Some results on invariant approximation

SOME RESULTS

ON

INVARIANT

APPROXIMATION

ANTONIO CARBONE Dipartimento di Matematica

UniversitidellaCalabria

87036

Arcavacata

di Rende

(Cosenza)

Italy

(Received December 4, 1992 and in revised form May 10, 1993)

ABSTRACT. In

this paper results on Invariant approximations, extending and unifying

earlierresults aregiven. Several interestingresultsarederived as corollaries.

KEY

WORDS AND

PHRASES.

Nonexpansive map, set of best approximants, starshaped set, demiclosed,approximativelycompact set.

1991

AMS

MATHEMATICS SUBJECT CLASSIFICATION

CODES.

Secondary 54H25.

Primary 47H10,

1.

INTRODUCTION

Severalmathematicians have given results dealingwith invariant approximation. The

earlierresultsdue to Meinardus

[7]

and Brosowski

[1]

have beenthemain sourceof inspiration

for researchers in the field.

The followingisdue to Brosowski.

Let

f

beacontractive//nearoperator ofanormedspaceX.

Let

Xo E

X

beaninvariant point and

C

aninvariantset under

f

I[the set of best C-approximants to

xo

isnonempty compact andconvexthen it containsan

f-invariaaat

point.

Recently Carbone

[2], [3],

Habiniak

[5],

Hicks and Humphries

[6],

Narang

[8],

Po

and Mariadoss

[9]

Sahab,

Khan and

Sessa

[10],

K.L.Singh

[11],

Singh

[12],

[13]

and

Subrahmanyaan

[15]

gaveextensionsof the above

theorem,

andprovedsomefurther related results.

The purpose of this paperis toextendand unifyearlierresults.

We

need the

following

preliminaries.

If

f:

X

X

where

X

is anormed linear space, and

Ilfx

fYl[

<-

x

Yll

for x,yE

X,

then

f

is calledanonexpansive map.

We

denote by

F(f),

the setof fixed points of

f.

Ifx0

fxo,

and

Ilfx-

f011

<

I1 x011

forallx 5

X

then

f

is said to bequasi-nonexpansive.

A

_{quasi-nonexpansive map}need not

becontinuous.

Let

C

be a

closed,

convex subset ofa normed linear space

X.

The set of best

C-approximants tox consistsof all y

C

suchthat

Ilu

11

d(, C)

inf{llz

zll

z

e

c}.

set

C

is said to _{be star-shapedifthere exists at least onepoint p}

C

_{such that}

A)p

C

for allx E

C

and0

< A <

1.

In

thiscasepis calledthestar-centerof

C.

is said tobe demiclosedif x, x weaklyand

fx,

ystrongly imply that y

fx.

The

following

result will beused inthe proof ofour

theorem

(see

Subrahmanyam

[15],

L(’t

X

bea Banach space"ad

C

o.mpt.r w(.,klycouq)act

starshaped

s,b,’tof

X

If

f

C Cisonexp,si’’ and

I-

.f

isdemiclosed then

f

hasafixed point

Note Theall,wetheoremh;beenals

pr(,ved

when

.f

C

X

isnonexpansive aleith,’,

f(0C)

C

(,

f

isan inwar(lmap.

n.elllark

1.

In

case

C

s acnnpact, convex

sutset

of

X

and

f

C

X

anonexpansive ap witlt

(OC)

C_

C,

then

f

has afixed point. This is valid even when

C

is

star-shaped

lint n,t

n.cessarily cmvex.

2 If Cisxveakly compactconvexand

I-f

isdemiclosedthenanonexpansivemap

f

C

-\

with

f(0C)

C_

C

hasafixed point.

Wecan replaceconvexity

by star-shaped

condition withutchanging the conclusion.

If

X

isa normedlinear spaceand

M"

is asubspace of

X.

andx0 E

X,

then

d(.r0,

M)

,,,/{llx0

yl]

y

e M}

denotes the distance of x0 to the set

M.

PM(xo)

{tl

E

M

II’ro

11

d(xo,M)}

is the set of best approximations of.r0 in

M.

It is wellkm,wn that

theset

P.u(x)

is closed andconvexfor anysubspace

M

of

X

(for

details see

[41).

2.

MAIN RESULTS

Now

westateourresult.

THEOREM Let

X

bea Banach space.

Suppose

f"

X

X

issuch that

f(OC)

C_ (’ and

f

xo

zo

forZo

X.

I D,

theset of best C-approximants toZo, is eat,’ly compact

star-shapedand

i)

f

isnonexpansiveon

D;

iii) I-

f

isdemiclosedon

D

then

f

hasatxedpoint closest to3:o.

PROOF

First weshow that

f"

D

D,

i.e.

f

isa self-mapon

D.

Let

yGD. Then

Thisimplies that fy isalso closest tox0 so fyco__

D.

Let

pbethe star centerof

D.

Define

f,,.x

k,,fx +(1-k,)p,

where

{k,

}

isasequence of

reals,

0

<

k,,

<

1converging

to 1 as n cxfor x

D.

Each

f,

is contraction on

D

with contraction constant

k, <

1, so

ft:,.x,,.

x,,,

for n 1,2,.

(see

Subrahmanyam

[15]

for

details).

Since

D

is weakly compact therefore

{xk.}

has a

convergent

subsequence convergint weaklyto

:

say.

For

the sake ofconveniencelet

x,,

:.

Now

Ilxt,

fx.,ll

Ilft,x,

Yx:,ll

(k,

1)llfx,,ll

+

(1

k,)p

0as

f

x

,

is

bounded).

Since

I-

f

is demiclosed so 0

(I-

f):,

i.e., 2

f’.

Thus

f

has a fixed point .r closest to x0.

Note

same conclusion using Schauder fixed point theorem.

In

this case, we have to assume that

f:

DD. In

ourtheorem

f

isnonexpansiveso

f:

DD

follows.

\Ve derivethe fillwingrcsldtsa. corollaries.

Corollary 1. If

C

iscnnpact, cmvex then

I-

f

dcmclsed

is not needed and we the esult (see

[19]).

Corollay 2. If

C

isweakly compact, convex and other conditions are the same tlwn weget the result.

Recall that everyconvexset isstarshaped.

However,

theconverseis not true.

In

thissection we discuss a result of

Rao

ad Mariadoss

[9]

givenfor approximatively

compactsets.

A

set

C

C_

X

is calledapproximatively compactif for each x G

X

and every sequence

{x,

in

C

withlim,,_oo

IIz

z,,

d(z,

C),

thereexistsasubsequence

{x,,

converging to anelement of

C.

A

compact set is always approximatively compact but not conversely.

A

closed unit

ball in infinite dimensional Hilbert space isapproximatively compact but not compact.

Let C

beanonempty approximatively compact subset of

X

and

P"

X

2 c ametric

projection of

X

onto

C

defined by

Pc(x)

{g

C

[[x

g[I

d(x,

C)}.

Then

Pc(x)

7

O,

closed subset of

C.

In fact,

Pc(x)

is compact.

Thefollowingis dueto

Rao

and Mariadoss

[9].

Let

X

beaBanachspaceand

f

X

X

acontinuous

map.Let

C

beanapproximatively compact subset of

X

and an

f-invariant

set.

Further,

let

f

zo

zo,

zo

X

and

f

satisfy thefollowingconditions:

Ilfz

fll

<

cl[

l[

/

(11

fzll

/

II

f[[)

/

(11

fy[[

/

I1

fx[[)

for

or,/3,

3’positive witha

+

2/3

+

27

<

1,for allz,g

X.

Ifthe set of best C-approximants to

xo

isnonempty

starshaped

then it hasan

f-invariant

point closest to

zo.

Note

It

is easytoderive that

f

satisfiesthecondition

Ilfm-

fm0[[

=<

II

m0ll

for 1

x,

(fXo

x0),

since c

+

2/3

+

27

<

1, and since

C

is approximatively compact so the set of best C-approximants toz iscompact.

Recently Carbone

[3]

proved thefollowing:

THEOREM

C

Let

f"

X

X

beamappingon aBanach space

X

satisfying the following

condition:

[Ifx

fll

<

11

x

ull

+

(ll

x

fxl[

/

IlY

fYll)/ 7(11

x

fYl[

+ IlY

fzll)

for

a,/3,

7positive witha,

+

2/3

+

27

_<

1.

Let

fxo

z0, forx0 G

X

and

C

asubset of

X

such that

f(OC) G

C.

Ifthe set

D

of

best C-approximants to

xo

is

dosed, strashaped

and

f(D)

is

compact

with

f

continuouson

D,

then

f

hasafixed point closest to

xo.

We givethefollowingto derive

[91

as aspecialcase of

[31.

Remarks

2. If

C

is approximatively compact then

D

is compact and

f(D)

iscompactsincea contin-uousinageofacompactset iscompact.

3.

C

neednot be

f-invariant.

Only

f(c3C)

C_

C

is required.

In

this sectinresultsinalocallyconvexHausdorfftopologicalvector space

E

aregiven.

L’t C

beanonempty subset of

E

and let p beacontinuous seminormon

E. For

.r

E

define

d,,(z,C)

inf

{p(:r

V)

V

C}

and

Pc()

{, e

c:

p(

u)

a,(, c)}.

A

mapping

f

C

C

issaidtobe p-nonexpansive if for all z,//

C

p(fx-yy) <p(z-y),

pe P

(P:

fanily of continuous

seminorms);

f

C

C

is asymptotically nonexpansive ifthere

exists

{k,}

with

k,

n such that

p(f"

z

f"

y)

k,p(z

V)

for all z,

C,

p

P,

n

1,2,.

It

is assumed that

k,

d

k,

k,+

for

n 1,2,.

[17].

f

isuniformlyasymptotically

regular

ifforeach p

P

and

>

0there exists a

N(p,

) (=N

say)

such that

p(f"z-

f"+z)

<

foall

n

>

N

andz

C.

Vijayaraju

[18]

proved

the following:

THEOREM

V

Let C

beanonemptysubset of

E.Let

Zo

E

d

f

C

C

ymptot-icMlynonexpsive,

uNformly

ymptoticM1y

rel

map.

Suppose

that theset

D

of best C-approximts to

zo

isnonempty, wely compact, stshaped dinvtunder

f.

further,

I-

f

isdemidosedthen

f

haed point closest to

zo.

For

further resultsonfixed points oneshould refer to

[141, [161

We

givethefollowinginlocallyconvexHausdorff topologicalvector space

E

improving Theorem

V.

THEOREM

3

Let C

beasubset of

E,

f

xo

zo,

:co

E

and

f(OC)

C_

C. Let

f

E

E

beanasymptoticM1ynonexpansiveuniformly asymptotically regularmap.

Assume

that the

set

D

of bestC-approximantsto

xo

isnonemptyweakly compact, starshapedand invariant

under

f.

If

I-

f

isdemiclosed then

f

hasatixed point closest to

:co.

Proofisonthesamelinesas in

[18].

Note

We

do not need

f"

C

C

asisgivenin

[18],

only

f(C)

_

C

servesthepurpose. Furtherrelated resultsare alsogivenin

[12].

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