数理解析研究所講究録
第 1963 巻 2015 年 9-159完備測地距離室間上の二写像に対する近似法

An iterative scheme for two mappings defined on acomplete geodesic space
東邦大学理学部木村泰紀

Yasunori Kimura
Department of Information Science

Faculty of ScienceToho University
Email: yasunori@is.sci.toho-u.ac.jp

1Introduction

Let $X$ be a metric space and $T:Xarrow X$ a nonexpansive mapping, that is,$T$satisfiesthat $d(Tx, Ty)\leq d(x, y)$ for$y\in X$ . A point $z\in X$ such that $Tz=z$ is called
a fixed point of $T$ . Approximation of fixed points of $T$ is one of the central topics in$a1\Gamma x,$

fixed point theory because it includes various types of problems in nonlinear analysis.
In particular, approximation of fixed points of a mapping defined on a complete
space is a trend of this study and there are a large number of researchesCAT
related to this problem. For example, the following result is a convergence theoremof an iterative scheme called the shrinking projection method on a CAT(I) space.
$(\kappa)$

Theorem 1 (Kimura-Sat\^o [5]). Let $X$ be a complete CAT(I) space such that $d(u, v)<$for every $u,.v\in X$ and suppose that the subset $\{z\in X : d(z, u)\leq d(z, v)\}$ of $X$
is convex for every $v\in X.$ Let : $Xarrow X$ be a nonexpansive mapping such thatthe set of fixed points $F=\{z\in X : Tz=z\}$ is nonempty. For a given initial point$x_{0}\in X$ and $C_{0}=X$ , generate aequence {as follows:$\pi/2$$T$$u,$$\mathcal{S}$$x_{n}\}$$C_{n+1}=\{z\in X:d(Tx_{n}, z)\leq d(x_{n}, z)\}\cap C_{n},$$x_{n+1}=P_{C_{n+1}}x_{0},$for each
$P_{C}$of. Thenis well defined and converges to $P_{F}x_{0}\in X,$ where: $Xarrow C$ is the metric projection of $X$ onto a nonempty closed convex subset$n\in \mathbb{N}$$\{x_{n}\}$$\cdot$$C$$X.$The shrinking projection method was first proposed by Takahashi, Takeuchi, andKubota [10], and it has been generalized to various directions. See, for instance,

10Takahashi and Zembayashi [11], Plubtieng and Ungchittrakool [8], Inoue, Takahashi,and Zembayashi [2], Qin, Cho, and Kang [9], Wattanawitoon and Kumam [13, 12],Kimura, Nakajo, and Takahashi [4], Kimura and Takahashi [7], Kimura [3], Kimura
and Sat\^o [6], and others.
In this paper, we deal with an approximation of common fixed points for twomappings. We attempt to prove tlie main result without using the notion of
convergence because it is not easy to understand for the beginners of this study. Theproof shown in this paper only uses basic notions.
$\triangle_{-}$2 Preliminaries
Let $X$ be a metric space. We say that $X$ is a geodesic space if, for any $v\in X$ , thereexists a mapping : $[0, d(u, v)]arrow X$ , which is called a geodesic between endpoints
and , such that $c(O)=u,$ $c(d(u, v))=v$ , and $d(c(s), c(t))=|s-t|$ for every$u,$$c$$u$$v$$s,$$t\in[0,$ $d(u,$ $v$

If a geodesic is unique for each pair of endpoints, $X$ is said to be uniquely geodesic.
In what follows, we always assume that $X$ is a complete uniquely geodesic space suchthat $d(u, v)<\pi/2$ for every $v\in X$ . On a uniquely geodesic space, the convex
combination of two points $v\in X$ can be defined in a natural way and we denote it, 1 . For $C\subset X$ , if every geodesics having the endpoints, wherebyin is contained in , then is said to be convex.
be the sphericalandLet be a unit sphere of 3-dimensional Euclidean space
metric defined on . A geodesic space $X$ is called a CAT(I) space if for each geodesic
triangle on $X$ is thinner than or equal to its comparison triangle on . Namely, every
satisfy the following which
and their comparison points
is called CAT(I) inequality:
$u,$

$u,$

$\alpha\in[0$$\alpha u\oplus(1-\alpha)v$$C$$C$$]$$C$$\mathbb{R}^{3}$$\mathbb{S}^{2}$$d_{\mathbb{S}^{2}}$$\mathbb{S}^{2}$$\mathbb{S}^{2}$$\overline{q}\in\overline{\triangle}\subset S^{2}$$p,$ $q\in\triangle\subset X$$\overline{p},$$d(p, q)\leq d_{\mathbb{S}^{2}}(\overline{p}, \overline{q})$If $X$ is a CAT(I) space, then for $x,$holds; see [5].

$y,$ $z\inX$.

and $t\in[0$ , 1 , the following inequality$]$$\cos d(tx\oplus(1-t)y, z)\sin d(x, y)$$\geq\cos d(x, z)\sin(td(x, y))+\cos d(y, z)\sin((1-t)d(x, y$

Let $C$ be a nonempty closed convex subset $C$ of $X$ . Since $X$ satisfies in our settingthat $d(u, v)<\pi/2$ for every $v\in X$ , we know that for every $x\in X$ , there exists
a unique $y_{x}\in C$ such that $d(x, y_{x})=d(x, C)$ , where $d(x, C)= \inf_{y\in C}d(x, y)$ . We
: $Xarrow C$ by $P_{C}x=y_{x}$ for $x\in X$ and we call it the metricdefine a mapping$C.$$X$ofontoprojection
For more details of CAT(I) spaces and related notions, see [1].
: $Xarrow X$ is quasinonexpansive if the set $F(T)=\{z\in X$ :We say a mapping$Tz=z\}$ of fixed points is nonempty and $d(Tx, z)\leq d(x, z)$ for every $x\in X$ and$z\in F(T)$ . We also know that if $X$ is CAT(I) space with $d(u, v)<\pi/2$ for every$v\in X$ , then $F(T)$ is closed and convex.
$u,$$P_{C}$

$T$

$u,$

11

3 Approximation of a common$fi\cross ed$point

In this section, we prove a convergence theorem of an iterative sequence generated
by the shrinking projection method for two quasinonexpansive mappings defined on
a complete CAT(I) space.

Theorem 2. Let $X$ be a complete CAT(I) space such that $d(u, v)<\pi/2$ for every
X$ andthat the subset $\{z\in X : d(z, u)\leq d(z, v)\}$ of $X$ is convex forevery $v\in X$ . Let and be continuous quasinonexpansive mappings of $X$ tosuch that the set of common fixed points $F=\{z\in X : Sz=z=Tz\}$ is nonempty.
Letbe a real sequence in , 1 such that there exists a subsequenceofconverging to] , 1 [. For a given initial point $x_{0}\in C$ and $C_{0}=X$ , generate asequence
as follows:$u,$ $v\in$s\prime\psi ppose$)

$S$

$u,$

$T$

$it_{\mathcal{S}}elf$$\{\alpha_{n}\}$$[0$$\alpha_{\infty}\in$$]$$\{\alpha_{n_{i}}\}$$\{\alpha_{n}\}$$0$$\{x_{n}\}$$y_{n}=\alpha_{n}Sx_{n}\oplus(1-\alpha_{n})Tx_{n},$$C_{n+1}=\{z\in X:d(y_{n}, z)\leq d(x_{n}, z)\}\cap C_{n},$$x_{n+1}=P_{C_{n+1}}x_{0},$for each
$P_{C}$

of:$Xarrow C$.

Then
is well defined and converges to $P_{F}x_{0}\in X$ , whereis the metric projection of $X$ onto a nonempty closed convex subset $C$$n\in \mathbb{N}$$\{x_{n}\}$$X.$

To prove this type of convergence theorems, one tends to make use of the following

theorem.
Theorem 3 (Kimura-Sat\^o [5]). Let $X$ be a complete CAT(I) space and $\{C_{n}\}a$sequence of nonempty closed -convex subsets of X. Let
be a nonempty closed-convex subset of X. Then the following are equivalent:$\pi$$C_{\infty}$$\pi$(i);$x\inX$(ii) forand a subsequence, if one of $\lim\sup_{iarrow\infty}d(x, C_{n_{i}})$ofand $d(x, C_{\infty})$ is less than $\pi/2$ , then the other is also less than $\pi/2$ andconverges to$C_{\infty}= \triangle_{1}M-\lim_{narrow\infty}C_{n}$$\{C_{n_{i}}\}$$\{C_{n}\}$$\{P_{C_{n_{i}}}x\}$$P_{C_{\infty}}x.$

Although this result is useful, one may think that it is rather diﬃcult to understandbecause it requires the notion of -Mosco convergence of a sequence of subsets in $X.$
We actually do not need to use this concept since we only use the result for the casewhere a sequenceof subsets of $X$ is decreasing with respect to inclusion. Here,
we show the proof of Theorem 2 without using the notion of -Mosco convergence.
$\triangle$$\{C_{n}\}$$\triangle$

Proof of Theorem 2. We first prove the well-definedness ofby showing that every
is closed, convex, and it includes
by induction. It is trivial that $C_{0}=X$
is a closed convex set such that, and a point $x_{0}\in X$ is given. Suppose thatis defined as a closed convex subset of $X$ which includes $F$ for some. Then,$x_{k}=P_{C_{k}}x_{0}$ is defined. Since
and are quasinonexpansive andis concave on$C_{n}$$\{x_{n}\}$$F\neq\emptyset$$F\subset C_{0}$$C_{k}$$k\in \mathbb{N}$

$S$

$T$

$\sin t$12
$t\in[0, \pi/2]$

with $\sin 0=0$ , for $z\in F$ we have that$\cos d(y_{k}, z)$s.in $d(Sx_{k}, Tx_{k})$$=\cos d(\alpha_{k}Sx_{k}\oplus(1-\alpha_{k})Tx_{k}, z)\sin d(Sx_{k}, Tx_{k})$$\geq\cos d(Sx_{k}, z)\sin(\alpha_{k}d(Sx_{k}, Tx_{k}))+\cos d(Tx_{k}, z)\sin((1-\alpha_{k})d(Sx_{k}, Tx_{k}))$$\geq\cos d(x_{k}, z)(\sin(\alpha_{k}d(Sx_{k}, Tx_{k}))+\sin((1-\alpha_{k})d(Sx_{k},$ $Tx_{k}$$\geq\cos d(x_{k}, z)(\alpha_{k}\sin d(Sx_{k}, Tx_{k})+(1-\alpha_{k})\sin d(Sx_{k}, Tx_{k}))$$=\cos d(x_{k}, z)\sin d(Sx_{k}, Tx_{k})$,and thus $d(y_{k}, z)\leq d(x_{k}, z)$ . This implies that
$F\subset\{z\in X:d(y_{k}, z)\leq d(x_{k}, z)\}\cap C_{k}=C_{k+1}.$

It is obvious from the continuity of the metric and the assumption of the space thatis a sequence of closed
is well defined andis closed and convex. Hence$X$
everyforconvex subsets of satisfying thatis decreasing with respect to inclusion andIt holds by definition that
, we. Since $x_{n}=P_{C_{n}}x_{0}$ for every
is nonempty sincehave that $\{d(x_{n}, x_{0})\}$ is nondecreasing and bounded above. Thus there exists $d=$.
and since
andbelong toLet $m,$such that $m\leq n$ . Then, both
is convex, we have that$\{C_{n}\}$$\{x_{n}\}$$C_{k}$$n\in \mathbb{N}.$$F\subset C_{n}$$C_{\infty}=$$\{C_{n}\}$$n\in \mathbb{N}$$C_{\infty}\supset F$$\bigcap_{n=1}^{\infty}C_{n}$$\lim_{narrow\infty}d(x_{n}, x_{0})$$n\in \mathbb{N}$$x_{m}$$x_{n}$$C_{m}$$C_{m}$$\cos d(x_{m}, x_{0})\sin d(x_{m}, x_{n})$$\geq\cos d(\frac{1}{2}x_{m}+\frac{1}{2}x_{n}, x_{0})\sin d(x_{m}, x_{n})$$\geq\cos d(x_{rn}, x_{0})\sin(\frac{1}{2}d(x_{m}, x_{n}))+\cos d(x_{n}, x_{0})\sin(\frac{1}{2}d(x_{m}, x_{n}))$.

Since$\cos d(x_{m}, x_{0})\sin d(x_{rn}, x_{n})=2\cos d(x_{m}, x_{0})\sin(\frac{1}{2}d(x_{rn}, x_{n}))\cos(\frac{1}{2}d(x_{m}, x_{n}))$we have that
$2 \cos d(x_{m}, x_{0})\cos(\frac{1}{2}d(x_{m}, x_{n}))\geq\cos d(x_{m}, x_{0})+\cos d(x_{n}, x_{0})$and since $d(x_{7n}, x_{0})\leq d(x_{n}, x_{0})$ , we get that$\cos(\frac{1}{2}d(x_{m}, x_{n}))\geq\frac{\cos d(x_{rn},x_{0})+\cos d(x_{n},x_{0})}{2\cos d(x_{m},x_{0})}$$\geq\frac{\cos d(x_{n},x_{0})}{\cos d(x_{m},x_{0})},$,

13
which is equivalent to that.

$- \log\cos(\frac{1}{2}d(x_{m}, x_{n}))\leq\log\cos d(x_{m}, x_{0})-\log\cos d(x_{n}, x_{0})$Since $\{\log\cos d(x_{n}, x_{0})\}$ is a convergent sequence to $\log\cos d$ , there exists a sequence
converging to such that$0$$\{t_{n}\}$$0\leq\log\cos d(x_{m}, x_{0})-\log\cos d(x_{n}, x_{0})\leq t_{n}$for all $m,$$n\in \mathbb{N}$with $m\leq n$ . Then we have that$d(x_{m}, x_{n})\leq 2\arccos e^{-t_{n}}$for all $m,$with $m\leq n$ and $\lim_{narrow\infty}2\arccos e^{-t_{n}}=$ O. It shows that
is aCauchy sequence and therefore it has a limit$\{x_{n+k}\}$ is a sequence in
For fixed. It follows from the closedness ofthat
is a point inand thus we have that$n\in \mathbb{N}$$\{x_{n}\}$$x_{\infty}\in X.$$k\in \mathbb{N},$$C_{k}$$C_{k}$$C_{k}$$x_{\infty}$.

$d(y_{k}, x_{\infty})\leq d(x_{k}, x_{\infty})$Tending $karrow\infty$ , we obtain that
also converges to. In addition, we also havethat
. We next show thatbelongs to $F$ . For $z\in F$ , we havethat
and$\{y_{k}\}$$x_{\infty}$$x_{\infty} \in\bigcap_{k=1}^{\infty}C_{k}=C_{\infty}$$x_{\infty}$$z\in C_{\infty}$$\cos d(y_{n}, z)\sin d(Sx_{n}, Tx_{n})$$=\cos d(\alpha_{n}Sx_{n}\oplus(1-\alpha_{n})Tx_{n}, z)\sin d(Sx_{n}, Tx_{n})$$\geq\cos d(Sx_{n}, z)\sin(\alpha_{n}d(Sx_{n}, Tx_{n}))+\cos d(Tx_{n}, z)\sin((1-\alpha_{n})d(Sx_{n}, Tx_{n}))$$\geq\cos d(x_{n}, z)(\sin(\alpha_{n}d(Sx_{n}, Tx_{n}))+\sin((1-\alpha_{n})d(Sx_{n},$ $Tx_{n}$$=2 \cos d(x_{n}, z)\sin(\frac{1}{2}d(Sx_{n}, Tx_{n}))\cos((\frac{1}{2}-\alpha_{n})d(Sx_{n}, Tx_{n}))$

.

Since
$\sin d(Sx_{n}, Tx_{n})=2\sin(\frac{1}{2}d(Sx_{n}, Tx_{n}))\cos(\frac{1}{2}d(Sx_{n}, Tx_{n}))$,we have that$\cos d(y_{n}, z)\cos(\frac{1}{2}d(Sx_{n}, Tx_{n}))$$\geq\cos d(x_{n}, z)\cos((\frac{1}{2}-\alpha_{n})d(Sx_{n}, Tx_{n}))$for all$n\in \mathbb{N}$. Then, for a subsequence$\{\alpha_{n_{i}}\}$

of$\{a_{n}\}$.

whose limit is$\cos d(x_{\infty}, z)\cos(\frac{1}{2}\lim_{iarrow}\sup_{\infty}d(Sx_{n_{i}}, Tx_{n_{i}}))$$\alpha_{\infty}\in$] , 1 [,
$0$

14$\geq\cos d(x_{\infty}, z)\cos((\frac{1}{2}-\alpha_{\infty})\lim_{iarrow}\sup_{\infty}d(Sx_{n_{i}}, Tx_{n_{i}}))$,

which implies that $\lim_{iarrow\infty}d(Sx_{n_{i}}, Tx_{n_{i}})=0$ . Hence we have that$d(x_{\infty}, Sx_{\infty})= \lim_{iarrow\infty}d(y_{n_{i}}, Sx_{n_{i}})$$= \lim_{iarrow\infty}d(\alpha_{n_{i}}Sx_{n_{i}}\oplus(1-\alpha_{n_{i}})Tx_{n_{i}}, Sx_{n_{i}})$$= \lim_{iarrow\infty}(1-\alpha_{n_{i}})d(Tx_{n_{i}}, Sx_{n_{i}})$$=(1- \alpha_{\infty})\lim_{iarrow\infty}d(Tx_{n_{i}}, Sx_{n_{i}})$$=0,$and, in a similar fashion, we get that $d(x_{\infty}, Tx_{\infty})=0$ . Thus $x_{\infty}\in F(S)\cap F(T)=F.$, we have thatSince$F\subset C_{\infty}$$d(x_{0}, x_{\infty})= \lim_{iarrow\infty}d(x_{0}, P_{C_{i}}x_{0})\leq d(x_{0}, P_{F}x_{0})\leq d(x_{0}, x_{\infty})$and, from the uniqueness of the minimizing point of the distance between
we have $x_{\infty}=P_{F}x_{0}$ . This is the desired result.

$x_{0}$

and $F,$$\square$

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$elemenl\mathcal{S}$$space\mathcal{S}$