ON THE KKM THEORY OF LOCALLY $p$-CONVEX SPACES (Nonlinear Analysis and Convex Analysis)

ON THE KKM THEORY OF LOCALLY _r‐CONVEX SPACES

SEHIE PARK

ABSTRACT. Ina_{recent paper [5],}Gholizadehetal. investigatedthe existence ofafixedpointof_multimapson_{almost p‐‐convex}or_p‐convexsubsets of\mathrm{t}\mathrm{o}\mathrm{p}\mathrm{c}\succ logicalvector spaces. Most of their resultsare_originatedfromsome_previous

works ofParkon_analyticalfixed_{point theory. In this surveyarticle,}werecall such works and comparethem withthecorresponding onesin _[5]. Finally, some_generalcomments to_[5]areadded.

1. Introduction

Inour

_previous

talk

[23]

atthe

_{NACA, Chiang Rai, January 2015,}

we introduced some recentresults in

_analytical

fixed point

theory

based on our

_previous

works.

After

that,

wefounda_paper

by

Gholizadehetal.

_[5],

whereanumber of fixed_point

theorems duetothe_present authorwereclaimedtobe

_generalized.

Our

_principal

aim inthis articleis tointroduceour

previous

works relatedtothose in

_[5].

Let

_{0<p\leq 1}

. In

[5],

its authors

investigated

the existence ofa fixed point

of

_multimaps

onalmost p‐convex or_p‐convexsubsets of

topological

_{vector spaces.}

Mostof their resultsare

originated

fromsome

previous

worksof Parkonthe KKM

theory

and

_analytical

fixed

_point

_theory.

In

_fact,

in

_[5]

and

_[3],

their authors extendedourresultsin

[7], [9],

and

[10].

Note that these three_papersarebasedon

the KKM

_theory.

In this _survey

_article,

we recall such works and _compare them

with the

_{corresponding}

onesin

_[3]

and

_[5].

_Finally,

some

_general

commentson

[5]

areadded.

This_paper is

_organized

asfollows. Section 2 isa

_preliminary

onbasic _concepts ofourKKM

theory

of abstract convex_spaces. We recall there that

_{$\phi$_{A}}

_‐spaces are

KKM _spaces. Section 3 devotes to definitions related _{to p}‐convex _spaces, which

are shown to be new

$\phi$_{A}

_{‐spaces. In} Section _4, we introduce

general

forms of the KKM_typetheorems duetoourselves. Oneof them istoobtainaKKM theorem for p-‐convex_spaces and a

general

Alexandroff‐Pasynkoff

theorem for abstract convex spaces. Section 5 devotesto compare our _previous fixed _pointtheorems with the

extendedp‐‐convex space versions in

[5]

and

[3].

Finally,

inSection_6,we

give

some

furthercommentsonthe_paper

[5].

2. Abstract convex _spaces

Multimaps

arealso called

simply

_maps. Let

\langle D\rangle

denote the setof all _nonempty finite subsets ofaset D. Recall the

following

in

[16]:

Definition. An abstractconvex_space

(E, D; $\Gamma$)

consistsofa

topological

_space

E,

\mathrm{a}

nonemptyset D, and a

multimap

$\Gamma$ :

\langle D\rangle\rightarrow E

withnonempty values$\Gamma$_{A}

:= $\Gamma$(A)

for

_A\in(D\}

, such that the $\Gamma$‐convexhull ofany D'\subseteq Disdenoted and defined

by

\mathrm{c}\mathrm{o}_{ $\Gamma$}D':=\cup\{$\Gamma$_{A}|A\in\{D')\}\subset E.

2010Mathematics _{Subject Classification. Primary47\mathrm{H}04,} 47\mathrm{H}10; Secondary 46\mathrm{A}16, 46\mathrm{A}55, 49\mathrm{J}27, 49\mathrm{J}35,52\mathrm{A}07, 54\mathrm{C}60, 54\mathrm{H}25, 55\mathrm{M}20,91\mathrm{B}50.

Key words and phrases. KKM _theorem, abstract convex _{space, p-}‐convex _space, \mathrm{G}-‐convex

SEHIE PARK

AsubsetXofE is calleda $\Gamma$‐convexsubset of

(E, D; $\Gamma$)

relativetoD' _{if for any}

N\in\{D'\}

,we have

$\Gamma$_{N}\subseteq X

,that

is, \mathrm{c}\mathrm{o}_{ $\Gamma$}D'\subset X.

IncaseE=D,let

(E; $\Gamma$) :=(E, E; $\Gamma$)

.

Recall thatsomecorrections on

[16]

appeared

in

_[22].

Definition. Let

_{(E, D; $\Gamma$)}

be anabstractconvex_spaceand Za

topological

_space.

For a

_multimap

F : E\rightarrow Z with _nonempty

values,

ifa

multimap

G : D\rightarrow Z

satisfies

F($\Gamma$_{A})\displaystyle \subset G(A):=\bigcup_{y\in A}G(y)

for all

_{A\in\langle D\rangle,}

then G is called a KKM mapwith _respect to F. A KKM map G : D\rightarrow E is a

KKM _mapwith_respecttothe

_identity

_{map 1_{E}.}

A

_multimap

F : E\rightarrow Z is called a \mathfrak{K}\mathrm{C}_‐map

[resp.

\mathrm{a}\mathfrak{K}\mathrm{D}

‐map]

if,

for _any

closed‐valued

_{[resp. open‐valued]}

KKM map G:D\rightarrow Z with _respect to F, the

family

\{G(y)\}_{y\in D}

has the finite intersection _property. In this _case, we denote

F\in \mathfrak{K}C(E, Z) [resp. F\in \mathfrak{K}\mathrm{D}(E,

Z

Definition. The

_partial

KKM

_principle

foran abstract convex _space

(E, D; $\Gamma$)

is thestatement

_{1_{E}\in \mathfrak{K}\mathrm{C}(E, E)}

_; thatis,foranyclosed‐valued KKMmap

G:D\rightarrow E,

the

_family

_{\{G(y)\}_{y\in D}}

has thefinite intersection _property. The KKM

_principle

is thestatement

_{1_{E}\in \mathrm{R}\not\subset(E, E)\cap \mathrm{f}\mathrm{l}\mathrm{D}(E, E)}

;that

is,

thesame_propertyalso holdsfor

any

open‐valued

KKM map.

An abstract convex _{space is} called \mathrm{a}

(partial)

KKM space if it satisfies the

(partial)

KKM

_principle,

_resp.

In our recent works

_I11‐13],

we studied elements or foundations of the KKM

theory

on abstract convex _spaces and noticed there that many

important

results thereinarerelatedtothe

_partial

KKM

_principle.

Example.

We gave known

examples

of

(partial)

KKM spaces in

[16]

and the references therein. The

_following

isoneof them.

Definition. A

_{$\phi$_{A}}

_‐space

_{(X, D;\{$\phi$_{A}\}_{A\in\langle D\rangle})}

consists ofa

_topological

_space

_X,

\mathrm{a}

nonempty set D, and a

family

of continuous functions

$\phi$_{A}

:

$\Delta$_{n}\rightarrow X

(that

is,

singular

n

‐simplices)

for

A\in\{D\}

with

|A|=n+1

.

By putting

$\Gamma$_{A}

:=$\phi$_{A}($\Delta$_{n})

for

each

_{A\in\{D\rangle}

, the

triple

(X, D; $\Gamma$)

becomesan abstractconvexspace.

Definition. Fora

$\phi$_{A}

_‐space

(X, D;\{$\phi$_{A}\})

, any

multimap

G:D\rightarrow X

satisfying

$\phi$_{A}($\Delta$_{J})\subset G(J)

for each

_A\in\{D\}

and

_{J\in\langle A\rangle}

iscalledaKKM map.

We show that_every

_{$\phi$_{A}}

_‐spaceisaKKM space:

Lemma 1. Let

_{(X, D; $\Gamma$)}

be a

_{$\phi$_{A}}

_‐space and G : D\rightarrow X a

_multimap

with

nonempty closed

_{[resp. open]}

values.

_Suppose

that G is a KKM map. Then

\{G(a)\}_{a\in D}

has the finite intersection_property.

Proof

Let

_{A=\{a_{0}, a_{1}, . . ., a_{n}\}\in\{D\rangle}

. Then there exists acontinuous function

$\phi$_{A}:$\Delta$_{n}\rightarrow$\Gamma$_{A}

such

_that,

forany0\leq i_{0}<i_{1}<\cdots<i_{k}\leq n,we have

$\phi$_{A}(\mathrm{c}\mathrm{o}\{e_{i\mathrm{o}}, e_{i_{1}}, \ldots, e_{i_{k}}\})\subset $\Gamma$(\{a_{i_{0}}, a_{i_{1}}, \ldots, a_{i_{k}}\})\cap$\phi$_{A}($\Delta$_{n})

.

Since Gisa KKM map, itfollows that

\mathrm{c}\mathrm{o}\{e_{i_{\mathrm{O}}}, e_{i_{1}}, . . . , e_{i_{k}}\}\subset$\phi$_{A}^{-1}( $\Gamma$(\{\%, a_{i_{1}}, \ldots, a_{i_{k}}\})\cap$\phi$_{A}($\Delta$_{n}))

\displaystyle \subset\bigcup_{j=0}^{k}$\phi$_{A}^{-1}(G(a_{i_{j}})\cap$\phi$_{A}(\triangle_{n}))

.

Since

_{G(a_{i_{j}})\cap$\phi$_{A}(\triangle_{n})}

isclosed

_{[resp. open]}

in_{the compact} subset

_{$\phi$_{A}($\Delta$_{n})}

of

_{$\Gamma$_{A},}

$\phi$_{A}^{-1}(G(a_{i_{j}})\cap$\phi$_{A}($\Delta$_{n}))

is closed

_{[resp. open]}

in

_{\triangle_{n}}

. Note that ei

-\neq$\phi$_{A}^{-1}(G(a_{i})\cap

$\phi$_{A}(\triangle_{n}))

isaKKM _mapon

\{e_{0}, e_{1}, \cdots, e_{n}\}

.

Hence, uy

the

original

KKM

theorem,

we have

\displaystyle \bigcap_{i=0}^{n}$\phi$_{A}^{-1}(G(a_{i})\cap$\phi$_{A}($\Delta$_{n}))\neq\emptyset,

which

_{readily implies}

_{\displaystyle \bigcap_{i=0}^{n}G(a_{i})\neq\emptyset}

. This

completes

the

proof.

\square

Nowwe have the

following diagram

for

triples

(E, D; $\Gamma$)

:

Simplex

\Rightarrow Convex subset ofat.v.\mathrm{s}. \RightarrowLassonde _typeconvex_space

\Rightarrow \mathrm{H}-_space \Rightarrow \mathrm{G}‐convex_space

\Rightarrow$\phi$_{A}-

_space\Rightarrow \mathrm{K}\mathrm{K}\mathrm{M}_space

\Rightarrow Partial KKM space\RightarrowAUstractconvex_space.

3. New KKMspaces

Let

_{0<p\leq 1}

. Recallthedefinitions

given

by Bayoumi

[4, 5]:

Definition.

₍

p‐convex

set)

Aset Ain a_{vector space} Vis said tobe ‐convex

if,

for_{any x,}

_{y\in A,}

_{s,t\geq 0},wehave

(1-t)^{1/p}x+t^{1/p}y\in A

, whenever0\leq t\leq 1.

Definition.

₍

_p‐convex

hull)

If X is a

topological

_{vector space} and A\subset X, the

closed _p‐convex hull of A denoted

_by

_{\overline{C}_{p}(A)}

is the smallest closed r‐convex set

containing

A.

Definition.

₍

p‐convex

combination)

Let A be_{p‐‐convex}and _{x_{1},}\cdots,

x_{n}\in A

, and

t_{i}\geq 0,

\displaystyle \sum_{1}^{n}t_{i}^{\mathrm{p}}=1

. Then

\displaystyle \sum_{1}^{n}t_{i}x_{i}

is called a p‐‐convex combination of

{xi}.

If

\displaystyle \sum_{1}^{n}|t_{i}|^{\mathrm{p}}\leq 1

,then

\displaystyle \sum_{1}^{n}t_{i}x_{i}

is calledan

absolutely

‐convexcombination. Itis easy

toseethat

\displaystyle \sum_{1}^{n}t_{i}x_{i}\in A

fora_{p‐‐convex set}A.

Definition.

_(locally

_p‐convex

space)

A

topological

_{vector space is}saidtobe

_locally

p‐‐convexif the

origin

hasafundamentalsetof

_absolutely

p‐‐convex0

_{‐neighborhoods.}

This

_topology

can be determined

by

_{p‐seminorms}whicharedefinedinthe obvious

way.

Using

these_concepts,in

_[5],

definitions of almostp\overline{-}convexsetsand the

_{p‐‐convexly}

almost fixed_{point property}areintroducedas

generalizations

ofalmostconvexsets

(due

to

_Himmelberg)

andthe almost fixed_{point property,resp.}

Nowwe havea newKKM_space:

Lemma 2.

_Suppose

that X is asubset ofa

topological

_{vector space E}and D isa

nonemptysubset ofXsuch that

_{C_{p}(D)\subset X}

. Let$\Gamma$_{N}

:=C_{p}(N)

For each

N\in\langle D\rangle.

Then

_{(X, D; $\Gamma$)}

isa

$\phi$_{A}

_‐space.

Proof.

Since

_{C_{p}(D)\subset X,}

$\Gamma$_{N} iswell‐defined. For each

_{N=\{x_{0}, x_{1}, \cdots, x_{n}\}\subset D,}

define

_{$\phi$_{N}}

:

$\Delta$_{n}\rightarrow$\Gamma$_{N} by

\displaystyle \sum_{i=0}^{n}t_{i}e_{i}\mapsto\sum_{i=0}^{n}(t_{i})^{\frac{1}{\mathrm{p}}}x_{i}.

SEHIE PARK

4. General KKM theorems

The

_following

whole intersection_property for the

_map‐values

ofaKKM_{map is} a

standard form of theKKM_typetheorems

_[15,16,18]:

Theorem 1. Let

_{(E, D; $\Gamma$)}

be a

partial

KKM space

[resp.

a KKM

space]

and

G:D\rightarrow E a

_{multimap satisfying}

(1)

G has closed

_{[resp. openJ}

_values;

and

(2) $\Gamma$_{N}\subset G(N)

for_any

_{N\in\langle D\rangle (that}

_is,

Gisa KKM

map).

Then

_{\{G(z)\}_{z\in D}}

has the finite intersection_property.

Further,

if

(3)

\displaystyle \bigcap_{z\in M}\overline{G(z)}

is_compactforsome

M\in\langle D\rangle,

then wehave

\displaystyle \bigcap_{y\in D}\overline{G(y)}\neq\emptyset.

Consider the

_following

relatedfour conditions for a_map G:D-\rightarrow E:

(a)

\displaystyle \bigcap_{z\in D}\overline{G(z)}\neq\emptyset

implies

_{\displaystyle \bigcap_{z\in D}G(z)\neq\emptyset.}

(b)

\displaystyle \bigcap_{z\in D}\overline{G(z)}=\overline{\bigcap_{z\in D}G(z)}(G

is

intersectionally

closed‐valuedinthe senseof

Lucet

_al).

(c)

\displaystyle \bigcap_{z\in D}\overline{G(z)}=\bigcap_{z\in D}G(z)

(

Gis

_transfer

_{closed‐valued).}

(d)

Gisclosed‐valued.

Fromthe

_partial

KKM

_principle

we havea whole intersection _property of the Fan

_type.

The

_following

is

_given

in

_[18,19]:

Theorem 2. Let

_{(E, D; $\Gamma$)}

bea

partial

KKM_spaceand G:D-\circ E a_mapsuch

that

(1)

\overline{G}

isaKKM_map

[that

_is,

$\Gamma$_{A}\subset\overline{G}(A)

for all

A\in\langle D\rangle ];

and

(2)

there existsa_{nonempty compact}subset K of E such that either

(i)

\displaystyle \bigcap_{z\in M}\overline{G(z)}\subset K

forsome

M\in\langle D};

or

(ii)

foreach

_{N\in\langle D},}

thereexistsa_compact $\Gamma$‐convexsubset L_{N} ofErelative

tosomeD'\subset Dsuch thatN\subset D' and

\displaystyle \overline{L_{N}}\cap\bigcap_{z\in D'}\overline{G(z)}\subset K.

Then wehave

K\displaystyle \cap\bigcap_{z\in D}\overline{G(z)}\neq\emptyset.

Furthermore,

( $\alpha$)

if Gistransfer

_{closed‐valued,}

then

_{K\cap\cap\{G(z)|z\in D\}\neq\emptyset}

_;

(

$\beta$

)

if Gis

_{intersectionally closed‐valued,}

then

_{\cap\{G(z)|z\in D\}\neq\emptyset.}

We

_give

some_consequencesof Theorem 1:

Theorem3.

_[5]

_Suppose

that Xisasubset ofa

topological

_{vector space E}andD isa_nonemptysubset of X such that

C_{p}(D)\subseteq X

. Alsosupposethat G:D-\triangleleft X

isa

multimap

satisfying

(a) G(x)

isclosed

_{[resp. openl}

inX for all x\in D.

(b) C_{\mathrm{p}}(N)\subset G(N)

foreach

_{N\in\langle D\rangle.}

Then

_{\{G(x)|x\in D\}}

has the finiteintersection_property.

Proof. By putting

$\Gamma$_{N}

_{:=C_{p}(N)}

,

(X, D; $\Gamma$)

isaKKMspace

by

Lemma 2. Now the

conclusion follows from Theorem 1. \square

From Theorem_1,wehave the

_{following generalization}

of the

Alexandroff‐Pasynkoff

Theorem 4. Let

_{(E, D; $\Gamma$)}

bea

_partial

_{KKM space}

[resp.

aKKM

_space],

A\subset E,

\{A_{0}, A_{1}, . .. , A_{N}\}

be a

family

ofclosed

[resp. open]

subsetsofE such that A\subset

\displaystyle \bigcup_{i=0}^{n}A_{i}

, and

N=\{z_{0}, z_{1}, \cdots, z_{n}\}

bea

family

of

points

inDsuch tlJat

$\Gamma$(N)\subset A.

If

_{$\Gamma$(N\backslash \{z_{i}\})\subset A_{i}}

for each i=0,1,..

.,n,

\displaystyle \mathfrak{t}he\mathrm{n}\bigcap_{i=0}^{n}A_{i}\neq\emptyset.

Proof.

Let

_{C_{0}= $\Gamma$(N\backslash \{z_{n}\})}

and fori=1,

2,

..._,n_,let

C_{i}= $\Gamma$(N\backslash \{z_{i-1}\})

. Define

a

multimap

F : D\rightarrow X

_by

F(z_{0})=A_{n}, F(z_{i})=A_{i-1}

for i=1,

2,

...

,n, and

F(z)=X

for all

_{z\in D\backslash N}

. We claim thatF is aKKM map. Tosee

this,

we note

that

_{$\Gamma$(N)\displaystyle \subset A\subset\bigcup_{i=0}^{n}A_{i}=F(N)}

and for _{any propersubset z_{i_{0}},z_{i_{1}}},..._,z_{i_{k}} of N

with0\leq k<nand_{0\leq i_{0}<i_{1}<\cdots<i_{k}\leq n},we have

$\Gamma$(\{z_{i_{0}}, z_{i_{1}}, \ldots, z_{i_{k}}\})\subset C_{i_{j}}\subset A_{i_{J}-1}=F(z_{i_{j}})

for some

j\in\{0, 1, . . . , k\}

. Note that

i_{j}=0

if and

only

if

i_{j}-1=n

_, and so

$\Gamma$(\displaystyle \{z_{i_{0}}, z_{i_{1}}, \ldots, z_{i_{k}}\})\subset\bigcup_{j=0}^{k}F(z_{i_{j}})

. Now

by

Theorem1 wehave

\displaystyle \bigcap_{i=0}^{n}A_{i}\neq\emptyset.

\square

Remarks. 1. Ifwe

adopt

Theorem 2instead of Theorem _1,we _mayhaveanother versionof Theorem4.

2. Note that

_[5,

Theorem

_2.2]

isa

generalized

minimal_spaceversionof Theorem 4motivated from the

_previous

work of Park

_[7].

3. Itiswell‐known that the

_{Alexandroff‐Pasynkoff}

theorem

_implies

the Brouwer fixed_pointtheorem

_(e.g.,

see

[24]).

Therefore,

Theorem 4 is also

equivalent

to the KKM theorem.

5.

_0riginal

results extended to p-‐convex _spaces

Recall

_that,

in

_[5]

and

_[3],

their authors extendedour results in

[7], [9],

and

[10]

to p‐‐convex spaces, and these three papersofours arebasedonthe KKM

theory.

Now,

we

_give

our

original

resultsin

there,

and indicatethe

corresponding

results

extended

_by

_[5]

and

_[3].

Theorem 5.

_[7]

Let X beasubset ofaHausdorff

topological

vector_{space E}and Y an almostconvexdense subset of X. LetT :X\rightarrow E be alower

lresp. upperl

semicontinuous

multimap

such that

_T(y)

is convexfor all

y\in Y.

If there is a

precompactsubset K of X such that

_{T(y)\cap K\neq\emptyset}

for each

_{y\in Y}

, then fora\mathrm{n}y^{r}

convex

_neighborhood

U of the

_origin

0 ofE, there existsa

point

x_{U}\in Ysuch that

T(x_{U})\cap(x_{U}+U)\neq\emptyset.

Note that Hausdorffness ofE is redundant. In

_[5,

Theorem

_2.7],

all ‘convex’ is

replaced

inTheorem5

_by

p‐‐convex.

Corollary

6.

_[7]

Let X beaconvexsubset ofaHausdorff

topological

vector_space

E. LetT : X\rightarrow E bea lower

lresp. upperl

semicontinuous

multimap

such that

T(x)

isconvexfor each x\in X. If thereisaprecompactsubset K ofX such that

T(x)\cap K\neq\emptyset

for each x\in X, thenforeveryconvex

neighborhood

U of the

origin

0 of E, thereexistsa

point x_{U}\in X

such that

T(x_{U})\cap(x_{U}+U)\neq\emptyset.

Note that Hausdorffness ofE isredundant. In

_[5,

_Corollary

_2.8],

all ‘convex’ in

Corollary

6 is

_{replaced by}

p‐convex.

Corollary

7.

_[7]

Let X Ueasubsetofa

locally

convexHausdorff

topological

vector

space Eand Yanalmostconvexdense subset ofX. Let T:X\rightarrow X bea_compact upper semicontinuous

multimap

with closed values such that

T(y)

is nonempty

convexfor all

y\in Y

. ThenThas\mathrm{a}fixedpoint x_{0}\in X; that

is,

x_{0}\in T(x_{0})

.

In

_[5,

Theorem

_2.12],

all ‘convex’ in

_Corollary

7 is

_{replaced by}

p‐convex and

Hausdorftness is not

_assumed,

but used in its

_proof.

This means

that,

in

_[5],

all

SEHIE PARK

Corollary

8.

_[7]

Let X beasubset ofaHausdorff

_topological

_{vector space}E and Yanalmost convexdense subset ofX. Let T:X\rightarrow E bea

multimap

such that

(1) T^{-}(z)

is openfor each z\in E;and

(2) T(y)

is convexfor for each

y\in \mathrm{y}.

If there isa_precompactsubsetKof X such that

T(y)\cap K\neq\emptyset

for each

y\in Y,

then for _any convex

neighborhood

U of the

origin

0 ofE, there exists a

point

x_{U}\in Y such that

_{T(x_{U})\cap(x_{U}+U)\neq\emptyset.}

In

_[5,

_Corollary

_2.9],

all ‘convex’ in

_Corollary

8 is

_{replaced by}

p‐‐convex, and

Hausdorffnessis not assumed.

Corollary

9.

_[7]

Let X bea convexsubset ofaHausdorff

topological

_{vector space} E, andT:X\rightarrow X beacompact

multimap

such that

(l) T(x)

is_nonemptyandconvexfor each x\in X_;

(2) T^{-}(y)

is openfor each

_{y\in X}

_;and

Then for_anyconvex

_neighborhood

U of the

_origin

0 ofE, thereexistsa

point

x_{U}\in X

suchthat

_{T(x_{U})\cap(x_{U}+U)\neq\emptyset.}

Here Hausdorffnessisredundant. In

_[5,

_Corollary

_2.10],

all‘convex’in

_Corollary

9 is

_{replaced by}

p‐‐convex, and added that Tcan be assumedu.s. \mathrm{c}. instead of

(2).

Theorem 10.

_[9]

Let X beastar

_shaped

subsetofaHausdorff

topological

vector space E with the

origin

O ofE as the center. Let

_f

: X\rightarrow X be a _compact continuous map. Thenoneof the

following

holds:

(i)

f

hasafixed_point

x_{0}=f(x_{0})\in X

_;

(ii)

there exista

point y_{0}\in X

anda

t_{0}\in(0,1)

such that

O\neq y_{0}=t_{0}f(y_{0})

_;or

(iii) f(O)\neq O.

In

_[3],

this isextendedto a_pstar

_shaped

subsets of a

topological

vector space

via Fan‐KKM

_principle

ina

generalized

convex_space.

Theorem 11.

_[10]

LetXbea convexsubset ofa

locally

convexHausdorfft.v.s. E.

Then anyclosedcompact

multimap

T:X\rightarrow X

_having

the almost fiXed_property hasafxed

point.

In

_[5,

Theorem

_2.14],

all ‘convex’inTheorem 11 is

replaced by

p‐‐convex. Theorem12.

_[10]

LetXbea_compactconvexsubsetofat.v.s. Eand T:X\rightarrow X

a

multimap

such that

(i)

T has thealmost fixed

_point

_property;

(ii)

Thas closed

_values;

and

(iii)

Tsatisfies condition

\displaystyle \bigcap_{U\in \mathcal{V}}\{x\in X|x\in T(x)+U\}=\bigcap_{U\in V}\mathrm{c}1\{x\in X|x\in T(x)+\mathrm{c}\mathrm{o}\mathrm{U}\},

where \mathcal{V}isalocal base of_open

neighborhoods

of0 in E. Then Thasafixed

point.

Note that

_[5,

Theorem 2.19 and Corollaries

_2.20‐2.22]

are all motivated from

Theorem 12 above

_{by replacing}

all ‘convex’

_by

_{p‐‐convex.}

Corollary

13.

_[10]

Let X bea_compactconvexsubset ofa

locally

convexHausdorff t.v.s. Thenanyclosed

multimap

T:X\rightarrow X

having

the almost fixedpoint

property

hasafixedpoint.

Moreover,

in

_[5,

Theorem

_2.24],

all ‘convex’ in

_Corollary

13 is

_{replaced Uy}

_p‐

convex and the almost fixed

point

_property

by

the_p

‐convexly

almost fixed

_point

6. Further comments on

[5]

1. In

_[5]

the authorsarebasedontheKKM_typetheorems

(Theorems

1.3 and1.4

there)

on

generalized

minimal_{spaces in}

[2],

and noted that

they

are

generalizations

of Theorem 1 inPark

_[8,6].

However theconcept of \mathrm{G}‐convex_spaces areobsolete

andwe established

already

muchmore

general theory

on abstract convex _spaces.

Moreover,

since anyminimalspacecanbe madeintoa

topological

_space,resultson

abstractconvexminimal_spacescanbe deduced from the

_theory

of abstractconvex spaces; see

[14, 31].

Note also that some authors are still

publishing

_papers on

minimal_spaces.

2. Notice thatnoconsiderationonthe Hausdorffnessof

topological

_{vector spaces} are

_given

in

[5].

Many

results there canhold without

_assuming

theHausdorffness.

This can be also stated the

_original

works of Park on which

[5]

has based. In

the _{present paper} we

clearly distinguish original

results where Hausdorffness is redundant.

3. The

_following

is

_given

in

_[17]:

Definition. A $\gamma$‐convex_space

(E, D; $\gamma$)

consistsofa

topological

_{space E},anonempty

setD,anda

multimap

$\gamma$: D\times D\rightarrow Ewithnonemptyvalues

$\gamma$(a, b)

forany a,b\in D.

For anyD'\subset D, the $\gamma$‐convexhult of D'isdenoted and defined

by

\mathrm{c}\mathrm{o}_{ $\gamma$}D' :=\cup\{ $\gamma$(a, b)|a, b\in D'\}\subset E.

A subset X ofE iscalleda _$\gamma$‐convexsubset of

(E, D; $\gamma$)

relativetoD' if forany

a,b\in D', wehave

$\gamma$(a, b)\subset X

,that

is,

\mathrm{c}\mathrm{o}_{ $\gamma$}D'\subset X.

Incase E\supset D, let

(E\supset D; $\gamma$) :=(E, D; $\gamma$)

and let

(E; $\gamma$) :=(E, E; $\gamma$)

.

Note thata_{p‐‐convex}subset X

(in

thesenseof

Bayoumi)

ofa

topological

vector space E is a _$\gamma$‐convexsubset of

(E, X; $\gamma$)

relativetoXitself.

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(Sehie Park) THENATIONAL ACADEMY OF SCIENCES, REPUBLIC0 _KOREA, SEOUL _137−044;

AND DEPARTMENTOFMATHEMATICAL _SCIENCES, SEOUL NATIONAL_UNIVERSITY, SEOUL_151‐747,

KOREA