Some fixed point theorems for nonconvex spaces

Internat. J. Math. & Math. Sci. VOL. 21 NO. (1998) 133-138

133

SOME

FIXED

POINT

THEOREMS

FOR

NONCONVEX SPACES

F.JAFARIandV.M. SEHGAL

Department

ofMathematics

University of Wyoming Laramie,

WY

82071-3036

(Received June 5, 1996 and in revised form November i, 1996)

ABSTRACT. We

give atheoremfor nonconvex topological vector spaces which yields the classical fixed pointtheorems of

Ky Fan,

Kim, Kaczynski, Kellyand Namiokaas immediate consequences, and proveanewfixedpoint theorem forset-valued mapsonarbitrarytopological

vector spaces.

KEY

WORDS

AND PHRASES:

Fixedpoints, nonconvextopological vectorspaces,

mul-tifunctions.

1991

AMS

SUBJECT CLASSIFICATION CODES: 47H10, 47H04,

46A16.

1.

INTRODUCTION.

In

1935, Tychonoff provedthefollowing celebrated result.

THEOREM

1.1.

If

X

is a nonempty convex and compact subset

of

a locally convex

topologicalvector space

E,

then anycontinuous map

f"

X

-+ X

hasa

fixed

point.

Even though

Theorem1.1 hasbeen the subjectof extensiveand sharp generalizations, the

question ofwhether Theorem1.1 istrue in

general

topologicalvector spaces still remains open.

Recently,the authorsencounteredpapersthat extend Theorem1.1 totopological vectorspaces

]E

having aseparating dual

]E*.

Thepurpose of this paper is to showthat this assumption

implieslocal convexity of

X

and consequently the results follow from known results.

Let

IF

be the scalar fields or

C

and

E

beatopologicalvector spaceover

F

withdual

*

(possibly

E*

{0)).

Let

T and

r

denote the original and weak topologies of ]E.

Note

that

r

is locallyconvexandif]E*

separates

points of

,

then

r

isHausdorff.

For

asubset

X

C_

]E,

(X, r,)

(resp.

(X, r))

denotes

X

withthe relative

topology

r,

(resp.

r).

2.

SOMETHING

OLD, SOMETHING NEW.

In

whatfollows

X

compactmeans that

X

iscompactin thetopology of ]E.

PROPOSITION

2.1.

Let

X

beanonempty compact subset

of

If(X,

r)

is

Hausdorff,

the,,,

(x,

-,)

(x,

-).

PROOF. By

definition, ’,

_c -.

Conversely, let

A

be ar-closed subset of

X.

Then

A

is

--compact,and hencer-compact. Since’,is

Hausdorff,

itfollowsthat

A

is

r-closed.

Thus

(X,

r)

C

(X, r)

and hence

(X,

154 F. JAFARI AND V. M. SEHGAL

Proposition 2.1 provides simpleproofs of the following results. Original proofs of these

resultsmakeuseofpartitions ofunityorother techniques.

COROLLARY

2.2

(Ky Fan

[2]).

Let

X

beanonempty compact andconvexsubset

of

_..

If IE*

separatesthe points

of

X,

then ever_{9 continuous} map

f

X

--+

X

hasa

fixed

poznt.

PROOF.

Since

*

separates the points of

X,

(X, r)

is

Hausdorff,

hence by Proposition

2.1,

(X,

r)=

(X,

r,o),

and thus

f: (X, r)

-+

(X, r)is

continuous. Since

X

isr-compact, by Theorem 1.1,

f

has afixedpoint.

The

following

resultis ageneralizationofCorollary2.2.

COROLLARY

2.3.

Let

X

beanonemptycompactandconvexsubset

ore

and

f

X

-->

be acontinuous

function.

If

IE*

separatesthe points

of

X,

then ezther

(a)

f

hasa

fixed

point, or

(b)

thereexists

:co

E

X

andav-continuous seminormpon

IE

suchthat 0

<

p(zo

f(xo))

min{p(z-

f(zo))

z

X}.

PROOF.

Since

(X,r,o)

is

Hausdorff,

by Proposition 2.1 it follows that

(X,

r)

(X,r).

Since%C_ r,

f

(X,

%)

-+

(E,

%)

iscontinuous. Since

X

is

r

compact,itfollowsby

Ky Fan

[2]

that either

(a)

f

hasafixed point,or

(b)

thereexistsz0

X

anda

r-continuous

seminorm pon

IE

with 0

<

p(zo-

I(zo))

min{p(z-

f(xo)):z

X}.

Note

that a

r

is continuous

seminormon

E

isar-continuous seminorm on

E.

o

As

animmediate consequence of the abovecorollary,wehave

COROLLARY

2.4

(Kaczynski

[4]).

Let X

be a nonempty compact convex subset

of

E

and

IE*

separate the points

of

X.

If

f

X

--4

E

is a continuous

function

such that

for

each z

e

X, f(z)

#

z, there ez,st

A

suchthat

lal

<

and

x

+

(I

A)f(x)

E

X,

(i)

then

f

has a

fixed

point.

PROOF. Assume

that

f

has no fixed points. Thenby Corollary 2.3, there exists z0E

X

andav-continuous seminorm pon

IE

satisfying0

<

p(xo-f(xo))

min{p(x-f(zo))"

x

X}.

By

assumption, there exists

A

with

IAI

<

such that u

Sx

+

(1

$)f(x)

X.

Thisimplies that

0

< p(xo

f(xo)) <_

p(u

f(xo))

[Alp(x0

f(xo))

< p(xo

f(xo)),

acontradiction.

Hence

f

hasafixed point.

DEFINITION

2.5.

Let

X

_C

]E.

A

_mapping

f

X

-

]E

is

weakly

continuousif forevery z*E

E*,

x*(f)’(X,

r,o)

--

IF

is continuous.

PROPOSITION

2.6.

If

f"

X

-->

IF_,

isweaklycontinuous, then

f

(X, r,)

--+

(E,

r,)

is continuous.

PROOF. For

e

>

0 let

N(0,

e)

denote the open neighborhoodof 0 ofradius e in

IF.

If

V

is a

r-basic

neighborhoodof 0 in

IE,

then

V

f’l,=l(x’)-l(N(0,

e,))

forsome

z,-.-,x

and1,.--,e

>

0.

Hence

f-(V)

,=,

f-(x*,)-l(N(O,,))

,%,(x*,(f))-(N(O,e,))

Thus

f

(X, r)

(IE,

r,,)

iscontinuous.

Let X

C

IE

andz

e

X.

The inward setof

X

isdefinedtobe

SOME FIXED POINT THEOREMS FOR NONCONVEX SPACES

COROLLARY

2.7

(Kim

[6];

also see Singh

[7],

Theorem 4.53, p.

206).

Let

X

be a nonempty compact convex subset

of

and

E*

separate the points

of

X.

If f"

X

-->

IE

zs weaklycontinuoussuch that

for

eachx

X

with

f(x)

5

x,

f(x)

7

closure(Ix(x)),

then

f

hasa

fixed

point.

PROOF. By

Proposition

2.1,

(X,r)

is a compact convex subset of the locally convex space

(E,T)

and by Proposition 2.6

f

(X,

rw)

-+

(E,r,o)

is continuous. Since the

r-closure(Ix(x))

C_

r-closure(Ix(x)),

itfollows that foranyx

#

f(x),

f(x)

r,o-closure(Ix(x)).

Theresultnowfollows from

Halpren

[3].

[]

COROLLARY

2.8

(Kelly-Namioka

[5,

p.

124]).

Let

X

be acompact convexsubset

of

E.

If/or

each nonzero z

X-

X,

there exists z* 6

I*

with

z*(z)

0 and

f"

X

-+

X

is a continuousmappin9satisfyzn9

f(]

a,z,)=

a,f(z,)

(2)

z=l t--1

for

each positive integer n, z, 6

X

for e {1,2,-", n},

anda,

>

0 with

],

a, 1, then

f

hasa

fixed

poznt.

PROOF.

Since

E*

separates the points of

X, (X,

r)

is

Hausdorff,

and hence

(X,r)

(X,

rw).

Since

(X,r)

is compact and

f

(X,r,o)

--+

(X,r)is

continuous the result follows from Theorem1.1.

REMARK. Note

that the condition

(:2)

isredundant in thepresentproof.

Theheoremsofthis sectionreadily imply that heHardyspaces

H,

0

<

p

<

1, have the fixed point property.

3.

A

FIXED

POINT

THEOREM FOR MULTIFUNCTIONS.

Let

2E denotethefamilyofnonempty subsets of and let

f

X

-

2E beamultifunction.

f

isupper semicontinuous ifforany closed set

F

C_

]E, f-l(F)

{x

q

X

f(x)

N

F

0}

isa closed subset of

X;

f

isclosed

(resp. compact)

valued if

f(x)

is closed

(resp.

compact)

subset of for eachx

X. It

is easy to show that if

f

isclosed-valued and ifanet

x

X,

xo

--4 x0

,

and y

f(x,)

with y -+ y0, then the uppersemicontinuity of

f

implies yo

f(xo).

Furthermore,

if

f

X

-+

2E is upper semicontinuousand

compact-valued,

then for

anycompact set

K,

the image

f(K)

(J{f(x)’x

K}

iscompact.

For

additional properties of multifunctionssee Dugundji and

Granas

[1]

or Smithson

[8],

for example. Thefollowing

proposition, which clearly implies Theorem 11.4 of Dugundji and

Granas

(see [1],

p.

97),

is equivalent to their theorem.

PROPOSITION

3.1.

Let ]

beatopologicalvector space and let

X

beanonempty compact convexsubset

of

.

If]E*

separatesthe points

of

X,

and

f"

X

--

2x is a

closed-valued,

upper semicontinuous and convex-valued

multi/unction,

then

f

hasa

fixed

point.

The followingresultismotivatedby Corollary 2.8.

THEOREM

3.2.

Let

]E bea topologicalvector space and let

X

be aclosed subset

of

and

f"

X

--+ 2E be a

closed-valued,

uppersemicontinuous,

multi/unction.

If

(i)

S

{x

y y6

f(x),x

6

X}

is convex,

(ii)

there existsasequence

(x,)

C_

X

with x,,+x

e f(x,) for

n 1,2,...,

(iii)

f(X)

is compact,

]__36 F. JAFARI AND V. M. SEHGAL

REMARK.

Condition

(i)

maybe replaced bythe somewhatmoregeneral hypothesisthat !

=1

_zk

zk+l E

S

foreverynE

N

where

(z)

areasin

(ii)

PROOF. We

firstshowthat

S

isaclosed subset of

.

Let

u

beanet in

S

with

uo

-+u0

]E

inthe’-topology of ]E. Thenbydefinition of

S, u

xo

y where yo

f(xo)

and

x

X.

Since

(y)

_C

f(X)

and

f(X)

iscompact,itfollowsthat

(yo)

hasasubnetyz

--

y0

e

E.

Hence

xz

uz

+

Yz ->Uo

+

yo,bydefinition.

However,

(xz)

C_

X

and

X

isclosed.

Hence uo

+

y0E

X.

Now

ifXo u0

+

y0, x0

X,

and since

xz

-+

x0, yz --+ y0 and y

f(x),

it follows that yo

f(Zo).

Consequently,

u0 z0-y0

S

and thus

S

isclosed.

By

the definitionof

(x)

in

(ii), x

xn+l E

S,

andsince

S

is convex, itfollowsthatforanypositiveintegern,

k=l

that is,

i(,

+,) e

s.

(3)

By

(ii),

x,+l

f(X)

for n 1,2,.--, andhencex,- z,+, x,-

f(X)

which isa compact

subset of ]E.

Consequently,

for anyneighborhood

U

of 0,

xx

f(X)

C_kU for somek

N.

Thus

-(xl

f(X))

_

U

foralln

>_

k.

In

particular,

(x-

f(x,,))

C_

U

for alln

>_

k. Letting n oc, since

S

is

closed,

by

(3)

we have0

S. Consequently,

thereis somex0, y0 E

f(x0)

withx0 y0 0. ThisimpliesXo yo

f(xo),

r

Note

thatcondition

(ii)

issatisfiedif

f(x)C X

(forallz

X;

inparticular,if

f(z)

C_

X.

Furthernotethat

I*

in Theorem3.2may be just

{0}.

That is to say, no assumption on

*

separatingpoints of

E

ismade here. The following corollary followsimmediately.

COROLLARY

3.3.

Let

X

be a closed convex subset

of

a topological vector space and

f"

X

-->

X

becontinuous.

If

f(X)

is compact and

f

satisfies

(2)

(see

Corollary

.8),

then

f

has a

fixed

point. []

It

is interesting to notethat the conditions on convexity of

S

canbe replaced by various

other usefulconditionson

f

and

X.

The remark andnotefollowing Theorem3.2 andCorollary 3.3 provide somesuchconditions. The following proposition gives acondition on

f

which is equivalent to the convexity assumptionon

S.

PROPOSITION

;3.4.

Let

X

be aclosed convexsubset

of

the topological vector space

]

and

f

X

--

2E be a

closed-valued,

upper semicontinuous,

multifunction

such that

f(X)

is

compact. Then

S

{x

y y

f(z),x

X}

is convez

if

andonly

if

1/2f(x)

+

f(y)

C_f(x

+

y)

vx,

ye

x.

(4)

PROOF. Suppose S

is convex. Then for z,z2

S,

7zl

+

z2

S.

In

particular, if

z, x,-y,, with x,

X,

and y,

f(x,)

for 1,2, we have

(xl

+

zz)-

1/2(yx

+

y)

S.

That is,

7(y

+

y)

f((x

+

x)). Now,

since this holdsfor every choice of y,

f(x,),

E

{1,2}, (4)

readily follows. Conversely, suppose

f

satisfies

(4),

and let z,z2,x,x,yx,y be chosen as above. Then since

S

isclosed

(by

proofofTheorem

3.2),

toshow

S

isconvexit is sufficient toshow

S

ismidpointconvex. Since

z

+

z

7(xl

+

z_)-

(y

+

y),

by

(4),

21-(yl-"

Y2) e

f(1/2(X, +

X2))

and thus

1/2zx

+

_iz=

S. Hence S

is midpoint convexand thus

SOME FIXED POINT THEOREMS FOR NONCONVEX SPACES 137

Note

that a multifunction is midpoint convex if

f(1

x

+

y)

C_

1/2f(x)

+

1/2f(y)

and

f

is

midpointconcaveifitsatisfies

(4).

Weconclude with asimpleexample.

EXAMPLE

3.i.

Let X

[0,

1]

C_

I,

:

[0,

cxz[-

[0,

oc[

beanondecreasing,continuous functionsuch that

(o(x)+

(y))

<

(1/2(x

+

y))

for all x,y

e

[0, cxz[.

For

example,let

(x)=

log(x

+

1).

Define

f(x)=

[0, T(x)]

for allz

e

X. Then

fX

[0,p(1)]

iscompact,andclearly

f

is a closed-valued and upper semicontinuous multifunction.

By

the hypothesis on

,

for

x,yE

X,

l(x

+y))

f(x)

+

-f(y)

[0,

((x)

+

(y))]

_

[0,

((x

+

y))]

f(

henceby Proposition 3.4,

S

is convex. Since

f(x)N

X

#

0

forany xE

X,

byTheorem3.2,

f

hasafixed point.

ACKNOWLEDGEMENT.

The authors wish to thank Professor

S. P.

Singh for providing

them with aworking draft of his noteson partitions of unity andfor somereferences herein cited.

REFERENCES

[1]

DUGUNDJI,

J.

and

GRANAS, A.,

Fixed Point Theory,

Vo.

1, Polish Scientific

Publishers,

Warsaw,

1982.

[2]

FAN, K.,

Sur

unthk:rme minimax,

C.

R.

Acad. Sci. Paris259

(1964),

3925-3928.

[3]

HALPREN, B.R.,

Fixed point theorems for set-valued maps ininfinite dimen-sional spaces, Math. Ann. 189

(1970),

87-89.

[4]

KACZYNSKI, T.,

Quelques thorme de points fixes dans des espaces ayant suffisammentdefonctionnelles linaires,

C.

R.

Acad. Sci.

Paros

Sdr.

I

Math.

296

(1983),

873-874.

[5]

KELLY,

J.

and

NAMIOKA, I.,

Linear Topological

Spaces, D. Van

Nostrand

Company,

Inc.,

Princeton, New

Jersey,

1963.

[6]

KIM, S.Y.,

Ph.D. Thesis, SeoulNational University,Korea.

[7]

SINGH,

S.P.,

Private communications

(1995).