fixed point theorems for nonexpansive mappings ... - Semantic Scholar

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, July 1975

FIXED POINT THEOREMS FOR NONEXPANSIVEMAPPINGS SATISFYING CERTAIN BOUNDARYCONDITIONS1 W. A. KIRK ABSTRACT. space

Let

X with

K be a bounded

int K 40,

respect

to nonexpansive

\\U(x)

- U(y)\\ < ||* - y||,

closed

convex

and suppose

K has

self-mappings

(i.e.,

x,y

€ K).

Let

the

subset

fixed

mappings

T: K -X

of a Banach

point

property

U: K^K

with

such that

be nonexpansive

and

satisfy

inf{||* - T(x)\\: x e boundary K, T'x) /Kl It is shown boundary

that

if in addition,

condition:

there

either

exists

(i)

T satisfies

z £ int K such

for all * e boundary K, A< 1, or (ii) infj||* fied, then T has a fixed point in K.

1. Introduction. this paper

Let

we consider

K —>X satisfying following

fixed

the domain

X be a Banach

the Leray-Schauder that

T(x) - z 4 M* - z)

- 7X*)||: * s K\ =0, is satis-

space

the nonexpansive

> 0.

and

K a subset

mappings,

a class

of X. In

of mappings

T:

||7T» - T(y)\\ < \\x - y\\, x, y £ K. In 1965 we obtained the

point

possess

theorem

for this

a property

class

Brodskil

by invoking

the assumption

and Mil man [2] call

'normal

that

struc-

ture'.

Theorem

1.1 [10].

of the Banach

space

nonexpansive

mapping

The assumption that

every

a point has been sets

studied

as well

K be a nonempty

subset

that

weakly

K has normal

T: K —►K has a fixed of normal

convex

x such

Let

X and suppose

structure

compact

has positive

one which diameter

sup |||x - y||: y £ A] < diam A. This

in some detail

as for bounded

by several

convex

subsets

Received by the editors March 18, 1974. AMS (MOS) subject classifications (1970).

Then

subset

every

point.

is a technical

A of X which

convex

structure.

authors,

must

contain

assumption,

always

of uniformly

asserts

holds

convex

which for convex

spaces

Primary 47H10.

Key words and phrases. Nonexpansive mapping, fixed Schauder boundary condition. ' Research supported by National Science Foundation

point

theorem,

grant

GP-18045.

Leray-

Copyright © 1975, American Mathematical Society

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143

144

W. A. KIRK

(cf. [2], [13]). It was in this latter setting also

in 1965, independently

obtained

that F. Browder and D. Göhde,

the same

result

as Theorem

1, 1 (see

[3], [8]). A number derived

of the subsequent

by weakening

it is shown

that

any convex

set containing

H back

into

mappings'

this

that

scribed

of the above

in Theorem

fixed

tions

of K —' K, the methods

also

such

weakening

2. The

of the assumption

same

condition is a weakening

the stronger

boundary

condition

The mapping (L-S)

in this

There

of K distinct

from

as pre-

result,

semigroup

paper

T:

if x £ K

may be taken

to a reduction in this

which

of transforma-

of the problem is to note

T: K —' K is possible,

that

to a

one which

setting.

(L-S).

In contrast

to the weakenings

of the assumption

uniformly

imposed

convex

z £ int K such

that

de-

T: K —' K which

setting;

by Browder

T: K —>X is said to satisfy exists

1.Î that

of Crandall's

amount

for the pseudo-

[9] (i.e.,

a point

for a commutative

Our purpose

there

to require

contains

of K in

Even more

(also

in Theorem

the setting

of proof

1.1.

boundary

above

Schauder

theorem

of Theorem

may be formulated

scribed pears

point

[6].

shown

and Bergman

1.1 and with the exception

a common

further

[x, T(x)]

generalizations

uses

an application

assume

of Halpern

T(x) 4 x then the segment

x). In each

boundary

H is

'pseudo-contractive

[18] have

one only need

been

in [ll]

T: K —' H, where

[l] and M. G. Crandall

in the sense

1.1 have

T map the relative

and G. Vidossich

mappings)

to

for the more general

in Assad-Kirk

K —• X be 'inward'

of Theorem

T: K —' K. For example

may be weakened

results

[l6]

that

K, provided

K. Similar

S. Reich

contractive

and

assumption

are found

recently,

generalizations

the assumption

this

ap-

is the Leray-

in [A] and defined

as follows:

(L-S) on d C K if:

T(x) - z 4 p(x - z) for x £ d

and p y I. This

assumption

fixed point

theorems

[12], [14], [15],

has been

imposed

for 'condensing'

[20]).

Browder's

by several

authors

recently

and 'zs-set contractive'

original

result

in proving

mappings

(for nonexpansive

(e.g.

mappings)

is the following: Theorem uniformly

2.1 [A]. Suppose

convex

is nonexpansive

Banach

K is a bounded

space

and satisfies

X with

closed

convex

0 £ int K, and suppose

subset

of a

T: K —• X

Tx 4 px for x in the boundary of K and p > 1.

Then T has a fixed point in K. The

above

theorem

(stated

in [4] for the more general


semicontractive

FIXED POINT THEOREMS mappings)

is not derived

from the earlier

but rather

it is a consequence

(Theorem

3 of [4]).

crucial

use

of the structure

3. Results. tially

pose

of the condition

this

(Theorem

theorem

(L-S) in Theorem

report.

in addition

3.1.

K be a bounded

that

on K

Browder

makes

on X.

3.1) we find

condition

T: K —' K,

/ - T is demiclosed latter

the assumption

motivate

below

for mappings

convexity

to replace

assumption

another

of uniform

attempt

orem 1.1 we give

that

of this

The use

successful

1.1 with this

theorems

of the fact

It is in the proof

145

2.1 and our par-

T: K —' K in Theorem

In the generalization

of The-

it necessary

to im-

to the condition

(L-S)

however

on the boundary,

dK, of K. Theorem space

X with

respect

Let

int K 4 0, and suppose

to nonexpansive

closed K has

s elf-map pings.

convex

subset

the fixed

point

Suppose

of a Banach property

with

T: K —* X is nonexpansive

and suppose:

(i) T satisfies

(L-S) 072 dK, and

(ii) infi||*- T(*)||: x £ dK, T(x) 4 K] > 0. Then T has a fixed point in K. The question the absence

remains

of this

the assumption

extra

Theorem

Theorem

assumption

of uniform

with the assumptions

int K 40,

open as to whether

on K and

3.1 does

1.1 because

X of Theorem

Theorem space

X with

respect

3.2.

Let

int K 40,

true in

1.1.

whether

2.1 can be replaced

Note,

however,

that

assumptions

(i) and (ii) of Theorem

if

on T in.

3-1 hold if

if T: dK —*K.

consequence

of Theorem

K be a bounded

to nonexpansive

remains

it is not known

the domain/range

both conditions

as a direct

and thus

theorem

on X in Theorem

weaken

T: K —' K, or even more generally We derive

(ii),

convexity

this

and suppose

closed K has

self-mappings.

3.1 the following

convex the fixed

subset point

result.

of the Banach property

with

If T: K —"X is nonexpansive

and

satisfies

(*)

infill*- 7*||: x £ K\ < infill*- 7x||: * e dK, T* //Ci,

then T has a fixed point in K. Proof

of Theorem

3.1.

T(x) - z 4 p(x - z) tot in assuming

z = 0.

First

By assumption

there

exists

x £ d = dK, p > 1, and there we show

(iii) z-=infi||T(*)-zM(*)||:

that

together

this

z £ int K such

is no loss fact

* £ d, T(x) 4 K, /i> l}>0.


that

in generality

and (ii)

imply

146

W. A. KIRK

For,

suppose

r = 0. Then

(1, oo) such

there

that

exists

bounded

||11T(x

p > 0 such

there

exists

there

exist

i*

i C d with

) - a' 72x 22"|| —> 0 as 7Z

that

||x

a number

T(x ) 4 K, and

n —> oo. However,

since

|| > p, n = 1, 2, • • • . Also M such

that

\p

iC

0 4 d

because

K is

if x £ K, ||*|| < M; thus

||r(*)|| < ||t(x) - t(o)\\ + ||r(o)|| < ||*|| + ||r(o)|| oo. It follows

(ii)

72=1,2,...,

is contradicted.

we may suppose

that

So we may suppose

to a contradiction,

observe

\\px

\p

i converges,

say

- T(x )|| —>0, and if p = 1 then

p > 1. To show

that

this

also

leads

that

||zi* nr „ - px r- mii|| = \\px ur~ „ - T(x n ) + T(x „ ) - T(x m ) + T(x m ) - rax m"\

pxQ - T(x

as

72, 772-,

which

oo.

must con-

) = 0, and this

contra-

is proved.

the proof,

let

r, = (M + ||T(0)||)/p

with

M and

p as above,

and let À £ (0, 1) satisfy

Xr-(1Let y* £ (1 - X)K and define

\)M(r, + A) > 0.

Ux: K —>X by Ux(x) = kT(x) + y*. Suppose

T(x) 4 K. Then if Ux(x) = a* for * e d and a y 1 we have

a= ||AT(*) + y*||/||*|| and hence

(using

\\Ux(x)-

< (M + || T(0)||)/p = r,,

(iii))

ax\\ y A||T(*)-

> Ar-(1

a*|| - ||a*-Aa*||

- ||y*||

- A)a||*|| - ||y*|| > Ar-(l

- A)aM - (l - \)M

= Ar - (1 - A)M(a + 1) > Ar - (1 - A)M(t/+ l) > 0. Also,

if T(x)

£ K then,

Ux(x) £ K. Thus

since

in either

case

y* = (1 - X)z fot some

Ux(x) 4 ax

z £ K, it follows

if x £ d and


a > 1. Since

that

Ux:

FIXED POINT THEOREMS K

'X

is a contraction

there exists

mapping,

Theorem

147

5 of [4] (also

see [14]) implies

that

x* £ K such that Ux(x*) = **; hence AT(**) + y* = ** and (/ - AT)(**) =

y*, proving

/ - AT maps

expansiveness

K onto

(1 -X)K.

But (/ —AT)-

exists

because

non-

of T implies

||(/ - XT)(u) -(I-

XT)(v)\\ > \\u - v\\ - A||7t«) - 7tv)|| > (1 - A)||zv- v\\, u,v

£ K.

Thus (/ - AT)-1: (1 - X)K — K, and it follows that if H = (1 - A)(/ - AT)"1 then

II: (1 - A)K -> (1 - A)K, and moreover

point

(+) shows

property

that

with respect

// is nonexpansive.

to nonexpansive

Since

K has

self-mappings,

the fixed

the mapping

H:

K — K defined by

H(x) = (1 - A)" ^((l has

a nonempty

point

set

fixed

point

- A)*),

set in K. It follows

S in (1 - X)K. Thus >T has

(1 - A)-

x £ K, that

H has a nonempty

5 as a fixed

point

set,

fixed for

if H(z) = z then

(/ - AT)U/(1 - A))= z, yielding

z/(l - A)- ATU/U - A))= z, and hence

T(z/(1 - A)) = z/(l

Theorem this

3.2 actually

observe

shows

that

that(*) it also

Proposition.

is a direct

implies implies

Let

- A). consequence

condition condition

of Theorem

(ii) immediately,

K be a closed

convex

subset

of a Banach

and satisfies

(*), then

(L-S) 072 dK. Choose z £ int K so that \\z - T(z)|| < r where

r= infill*- T(*)||: * e dK, T(x) 1. Then

|| T(x) - *|| = ||(i - ß)(z - *)|| = (p - l)\\z - *||.

But

|| T(x) - z\\ < || T(x) - T(z)\\ + || TU) - z\\ < ||* -z\\+r License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

X with T satisfies

148

W. A. KIRK

and

||T(*)

- z\\ = p\\x - z||,

so p\\x - z\\ < \\x - z\\ + r. Thus

(pand this

implies

||T(*)

l)||*-z||

- *|| < r. This

T(x) --z = p(x - z), p y 1, implies

As a corollary noticed

more.

of Theorem

by Browder[5].

It actually

ciently

only if T(x)

£ K, but

4 K — a contradiction.

3.1 we have result

the following

which

is also

that for T lipschitzian,

known given

result,

in [19],

first yields

p £ int K, and t > 0 suffi-

small, tT + (1 - t)p maps K into K (if T: d/C — K).

Corollary. the Banach

spect

can happen

T(x)

Browder's

shows