{narrow\infty}px_{n}$ $P$ $A^{-1}0$ $v$ $H$ $A^{-1}0$ n} $\alpha_{n}$ \Sigma - [8]

8 38 4 Banach m- Banach $\mathrm{m}$- - [9] $\mathrm{m}$ Ballach 41 ( - [9]) $E$ Banach $E$ \ echet $E$ Opial $A\subset E\mathrm{x}E$ m- $x_{0}=x\in E$ $\{x_{n}\}$ $x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})\text{_{}\lambda_{\iota}x_{\overline{n}}+e_{n}}$ $n\in N$ $\{\alpha_{n}\}\subset[01]$ $\{\lambda_{n}\}\subset(0 \infty)$ $\{e_{n}\}\subset E$ $\lim_{narrow}\sup_{\infty}\alpha_{n}<1$ $\lim_{narrow}\inf_{\infty}\lambda_{n}>0\sum_{n=0}^{\infty} e_{n} <\infty$ $A^{-1}0\neq\emptyset$ $\{x_{n}\}$ $A^{-1}0$ $u$ 42 ( - [9]) $E$ G\ atea-- Banach $E$ $A\subset ExE$ $m$- $x_{0}=x\in E$ $\{x_{n}\}$ $x_{n+1}=\alpha_{n}x+(1-\alpha_{n})i_{\lambda_{\mathfrak{n}}}x_{n}$ $e_{n}$ $71\in N$ $\{\alpha_{n}\}\subset[01]$ $\{\lambda_{n}\}\subset(0 \infty)$ $\{e_{n}\}\subset E$ $\lim_{narrow\infty}\alpha_{n}=0\sum_{n\approx 0}^{\infty}\alpha_{n}=\infty$ $n arrow\infty 1\mathrm{i}\ln\lambda_{n}=\infty\sum_{narrow 0}^{\infty} e_{n} <\infty$ $A^{-1}0\neq\emptyset$ $P$ $E$ $Px=v$ $\{x_{n}\}$ $A^{-1}0$ $v$ $A^{-1}0$ sunny 43 $E$ Banach $E$ Fr\ echet $E$ Opial $A\subset E\mathrm{x}E$ $n$- $\{x_{n}\}$ $x_{0}=x\in E$ $x_{n+1}=\alpha_{n}x+\beta_{n}x_{n}+\gamma_{n^{\text{}}\lambda}x_{n}+e_{n}$ $n\in N$

9 39 $\{\alpha_{n}\}$ $\{\beta_{n}\}$ $\{\gamma_{n}\}\subset[01]$ $\{\lambda_{n}\}\subset(0 \infty)$ $\{e_{n}\}\subset E$ $1 \mathrm{i}\mathrm{n}1\sup_{narrow\infty}\alpha_{n}<11\mathrm{i}\mathrm{n})\sup_{narrow\infty}\beta_{n}<1$ $\lim_{narrow}\inf_{\infty}\lambda_{n}>0\sum_{n=0}^{\infty} e_{n} <\infty$ $A^{-1}0\neq\emptyset$ $\{x_{n}\}$ $A^{-1}0$ $u$ 44 $E$ G\ ateaux \urcorner p $\prime \mathit{2}$ Banach $k$ $E$ $A\subset ExE$ $m$- $x_{0}=x\in E$ $\{x_{n}\}$ $x_{n+1}=\alpha_{n}x+\beta_{n}x_{n}+\gamma_{n}\text{_{}\lambda_{l}x_{n}+6_{n}}$ $n\in N$ $\{\alpha_{n}\}$ $\{\beta_{n}\}$ $\{\gamma_{n}\}\subset[01]$ $\{\lambda_{n}\}\subset(0 \infty)$ $\{e_{n}\}\subset E$ $n arrow\infty 1\mathrm{i}\mathrm{l}11\alpha_{n}=0\sum_{n\approx 0}^{\infty}\alpha_{n}=\infty\lim_{narrow\infty}\beta_{n}=0$ $n arrow\infty 1\mathrm{i}\iota \mathrm{n}\lambda_{n}=\infty\sum_{n=0}^{\infty} e_{n} <$ $A^{-1}0\neq\emptyset$ $Px=v$ $P$ $E$ $\{x_{n}\}$ $A^{-1}0$ $A^{-1}0$ $v$ sunny 45 $E$ Banach $E$ G\ ateanx $E$ $A\subset E\mathrm{x}E$ $m$- $x$ $u\in E$ $\{x_{n}\}$ $\{$ $x_{0}=x\in E$ $x_{n+1}=\alpha_{n}u+\beta_{n}x_{n}+\gamma_{n^{\text{}}\lambda_{1}}x_{n}+e_{n}$ $n\in N$ $\{\alpha_{n}\}$ $\{\beta_{n}\}$ $\{\gamma_{n}\}\subset[01]$ $\{\lambda_{n}\}\subset(0 \infty)$ $\{e_{n}\}\subset E$ $n arrow\infty 1\mathrm{i}\ln\alpha_{n}=0\sum_{n=1}^{\infty} \alpha_{n}-\alpha_{n-1} <\infty\sum_{n=0}^{\infty}\alpha_{n}=$ $\lim_{narrow}\sup_{\infty}\beta_{n}<1\sum_{n\approx 1}^{\infty} \beta_{n}-\beta_{n-1} <\infty$ $\lim \mathrm{i}11\mathrm{f}\lambda_{n}>0narrow\infty \sum_{n=1}^{\infty} \lambda_{n}-\lambda_{n-1} <\infty\sum_{n=0}^{\infty} e_{n} <\infty$ $A^{-1}0\neq\emptyset$ $\{x_{n}\}$ $A^{-1}0$ $v$ Banach Baxtadh $\ovalbox{\tt\small REJECT}$

10 40 5 Theorem 43 Theorem 44 Hilbert 51 $H$ Hilbert : $f$ $Harrow(-\infty \infty]$ $x_{0}=x\in H$ $\{x_{n}\}$ $y_{n} \approx\arg\iota \mathrm{n}\mathrm{i}\mathrm{n}z\in H\{f(z)+\frac{1}{2\lambda_{n}} z-x_{n} ^{2}\}=\text{_{}\lambda_{l}x_{n}}$ $x_{n+1}=a_{n}x_{n}+\beta_{n}x_{n}+\gamma_{n}j_{\lambda_{1}}x_{n}+e_{n}$ $n\in N$ $ y_{n^{-j_{\lambda_{\iota}}}}x_{n} \leq\delta_{n}$ $(0 \infty)$ $\{e_{n}\}\subset E$ $\{a_{n}\}$ $\{\beta_{n}\}$ $\{\lambda_{n}\}\subset$ $\{\gamma_{n}\}\subset[01]$ $1 \mathrm{i}_{\mathrm{l}}\mathrm{n}\sup_{narrow\infty}\alpha_{n}<11\mathrm{i}_{\mathrm{l}}\mathrm{n}\sup_{narrow\infty}\beta_{n}<1$ $\mathrm{l}\mathrm{i}\ln \mathrm{i}1\mathrm{l}\mathrm{f}\lambda_{n}>0narrow\infty \sum_{n=0}^{\infty} * <\infty$ $(\partial f)^{-1}0\neq\emptyset$ $u=1\mathrm{i}\iota \mathrm{n}_{narrow\infty}px_{n}$ $P$ $\{x_{n}\}$ $(\partial f)^{-1}0$ $u$ $H$ $(\partial f)^{-1}0$ 52 $H$ Hilbert ; $f$ $Harrow(-\infty \infty]$ $x_{0}=x\in H$ $\{x_{n}\}$ $y_{n} \approx \mathrm{a}1^{\cdot}z\in H\mathrm{g}m\mathrm{i}\mathrm{n}\{f(_{\wedge}^{\sim})+\frac{1}{2\lambda_{n}} z-x_{n} ^{2}\}=\text{_{}\lambda_{n}x_{n}}$ $x_{n+1}=\alpha_{n}x+\beta_{n}x_{n}+\gamma_{n}\text{_{}\lambda_{\iota}x_{n}+\epsilon_{n}}$ $n\in N$ $ y_{n^{-j_{\lambda}}}x_{n} \leq\delta_{n}$ $(0 \infty)$ $\{e_{n}\}\subset E$ $\{a_{n}\}$ $\{\beta_{n}\}$ $\{\gamma_{n}\}\subset[01]$ $\{\lambda_{n}\}\subset$ $n arrow\infty 1\mathrm{i}_{\mathrm{l}}\mathrm{n}\alpha_{n}=0\sum_{n=0}^{\infty}\alpha_{n}=\infty\lim_{narrow\infty}\beta_{n}=0$ $n arrow\infty 1\mathrm{i}111\lambda_{n}=\infty\sum_{n=0}^{\infty} e_{n} <\infty$ $(\partial f)^{-1}0\neq\emptyset$ $Px=v$ $P$ $H$ $(\partial f)^{-1}0$ $\{x_{n}(\}$ $(\partial f)^{-1}0$ $v$ [1] H Br\ ezis and P L Lions Produits infinis de resolvants Israel J Math 29 (1978)

11 Y 41 $ {}^{t}a$ [2] R E Bruck strongly convergent iterative solution of for a maximal mmonotone operator in Hilbert space J Math Anal Appl $U$ 48 (1974) $\mathrm{o}\in U(x)$ [3] R E Bruck and G B Passty $ttalmost$ convergence of the infinite product of resolvents in Banach spaces Nonlinear Anal 3 (1979) [4] O G\"uler On the convergence of the proximal point algorithm for convex $\min-$ $imization^{n}$ SIAM J Control Optim 29 (1991) [5] B Halpern $ \prime pixed$ points of nonexpansive maps Bull Amer Math Soc 73 (1967) $\mathrm{a}\mathrm{s}\mathrm{h}\mathrm{i}$ [6] S Iemoto and W $\mathrm{t}\mathrm{a}\mathrm{m}$) Strong and Weak Convergence Theorems for Resolvents of Maximal Monotone Operators in Hilbert Spaces Proceedings $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\iota\cdot$ of the International Conference on Analysis and Convex Analysis Yokohalna Publishers 2004 [7] J S Jung and W Takallashi Dual convergence theorems for the infinite products of resolvents in Banach spaces Kodai Math J 14 (1991) [8] S Kamimura and W Takaliashi $uapproximating$ solutions of masimal monotone operators in Hilbert spaces J Approx Theory 106 (2000) [9] S Kamimura and W Takahashi $u_{weak}$ and Strong Convergence of Solutions to Accretive Opefator Inclusions and Applications Set-Valued Anal 8 (2000) $\mathrm{k}\mathrm{i}m$ [10] T H and H K Xu $ {}^{t}strong$ convergence of Mann modifi$\mathrm{e}d$ Nonlinear Anal 61 (2005) iterations [ $11_{\mathrm{J}}^{]}$ Kimura W Takaliashi and M Toyoda Convergence to common fixed points of a finite family of nonexpansive mappings Arch Math 84 (2005) [12] F Kohsaka and W Takallashi Strong Convergence of an Iterative Sequence for Maximal Monotone Operators in a Banach Space Abstr Appl Anal 3 (2004) in\ equation variation- [13] P L Lions $u_{une}$ m\ ethode iterative de r\ esolutio $\mathrm{t}7$ d une nelle Israel J Math 31 (1978) [14] W R Mann $u_{\lambda fean}$ value meth $ods$ in iteration Proc Amer Math Soc 4 (1953)

12 42 [15] O Nevanlinna and S Reich $ttstrong$ convergence of contraction semigroups an$d$ of iterative methods for accretive operators in Banach spaces Israel J Math 32 (1979) [16] G B Passty $ {}^{t}ergodic$ convergence to a zero of the sum of monotone operators in Hilbert space ) J Math Anal Appl 72 (1979) [17] A Pazy $ttremarks$ on nonlinear ergodic theory in Hilbert space Nonlinear Anal 6 (1979) [18] S Reich On infinite products of resolvents Atti Acad Naz Lincei 63 (1977) [19] S Reich An iterative producedure for constructiong zeros of accretive sets in Banach spaces Nonlinear Anal 2 (1978) [20] S Reich $ constrw$ ction zeros of accretive operators Appl Anal 8 (1979) [21] S Reich $ {}^{t}constmction$ zeros of accretive operators II Appl Anal 9 (1979) [22] S Reich Weak convergence theorems for $none\varphi ansive$ mappings in Banach spaces J Math Alal Appl 67 (1979) [23] S Reich $ {}^{t}strong$ convergence theorems for resolvents of accretive operators in Banach spaces J Math Anal Appl 75 (1980) [24] R T Rockafellar Characterizationn of the subdifferentials of convex functions Pacific J Math 17 (1966) [25] R T Rockafellar $ {}^{t}on$ th $e$ maximal monotonicity of subdefferential mappings Pacific J Math 33 (1970) [26] R T Rockafellar Monotone operators and the proximal point algoritllm SIAM J Control Optim 14 (1976) [27] N Shioji and W Takahashi Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces Proc Amer Math Soc 125 (1997) [28] M V Solodov and B F Svaiter Forcing strong convergence of proxfmal point iterations in a Hilbert space Math Prograni 87 (2000)

13 43 [29] W Takahashi Nonlinear Functional Analysis Yokohama Publishers 2000 [30] W Takahashi $ttintroduction$ to Nonlinear & Convex Analysis (Japanese) Yokohama Publishers 2005 [31] W Takahashi and G E Kim Approximating fixed points of nonexpansive mappings in Banach spaces Math Japon 48 (1998) 1-9 [32] W Takahashi and Y Ueda On Reich s strong convergence theorems for resolvents of accretive operators J Math Anal Appl 104 (1984) [33] K K Tall and H K Xu Approximating fixed points of $nonexpansive$ mappings by the iteration process J Math Anal Appl 178 $is1\iota ikau a$ (1993) [34] R Wittmann $ {}^{t}approximation$ of fixed points of $non\epsilon xpansive$ mappings Arch Math 58 ( $1992\rangle$ [35] H K Xu $ {}^{t}an$ iterative algorithms for nonlinear operators J Optimiz Theory Appl 116 (2003)

Title 凸計画問題と関連する反復法 ( 最適化数理の手法と実際 ) Author(s) 高阪, 史明 ; 高橋, 渉 Citation 数理解析研究所講究録 (2005), 1461: Issue Date URL

Title 凸計画問題と関連する反復法 ( 最適化数理の手法と実際 ) Author(s) 高阪, 史明 ; 高橋, 渉 Citation 数理解析研究所講究録 (2005), 1461: 47-61 Issue Date 2005-12 URL http://hdl.handle.net/2433/47969 Right Type Departmental Bulletin Paper Textversion