Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
ZERO
EPI
KKM
TYPE
MAPS
AND
FIXED
POINTS
∗
Donal
O’Regan
†
Dedicated
with
admiration
to
Professor
Constantin
Fetecau
DOI 10.56082/annalsarscimath.2026.4.233
Abstract
In
this
paper,
we
present
two
approaches
to
fixed
point
theory
for
compact
KKM
type
maps.
In
particular,
we
consider
Leray–Schauder
type
alternatives
for
these
maps.
Keywords:
fixed
points,
set–valued
maps,
zero
epi
maps.
MSC:
47H10,
54H25.
1
Introduction
This
paper
discusses
subclasses
of
KKM
maps
(also
called
KKM
maps
for
simplicity)
or
maps
with
a
KKM
type
selection
(motivated
from
[5,
6,
11])
and
we
present
two
approaches
to
fixed
point
theory.
In
the
first
approach
using
results
of
the
author
[9]
(which
uses
the
idea
of
null
homotopy)
we
present
fixed
point
results
(see
Theorem
8)
for
compact
KKM
type
maps.
This
motivates
our
second
approach
which
is
based
on
the
notion
of
a
zero
epi
map
(see
[4,
7])
which
enables
us
to
present
coincidence
and
Leray–
Schauder
type
results
for
the
various
classes
of
maps
mentioned
above.
Both
approaches
are
based
on
composition
and
product
rules
so
our
results
are
new
and
complement
previous
results
in
the
literature
(see
[1,
2,
8,
10,
11]
and
the
references
therein).
∗
Accepted
for
publication
on
August
07,
2026
†
donal.oregan@nuigalway.ie
,
School
of
Mathematical
and
Statistical
Sciences,
Uni-
versity
of
Galway,
Ireland
233
Zero
epi
KKM
type
maps
and
fixed
points
234
Now,
we
describe
the
KKM
maps
which
we
consider
in
this
paper.
Let
X
be
a
convex
subset
of
a
Hausdorff
topological
vector
space
and
Y
a
Hausdorff
topological
space.
If
S,
T
:
X
→
2
Y
(nonempty
subsets
of
Y
)
are
two
set
valued
maps
such
that
T
(
co
(
A
))
⊆
S
(
A
)
for
each
finite
subset
A
of
X
then
we
call
S
a
generalized
KKM
mapping
w.r.t.
T
.
Now
the
set
valued
map
T
:
X
→
2
Y
is
said
to
have
the
KKM
property
if
for
any
generalized
KKM
map
S
:
X
→
2
Y
w.r.t.
T
the
family
{
S
(
x
)
:
x
∈
X
}
has
the
finite
intersection
property
(the
intersection
of
each
finite
subfamily
is
nonempty).
We
let
KKM
(
X,
Y
)
=
{
T
:
X
→
2
Y
|
T
has
the
KKM
property
}
.
First,
we
recall
the
following
result
[2].
Theorem
1.
Let
X
be
a
convex
subset
of
a
Hausdorff
topological
vector
space
and
Y,
Z
be
Hausdorff
topological
spaces.
(i).
T
∈
KKM
(
X,
Y
)
iff
T
|
∈
KKM
(
,
Y
)
for
each
polytope
in
X
;
(ii).
if
T
∈
KKM
(
X,
Y
)
and
f
∈
C
(
Y,
Z
)
then
f
T
∈
KKM
(
X,
Z
)
;
(iii).
if
Y
is
a
normal
space,
a
polytope
of
X
and
if
T
:
→
2
Y
is
a
set
valued
map
such
that
for
each
f
∈
C
(
Y,
)
we
have
that
f
T
has
a
fixed
point
in
,
then
T
∈
KKM
(
,
Y
)
.
Next
we
note
a
fixed
point
result
for
KKM
maps
in
[1].
Recall
a
nonempty
subset
W
of
a
Hausdorff
topological
vector
space
E
is
said
to
be
admissible
if
for
any
nonempty
compact
subset
K
of
W
and
every
neigh-
borhood
V
of
0
in
E
there
exists
a
continuous
map
h
:
K
→
W
with
x
−
h
(
x
)
∈
V
for
all
x
∈
K
and
h
(
K
)
is
contained
in
a
finite
dimensional
subspace
of
E
(for
example
every
nonempty
convex
subset
of
a
locally
convex
space
is
admissible).
Theorem
2.
Let
X
be
an
admissible
convex
set
in
a
Hausdorff
topological
vector
space
E
and
T
∈
KKM
(
X,
X
)
be
a
closed
compact
map.
Then
T
has
a
fixed
point
in
X
.
From
Theorem
1
and
Theorem
2
one
has
the
following
result.
Theorem
3.
Let
X
be
an
admissible
convex
set
in
a
Hausdorff
topologi-
cal
vector
space
and
Z
a
subset
of
a
Hausdorff
topological
space.
If
T
∈
KKM
(
X,
Z
)
is
a
upper
semicontinuous
compact
map
with
compact
values
and
f
∈
C
(
Z,
X
)
then
T
f
has
a
fixed
point
in
Z
.
Proof.
Now
T
∈
KKM
(
X,
Z
),
f
∈
C
(
Z,
X
)
and
Theorem
1
(ii)
implies
f
T
∈
KKM
(
X,
X
).
Also
f
T
is
a
compact
upper
semicontinuous
map
with
D.
O’Regan
235
compact
values
(so
f
F
is
a
closed
map).
Now,
Theorem
2
guarantees
that
f
T
has
a
fixed
point
in
X
and
consequently
T
f
has
a
fixed
point
in
Z
.
Next,
we
present
an
analogue
of
Theorem
1
(ii)
for
T
f
and
this
compo-
sition
will
be
needed
later.
Theorem
4.
(i).
Let
X
be
an
admissible
convex
set
in
a
Hausdorff
topologi-
cal
vector
space,
Y
a
convex
set
in
a
Hausdorff
topological
vector
space
and
Z
a
normal
topological
space.
If
T
∈
KKM
(
X,
Z
)
is
an
upper
semicontinuous
map
with
compact
values
and
f
∈
C
(
Y,
X
)
then
T
f
∈
KKM
(
Y,
Z
)
.
(ii).
Let
X
be
an
admissible
convex
set
in
a
Hausdorff
topological
vec-
tor
space,
Y
a
convex
set
in
a
Hausdorff
topological
vector
space
and
Z
a
Hausdorff
topological
space.
Suppose
T
∈
KKM
(
X,
Z
)
is
an
upper
semicon-
tinuous
map
with
compact
values,
f
∈
C
(
Y,
X
)
and
there
exists
a
compact
subset
K
of
Z
with
T
(
X
)
⊆
K
.
Then
T
f
∈
KKM
(
Y,
K
)
.
Proof.
(i).
Note
T
f
:
Y
→
2
Z
.
From
Theorem
1
(i),
(iii)
we
need
to
show
that
for
each
polytope
in
Y
that
g
(
T
f
)
has
a
fixed
point
in
for
any
g
∈
C
(
Z,
).
Note
from
Theorem
1
(ii)
since
T
∈
KKM
(
X,
Z
)
and
g
∈
C
(
Z,
)
that
g
T
∈
KKM
(
X,
).
Now
from
Theorem
3
(note
there
Z
=
is
compact
and
g
T
:
X
→
2
is
a
upper
semicontinuous
compact
map
with
compact
values)
guarantees
that
(
g
T
)
f
has
a
fixed
point
in
.
(ii).
Note
T
f
:
Y
→
2
K
and
from
Theorem
1
(i),
(iii)
(note
K
is
normal)
we
need
to
show
that
for
each
polytope
in
Y
that
g
(
T
f
)
has
a
fixed
point
in
for
any
g
∈
C
(
K,
).
The
proof
is
exactly
the
same
as
in
(i)
with
Z
replaced
by
K
.
Note
the
following
two
properties
(see
[8]).
Let
C
and
X
be
convex
subsets
of
a
Hausdorff
topological
vector
space
E
with
C
⊆
X
and
Y
a
Hausdorff
topological
space.
(i).
If
T
∈
KKM
(
X,
Y
)
then
G
≡
T
|
C
∈
KKM
(
C,
Y
).
(ii).
If
T
∈
KKM
(
X,
Y
),
T
(
X
)
⊆
Z
⊆
Y
and
Z
is
closed
in
Y
then
T
∈
KKM
(
X,
Z
).
A
product
rule
for
the
KKM
class
is
unknown
and
we
note
the
argu-
ment
given
in
[10]
was
incorrect.
However
a
product
rule
is
known
for
most
subclasses
of
KKM
maps
(so
the
results
in
[10]
are
correct
for
these
sub-
classes).
As
a
result
in
this
paper
we
consider
subclasses
of
KKM
maps
(for
simplicity
in
notation
we
call
these
KKM
also)
where
the
following
Property
A
holds.
We
also
assume
throughout
the
paper
that
the
analogue
of
Theorem
1
(ii),
Theorem
4
(ii),
properties
(i)
and
(ii)
above
hold
(in
the
relevant
setting)
for
this
subclass
of
KKM
maps
(see
Remark
1).
Zero
epi
KKM
type
maps
and
fixed
points
236
Property
A.
Let
I
be
a
finite
set,
X
be
a
convex
set
in
a
Hausdorff
topo-
logical
vector
space
and
{
Y
i
}
i
∈
I
be
a
family
of
Hausdorff
topological
spaces.
For
each
i
∈
I
suppose
T
i
∈
KKM
(
X,
Y
i
)
is
an
upper
semicontinuous
map
with
compact
values
and
let
T
:
X
→
2
Y
(here
Y
=
i
∈
I
Y
i
)
be
defined
by
T
(
x
)
=
i
∈
I
T
i
(
x
)
for
x
∈
X
.
Also
assume
for
each
i
∈
I
that
there
exists
a
compact
set
K
i
⊆
Y
i
with
T
i
(
X
)
⊆
K
i
.
Then
T
∈
KKM
(
X,
K
)
where
K
=
i
∈
I
K
i
.
Remark
1.
Note
we
also
have
T
∈
KKM
(
X,
Y
)
;
consider
i
◦
T
where
i
:
K
→
Y
is
the
inclusion
map.
Note
Property
A
holds
(in
the
relevant
setting)
for
the
usual
classes
of
maps
in
the
literature,
for
example
the
maps
admissible
with
respect
to
Gorniewicz
[3],
selectionable
maps
[3],
approx-
imable
maps
etc
(there
are
also
analogues
of
Theorem
1
(ii),
Theorem
4
(ii),
properties
(i)
and
(ii)
above
for
these
maps).
Note
Theorem
1
(ii),
Theorem
4
(ii),
and
property
(i)
are
a
lot
less
restrictive
for
Gorniewicz
maps,
selectionable
maps
and
approximable
maps
(i.e.,
the
analogue
of
The-
orem
4
(ii)
is
a
lot
more
general
for
these
maps).
Property
(ii)
above
is
immediate
for
the
Gorniewicz
maps
but
the
set
Z
is
more
special
for
the
selectionable
and
approximable
maps.
As
a
result
some
of
the
conditions
on
the
spaces
can
be
removed
in
Section
2
for
these
subclasses.
This
will
be
discussed
in
a
future
paper.
2
Fixed
point
theory
We
begin
by
recalling
some
results
of
the
author
established
in
[9].
Let
E
be
a
Hausdorff
topological
vector
space.
Definition
1.
We
say
F
∈
KKMC
(
E,
E
)
if
F
∈
KKM
(
E,
E
)
is
an
upper
semicontinuous
compact
map
with
compact
values.
Definition
2.
Let
F
∈
KKMC
(
E,
E
)
.
We
say
F
∼
=
{
0
}
in
KKMC
(
E,
E
)
if
there
exists
an
upper
semicontinuous
compact
map
Ω
:
E
×
[0
,
1]
→
K
(
E
)
with
Ω(
.
,
η
(
.
))
∈
KKM
(
E,
E
)
for
any
continuous
map
η
:
E
→
[0
,
1]
,
Ω
1
=
F
and
Ω
0
=
{
0
}
;
here
K
(
E
)
denotes
the
family
of
nonempty,
com-
pact
subsets
of
E
and
Ω
t
(
x
)
=
Ω(
x,
t
)
.
In
[9],
we
used
the
idea
of
null
homotopy
to
establish
the
following
result.
Theorem
5.
Let
E
be
a
Hausdorff
topological
vector
space
and
U
an
open
subset
of
E
with
0
∈
U
.
Suppose
there
exists
an
upper
semicontinuous
compact
map
H
:
E
×
[0
,
1]
→
K
(
E
)
with
H
(
.
,
η
(
.
))
∈
KKM
(
E,
E
)
for
D.
O’Regan
237
any
continuous
map
η
:
E
→
[0
,
1]
,
H
(
x,
0)
=
{
0
}
for
each
x
∈
E
and
x
/
∈
H
t
(
x
)
for
x
∈
∂U
and
t
∈
(0
,
1]
;
here
∂U
denotes
the
boundary
of
U
in
E
.
In
addition
assume
the
following
holds:
(2
.
1)
for
any
Φ
∈
KKMC
(
E,
E
)
with
Φ
∼
=
{
0
}
in
KKMC
(
E,
E
)
we
have
that
Φ
has
a
fixed
point
in
E.
Then
H
1
has
a
fixed
point
in
U
.
A
natural
question
to
ask
in
Theorem
5
is
if
we
had
an
upper
semicon-
tinuous
compact
map
Λ
:
U
×
[0
,
1]
→
K
(
E
)
(here
U
denotes
the
closure
of
U
in
E
)
would
it
be
possible
to
construct
the
map
H
in
the
statement
of
Theorem
5.
In
[9],
we
obtained
the
following
result.
Theorem
6.
Let
E
be
a
Hausdorff
topological
vector
space,
U
an
open
subset
of
E
with
0
∈
U
,
and
suppose
there
exists
a
retraction
(continuous)
r
:
E
→
U
.
Also
assume
there
exists
an
upper
semicontinuous
compact
map
Λ
:
U
×
[0
,
1]
→
K
(
E
)
with
Λ(
r
(
.
)
,
η
(
.
))
∈
KKM
(
E,
E
)
for
any
continuous
map
η
:
E
→
[0
,
1]
,
Λ(
x,
0)
=
{
0
}
for
each
x
∈
U
and
x
/
∈
Λ
t
(
x
)
for
x
∈
∂U
and
t
∈
(0
,
1]
.
In
addition
assume
(2
.
1)
holds.
Then
Λ
1
has
a
fixed
point
in
U
.
Remark
2.
In
the
proof
of
Theorem
6
we
used
Theorem
5
with
H
(
x,
t
)
=
Λ(
r
(
x
)
,
t
)
for
(
x,
t
)
∈
E
×
[0
,
1]
.
We
consider
the
subclass
of
KKM
maps
(again
for
simplicity
we
call
these
KKM
)
for
the
remainder
of
Section
2
and
we
assume
Theorem
1
(ii),
Theorem
4
(ii),
properties
(i)
and
(ii)
in
Section
1
and
Property
A
hold
for
the
subclass.
The
new
idea
here
is
to
consider
the
natural
homotopy
between
H
1
and
the
zero
map
in
Definition
2
and
Theorem
5
(and
the
corresponding
one
in
Theorem
6)
to
generate
applicable
fixed
point
theory
for
the
map
H
1
in
Theorem
5
(and
also
Λ
1
in
Theorem
6).
This
is
possible
because
of
Section
1
where
the
idea
here
is
to
use
appropriately
Theorem
1
(ii),
Theorem
4
(ii)
and
Property
A
(for
the
subclasses
of
KKM
mentioned
above
and
we
use
the
appropriate
∼
=
in
that
subclass;
note
the
proof
of
the
analogue
of
Theorem
5
and
Theorem
6
for
this
subclass
is
exactly
the
same
as
in
[9]).
We
first
consider
Theorem
5
and
then
Theorem
6.
Theorem
7.
Let
E
be
a
Hausdorff
topological
vector
space
and
U
an
open
subset
of
E
with
0
∈
U
.
Let
F
∈
KKMC
(
E,
E
)
and
assume
the
following
conditions
hold:
(2
.
2)
x
/
∈
t
F
(
x
)
for
x
∈
∂U
and
t
∈
(0
,
1]
Zero
epi
KKM
type
maps
and
fixed
points
238
and
(2
.
3)
any
Φ
∈
KKMC
(
E,
E
)
has
a
fixed
point
in
E.
Then
F
has
a
fixed
point
in
U
.
Proof.
Let
H
(
x,
t
)
=
t
F
(
x
)
for
(
x,
t
)
∈
E
×
[0
,
1].
Note
H
:
E
×
[0
,
1]
→
K
(
E
)
is
an
upper
semicontinuous
compact
map
with
compact
values.
Let
η
:
E
→
[0
,
1]
be
a
continuous
map
(i.e.
η
∈
C
(
E,
[0
,
1])).
We
claim
H
(
.,
η
(
.
))
=
η
(
.
)
F
(
.
)
∈
KKM
(
E,
E
).
To
see
this
first
note
that
there
exists
a
compact
subset
K
of
E
with
F
(
E
)
⊆
K
.
Let
Ψ
:
E
→
2
[0
,
1]
×
E
be
given
by
Ψ(
x
)
=
J
(
x
)
×
F
(
x
),
x
∈
E
where
J
(
x
)
=
η
(
x
).
From
Property
A
(with
T
1
=
J
,
T
2
=
F
,
X
=
E
,
K
1
=
[0
,
1]
and
K
2
=
K
)
we
have
that
Ψ
∈
KKM
(
E,
[0
,
1]
×
K
)
is
an
upper
semicontinuous
compact
(note
Ψ(
E
)
⊆
[0
,
1]
×
K
)
map
with
compact
values.
Let
p
:
[0
,
1]
×
K
→
E
be
given
by
p
(
t,
x
)
=
t
x
for
(
t,
x
)
∈
[0
,
1]
×
K
and
note
p
∈
C
([0
,
1]
×
K,
E
).
From
Theorem
1
(ii)
we
see
that
H
(
.,
η
(
.
))
=
η
(
.
)
F
(
.
)
=
p
Ψ(
.
)
∈
KKM
(
E,
E
),
so
our
claim
is
true.
Also
note
H
0
=
{
0
}
,
H
1
=
F
and
x
/
∈
H
t
(
x
)
for
x
∈
∂U
and
t
∈
[0
,
1]
from
(2
.
2).
Thus
we
have
constructed
a
H
as
in
Theorem
5.
We
next
note
if
Φ
∈
KKMC
(
E,
E
)
then
Φ
∼
=
{
0
}
in
KKMC
(
E,
E
)
(i.e.
we
need
to
show
that
there
exists
an
upper
semicontinuous
compact
map
Ω
:
E
×
[0
,
1]
→
K
(
E
)
with
Ω(
.,
η
(
.
))
∈
KKM
(
E,
E
)
for
any
continuous
η
:
E
→
[0
,
1],
Ω
1
=
Φ
and
Ω
0
=
{
0
}
).
To
see
this
let
Ω(
t,
x
)
=
t
Φ(
x
)
for
(
x,
t
)
∈
E
×
[0
,
1]
and
exactly
the
same
argument
as
above
with
H
replaced
by
Ω
and
F
replaced
by
Φ
guarantees
this.
Thus
(2
.
3)
guarantees
(2
.
1).
Thus,
our
result
now
follows
from
Theorem
5.
Remark
3.
Note
if
E
was
admissible
then
Theorem
2
guarantees
that
(2
.
3)
holds.
Let
E
be
a
Hausdorff
topological
vector
space
and
U
an
open
convex
subset
of
E
.
Definition
3.
We
say
F
∈
KKMC
(
U,
E
)
if
F
∈
KKM
(
U,
E
)
is
an
upper
semicontinuous
compact
map
with
compact
values.
Theorem
8.
Let
E
be
a
Hausdorff
topological
vector
space,
U
an
open
convex
subset
of
E
,
0
∈
U
,
U
an
admissible
subset
of
E
and
assume
there
exists
a
retraction
(continuous)
r
:
E
→
U
.
Let
F
∈
KKMC
(
U,
E
)
and
suppose
(2
.
2)
and
(2
.
3)
hold.
Then
F
has
a
fixed
point
in
U
.
Proof.
Let
Λ(
x,
t
)
=
t
F
(
x
)
for
(
x,
t
)
∈
U
×
[0
,
1].
Note
Λ
:
U
×
[0
,
1]
→
K
(
E
)
is
an
upper
semicontinuous
compact
map
with
compact
values.
Let
η
:
E
→
D.
O’Regan
239
[0
,
1]
be
a
continuous
map
and
we
claim
Λ(
r
(
.
)
,
η
(
.
))
=
η
(
.
)
F
r
(
.
)
∈
KKM
(
E,
E
).
First
note
that
there
exists
a
compact
subset
K
of
E
with
F
(
U
)
⊆
K
and
note
F
∈
KKM
(
U,
E
)
(is
an
upper
semicontinuous
com-
pact
map
with
compact
values),
r
∈
C
(
E,
U
)
so
Theorem
4
(ii)
(note
U
is
admissible)
guarantees
that
F
r
∈
KKM
(
E,
K
)
is
an
upper
semicontin-
uous
compact
map
with
compact
values.
Let
Ψ
:
E
→
2
[0
,
1]
×
E
be
given
by
Ψ(
x
)
=
J
(
x
)
×
F
r
(
x
),
x
∈
E
where
J
(
x
)
=
η
(
x
).
From
Property
A
(with
T
1
=
J
,
T
2
=
F
r
,
X
=
E
,
K
1
=
[0
,
1]
and
K
2
=
K
)
we
have
that
Ψ
∈
KKM
(
E,
[0
,
1]
×
K
)
is
an
upper
semicontinuous
compact
with
compact
values.
Let
p
:
[0
,
1]
×
K
→
E
be
given
by
p
(
t,
x
)
=
t
x
for
(
t,
x
)
∈
[0
,
1]
×
K
and
note
p
∈
C
([0
,
1]
×
K,
E
).
From
Theorem
1
(ii)
we
see
that
Λ(
r
(
.
)
,
η
(
.
))
=
η
(
.
)
F
r
(
.
)
=
p
Ψ(
.
)
∈
KKM
(
E,
E
),
so
our
claim
is
true.
Also
note
Λ
0
=
{
0
}
,
Λ
1
=
F
and
x
/
∈
Λ
t
(
x
)
for
x
∈
∂U
and
t
∈
[0
,
1]
from
(2
.
2).
Thus
we
have
constructed
a
Λ
as
in
Theorem
6.
Also
(2
.
1)
holds
as
in
Theorem
7.
Thus,
our
result
follows
from
Theorem
5.
Remark
4.
If
E
is
a
locally
convex
topological
vector
space
and
U
is
an
open
convex
subset
of
E
then
U
is
a
retract
of
E
and
also
note
U
is
an
admissible
subset
of
E
.
We
either
note
Dugundji’s
extension
theorem
[3]
or
alternatively
let
r
:
E
→
U
be
given
by
r
(
x
)
=
x
max
{
1
,
η
(
x
)
}
for
x
∈
E
where
η
is
the
Minkowski
functional
on
U
i.e.
η
(
x
)
=
inf
{
α
:
x
∈
α
U
}
.
The
theory
above
now
motivates
a
theory
of
zero
epi
compact
KKM
maps
which
we
now
present.
The
classes
of
maps
considered
was
motivated
from
the
KLU
[5],
HLPY
[6]
and
Wu
[11]
maps.
We
will
consider
three
classes
(the
arguments
are
very
similar
in
each
class).
Let
E
be
a
Hausdorff
topological
vector
space
(so
completely
regular)
and
U
an
open
convex
subset
of
E
.
Definition
4.
We
say
F
∈
A
(
U,
E
)
if
F
:
U
→
C
(
E
)
is
upper
semicontin-
uous;
here
C
(
E
)
denotes
the
family
of
nonempty
closed
subsets
of
E
.
Definition
5.
We
say
F
∈
A
∂U
(
U,
E
)
if
F
∈
A
(
U,
E
)
and
0
/
∈
F
(
x
)
for
x
∈
∂U
.
Definition
6.
We
say
G
∈
csKKMC
(
U,
E
)
if
G
:
U
→
2
E
and
there
exists
a
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
co
(
G
(
x
))
for
x
∈
U
.
Zero
epi
KKM
type
maps
and
fixed
points
240
Definition
7.
We
say
G
∈
csKKMC
0
(
U,
E
)
if
G
:
U
→
2
E
and
there
exists
a
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
co
(
G
(
x
))
for
x
∈
U
and
Φ(
x
)
=
{
0
}
for
x
∈
∂U
.
Now
we
present
the
notion
of
0–epi
maps
[4,
7].
Definition
8.
Let
F
∈
A
∂U
(
U,
E
)
.
We
say
F
is
0
–epi
if
for
any
map
G
∈
csKKMC
0
(
U,
E
)
and
any
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
co
(
G
(
x
))
for
x
∈
U
and
Φ(
x
)
=
{
0
}
for
x
∈
∂U
there
exists
a
x
∈
U
with
F
(
x
)
∩
Φ(
x
)
=
∅
.
Remark
5.
(i).
In
Definition
8
note
∅
=
F
(
x
)
∩
Φ(
x
)
⊆
F
(
x
)
∩
co
(
G
(
x
))
so
there
exists
a
x
∈
U
with
F
(
x
)
∩
co
(
G
(
x
))
=
∅
.
(ii).
In
Definition
8
(take
G
=
0
)
note
there
exists
a
x
∈
U
with
0
∈
F
(
x
)
.
We
begin
with
a
coincidence
result.
Theorem
9.
Let
E
be
a
Hausdorff
topological
vector
space,
U
an
open
convex
subset
of
E
,
G
∈
csKKMC
(
U,
E
)
and
F
∈
A
∂U
(
U,
E
)
be
0
–epi.
In
addition
for
any
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
co
(
G
(
x
))
for
x
∈
U
suppose
x
∈
U
:
F
(
x
)
∩
t
Φ(
x
)
=
∅
for
some
t
∈
[0
,
1]
does
not
intersect
∂U
.
There
there
exists
an
x
∈
U
with
F
(
x
)
∩
co
(
G
(
x
))
=
∅
.
Proof.
Let
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
co
(
G
(
x
))
for
x
∈
U
and
let
K
=
x
∈
U
:
F
(
x
)
∩
t
Φ(
x
)
=
∅
for
some
t
∈
[0
,
1]
.
Note
K
=
∅
(take
t
=
0
and
note
Remark
5
(ii)),
K
is
closed
(recall
F
and
Φ
are
upper
semicontinuous)
and
compact
(note
Φ
is
compact).
By
assumption
K
∩
∂U
=
∅
.
Now
since
E
is
completely
regular
then
there
exists
a
continuous
map
µ
:
U
→
[0
,
1]
with
µ
(
∂U
)
=
0
and
µ
(
K
)
=
1.
Let
J
(
x
)
=
µ
(
x
)
G
(
x
)
and
R
(
x
)
=
µ
(
x
)
Φ(
x
)
for
x
∈
U
.
Note
R
(
x
)
⊆
co
(
G
(
x
))
for
x
∈
U
(since
for
a
fixed
x
∈
U
note
µ
(
x
)
Φ(
x
)
⊆
µ
(
x
)
co
(
G
(
x
))
=
co
(
µ
(
x
)
G
(
x
))
because
co
(
t
A
)
=
t
co
(
A
)).
Also
note
R
:
U
→
2
E
is
an
upper
semicontinuous
compact
map
with
compact
values
and
R
∈
KKM
(
U,
E
).
To
see
this
note
that
there
exists
a
compact
subset
K
0
of
E
with
Φ(
U
)
⊆
K
0
.
Let
Ψ
:
U
→
2
[0
,
1]
×
E
be
given
by
Ψ(
x
)
=
J
0
(
x
)
×
Φ(
x
),
x
∈
U
where
J
0
(
x
)
=
µ
(
x
).
From
Property
A
(with
T
1
=
J
0
,
T
2
=
Φ,
X
=
U
,
K
1
=
[0
,
1]
and
K
2
=
K
0
)
we
have
that
Ψ
∈
KKM
(
U,
[0
,
1]
×
K
0
)
is
an
upper
semicontinuous
compact
(note
Ψ(
U
)
⊆
[0
,
1]
×
K
0
)
map
with
compact
values.
Let
p
:
[0
,
1]
×
K
0
→
E
be
given
by
p
(
t,
x
)
=
t
x
for
(
t,
x
)
∈
[0
,
1]
×
K
0
D.
O’Regan
241
and
note
p
∈
C
([0
,
1]
×
K
0
,
E
).
From
Theorem
1
(ii)
we
see
that
R
(
.
)
=
µ
(
.
)
Φ(
.
)
=
p
Ψ(
.
)
∈
KKM
(
U,
E
)
is
an
upper
semicontinuous
compact
map
with
compact
values.
Next
notice
J
∈
csKKMC
0
(
U,
E
)
since
if
x
∈
∂U
then
R
(
x
)
=
µ
(
x
)
Φ(
x
)
=
{
0
}
(note
µ
(
∂U
)
=
0).
Now
since
F
is
0–epi
then
there
exists
a
y
∈
U
with
F
(
y
)
∩
R
(
y
)
=
∅
.
Note
y
∈
K
so
µ
(
y
)
=
1
i.e.
F
(
y
)
∩
Φ(
y
)
=
∅
so
F
(
y
)
∩
co
(
G
(
y
))
=
∅
.
Next,
we
present
a
normalization
result.
Theorem
10.
Let
E
be
an
admissible
Hausdorff
topological
vector
space,
U
an
open
convex
subset
of
E
,
0
∈
U
,
U
an
admissible
subset
of
E
and
assume
there
exists
a
retraction
r
:
E
→
U
.
Then
the
identity
mapping
i
is
0
–epi.
Proof.
Note
i
(
x
)
=
x
for
x
∈
U
and
i
∈
A
∂U
(
U,
E
)
(note
0
/
∈
i
(
x
)
=
x
for
x
∈
∂U
since
0
∈
U
).
Consider
any
G
∈
csKKMC
0
(
U,
E
)
and
any
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
co
(
G
(
x
))
for
x
∈
U
and
Φ(
x
)
=
{
0
}
for
x
∈
∂U
.
Let
Ω
=
x
∈
U
:
x
∈
λ
Φ(
x
)
for
some
λ
∈
[0
,
1]
.
Now
Ω
=
∅
(note
0
∈
U
)
and
Ω
is
closed
and
compact.
Note
Ω
∩
∂U
=
∅
since
if
there
exists
a
x
∈
∂U
and
x
∈
Ω
then
for
some
λ
∈
[0
,
1]
we
have
x
∈
λ
Φ(
x
)
=
{
0
}
,
so
x
=
0
which
is
a
contradiction
since
0
∈
U
.
Now
there
exists
a
continuous
map
µ
:
E
→
[0
,
1]
with
µ
(
E
\
U
)
=
0
and
µ
(Ω)
=
1.
Let
J
(
x
)
=
µ
(
x
)
Φ(
r
(
x
)).
Note
J
:
E
→
2
E
is
an
upper
semicontinuous
compact
map
with
compact
values
and
J
∈
KKM
(
E,
E
).
To
see
this
first
note
that
there
exists
a
compact
subset
K
of
E
with
Φ(
U
)
⊆
K
so
Φ
r
(
E
)
⊆
Φ(
U
)
⊆
K
.
Note
Φ
∈
KKM
(
U,
E
),
r
∈
C
(
E,
U
)
so
Theorem
4
(ii)
(note
U
is
admissible)
guarantees
that
Φ
r
∈
KKM
(
E,
K
)
is
an
upper
semicontinuous
compact
map
with
compact
values.
Let
Ψ
:
E
→
2
[0
,
1]
×
E
be
given
by
Ψ(
x
)
=
J
0
(
x
)
×
Φ
r
(
x
),
x
∈
E
where
J
0
(
x
)
=
µ
(
x
).
From
Property
A
(with
T
1
=
J
0
,
T
2
=
Φ
r
,
X
=
E
,
K
1
=
[0
,
1]
and
K
2
=
K
)
we
have
that
Ψ
∈
KKM
(
E,
[0
,
1]
×
K
)
is
an
upper
semicontinuous
compact
(note
Ψ(
E
)
⊆
[0
,
1]
×
K
)
with
compact
values.
Let
p
:
[0
,
1]
×
K
→
E
be
given
by
p
(
t,
x
)
=
t
x
for
(
t,
x
)
∈
[0
,
1]
×
K
and
note
p
∈
C
([0
,
1]
×
K,
E
).
From
Theorem
1
(ii)
we
see
that
J
(
.
)
=
µ
(
.
)
Φ
r
(
.
)
=
p
Ψ(
.
)
∈
KKM
(
E,
E
)
is
an
upper
semicontinuous
compact
with
compact
values.
Now,
Theorem
2
guarantees
that
there
exists
a
x
∈
E
with
x
∈
J
(
x
).
If
x
∈
E
\
U
then
µ
(
x
)
=
0,
so
x
∈
{
0
}
i.e.
x
=
0,
a
contradiction
since
0
∈
U
.
Thus
x
∈
U
so
x
∈
µ
(
x
)
Φ(
r
(
x
))
=
µ
(
x
)
Φ(
x
),
so
x
∈
Ω
and
so
µ
(
x
)
=
1.
Thus
x
∈
Φ(
x
)
i.e.
i
(
x
)
∩
Φ(
x
)
=
∅
.
Consequently
i
is
0–epi.
Zero
epi
KKM
type
maps
and
fixed
points
242
Now,
Theorem
9
and
Theorem
10
immediately
gives
us
a
Leray–Schauder
type
alternative.
Theorem
11.
Let
E
be
an
admissible
Hausdorff
topological
vector
space,
U
an
open
convex
subset
of
E
,
0
∈
U
,
U
an
admissible
subset
of
E
and
assume
there
exists
a
retraction
r
:
E
→
U
.
Suppose
G
∈
csKKMC
(
U,
E
)
and
x
/
∈
t
co
G
(
x
)
for
x
∈
∂U
and
t
∈
(0
,
1)
.
Then
there
exists
a
x
∈
U
with
x
∈
co
G
(
x
)
.
Proof.
Suppose
x
/
∈
co
G
(
x
)
for
x
∈
∂U
(otherwise
we
are
finished).
Then
x
/
∈
t
co
G
(
x
)
for
x
∈
∂U
and
t
∈
[0
,
1].
Consider
any
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
co
(
G
(
x
))
for
x
∈
U
.
Then
x
∈
U
:
i
(
x
)
∩
t
Φ(
x
)
=
∅
for
some
t
∈
[0
,
1]
does
not
intersect
∂U
since
if
there
exists
a
x
∈
∂U
with
i
(
x
)
(=
x
)
∈
t
Φ(
x
)
for
some
t
∈
[0
,
1]
then
x
∈
t
co
G
(
x
),
which
is
a
contradiction.
Now,
i
is
0–epi
(Theorem
10),
so
Theorem
9
(with
F
=
i
)
guarantees
a
x
∈
U
with
i
(
x
)
∩
co
G
(
x
)
=
∅
i.e.
x
∈
co
G
(
x
).
From
the
ideas
above
we
could
also
consider
the
following
class
of
maps.
Definition
9.
We
say
G
∈
sKKMC
(
U,
E
)
if
G
:
U
→
2
E
and
there
exists
a
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
G
(
x
)
for
x
∈
U
.
Definition
10.
We
say
G
∈
sKKMC
0
(
U,
E
)
if
G
:
U
→
2
E
and
there
exists
a
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
G
(
x
)
for
x
∈
U
and
Φ(
x
)
=
{
0
}
for
x
∈
∂U
.
Definition
11.
Let
F
∈
A
∂U
(
U,
E
)
.
We
say
F
is
0
–epi
if
for
any
map
G
∈
sKKMC
0
(
U,
E
)
and
any
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
G
(
x
)
for
x
∈
U
and
Φ(
x
)
=
{
0
}
for
x
∈
∂U
there
exists
a
x
∈
U
with
F
(
x
)
∩
Φ(
x
)
=
∅
.
The
same
proofs
with
minor
adjustments
(i.e.
removing
”
co
”)
in
Theo-
rem’s
9,
10
and
11
establish
the
following
results.
Theorem
12.
Let
E
be
a
Hausdorff
topological
vector
space,
U
an
open
convex
subset
of
E
,
G
∈
sKKMC
(
U,
E
)
and
F
∈
A
∂U
(
U,
E
)
be
0
–epi
(in
the
sense
of
Definition
11).
In
addition
for
any
Φ
∈
KKMC
(
U,
E
)
with
Φ(
x
)
⊆
G
(
x
)
for
x
∈
U
suppose
x
∈
U
:
F
(
x
)
∩
t
Φ(
x
)
=
∅
for
some
t
∈
[0
,
1]
does
not
intersect
∂U
.
There
there
exists
an
x
∈
U
with
F
(
x
)
∩
G
(
x
)
=
∅
.
D.
O’Regan
243
Theorem
13.
Let
E
be
an
admissible
Hausdorff
topological
vector
space,
U
an
open
convex
subset
of
E
,
0
∈
U
,
U
an
admissible
subset
of
E
and
assume
there
exists
a
retraction
r
:
E
→
U
.
Then
the
identity
mapping
i
is
0
–epi
(in
the
sense
of
Definition
11).
Theorem
14.
Let
E
be
an
admissible
Hausdorff
topological
vector
space,
U
an
open
convex
subset
of
E
,
0
∈
U
,
U
an
admissible
subset
of
E
and
assume
there
exists
a
retraction
r
:
E
→
U
.
Suppose
G
∈
sKKMC
(
U,
E
)
and
x
/
∈
t
G
(
x
)
for
x
∈
∂U
and
t
∈
(0
,
1)
.
Then
there
exists
a
x
∈
U
with
x
∈
G
(
x
)
.
One
could
also
consider
the
following
class
of
maps.
Definition
12.
We
say
G
∈
KKMC
0
(
U,
E
)
if
G
∈
KKMC
(
U,
E
)
with
G
(
x
)
=
{
0
}
for
x
∈
∂U
.
Definition
13.
Let
F
∈
A
∂U
(
U,
E
)
.
We
say
F
is
0
–epi
if
for
any
map
G
∈
KKMC
0
(
U,
E
)
there
exists
a
x
∈
U
with
F
(
x
)
∩
G
(
x
)
=
∅
.
Then
we
have
the
following
results.
Theorem
15.
Let
E
be
a
Hausdorff
topological
vector
space,
U
an
open
convex
subset
of
E
,
G
∈
KKMC
(
U,
E
)
and
F
∈
A
∂U
(
U,
E
)
be
0
–epi
(in
the
sense
of
Definition
13).
In
addition
suppose
x
∈
U
:
F
(
x
)
∩
t
G
(
x
)
=
∅
for
some
t
∈
[0
,
1]
does
not
intersect
∂U
.
There
there
exists
an
x
∈
U
with
F
(
x
)
∩
G
(
x
)
=
∅
.
Theorem
16.
Let
E
be
an
admissible
Hausdorff
topological
vector
space,
U
an
open
convex
subset
of
E
,
0
∈
U
,
U
an
admissible
subset
of
E
and
assume
there
exists
a
retraction
r
:
E
→
U
.
Then
the
identity
mapping
i
is
0
–epi
(in
the
sense
of
Definition
13).
Theorem
17.
Let
E
be
an
admissible
Hausdorff
topological
vector
space,
U
an
open
convex
subset
of
E
,
0
∈
U
,
U
an
admissible
subset
of
E
and
assume
there
exists
a
retraction
r
:
E
→
U
.
Suppose
G
∈
KKMC
(
U,
E
)
and
x
/
∈
t
G
(
x
)
for
x
∈
∂U
and
t
∈
(0
,
1)
.
Then
there
exists
a
x
∈
U
with
x
∈
G
(
x
)
.
Remark
6.
It
is
also
easy
to
extend
Theorem
5
,
Theorem
6
,
Theorem
7
and
Theorem
8
to
csKKMC
and
sKKMC
maps.
Zero
epi
KKM
type
maps
and
fixed
points
244
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