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数理解析研究所講究録
第2065巻 2018年 100-110

COMMON ACUTE POINTS AND CONVERGENCE
THEOREMS FOR FAMILIES OF NONLINEAR
MAPPINGSSACHIKO ATSUSHIBAABSTRACT. In this paper, we prove an attractive points theorem

and strong convergence theorems of Halpern’s type [21] for uniformly

asymptotically regular $\lambda$‐hybrid mappings in a star‐shaped subset ofa Hilbert space. Using these results, we obtain a fixed point theorem
and some strong convergence theorems. lburther, we prove conver‐gence theorems by using the concept of acute points of nonlinear
mappin\mathrm{g}\mathrm{s}.
1. INTRODUCTION

Let H be a real Hilbert space with inner product\rangle and norm \Vert \VertH
C
and letbe a nonempty subset of . For a mapping T : C \rightarrow C,
we denote by F(T) the set of fixed points of T and by A(T) the set of

attractive points [29] of T , i.e.,(i) F(T)=\{z\in C: Tz=z\} ;(ii) A(T)=\{z\in H : \Vert Tx-z\Vert\leq\Vert x-z\Vert, \forall x\in C\}.
A mapping T : C\rightarrow C is called nonempansive if \Vert Tx-Ty\Vert \leq \Vert x-y\Vertfor all x, y\in C.

In 1975, Baillon [14] proved the following first nonlinear ergodic the‐orem in a Hilbert space: Let C be a nonempty bounded closed convex

subset of a Hilbert space H and let T be a nonexpansive mapping of C
into itself. Then, for any x\in C,

fixed point of T (see also [27]).

S_{n}x=\displaystyle\frac{1}{n}\sum_{i=0}^{n-1}\dot{T}xconverges weakly to a2010 Mathematics Subject Classification. Primary 47\mathrm{H}09, 47\mathrm{H}10.
Key words and phruses. Fixed point, attractive point, acute point, iteration, non‐expansive mapping, nonexpansive semigroup, strong convergence.

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Kocourek, Takahashi and Yao [23] introduced a broad class of non‐linear mappings called generalized hybrid which containing nonexpansive
mappings, nonspreading mappings, and hybrid mappings in a Hilbert
space. They proved a mean convergence theorem for generalized hy‐

brid mappings which generalizes Baillon’s nonlinear ergodic theorem [14].
Aoyama, Iemoto, Kohsaka and Takahashi [4] introduced the class of $\lambda$-

hybrid mappings in a Hilbert space. This class obtain the classes of non‐expansive mappings, nonspreading mappings, and hybrid mappings in a
Hilbert space. They proved fixed point theorems and mean convergence

theorems for such mappings. Motivated by Baillon [14], and Kocourek,
Takahashi and Yao [23], Takahashi and Takeuchi [29] introduced the con‐

cept of attractive points of a nonlinear mapping in a Hilbert space andthey proved a mean convergence theorem of Baillon’s type without con‐vexity for generalized hybrid mappings. In 1992, Wittmann [30] proved
the following strong convergence theorems of Halpern’s type [21] in a
Hilbert space;

Theorem 1.1. Let C be a nonempty closed convex subset of a Hilbert
space H. Let T be a nonexpansive mapping of C into itself with F(T)\neq\emptyset.
For any x_{1}=x\in C , define a sequence \{x_{n}\} in C by

x_{n+1}=$\alpha$_{n}x+(1-$\alpha$_{n})Tx_{n}, \forall n\geq 1
where \{$\alpha$_{n}\}\subset[0 , 1 ] satisfies\displaystyle\lim_{n\rightar ow\infty}$\alpha$_{n}=0,\sum_{n=1}^{\infty}$\alpha$_{n}=\infty,\sum_{n=1}^{\infty}|$\alpha$_{n}-$\alpha$_{n+1}|<\infty.
Then, \{x_{n}\} converges strongly to P_{F\langle T)}x , where P_{F(T)} is the metric pro‐jection from H onto F(T) .

Motivated by Takahashi and Takeuchi [29], Akashi and Takahashi [2]proved a strong convergence theorem of Halpern’s type [21] for nonex‐

pansive mappings in a star‐shaped subset of a Hilbert space. On theother hand, Domingues Benavides, Acedo and Xu [18] proved strong
convergence theorems of Halpern’s type [21] for uniformly asymptoticallyregular one‐parameter nonexpansive semigroups. The author [8] studied
Halpern’s type iterations for nonexpansive semigroups and proved strong
convergence theorems for uniformly asymptotically regular nonexpansive
semigroups in Hilbert spaces (see also [1, 7, 9, 18, 26, 27

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In this paper, we prove an attractive points theorem and strong con‐vergence theorems of Halpern’s type [21] for uniformly asymptoticallyregular $\lambda$‐hybrid mappings in a star‐shaped subset of a Hilbert space.
Using these results, we obtain a fixed point theorem and some strong
convergence theorems. Further, we prove convergence theorems by using
the concept of acute points of nonlinear mappings.
2. PRELIMINARIES AND NOTATIONS

Throughout this paper, we denote by \mathrm{N} and \mathbb{R} the set of all positive
integers and the set of all real numbers, respectively. We also denote by\mathbb{Z}^{+} and \mathbb{R}^{+} the set of all nonnegative integers and the set of all nonnega‐tive real numbers, respectively. Let H be a real Hilbert space with innerproduct\rangle and norm \Vert\cdot\Vert . We know the following basic equality from[27]. For x, y\in H and $\lambda$\in \mathbb{R} , we have

\Vert x+y\Vert^{2}\leq\Vert x\Vert^{2}+2\langle y, x+y\rangle

(2.1)

and

\Vert $\lambda$ x+(1- $\lambda$)y\Vert^{2}= $\lambda$\Vert x\Vert^{2}+(1- $\lambda$)\Vert y\Vert^{2}- $\lambda$(1- $\lambda$)||x-y\Vert^{2} .

(2.2)

Furthermore, we obtain that for all x, y, w\in H,\langle(x-y)+(x-w),y-w\rangle=\Vert x-w\Vert^{2}-\Vert x-y\Vert^{2}(2.3)

In fact, we have that\langle(x-y)+(x-w),y-w\rangle=\{(x-y)+(x-w) , (y-x)+(x-\mathrm{w})\rangle=\Vert x-w\Vert^{2}-\Vert x-y\Vert^{2}+\langle x-y, x-w\}+\langle x-w, y-x\rangle=\Vert x-w\Vert^{2}-\Vert x-y||^{2}
Let C be a closed and convex subset of H . For every point x\in H , there

exists a unique nearest point in C , denoted by P_{C}x , such that

\Vert x-P_{C}x\Vert\leq\Vert x-y\Vertfor all y\in C . The mapping P_{C} is called the metric projection of H onto
C . It is characterized by\langle P_{C}x-y,x-P_{C}x\rangle\geq 0

for all y\in C . See [27] for more details. The following result is well‐known(see [27]).

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Lemma 2.1. Let C be a nonempty, bounded, closed and convex subset ofa Hilbert space H and let T be a nonempansive mapping of C into itself.
Then, F(T)\neq\emptyset.
x ) to indicate that the sequence
We write x_{n} \rightarrow x (or \displaystyle \lim_{n\rightar ow\infty}x_{n}x\{x_{n}\} of vectors in H converges strongly to x . We also write x_{n}H=x(or \displaystyle \mathrm{w}-\lim_{n\rightar ow\infty}x_{n}) to indicate that the sequence \{x_{n}\} of vectors in=\rightarrow

converges weakly to x . In a Hilbert space, it is well known that x_{n}\rightarrow x
and \Vert x_{n}| \rightarrow\Vert x\Vert imply x_{n}\rightarrow x.
A mapping T:C\rightarrow C is called nonexpansive if \Vert Tx-Ty\Vert\leq\Vert x-y\Vertfor all x, y\in C. Let $\lambda$\in \mathbb{R} be given. Following [4], we say that a mapping
T:C\rightarrow C is $\lambda$‐hybrid if\Vert Tx-Ty\Vert^{2}\leq\Vert x-y\Vert^{2}+2(1- $\lambda$)\langle x-Tx,y-Ty\rangle
for all x, y \inC.

It is obvious that T is 1‐hybrid if and only if T isnonexpansive; T is ‐hybrid if and only if T is nonspreading [24]; T is
1/2‐hybrid if and only if T is hybrid [28]); If $\lambda$>1 , then T is $\lambda$‐hybrid
if and only if T=I . It is known [3, Proposition 2.2] that if $\lambda$<2 and$\alpha$=(1- $\lambda$)/(2-\mathrm{A}) , then T is $\lambda$‐hybrid if and only if it is $\alpha$‐nonexpansive[3], i.e.,\Vert Tx-Ty\Vert^{2}\leq $\alpha$(\Vert x-Ty\Vert^{2}+\Vert Tx-y||^{2}+(1-2 $\alpha$)\Vert x-y\Vert^{2}
for all x, y \in C . In general, nonspreading and hybrid mappings are
not continuous mappings. A mapping T : C \rightarrow C is called quasi‐
nonexpansive if F(T) is nonempty and \Vert \mathrm{w}-Tx\Vert \leq \Vert \mathrm{w}-y\Vert for all

w\in F(T) and x\in C . By Dotson [17, Theorem 1] and Ithoh and Takanhashi [22, Corollary 1], we know that F(T) is closed convex whenever T

is quasi‐nonexpansive. Every $\lambda$‐hybrid with a fixed point is cleary quasi‐
nonexpansive. Thus, the set of fixed point of each $\lambda$‐hybrid mapping isclosed convex. The mapping T is said to be firmly nenexpansive if\Vert Tx-Ty||^{2}+\Vert(I-T)x-(I-T)y\Vert^{2}\leq\Vert x-y\Vert^{2}
for all x, y\in C(\mathrm{s}\mathrm{e}\mathrm{e}[15,16,19,20] . It is known [4, Lemma 3.1] that if T
is firmly nenexpansive, then T is $\lambda$‐hybrid for each $\lambda$\in[0 , 1 ].

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3. LEMMAS

In this section, we give some lemmas which are used in the proofsof our main theorems. We have basic properties of attractive points ofnonlinear mappings in a Hilbert space (see [29]).

Lemma 3.1 ([29]). Let H be a Hilbert space, let C be a nonempty,

closed and convex subset of H. Let T be a mappings of C into itself. If

A(T)\neq\emptyset , then F(T)\neq\emptyset.

Lemma 3.2 ([29]). Let H be a Hilbert space, let C be a nonempty subsetof H. Let T be a mappings of C into H. Then, A(T) is a closed andconvex subset of H.

We also have the following lemma (see also [12, 29

Lemma 3.3 ([29]). Let H be a Hilbert space, let C be a nonempty subsetof H. Let T be a mappings of C into H. Let \{u_{n}\} be a sequence in H
such that\varlimsup_{n\rightarrow\infty}\{(u_{n}-y)+(u_{n}-Ty), y- Ty\}\leq 0

for all y\in C . If a subsequence {un:} of \{u_{n}\} converges weakly to u\in H,
then u\in A(T) .
To prove our main results, we need the following lemma (see [5]; seealso [31]).
Lemma 3.4. Let \{s_{n}\} be a sequence of nonnegative real numbers, let\{$\alpha$_{n}\} be a sequence of [0 , 1] with \displaystyle \sum_{n=1}^{\infty}$\alpha$_{n}=\infty . Let \{$\beta$_{n}\} be a sequence ofnonnegative real numbers with \displaystyle \sum_{n=1}^{\infty}$\beta$_{n}<\infty and let \{$\gamma$_{n}\} be a sequenceof real numbers with \overline{\mathrm{h}\mathrm{m} _{n\rightar ow\infty}$\gamma$_{n}\leq 0 . Suppose thats_{n+1}\leq(1-$\alpha$_{n})s_{n}+$\alpha$_{n}$\gamma$_{n}+$\beta$_{n}for all n\in \mathrm{N} . Then, \mathrm{h}\mathrm{m}_{n\rightarrow\infty}s_{n}=0.
4. ACUTE POINTS AND CONVERGENCE THEOREMSIn this section, we prove convergence theorems by using the conceptof k‐acute points of a mapping for k \in [0 , 1 ] . Let C be a subset of a
Hilbert space H and let T be a mapping of C into H . A mapping T issaid to be L‐‐Lipschitzian if \Vert Tx-Ty\Vert \leq L\Vert x-y\Vert for any x, y \in C,

where L\in[0, \infty). Usually, T is said to be quasi‐nonexpansive if(1) F(T)\neq\emptyset, (2) \Vert Tx-v||\leq\Vert x-v\Vert for x\in C, v\in F(T) .

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Let I be the identity mapping on C. Usually, T is said to be hemi‐
contractive if(2) \Vert Tx-v\Vert^{2}\leq\Vert x-v\Vert^{2}+\Vert x-Tx\Vert^{2}(1) F(T)\neq\emptyset ,for x\in C, v\in F(T) .

These concepts depend on the condition F(T)\neq\emptyset . Usually, T is said tobe k‐demi‐contractive if(1)

F(T)\neq\emptyset ,(2)\Vert Tx-v||^{2}\leq\Vert x-v\Vert^{2}+k\Vert x-Tx\Vert^{2} for x\in C, v\in F(T) .

We also call T a demi‐contraction if T is a k-\mathrm{d}\mathrm{e}\mathrm{m}\mathrm{i}\precontraction for somek\in[0 , 1 ) . Assume F(T)\neq\emptyset.
Let k\in[0 , 1 ] . We define the set of k‐acute points \mathcal{A}_{k}(T) of T by\mathcal{A}_{k}(T)= { v\in H : \Vert Tx-v\Vert^{2}\leq\Vert x-v\Vert^{2}+k||x-Tx\Vert^{2}

for all x\in C }.

We denote A_{0}(T) by A(T) because A_{0}(T) and attractive points set of T
are the same. We denote \mathcal{A}_{1}(T) by A(T) , that is,\mathcal{A}(T)= { v\in H : \Vert Tx-v\Vert^{2}\leq\Vert x-v\Vert^{2}+\Vert x-Tx\Vert^{2}

for all x\in C }.

Now, we get the following convergence theorems [13]. We consider weak
convergence theorems in the case A(S)\neq\emptyset and F(S)\subset A(S) . To havethe following results, we have to assume demicloseness at 0 of I-S.

Theorem 4.1 ([13]). Let a, b\in(0,1) with a\leq b and \{a_{n}\} be a sequencein [a, b] . Let C be a weakly closed subset of a Hilbert space H. Let S bea self‐mapping on C such that F(S) \subset A(S) , A(S)\neq\emptyset , and I-S isdemiclosed at 0 . Suppose there is a sequence \{u_{n}\} in C such thatu_{n+1}=a_{n}u_{n}+(1-a_{n})Su_{n} forn\in \mathrm{N}.
Then, \{u_{n}\} converges weakly to some u\in F(S) .

Theorem 4.2 ([13]). Let a, b\in(0,1) with a\leq b and \{a_{n}\} be a sequencein [a, b] . Let C be a weakly closed subset of a Hilbert space H and T be aself‐mapping on C such that I-T is demiclosed at 0 . Assume that oneof the followings hold.

(1) T is hemi‐contractive with A(T)\neq\emptyset. S is the mapping definedby S=T.

(2) T is k‐demi−contractive. S is the mapping defined by S=kI+(1-k)T.
T(3) is quasi‐nonexpansive. S is the mapping defined by S=T.

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Suppose S is a self‐mapping on C and there \upar ow\dot{s} a sequence \{u_{n}\} in C

such thatfor n\in \mathrm{N}.
u_{n+1}=a_{n}u_{n}+(1-a_{n})Su_{n}Then, \{u_{n}\} converges weakly to some u\in F(T) .

Now, we get a nonlinear mean ergodic theorem (see also [14]).

Theorem 4. \mathrm{S} ([13]). Let k \in [0 , 1). Let C be a bounded subset of a
Hilbert space H. Let T be a k ‐strictly pseudo‐contractive self‐mappingon C. Let S be the mapping defined by Sx=(kI+(1-k)T)xforx\in C.
Assume that S is a self‐mapping on C. Let \{v_{n}\} and \{b_{n}\} be sequences
defined by v_{1}\in C andv_{n+1}=Sv_{n},Then the followings hold.

b_{n}=\displaystyle\frac{1}{n}\sum_{t=1}^{n}v_{t}forn\in N.

(1) \mathcal{A}_{k}(T) is non‐empty, closed and convex.
(2) \{b_{n}\} converges weakly to some u\in A_{k}(T) .
fUrthermore, if C is closed and convex then the followings hold.

(3) F(T) is non‐empty, closed and convex.
(4) \{b_{n}\} converges weakly to u\in F(T) .
5. STRONG CONVERGENCE THEOREMS FOR $\lambda$ ‐HyBRID MAPPINGSIn this section, we prove an attractive points theorem and strong
convergence to attractive points of uniformly asymptotically regular $\lambda$-

hybrid mappings in Hilbert spaces (see also [2, 7, 12, 18, 25, 26, 27, 29

Let C be a nonempty subset of H . Then, C is called star‐shaped if
there exists z\in C such that for any x\in C and any $\gamma$\in(0,1) ,$\gamma$ z+(1- $\gamma$)x\in C.
We say that a mapping T of C into itself is asymptotically regular if

n\rightar ow\infty 1\dot{\mathrm{r} \mathrm{m}\Vert T^{n+1}x-\mathcal{I}^{m}x\Vert=0
for all x\in C (see also [27]). We also say that a mapping T of C into

itself is uniformly asymptotically regular if for every bounded subset K

of C,\displaystyle \lim_{n\rightar ow\infty}\sup_{x\in K}| \mathcal{I}^{m+1}x-T^{n}x\Vert=0

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holds.

Lemma 5.1 ([6]). Let C be a nonempty subset of a Hilbert space H.

Let $\lambda$ \in \mathbb{R} be given. Let T be a $\lambda$ ‐hybrid mapping of C into itself. If
A(T)\neq\emptyset , {Tnx} is bouded for each x\in C.

We also get the following attractive point theorems (see also [12, 29Theorem 5.2 ([6]). Let H be a Hilbert space and let C be a nonempty

subset of H. Let $\lambda$ be a real number. Let T be a uniformly asymptoticallyregular $\lambda$ ‐hybrid mapping of C into itself. Suppose that {Tnx} is bounded
for some x\in C . Then, A(T)\neq\emptyset.

We obtain a strong convergence theorem of Halpern’s [21] type for$\lambda$‐hybrd mappings on a star‐shaped subset of H (see [6]).

Theorem 5.3 ([6]). Let H be a Hilbert space, let C be a star‐shaped
subset of H with center z \in C. Let $\lambda$ be a real number. Let T be a
uniformly asymptotically regular $\lambda$ ‐hybrid mapping of C into itself suchthat A(T)\neq\emptyset . Let \{m_{n}\} be a sequence in \mathrm{N} such that m_{m} \rightarrow\infty . Let\{x_{n}\} be a sequence in C defined by x_{1}\in C and

x_{n+1}=$\alpha$_{n}z+(1-$\alpha$_{n})T^{m_{n}}x_{n}for each n\in \mathrm{N} , where \{$\alpha$_{n}\}\subset[0 , 1 ] satisfies\displaystyle\lim_{n\rightar ow\infty}$\alpha$_{n}=0,\sum_{n=1}^{\infty}$\alpha$_{n}=\infty.

Then, \{x_{n}\} converges strongly to P_{A(T)}z , where P_{A(T)} is the metric pro‐jection from H onto A(T) .
Using Theorem 5.2, we obtain the following fixed point theorem,

Theorem 5.4 ([6]). Let H be a Hilbert space and let C be a closed andstar‐shaped subset of H. Let $\lambda$ be a real number. Let T be a uniformly
asymptotically regular $\lambda$ ‐hybrid mapping of C into itself. Suppose that

{Tnx} is bounded for some x\in C . Then, F(T)\neq\emptyset_{-}Using Theorem 5.3, we also get the following strong convergence theo‐

rem for $\lambda$‐hybrid mappings on a star‐shaped subset of H (see [21, 30, 31

Theorem 5.5 ([6]). Let H be a Hilbert space, let C be a closed and star‐
shaped subset of H with center z\in C. Let $\lambda$ be a real number. Let T

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be a uniformly asymptotically regular $\lambda$ ‐hybrid mapping of C into itself
such that F(T)\neq\emptyset . Let \{m_{n}\} be a sequence in \mathrm{N} such that m_{n}\rightarrow\infty.
Let \{x_{n}\} be a sequence in C defined by x_{1}\in C and
x_{n+1}=$\alpha$_{n}z+(1-$\alpha$_{n})T^{m_{n}}x_{n}for each n\in \mathrm{N} , where \{$\alpha$_{n}\}\subset[0,\cdot 1] satisfies\displaystyle\lim_{n\rightar ow\infty}$\alpha$_{n}=0,\sum_{n=1}^{\infty}$\alpha$_{n}=\infty.

Then, \{x_{n}\} converges strongly to u_{0} , where \displaystyle \Vert u_{0}-z\Vert=\min\{\Vert u-z\Vert : u\in

F(T)\}We also have the following strong convergence theorem.

Theorem 5.6 ([6]). Let H be a Hilbert space, let C be a nonempty subset

ofH . Let $\lambda$ be a real number. Let T be a uniformly asymptotically regular
Let \{m_{n}\} be asequence in \mathrm{N} such that m_{n}\rightarrow\infty . Let \{x_{n}\} be a sequence in C definedby x_{1}EC and$\lambda$ ‐hybnd mapping of C into itself such that A(T) \neq\emptyset .

x_{n+1}=$\alpha$_{n}z+(1-$\alpha$_{n})T^{m_{n}}x_{n}for each n\in \mathrm{N} , where \{$\alpha$_{n}\}\subset[0 , 1 ] satisfies\displaystyle\lim_{n\rightar ow\infty}$\alpha$_{n}=0,\sum_{n=1}^{\infty}$\alpha$_{n}=\infty.

If \{x_{n}\} is in C , then \{x_{n}\} converges strongly to u_{0}\in A(T) , where u_{0}=

P_{A(T)}.

ACKNOWLEDGEMENTS

The author is supported by Grand‐in‐Aid for Scientific Research No. 26400196
from Japan Society for the Promotion of Science.
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(S. Atsushiba) DEPARTMENT OF MATHEMATrCS, GRADUATE SCHOOL OF EDUCA‐
TION, UNIVERSITY OF YAMANASHI, 4‐4‐37, TAKEDA KOFU, YAMANASHI 400‐8510,
JAPAN
E‐mail address:

asachikoeyamanashi. ac. jp