ATTRACTIVE POINTS, ACUTE POINTS AND APPROXIMATION OF COMMON FIXED POINTS OF FAMILIES OF NONLINEAR MAPPINGS RELATED TO HYBRID MAPPINGS (The structure of function spaces and its environment)

ATTRACTIVE

_POINTS,

ACUTE POINTS AND

APPROXIMATION OF COMMON FIXED POINTS OF

FAMILIES OF NONLINEAR MAPPINGS RELATED TO

HYBRID MAPPINGS

SACHIKO ATSUSHIBA

ABSTRACT. In this _paper, we _prove an attractive points theorem

and_strongconvergencetheorems ofHalpern’stype

[20]

for_uniformly asymptotically regular $\lambda$_{‐hybrid mappings}inastar‐shapedsubset of aHilbert _space. Usingtheseresults, weobtainafixedpointtheorem

andsome_{strongconvergence}theorems.

1. INTRODUCTION

Let H be areal Hilbert _space with inner

product

\rangle

and norm

\Vert\cdot\Vert

and let C be a _nonempty subset of H. 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

_[28]

ofT,

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 Cis called

_nonexpansive

if

_{\Vert Tx-Ty\Vert}

\leq

_{\Vert x-y\Vert}

for all _x,

_{y\in C}

.

Kocourek,

Takahashi andYao

[22]

introduced abroad class of nonlinear

_mappings

called

_generalized

_hybrid

which

_containing

non‐

expansive mappings,

nonspreading

mappings,

and

_hybrid

_mappings

in a

Hilbert _space.

_They

_proved

a mean _convergence theorem for

generalized

hybrid

mappings

which

_generalizes

Baillon’s nonlinear

_ergodic

theorem

[13].

Aoyama, Iemoto,

Kohsaka and Takahashi

_[4]

introduced the class of $\lambda$

_‐hybrid

_mappings

in a Hilbert_space. This class obtain the classes of

nonexpansive mappings,

nonspreading

mappings,

and

_hybrid

_mappings

2010 Mathematics _{Subject Classification. Primary 47\mathrm{H}09,} 47\mathrm{H}10.

Keywords and_phrases. Fixed_point,attractive_{point, iteration, nonexpansivemap‐}

in a Hilbert _space.

They

proved

fixed

point

theorems and mean con‐

vergence theorems for such

mappings.

Motivated

by

Baillon

[13],

and

Kocourek,

Takahashi and Yao

_[22],

Takahashi and Takeuchi

_[28]

intro‐

ducedthe_concept ofattractive

_points

ofanonlinear

mapping

inaHilbert

space and

they proved

a mean _convergence theorem of Baillon’s type

without

_convexity

for

_generalized

_hybrid

_mappings.

In

_1992,

Wittmann

[29]

proved

the

_following

_{strong convergence}theorems of

_Halpern’s

_type

[20]

in a Hilbert space;

Theorem 1.1. Let C be a

nonempty

closed convex subset

of

a Hilbert

spaceH. Let T be a

nonexpansive mapping

ofC

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\rightarrow\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(T)}x

, where

P_{F(T)}

is the metricpro‐

jection

from

H onto

_F(T)

.

Motivated

_by

Takahashi and Takeuchi

_[28],

Akashi and Takahashi

_[2]

proved

a strong _convergence theorem of

Halpern’s

type

[20]

for nonex‐

pansive

mappings

in a

star‐shaped

subset of a Hilbert _space. On the

other

_hand,

_{Domingues Benavides,}

Acedo and Xu

_[17]

_proved

_strong convergencetheoremsof

Halpern’s

type

[20]

for

uniformly

asymptotically

regular

one‐parameter

nonexpansive semigroups.

The author

_[8]

studied

Halpern’s

type

iterations for

_{nonexpansive semigroups}

and

_proved

_strong convergence theoremsfor

uniformly asymptotically regular

nonexpansive

semigroups

in Hilbert spaces

(see

also

[1,

7, 9, 17, 25,

26

In this _paper, we _prove an attractive

points

theorem and

strong

con‐

vergence theorems of

Halpern’s

type

[20]

for

uniformly

asymptotically

regular

$\lambda$

_‐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.

2. PRELIMINARIES AND NOTATIONS

Throughout

this _paper, we denote

by

\mathrm{N} and \mathbb{R} the set of all

_positive

\mathbb{Z}^{+} and\mathbb{R}^{+} theset of all

_{nonnegative integers}

and theset of all nonnega‐ tivereal

_{numbers, respectively.}

Let H be areal Hilbert _space with inner

product

\rangle

and norm

\Vert

. We know the

following

basic

_equality

from

[26].

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- $\lambda$)||y\Vert^{2}- $\lambda$(1- $\lambda$)\Vert 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

\{(x-y)+(x-w) , y-w\rangle

=\langle(x-y)+(x-w) , (y-x)+(x-w)\rangle

= \Vert x-w\Vert^{2}-||x-y\Vert^{2}+\{x-y, x-w\rangle+\{x-w, y-x\rangle

=\Vert x-w\Vert^{2}-\Vert x-y\Vert^{2}

Let C be aclosed 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\Vert

for all

_{y\in C}

. The

mapping P_{C}

is called the metric

projection

ofH onto C. It is characterized

by

\{P_{C}x-y, x-P_{C}x\rangle \geq 0

for all

_{y\in C}

. See

[26]

formoredetails. The

following

resultiswell‐known

(see [26]).

Lemma 2.1. LetC be a

nonempty,

bounded,

closed and convexsubset

of

a Hilbert _space H and letT be a

nonexpansive mapping

of

C into

itself.

Thenf

F(T)\neq\emptyset.

We write x_{n} \rightarrow x

(or

\displaystyle \lim_{n\rightarrow\infty}x_{n}

= _x

)

to indicate that the _sequence

\{x_{n}\}

of vectors _{in H converges}

_strongly

to x. We also write x_{n}

-\triangle _x

(or

_{\displaystyle \mathrm{w}-\lim_{n\rightarrow\infty}x_{n}}

= _x

)

to indicate that the

sequence

\{x_{n}\}

of vectors in H

converges

weakly

to x. In a Hilbert space, it is well known that x_{n}- $\Delta$ x and

_{\Vert x_{n}\Vert \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\Vert}

for all x,

y\in C

. Let $\lambda$\in \mathbb{R} be

given.

Following

[4],

we saythat 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 \in} C. It is obvious that T is

1‐hybrid

if and

only

if T is

nonexpansive;

T is 0

_‐hybrid

if and

_only

if T is

_nonspreading

_[23];

T is

1/2‐hybrid

if and

_only

if T is

_hybrid

_[27]);

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- $\lambda$)

, thenT is $\lambda$

‐hybrid

if and

only

ifit is $\alpha$

‐nonexpansive

[3],

i.e.,

\Vert Tx-Ty\Vert^{2}\leq $\alpha$(\Vert x-Ty\Vert^{2}+\Vert Tx-y\Vert^{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 w -Tx\Vert}

\leq

_{\Vert w -y\Vert}

for all

w\in

F(T)

and x\in C.

By

Dotson

[16,

Theorem

1]

and Ithoh and Taka‐ hashi

_[21,

_Corollary

_1],

weknow that

F(T)

is closed convex whenever T

is

_{quasi‐nonexpansive.}

_Every

$\lambda$

_‐hybrid

with afixed

point

is

cleary

quasi‐

nonexpansive.

Thus,

the set of fixed

_point

of each $\lambda$

_‐hybrid

_mapping

is closed convex. The

_mapping

T is said to be

_firmly

_nenexpansive

if

\Vert Tx-Ty\Vert^{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} [14}

,

15, 18,

19

]

. It is known

[4,

Lemma

3.1]

that ifT is

_firmly

_{nenexpansive,}

then T is $\lambda$

_‐hybrid

for each $\lambda$\in

_[0

,1

].

3. LEMMAS

In this

_section,

we

give

some lemmas which are used in the

proofs

ofour main theorems. We have basic

_properties

of_attractive

_points

of

nonlinear

_mappings

in a Hilbert _space

(see [28]).

Lemma 3.1

_([28]).

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

_([28]).

LetH be a Hilbert_space, let C be a

nonempty

subset

of

H. Let T be a

mappings

of

C into H.

Then,

A(T)

is a closed and

convex subset

of

H.

Lemma 3.3

_([28]).

Let H be aHilbert_space, letC be a _nonemptysubset

of

H. Let T be a

mappings

of

C into H. Let

\{u_{n}\}

be a _sequence in H

such that

\varlimsup\langle(u_{n}-y)+(u_{n}-Ty)

,

y—Ty

)

\leq 0

n\rightarrow\infty

for

all

_{y\in C}

.

If

a

subsequence

\{u_{n_{i}}\}

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];

see

also

_[30]).

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

of

nonnegative

real numbers with

_{\displaystyle \sum_{n=1}^{\infty}$\beta$_{n}<\infty}

and let

_{\{$\gamma$_{n}\}}

be a _sequence

of

real numbers with

_{\varlimsup_{n\rightarrow\infty}$\gamma$_{n}\leq 0}

.

Suppose

that

s_{n+1}\leq(1-$\alpha$_{n})s_{n}+$\alpha$_{n}$\gamma$_{n}+$\beta$_{n}

for

all n\in \mathbb{N}.

Then,

\displaystyle \lim_{n\rightarrow\infty}s_{n}=0.

4. MAIN THEOREMS

Inthis

_section,

we _provean attractive

points

theorem and _strong con‐

vergence tocommonattractive

points

of

uniformly

asymptotically

regular

$\lambda$

_‐hybrid

_mappings

inHilbert_spaces

_(see

also

_[2,

_{7, 12, 17, 24, 25, 26,}

28 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 Cinto itself_is

asymptotically regular

if

\displaystyle \lim_{n\rightarrow\infty}\Vert T^{n+1}x-T^{n}x\Vert=0

for all x \in C

(see

also

[26]).

We also _say that a

mapping

T of C into

itselfis

_{uniformly asymptotically}

_regular

if for _every bounded subset K of

_C,

\displaystyle \lim_{n\rightarrow\infty}\sup_{x\in K}\Vert T^{n+1}x-T^{n}x\Vert=0

holds.

Lemma 4.1

_([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

We also

_get

the

_following

attractive

_point

theorems

_(see

also

_[12,

28 Theorem 4.2

_([6]).

Let H be a Hilbert _space and let C be a nonempty

subset

_of

H. Let $\lambda$ be a realnumber. Let T be a

uniformly

asymptotically

regular

$\lambda$

_‐hybrid

_mapping

_of

C into

_{itself. Suppose}

that

_{Tnx}

is bounded

for

some x\in C.

Thenf

A(T)\neq\emptyset.

We obtain a strong _convergence theorem of

Halpern’s

[20]

type for

$\lambda$

_‐hybrd

_mappings

on a

star‐shaped

subset of H

(see

[6]).

Theorem 4.3

_([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

such that

_A(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

,1

]

satisfies

\displaystyle \lim_{n\rightarrow\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

_4.2,

weobtain the

following

fixed

point

theorem.

Theorem 4.4

_([6]).

Let H be a Hilbert _space and let C be a closed and

star‐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

_4.3,

we also getthe

following

strong

_convergencetheo‐ remfor $\lambda$

‐hybrid

mappings

on a

star‐shaped

subset of H

(see [20,

29,

30

Theorem 4.5

_([6]).

LetH be a Hilbert_space, letC be a closed andstar‐

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

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

for

each n\in \mathrm{N}, where

\{$\alpha$_{n}\}\subset [0

,1

]

satisfies

\displaystyle \lim_{n\rightarrow\infty}$\alpha$_{n}=0, \sum_{n=1}^{\infty}$\alpha$_{n}=\infty.

Then,

\{x_{n}\}

converges

strongly

to u_{0_{2}} where

\Vert u_{0}-z\Vert =\displaystyle \min\{\Vert u-z\Vert

: u\in

F(T)\}

We also have the

_following

_{strong convergence theorem.}

Theorem4.6

_([6]).

LetH bea Hilbert_space, letC be a

nonempty

subset

of

H. Let $\lambda$ be areal number. Let T be a

uniformly asymptotically regular

$\lambda$

_‐hybrid

_mapping

_of

C into

_itself

such that

_A(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

,1

]

satisfies

\displaystyle \lim_{n\rightarrow\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)

DEPARTMENTOF _MATHEMATICS, GRADUATE SCHOOL OFEDUCA‐

TiON, UNIVERSITYOF_{YAMANASHI, 4‐4‐37,} TAKEDA_KOFU, YAMANASHI_400‐8510, JAPAN