Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
ON
EKELAND
VARIATIONAL
PRINCIPLE
FOR
PREORDERED
UNIFORM
SPACE
∗
Lucas
Fresse
†
Viorica
V.
Motreanu
‡
DOI 10.56082/annalsarscimath.2026.4.365
Abstract
We
introduce
a
notion
of
preordered
uniform
space
which
extends
the
classical
notion
of
uniform
space
and
uniformizable
topological
space.
We
establish
two
versions
of
Ekeland
variational
principle
for
the
present
setting.
The
rst
version
is
for
a
real-valued
submonotone
map
in
a
general
preordered
uniform
space
while
the
second
version
is
for
a
vector-valued
map
but
within
an
at
most
countable
uniform
structure.
Keywords:
uniform
space,
preordered
space,
submonotone
map,
vari-
ational
principle.
MSC:
58E30,
06A75,
54E15,
49J52.
1
Introduction
Uniform
topological
spaces
appear
in
general
topology
in
relation
with
Ty-
chono’s
separation
axiom
(
T
3
1
2
).
Specically,
a
topological
space
X
is
said
to
satisfy
(
T
3
1
2
)
if
for
every
x
∈
X
and
every
closed
subset
C
⊂
X
with
x
/
∈
C
,
there
is
a
continuous
function
f
:
X
→
[0
,
1]
such
that
f
(
x
)
=
0
and
∗
Accepted
for
publication
on
October
18,
2026
†
lucas.fresse@univ-lorraine.fr
,
Universit´
e
de
Lorraine,
Institut
´
Elie
Cartan,
54506
Vandoeuvre-l`
es-Nancy,
France
‡
vmotreanu@gmail.com
,
Lyc´
ee
Varoquaux,
10
rue
Jean
Moulin,
54510
Tomblaine,
France
365
On
Ekeland
variational
principle
for
preordered
uniform
space
366
f
|
C
≡
1.
It
is
shown
in
[
1
]
that
this
axiom
is
equivalent
to
the
property
of
X
to
be
uniformizable,
i.e.,
the
topology
of
X
can
be
induced
by
a
so-called
uniform
structure
.
A
uniform
structure
consists
of
a
family
of
´
ecarts,
also
called
pseudometrics.
Endowed
with
a
uniform
structure,
X
is
called
a
uni-
form
space
.
We
refer
to
[
1
]
for
the
precise
denition
of
uniform
structure
and
uniform
space,
or
to
Section
3.2
below.
The
setting
of
the
present
paper
is
in
fact
more
general:
we
consider
a
set
P
endowed
with
a
preorder
≤
(a
binary
relation
which
is
reexive
and
transitive).
We
rst
extend
the
notion
of
uniform
structure
and
uniform
space
to
the
case
of
a
preordered
set
(
P,
≤
),
by
dening
a
notion
of
preordered
uniform
space
(
P,
≤
,
D
)
parametrized
by
a
family
D
of
≤
-pseudometrics
(see
Section
3.2
).
When
the
preorder
≤
is
trivial
(i.e.,
such
that
x
≤
y
for
all
x,
y
∈
P
),
we
retrieve
the
classical
notion
of
uniform
space.
Instead
of
focussing
on
purely
topological
properties,
our
aim
in
this
paper
is
to
do
variational
analysis
on
preordered
uniform
spaces.
Our
main
results
are
two
versions
of
the
classical
Ekeland
variational
principle
(which
is
usually
formulated
for
metric
spaces)
valid
for
a
preordered
uniform
space
(
P,
≤
,
D
):
•
The
rst
result
(Theorem
1
)
is
a
version
of
Ekeland
variational
princi-
ple
for
an
arbitrary
complete
preordered
uniform
space
(
P,
≤
,
D
)
and
concerns
a
map
f
:
P
→
R
∪
{
+
∞}
with
a
submonotonicity
property
(it
is
not
assumed
continuous
and
not
even
lower
semicontinuous).
•
The
second
result
(Theorem
2
)
is
a
version
of
Ekeland’s
principle
for
a
family
of
maps
F
=
{
f
d
}
d
∈D
with
f
d
:
P
→
R
∪{
+
∞}
,
instead
of
just
a
map
f
,
but
with
the
assumption
that
the
family
D
of
≤
-pseudometrics
is
(at
most)
countable.
For
showing
these
results,
the
main
ingredient
is
a
version
of
Ekeland
variational
principle
over
preordered
monoids;
see
[
3
].
For
the
rst
theorem
(Theorem
1
),
we
apply
this
result
to
the
monoid
K
=
R
∪
{
+
∞}
;
for
the
second
theorem
(Theorem
2
),
we
use
the
monoid
K
=
(
R
∪
{
+
∞}
)
N
.
In
order
to
be
able
to
apply
the
general
result
in
[
3
],
we
rely
on
a
thorough
analysis
of
preordered
uniform
spaces
and
their
properties:
key
lemmas
in
this
respect
are
Lemmas
1
(b),
4
,
and
5
.
The
paper
is
organized
as
follows.
In
Section
2
,
we
introduce
a
general
notion
of
preordered
topological
space.
In
Section
3
,
we
rst
introduce
the
notions
of
≤
-pseudometric
and
preordered
pseudometric
space
(Section
3.1
);
then
we
introduce
the
more
general
notion
of
preordered
uniform
space
L.
Fresse,
V.V.
Motreanu
367
(Section
3.2
).
Our
main
results
are
stated
in
Section
4
and
the
proofs
are
given
in
Section
5
.
2
Preordered
topological
space
Throughout
this
paper,
we
x
a
nonempty
set
P
endowed
with
a
preorder
≤
,
i.e.,
a
binary
relation
on
P
which
is
reexive
and
transitive
(but
not
necessarily
antisymmetric).
A
sequence
(
x
n
)
⊂
P
is
said
to
be
≤
-ascending
if
it
satises
x
n
≤
x
n
+1
for
all
n
.
Moreover,
we
say
that
x
is
an
upper
bound
of
a
sequence
(
x
n
)
⊂
P
if
x
n
≤
x
for
all
n
.
Assume
that
the
set
P
is
equipped
with
a
topology
τ
.
We
associate
to
this
topology
a
second
topology
˜
τ
≤
which
takes
the
preorder
≤
into
account
and
is
dened
in
the
following
way.
Hereafter,
the
notation
x
n
τ
→
x
means
that
x
is
a
limit
of
(
x
n
)
with
respect
to
τ
(the
limit
is
a
priori
not
unique).
•
A
subset
A
⊂
P
is
said
to
be
(
≤
,
τ
)
-closed
if
L
(
≤
,τ
)
(
A
)
:=
{
x
∈
P
:
∃
(
x
n
)
⊂
A
≤
-ascending
,
x
n
τ
→
x
}
=
A
(note
that
the
inclusion
A
⊂
L
(
≤
,τ
)
(
A
)
is
always
valid).
•
We
say
that
U
⊂
P
is
(
≤
,
τ
)
-open
if
P
\
U
is
(
≤
,
τ
)-closed.
Then
it
is
straightforward
to
check
that
˜
τ
≤
:=
{
U
⊂
P
:
U
is
(
≤
,
τ
)-open
}
is
indeed
a
topology
on
P
.
We
call
˜
τ
≤
the
≤
-ascending
extension
of
τ
.
Lemma
1.
(a)
τ
⊂
˜
τ
≤
.
(b)
Let
(
x
n
)
⊂
P
be
a
≤
-ascending
sequence
and
let
x
∈
P
.
Then
x
n
˜
τ
≤
→
x
⇔
x
n
τ
→
x.
Proof.
(a)
Let
A
⊂
P
be
a
closed
subset
for
the
topology
τ
and
let
us
show
that
A
is
also
closed
for
the
topology
˜
τ
≤
,
i.e.,
that
A
is
(
≤
,
τ
)-closed.
So
let
(
x
n
)
⊂
A
be
a
≤
-ascending
sequence
and
let
x
∈
P
be
such
that
x
n
τ
→
x
.
The
latter
fact,
combined
with
the
assumption
that
A
is
closed
for
the
topology
τ
,
already
ensures
that
x
∈
A
,
and
the
claim
is
established.
On
Ekeland
variational
principle
for
preordered
uniform
space
368
(b)
The
implication
⇒
easily
follows
from
part
(a).
Let
us
prove
the
reversed
implication
⇐
.
Arguing
by
contradiction,
assume
that
there
exists
U
∈
˜
τ
≤
such
that
x
∈
U
and
∀
n
0
∈
N
,
∃
n
≥
n
0
,
x
n
̸∈
U.
This
yields
a
subsequence
(
x
n
k
)
of
(
x
n
)
such
that
x
n
k
̸∈
U
for
all
k
.
Note
that
(
x
n
k
)
is
a
≤
-ascending
sequence
(since
(
x
n
)
is
≤
-ascending)
and
x
n
k
τ
→
x
(since
x
n
τ
→
x
).
The
subset
A
:=
P
\
U
is
(
≤
,
τ
)-closed
(since
U
∈
˜
τ
≤
)
and
x
n
k
∈
A
for
all
k
.
Whence
x
∈
A
=
P
\
U
,
a
contradiction.
Remark
1.
(a)
In
Lemma
1
(b),
the
assumption
that
(
x
n
)
is
≤
-ascending
is
not
needed
for
the
implication
⇒
,
but
is
required
for
the
reversed
implication
⇐
,
as
shown
by
the
following
example.
Let
P
=
R
be
endowed
with
the
standard
topology
τ
and
standard
order
≤
.
Let
(
x
n
)
n
≥
1
be
the
sequence
given
by
x
n
=
1
n
and
let
x
=
0.
Thus
x
n
τ
→
x
.
However,
(
x
n
)
does
not
converge
to
x
with
respect
to
˜
τ
≤
.
Indeed,
it
is
easy
to
see
that
(0
,
+
∞
)
is
(
≤
,
τ
)-closed,
hence
(
−∞
,
0]
is
an
open
neighborhood
of
x
=
0
in
˜
τ
≤
which
contains
no
term
of
the
sequence
(
x
n
).
(b)
If
we
dene
σ
=
˜
τ
≤
,
then
Lemma
1
implies
that
˜
σ
≤
=
σ
(=
˜
τ
≤
).
Indeed,
Lemma
1
(a)
already
yields
the
inclusion
σ
⊂
˜
σ
≤
.
For
the
reversed
inclusion,
let
A
⊂
P
be
(
≤
,
σ
)-closed
and
let
us
show
that
A
is
(
≤
,
τ
)-closed.
So
let
(
x
n
)
⊂
A
be
≤
-ascending
and
x
∈
P
such
that
x
n
τ
→
x
.
By
Lemma
1
(b)
we
get
x
n
σ
→
x
,
whence
x
∈
A
(since
A
is
(
≤
,
σ
)-closed),
and
the
argument
is
complete.
(c)
A
subset
A
⊂
P
is
sequentially
closed
with
respect
to
τ
if
for
every
sequence
(
x
n
)
⊂
A
and
x
∈
P
such
that
x
n
τ
→
x
,
we
have
x
∈
A
.
We
say
that
U
⊂
P
is
sequentially
open
with
respect
to
τ
if
P
\
U
is
sequentially
closed.
The
collection
˜
τ
of
all
sequentially
open
subsets
is
a
topology
on
P
which
contains
τ
and
diers
from
τ
in
general.
The
equality
˜
τ
=
τ
holds
if
every
x
∈
P
has
a
countable
local
basis
in
the
topology
τ
,
i.e.,
the
topological
space
(
P,
τ
)
satises
the
axiom
of
countability
(C1).
The
inclusion
of
Lemma
1
(a)
can
be
rened
as
the
chain
of
inclusions
τ
⊂
˜
τ
⊂
˜
τ
≤
.
If
≤
is
the
trivial
preorder
on
P
,
i.e.,
such
that
x
≤
y
for
all
x,
y
∈
P
,
then
˜
τ
≤
=
˜
τ
.
Remark
2.
It
can
be
noted
that
the
≤
-ascending
extension
preserves
in-
clusions
between
topologies,
in
the
following
sense.
If
τ
and
τ
′
are
two
topologies
on
P
such
that
τ
⊂
τ
′
,
then
we
obtain
that
˜
τ
≤
⊂
˜
τ
′
≤
.
Indeed,
if
A
⊂
P
is
(
≤
,
τ
)-closed,
then
whenever
(
x
n
)
⊂
A
is
a
≤
-ascending
sequence
such
that
x
n
τ
′
→
x
∈
P
,
then
the
inclusion
τ
⊂
τ
′
yields
x
n
τ
→
x
,
whence
L.
Fresse,
V.V.
Motreanu
369
x
∈
A
(using
that
A
is
(
≤
,
τ
)-closed),
and
nally
we
have
shown
that
A
is
(
≤
,
τ
′
)-closed.
We
say
that
the
preorder
≤
is
self-closed
for
the
topology
τ
if
the
set
P
(
≥
x
)
:=
{
y
∈
P
:
x
≤
y
}
is
closed
(in
τ
)
for
all
x
∈
P
.
Lemma
2.
(a)
If
≤
is
self-closed
for
the
topology
τ
,
then
it
is
self-closed
for
the
topology
˜
τ
≤
.
(b)
The
following
conditions
are
equivalent:
(i)
≤
is
self-closed
for
the
topology
˜
τ
≤
;
(ii)
for
every
≤
-ascending
sequence
(
z
n
)
⊂
P
and
every
z
∈
P
such
that
z
n
τ
→
z
we
have
z
n
≤
z
for
all
n
(i.e.,
any
limit
of
a
≤
-ascending
sequence
is
an
upper
bound
of
this
sequence).
Proof.
(a)
This
follows
from
Lemma
1
(a).
(b)
First
we
show
the
implication
(ii)
⇒
(i).
Let
x
∈
P
and
let
us
show
that
P
(
≥
x
)
is
closed
for
the
topology
˜
τ
≤
,
i.e.,
is
(
≤
,
τ
)-closed.
Let
(
z
n
)
⊂
P
(
≥
x
)
be
a
≤
-ascending
sequence
such
that
z
n
τ
→
z
∈
P
.
For
any
xed
n
,
condition
(ii)
yields
z
n
≤
z
,
while
we
know
that
x
≤
z
n
(since
z
n
∈
P
(
≥
x
)),
whence
z
∈
P
(
≥
x
),
which
establishes
(i).
Next
let
us
show
the
implication
(i)
⇒
(ii).
Let
(
z
n
)
⊂
P
be
a
≤
-
ascending
sequence
such
that
z
n
τ
→
z
.
For
every
n
,
the
sequence
(
z
m
)
m
≥
n
is
≤
-ascending,
contained
in
P
(
≥
z
n
),
and
has
limit
z
.
Since
P
(
≥
z
n
)
is
(
≤
,
τ
)-closed
(by
virtue
of
(i)),
we
obtain
that
z
∈
P
(
≥
z
n
);
whence
z
n
≤
z
for
all
n
,
and
the
lemma
is
established.
Remark
3.
(a)
Clearly,
≤
is
self-closed
for
˜
τ
≤
whenever
the
subset
P
(
≥
x
)
is
sequentially
closed
with
respect
to
the
topology
τ
for
all
x
∈
P
,
i.e.,
≤
is
self-closed
for
the
topology
˜
τ
(see
Remark
1
(c)).
The
latter
condition
is
a
priori
weaker
than
requiring
that
≤
is
self-closed
with
respect
to
τ
,
though
both
notions
coincide
if
for
instance
(
P,
τ
)
satises
the
axiom
of
countability
(C1).
(b)
Another
sucient
condition
for
having
that
≤
is
self-closed
with
respect
to
˜
τ
≤
is
that
P
(
≤
x
)
:=
{
y
∈
P
:
y
≤
x
}
∈
τ
for
all
x
∈
P
.
Indeed,
if
(
x
n
)
⊂
P
is
a
≤
-ascending
sequence
such
that
x
n
τ
→
x
,
then
we
obtain
that
x
n
∈
P
(
≤
x
)
for
all
large
enough
n
,
and
in
fact
x
n
≤
x
for
all
n
,
since
(
x
n
)
is
≤
-ascending.
The
desired
conclusion
now
comes
from
Lemma
2
(b).
On
Ekeland
variational
principle
for
preordered
uniform
space
370
(c)
The
implication
in
Lemma
2
(a)
is
not
an
equivalence
even
if
(
P,
τ
)
satises
axiom
(C1)
(see
part
(a)).
Take
for
instance
P
=
[0
,
1]
∪{
2
}
endowed
with
the
standard
metric
topology
τ
and
the
total
order
≤
whose
restriction
to
(0
,
1]
is
the
standard
total
order
and
such
that
0
<
2
<
x
for
all
x
∈
(0
,
1].
Then
≤
is
not
self-closed
with
respect
to
τ
since
P
(
≥
2)
=
(0
,
1]
∪
{
2
}
is
not
closed
for
the
metric
topology.
However,
P
(
≥
2)
is
(
≤
,
τ
)-closed,
as
well
as
P
(
≥
x
)
for
all
x
∈
[0
,
1],
hence
≤
is
self-closed
with
respect
to
˜
τ
≤
.
3
Preordered
uniform
space
In
this
section,
we
still
consider
a
nonempty
set
P
endowed
with
a
preorder
≤
.
First,
we
apply
the
denitions
and
results
of
Section
2
to
the
case
where
the
topology
of
P
is
induced
by
a
pseudometric
–
in
fact,
a
≤
-pseudometric.
Then
we
replace
the
single
≤
-pseudometric
by
a
family
of
≤
-pseudometrics.
3.1
Preordered
pseudometric
space
We
call
≤
-pseudometric
on
P
a
map
d
:
P
2
≤
:=
{
(
x,
y
)
∈
P
×
P
:
x
≤
y
}
→
[0
,
+
∞
]
such
that
•
d
(
x,
x
)
=
0
for
all
x
∈
P
,
•
d
(
x,
z
)
≤
d
(
x,
y
)
+
d
(
y,
z
)
for
all
x,
y,
z
∈
P
such
that
x
≤
y
≤
z
.
Then,
we
say
that
(
P,
≤
,
d
)
is
a
preordered
pseudometric
space
.
For
every
η
∈
R
∗
+
and
x
∈
P
,
we
set
P
d
η
(
≤
x
)
=
{
y
∈
P
:
y
≤
x,
d
(
y,
x
)
<
η
}
.
The
subsets
P
d
η
(
≤
x
),
for
all
η
and
x
,
form
the
basis
of
a
topology
τ
d
on
P
.
Given
a
sequence
(
x
n
)
⊂
P
,
we
write
x
n
d
→
x
if
x
is
a
limit
(possibly
not
unique)
of
(
x
n
)
with
respect
to
the
topology
τ
d
.
Note
that
this
implies
that
x
n
∈
P
d
η
0
(
≤
x
)
for
all
n
large
enough,
for
any
xed
η
0
∈
R
∗
+
,
whence
in
particular
x
n
≤
x
for
all
n
large.
If
(
x
n
)
is
≤
-ascending,
then
it
follows
that
x
n
≤
x
for
all
n
,
i.e.,
any
limit
of
(
x
n
)
in
the
topology
τ
d
is
also
an
upper
bound
of
this
sequence.
L.
Fresse,
V.V.
Motreanu
371
The
≤
-pseudometric
d
allows
us
to
dene
two
notions
of
distance
from
a
point
x
∈
P
to
a
subset
A
⊂
P
,
namely
dist
d
(
≤
x,
A
)
:=
inf
{
d
(
y,
x
)
:
y
∈
A,
y
≤
x
}
and
dist
d
(
≥
x,
A
)
:=
inf
{
d
(
x,
y
)
:
y
∈
A,
y
≥
x
}
.
The
former
one
is
more
closely
related
to
the
topology
τ
d
,
as
pointed
out
by
the
following
lemma.
Lemma
3.
The
closure
of
a
subset
A
⊂
P
with
respect
to
τ
d
is
S
(
≤
,d
)
(
A
)
:=
{
x
∈
P
:
dist
d
(
≤
x,
A
)
=
0
}
.
Proof.
Let
A
d
denote
the
closure
of
A
with
respect
to
τ
d
and
rst
let
us
show
that
A
d
⊂
S
(
≤
,d
)
(
A
).
Let
x
∈
A
d
.
For
every
η
∈
R
∗
+
,
the
set
P
d
η
(
≤
x
)
is
an
open
neighborhood
of
x
with
respect
to
τ
d
.
Hence
there
exists
y
∈
A
∩
P
d
η
(
≤
x
),
so
y
≤
x
and
d
(
y,
x
)
<
η
.
We
deduce
that
dist
d
(
≤
x,
A
)
<
η
for
all
η
∈
R
∗
+
,
whence
dist
d
(
≤
x,
A
)
=
0
and
so
x
∈
S
(
≤
,d
)
(
A
).
Conversely,
let
x
∈
S
(
≤
,d
)
(
A
).
For
showing
that
x
∈
A
d
,
it
is
sucient
to
show
that
any
neighborhood
V
of
x
with
respect
to
τ
d
intersects
A
.
By
denition
of
τ
d
,
there
is
η
∈
R
∗
+
such
that
P
d
η
(
≤
x
)
⊂
V
.
On
the
other
hand,
since
dist
d
(
≤
x,
A
)
=
0,
we
can
nd
y
∈
A
such
that
y
≤
x
and
d
(
y,
x
)
<
η
.
Thus
y
∈
A
∩
P
d
η
(
≤
x
)
⊂
A
∩
V
,
so
that
A
∩
V
̸
=
∅
,
and
the
proof
of
the
lemma
is
complete.
Remark
4.
(a)
The
topology
τ
d
satises
the
axiom
(C1)
(see
Remark
1
(c)).
(b)
In
view
of
Lemma
3
,
the
topology
τ
d
actually
coincides
with
the
S
≤
-topology
considered
in
[
2
,
§
2]
(with
the
slight
dierence
that
in
[
2
],
d
is
assumed
to
be
dened
on
P
×
P
instead
of
P
2
≤
,
with
values
in
[0
,
+
∞
)
instead
of
[0
,
+
∞
]).
Now
let
us
denote
by
˜
τ
(
≤
,d
)
the
≤
-ascending
extension
of
τ
d
,
in
the
sense
of
Section
2
.
The
closed
subsets
for
the
topology
˜
τ
(
≤
,d
)
are
the
so-called
(
≤
,
τ
d
)-closed
subsets
of
Section
2
;
here,
for
simplicity,
we
will
call
them
(
≤
,
d
)
-closed
subsets
.
Also
recall
that
τ
d
⊂
˜
τ
(
≤
,d
)
,
hence
P
d
η
(
≤
x
)
∈
˜
τ
(
≤
,d
)
for
all
η
∈
R
∗
+
,
x
∈
P
.
By
A
(
≤
,d
)
we
denote
the
closure
of
A
with
respect
to
the
topology
˜
τ
(
≤
,d
)
,
i.e.,
A
(
≤
,d
)
is
the
smallest
(
≤
,
d
)-closed
subset
in
P
which
contains
A
,
and
A
is
itself
(
≤
,
d
)-closed
if
and
only
if
A
(
≤
,d
)
=
A
.
On
Ekeland
variational
principle
for
preordered
uniform
space
372
In
the
next
lemma,
we
write
L
(
≤
,d
)
(
A
)
:=
L
(
≤
,τ
d
)
(
A
)
=
{
x
∈
P
:
∃
(
x
n
)
⊂
A
≤
-ascending
with
x
n
d
→
x
}
(see,
again,
Section
2
)
while
S
(
≤
,d
)
(
A
)
comes
from
Lemma
3
above.
Recall
that
P
(
≤
x
)
=
{
y
∈
P
:
y
≤
x
}
and
P
(
≥
x
)
=
{
y
∈
P
:
x
≤
y
}
.
Lemma
4.
(a)
For
every
x
∈
P
,
the
subset
P
(
≤
x
)
is
open
for
the
topology
τ
d
,
thus
also
for
˜
τ
(
≤
,d
)
;
the
subset
P
(
≥
x
)
is
closed
for
the
topology
τ
d
,
thus
also
for
˜
τ
(
≤
,d
)
.
In
particular,
≤
is
self-closed
with
respect
to
τ
d
and
to
˜
τ
(
≤
,d
)
.
(b)
For
every
A
⊂
P
,
we
have
the
inclusions
A
⊂
L
(
≤
,d
)
(
A
)
⊂
A
(
≤
,d
)
⊂
S
(
≤
,d
)
(
A
)
.
(c)
The
subset
A
is
closed
with
respect
to
˜
τ
(
≤
,d
)
if
and
only
if
the
rst
two
inclusions
of
(b)
are
equalities.
The
subset
A
is
closed
with
respect
to
τ
d
if
and
only
if
all
inclusions
of
(b)
are
equalities.
Proof.
(a)
Let
x
∈
P
.
Let
y
∈
P
(
≤
x
)
and
choose
any
η
0
∈
R
∗
+
.
Since
≤
is
transitive,
we
easily
have
the
inclusion
P
d
η
0
(
≤
y
)
⊂
P
(
≤
x
),
which
shows
that
y
is
an
interior
point
of
P
(
≤
x
),
by
the
denition
of
the
topology
τ
d
;
hence
P
(
≤
x
)
∈
τ
d
⊂
˜
τ
(
≤
,d
)
.
Next
let
y
∈
P
\
P
(
≥
x
)
and,
again,
we
choose
any
η
0
∈
R
∗
+
.
For
any
z
∈
P
d
η
0
(
≤
y
),
since
x
̸≤
y
and
z
≤
y
,
we
must
have
x
̸≤
z
,
hence
z
∈
P
\
P
(
≥
x
).
This
shows
that
P
(
≥
x
)
is
closed
with
respect
to
τ
d
as
well
as
with
respect
to
˜
τ
(
≤
,d
)
.
(b)
The
inclusion
A
⊂
L
(
≤
,d
)
(
A
)
is
immediate.
Since
A
⊂
A
(
≤
,d
)
,
we
easily
have
L
(
≤
,d
)
(
A
)
⊂
L
(
≤
,d
)
(
A
(
≤
,d
)
)
=
A
(
≤
,d
)
,
where
the
last
equality
comes
from
the
fact
that
A
(
≤
,d
)
is
(
≤
,
d
)-closed.
Finally,
note
that
S
(
≤
,d
)
(
A
)
contains
A
and
is
closed
with
respect
to
τ
d
(due
to
Lemma
3
)
thus
with
respect
to
˜
τ
(
≤
,d
)
(since
τ
d
⊂
˜
τ
(
≤
,d
)
);
this
yields
A
(
≤
,d
)
⊂
S
(
≤
,d
)
(
A
)
as
asserted.
(c)
The
rst
assertion
comes
from
(b)
and
the
fact
that
A
is
closed
with
respect
to
˜
τ
(
≤
,d
)
if
and
only
if
A
=
A
(
≤
,d
)
.
The
second
assertion
follows
from
(b)
and
the
fact
that
A
is
closed
with
respect
to
τ
d
if
and
only
if
A
=
S
(
≤
,d
)
(
A
)
(due
to
Lemma
3
).
Remark
5.
(a)
All
the
inclusions
in
Lemma
4
(b)
may
be
strict
in
general.
Take
for
instance
P
=
[0
,
1]
×
[0
,
1]
equipped
with
the
preorder
≤
given
by
(
x,
y
)
≤
(
x
′
,
y
′
)
if
x
=
x
′
<
1
and
y
≤
y
′
<
1
,
or
x
≤
x
′
<
1
and
y
′
=
1
,
or
x
′
=
1
,
L.
Fresse,
V.V.
Motreanu
373
and
the
map
d
:
P
2
≤
→
[0
,
+
∞
)
dened
as
the
restriction
of
the
standard
metric
on
P
.
If
A
=
[0
,
1)
×
[0
,
1),
then
it
is
easy
to
see
that
A
⊊
L
(
≤
,d
)
(
A
)
=
[0
,
1)
×
[0
,
1]
⊊
A
(
≤
,d
)
=
([0
,
1)
×
[0
,
1])
∪
{
(1
,
1)
}
⊊
S
(
≤
,d
)
(
A
)
=
P
=
[0
,
1]
×
[0
,
1]
.
(b)
For
x
∈
P
,
note
that
L
(
≤
,d
)
(
{
x
}
)
=
{
x
}
(
≤
,d
)
=
S
(
≤
,d
)
(
{
x
}
)
=
{
y
∈
P
:
x
≤
y,
d
(
x,
y
)
=
0
}
.
In
particular,
a
singleton
{
x
}
is
(
≤
,
d
)-closed
if
and
only
if
every
y
∈
P
(
≥
x
)
\
{
x
}
is
such
that
d
(
x,
y
)
̸
=
0.
Thus
the
topology
˜
τ
(
≤
,d
)
satises
the
separation
axiom
(
T
1
)
(that
is,
singletons
are
(
≤
,
d
)-closed
in
P
)
if
and
only
if
d
is
nondegenerate
,
that
is,
(
x,
y
)
∈
P
2
≤
and
x
̸
=
y
imply
d
(
x,
y
)
̸
=
0.
Note
also
that
if
x
is
minimal
with
respect
to
the
preorder
≤
,
which
equivalently
means
that
P
(
≤
x
)
=
{
x
}
,
then
Lemma
4
(a)
implies
that
the
singleton
{
x
}
is
(
≤
,
d
)-open
whatever
d
we
consider.
3.2
Preordered
uniform
space
We
call
≤
-uniform
structure
on
(
P,
≤
)
a
(nonempty)
family
D
of
≤
-pseudo-
metrics
on
P
,
and
then
we
call
the
triple
(
P,
≤
,
D
)
a
preordered
uniform
space
.
This
gives
rise
to
topologies
on
P
:
•
Let
τ
D
be
the
coarsest
topology
on
P
which
contains
τ
d
for
all
d
∈
D
.
In
other
words,
a
basis
for
this
topology
is
formed
by
nite
intersections
U
1
∩
.
.
.
∩
U
k
with
U
1
,
.
.
.
,
U
k
∈
d
∈D
τ
d
.
We
will
use
the
notation
x
n
D
−
→
x
if
(
x
n
)
⊂
P
converges
to
x
with
respect
to
the
topology
τ
D
.
•
Let
˜
τ
(
≤
,
D
)
denote
the
≤
-ascending
extension
of
τ
D
in
the
sense
of
Sec-
tion
2
.
In
other
words,
the
closed
subsets
for
the
topology
˜
τ
(
≤
,
D
)
are
the
(
≤
,
τ
D
)-closed
subsets
of
P
–
which
we
will
rather
call
(
≤
,
D
)-
closed
for
simplicity.
Recall
that
A
⊂
P
is
(
≤
,
D
)-closed
if,
for
every
≤
-ascending
sequence
(
x
n
)
⊂
A
such
that
x
n
D
→
x
,
we
get
x
∈
A
.
Remark
6.
(a)
We
have
the
inclusion
τ
D
⊂
˜
τ
(
≤
,
D
)
(see
Lemma
1
(a)).
More-
over,
since
d
∈D
τ
d
⊂
τ
D
,
we
have
d
∈D
˜
τ
(
≤
,d
)
⊂
˜
τ
(
≤
,
D
)
(due
to
Remark
2
).
(b)
The
preorder
≤
is
self-closed
with
respect
to
the
topologies
τ
D
and
˜
τ
(
≤
,
D
)
.
Indeed,
for
every
x
∈
P
,
taking
any
d
∈
D
,
we
know
that
P
(
≥
x
)
is
closed
for
the
topologies
τ
d
and
˜
τ
(
≤
,d
)
(see
Lemma
4
(a)).
In
view
of
the
On
Ekeland
variational
principle
for
preordered
uniform
space
374
inclusions
pointed
out
in
part
(a)
above,
we
conclude
that
P
(
≥
x
)
is
also
closed
for
τ
D
and
for
˜
τ
(
≤
,
D
)
.
(c)
Part
(b)
of
the
present
remark
tells
us
that
if
(
x
n
)
⊂
P
is
≤
-ascending
and
x
n
D
→
x
,
then
x
n
≤
x
for
all
n
(we
apply
Lemma
2
(b)
with
τ
=
τ
D
).
Remark
7.
If
≤
is
the
trivial
preorder
on
P
(i.e.,
x
≤
y
for
all
x,
y
∈
P
),
then
a
≤
-pseudometric
d
:
P
2
≤
=
P
×
P
→
[0
,
+
∞
]
is
an
´
ecart
in
the
sense
of
[
1
,
Chapitre
IX].
For
a
family
D
of
´
ecarts
on
P
,
the
pair
(
P,
D
)
is
called
a
uniform
space,
and
τ
D
is
the
topology
on
P
considered
in
[
1
,
Chapitre
IX].
Thus
we
retrieve
the
classical
notion
of
uniform
space
when
we
restrict
to
the
case
of
the
trivial
preorder
on
P
.
Lemma
5.
Let
(
x
n
)
⊂
P
be
a
≤
-ascending
sequence
and
let
x
∈
P
.
The
following
conditions
are
equivalent:
(i)
x
n
D
→
x
;
(ii)
x
n
→
x
with
respect
to
the
topology
˜
τ
(
≤
,
D
)
;
(iii)
x
n
d
→
x
for
all
d
∈
D
.
Proof.
The
equivalence
between
(i)
and
(ii)
comes
from
Lemma
1
(b).
The
implication
(i)
⇒
(iii)
is
easily
implied
by
the
fact
that
τ
d
⊂
τ
D
for
all
d
∈
D
.
It
remains
to
show
(iii)
⇒
(i).
So
assume
that
x
n
→
x
with
respect
to
τ
d
for
all
d
∈
D
.
Let
V
⊂
P
be
a
neighborhood
of
x
for
the
topology
τ
D
.
By
denition
of
that
topology,
V
contains
a
subset
of
the
form
U
1
∩
.
.
.
∩
U
k
where,
for
every
i
∈
{
1
,
.
.
.
,
k
}
,
there
is
d
i
∈
D
such
that
U
i
is
an
open
neigh-
borhood
of
x
for
the
topology
τ
d
i
.
For
every
i
,
since
x
n
d
i
→
x
,
there
is
a
rank
n
i
∈
N
such
that
x
n
∈
U
i
whenever
n
≥
n
i
.
Setting
n
0
=
max
{
n
1
,
.
.
.
,
n
k
}
,
we
obtain
that
x
n
∈
U
1
∩
.
.
.
∩
U
k
⊂
V
for
all
n
≥
n
0
.
This
shows
that
x
n
→
x
with
respect
to
τ
D
,
and
(i)
is
established.
We
need
a
notion
of
completeness
for
the
preordered
pseudometric
spaces
(
P,
≤
,
d
)
and
the
preordered
uniform
space
(
P,
≤
,
D
).
Let
us
present
it
now.
•
A
≤
-ascending
sequence
(
x
n
)
is
said
to
be
d
-Cauchy
if,
for
every
η
∈
R
∗
+
,
there
is
a
rank
n
0
such
that,
whenever
n
0
≤
m
≤
n
,
we
have
d
(
x
m
,
x
n
)
<
η
.
L.
Fresse,
V.V.
Motreanu
375
•
We
say
that
P
is
(
≤
,
d
)
-complete
if
every
d
-Cauchy
≤
-ascending
se-
quence
(
x
n
)
⊂
P
is
convergent
with
respect
to
the
topology
τ
d
(or,
equivalently,
with
respect
to
˜
τ
(
≤
,d
)
;
see
Lemma
1
(b)).
•
A
≤
-ascending
sequence
(
x
n
)
is
said
to
be
D
-Cauchy
if
it
is
d
-Cauchy
for
all
d
∈
D
.
•
We
say
that
P
is
(
≤
,
D
)
-complete
if
every
D
-Cauchy
≤
-ascending
se-
quence
is
convergent
with
respect
to
τ
D
(or
equivalently
to
˜
τ
(
≤
,
D
)
;
see
Lemma
5
).
Remark
8.
(a)
The
fact
that
P
is
(
≤
,
D
)-complete
does
not
imply
that
P
is
(
≤
,
d
)-complete
for
every
or
even
for
some
d
∈
D
,
as
shown
by
the
following
example.
Let
P
=
(0
,
+
∞
)
be
endowed
with
the
trivial
preorder
≤
and
let
D
=
{
d
1
,
d
2
}
with
d
1
(
x,
y
)
=
|
x
−
y
|
and
d
2
(
x,
y
)
=
1
x
−
1
y
for
all
x,
y
∈
P
.
Thus
P
is
neither
(
≤
,
d
1
)-complete
nor
(
≤
,
d
2
)-complete.
However,
we
claim
that
P
is
(
≤
,
D
)-complete.
Indeed,
if
(
x
n
)
⊂
P
is
D
-Cauchy,
then
there
must
be
n
0
∈
N
such
that
|
x
n
−
x
n
0
|
<
1
and
|
1
x
n
−
1
x
n
0
|
<
1
for
all
n
≥
n
0
,
which
easily
implies
that
there
must
be
a,
b
∈
(0
,
+
∞
)
such
that
a
≤
x
n
≤
b
for
all
n
.
The
metrics
d
1
,
d
2
are
equivalent
on
the
interval
[
a,
b
],
and
we
obtain
that
there
is
x
∈
[
a,
b
]
with
x
n
→
x
with
respect
to
both
d
1
and
d
2
,
thus
also
x
n
D
→
x
by
Lemma
5
.
(b)
The
fact
that
P
is
(
≤
,
d
)-complete
for
all
d
∈
D
is
also
not
sucient
for
ensuring
that
P
is
(
≤
,
D
)-complete.
Take
for
instance
P
=
[0
,
1]
∪
{
1
′
}
endowed
with
the
partial
order
≤
whose
restriction
to
[0
,
1]
is
the
standard
total
order
and
such
that
x
≤
1
′
for
all
x
∈
[0
,
1),
and
1
,
1
′
are
not
compa-
rable.
Let
d,
d
′
:
P
2
≤
→
[0
,
+
∞
)
be
the
≤
-pseudometrics
whose
restrictions
to
[0
,
1)
are
given
by
the
standard
metric
and
such
that
d
(
x,
1)
=
d
′
(
x,
1
′
)
=
1
−
x,
d
(
x,
1
′
)
=
d
′
(
x,
1)
=
2
−
x
for
all
x
∈
[0
,
1)
.
It
is
easy
to
see
that
every
≤
-ascending
sequence
(
x
n
)
⊂
P
converges
with
respect
to
both
d
and
d
′
,
and
this
guarantees
that
P
is
both
(
≤
,
d
)-complete
and
(
≤
,
d
′
)-complete.
However
the
sequence
(
x
n
)
given
by
x
n
=
1
−
1
n
is
D
-
Cauchy
(since
it
is
both
d
-Cauchy
and
d
′
-Cauchy)
but
not
convergent
with
respect
to
τ
D
:
this
follows
from
Lemma
5
combined
with
the
observation
that
1
(resp.,
1
′
)
is
the
only
limit
of
(
x
n
)
with
respect
to
d
(resp.,
d
′
)
and
1
̸
=
1
′
.
On
Ekeland
variational
principle
for
preordered
uniform
space
376
(c)
Here
we
assume
that
D
is
directed
in
the
sense
that,
for
all
d
1
,
d
2
∈
D
there
is
d
3
∈
D
with
d
1
≤
d
3
and
d
2
≤
d
3
.
We
assume
also
that,
for
all
d
∈
D
,
every
non-stationary
≤
-ascending
sequence
(
x
n
)
has
at
most
one
limit
with
respect
to
d
.
Then,
in
this
case,
the
implication
(
P
is
(
≤
,
d
)-complete
for
all
d
∈
D
)
⇒
P
is
(
≤
,
D
)-complete
(1)
holds.
For
showing
this,
let
(
x
n
)
⊂
P
be
a
D
-Cauchy
≤
-ascending
sequence.
If
(
x
n
)
is
stationary
then
it
is
immediately
convergent
with
respect
to
D
,
so
we
can
assume
that
(
x
n
)
is
not
stationary.
We
have
that
(
x
n
)
is
d
-Cauchy
for
every
d
∈
D
.
Since
P
is
(
≤
,
d
)-complete,
there
is
x
d
∈
P
with
x
n
d
→
x
d
:
this
limit
a
priori
depends
on
d
,
but
we
claim
that
x
d
1
=
x
d
2
for
all
d
1
,
d
2
∈
D
.
The
assumption
that
D
is
directed
yields
d
3
∈
D
such
that
d
1
≤
d
3
and
d
2
≤
d
3
.
We
have
x
n
d
3
→
x
d
3
thus
also
x
n
d
1
→
x
d
3
and
x
n
d
2
→
x
d
3
,
due
to
the
previous
inequalities.
Since
(
x
n
)
is
not
stationary,
it
has
at
most
one
limit
with
respect
to
d
1
,
whence
x
d
1
=
x
d
3
.
Similarly,
x
d
2
=
x
d
3
,
and
nally
x
d
1
=
x
d
2
,
our
claim
is
established.
That
claim
yields
x
∈
P
such
that
x
n
d
→
x
for
all
d
∈
D
and,
therefore,
x
n
D
→
x
by
virtue
of
Lemma
5
.
This
proves
that
P
is
(
≤
,
D
)-complete.
Even
under
the
additional
assumptions
considered
in
the
present
part
(c),
the
converse
of
(
1
)
is
not
true.
Take
for
instance
(
P,
≤
),
d
1
,
d
2
as
in
part
(a)
above,
and
set
in
addition
d
3
(
x,
y
)
=
max
{
d
1
(
x,
y
)
,
d
2
(
x,
y
)
}
for
all
x,
y
∈
P
,
and
D
=
{
d
1
,
d
2
,
d
3
}
.
Thus
D
is
directed
and
every
sequence
(
x
n
)
⊂
P
has
at
most
one
limit
with
respect
to
d
1
,
d
2
,
d
3
(since
these
are
metrics).
With
the
same
argument
as
in
part
(a),
P
is
(
≤
,
D
)-complete.
However,
as
shown
in
(a),
it
is
neither
(
≤
,
d
1
)-complete
nor
(
≤
,
d
2
)-complete.
4
Statement
of
main
results
Our
rst
main
result
concerns
a
map
f
:
P
→
R
∪
{
+
∞}
.
Let
dom(
f
)
=
f
−
1
(
R
)
denote
the
domain
of
f
.
•
We
say
that
f
is
(
≤
,
D
)
-submonotone
if,
for
every
≤
-ascending
se-
quence
(
x
n
)
⊂
P
and
x
∈
P
such
that
x
n
D
→
x
and
(
f
(
x
n
))
is
nonin-
creasing,
we
have
f
(
x
)
≤
f
(
x
n
)
for
all
n
.
•
We
say
that
f
is
≥
-locally
bounded
from
below
if,
for
every
x
∈
dom(
f
),
there
is
µ
∈
R
such
that
∀
y
∈
P,
x
≤
y
⇒
f
(
y
)
≥
µ.
L.
Fresse,
V.V.
Motreanu
377
Theorem
1.
Let
(
P,
≤
,
D
)
be
a
preordered
uniform
space
which
is
(
≤
,
D
)
-
complete.
Let
f
:
P
→
R
∪
{
+
∞}
be
(
≤
,
D
)
-submonotone
and
≥
-locally
bounded
from
below.
Let
x
∈
dom(
f
)
.
Then
there
is
y
∈
dom(
f
)
such
that
(a)
x
≤
y
;
(b)
d
(
x,
y
)
≤
f
(
x
)
−
f
(
y
)
for
all
d
∈
D
;
(c)
whenever
z
∈
dom(
f
)
and
z
′
∈
P
are
such
that
y
≤
z
≤
z
′
,
we
have
the
implication
∃
d
∈
D
,
d
(
z,
z
′
)
>
0
⇒
∃
d
∈
D
,
(
d
(
y,
z
)
>
f
(
y
)
−
f
(
z
)
or
d
(
z,
z
′
)
>
f
(
z
)
−
f
(
z
′
))
.
For
our
second
main
result,
instead
of
a
single
map
f
,
we
consider
a
family
F
=
{
f
d
}
d
∈D
of
maps
f
d
:
P
→
R
∪
{
+
∞}
.
•
We
say
that
F
is
(
≤
,
D
)
-submonotone
if,
for
every
≤
-ascending
se-
quence
(
x
n
)
⊂
P
and
x
∈
P
such
that
x
n
D
→
x
and
(
f
d
(
x
n
))
is
non-
increasing
for
all
d
∈
D
,
we
have
f
d
(
x
)
≤
f
d
(
x
n
)
for
all
n
and
all
d
∈
D
.
•
We
say
that
F
is
≥
-locally
bounded
from
below
if,
for
every
x
∈
dom(
F
)
:=
d
∈D
dom(
f
d
),
there
is
a
family
{
µ
d
}
d
∈D
⊂
R
such
that
∀
y
∈
P,
x
≤
y
⇒
(
∀
d
∈
D
,
f
d
(
y
)
≥
µ
d
)
.
Theorem
2.
Let
(
P,
≤
,
D
)
be
a
preordered
uniform
space
which
is
(
≤
,
D
)
-
complete.
Assume
that
D
is
(at
most)
countable.
Let
F
=
{
f
d
}
d
∈D
be
a
family
of
maps
f
d
:
P
→
R
∪
{
+
∞}
which
is
(
≤
,
D
)
-submonotone
and
≥
-
locally
bounded
from
below.
Let
x
∈
dom(
F
)
.
Then
there
is
y
∈
dom(
F
)
such
that
(a)
x
≤
y
;
(b)
d
(
x,
y
)
≤
f
d
(
x
)
−
f
d
(
y
)
for
all
d
∈
D
;
(c)
whenever
z
∈
dom(
F
)
and
z
′
∈
P
are
such
that
y
≤
z
≤
z
′
,
we
have
the
implication
∃
d
∈
D
,
d
(
z,
z
′
)
>
0
⇒
∃
d
∈
D
,
(
d
(
y,
z
)
>
f
d
(
y
)
−
f
d
(
z
)
or
d
(
z,
z
′
)
>
f
d
(
z
)
−
f
d
(
z
′
))
.
The
proofs
of
Theorems
1
and
2
are
given
in
Section
5
.
On
Ekeland
variational
principle
for
preordered
uniform
space
378
5
Proofs
5.1
Key
tool:
Ekeland
variational
principle
over
monoids
The
key
tool
for
proving
our
main
results
is
a
version
of
the
Ekeland
varia-
tional
principle
over
monoids,
established
in
[
3
].
We
give
the
full
statement
here
for
making
this
paper
as
self-contained
as
possible
(also,
the
result
presented
below
is
slightly
rephrased).
•
Let
(
K
,
+)
be
a
commutative
monoid,
with
neutral
element
0
and
we
denote
by
K
′
its
subset
of
invertible
elements.
The
set
K
is
also
equipped
with
a
preorder
≤
and
a
countable
collection
of
subsets
V
which
is
stable
under
nite
intersections
and
such
that
V
∈V
V
=
{
0
}
.
We
will
write
K
+
:=
{
α
∈
K
:
0
≤
α
}
,
K
′
+
:=
K
′
∩
K
+
.
For
(
α
n
)
n
∈
N
⊂
K
,
we
write
α
n
→
0
if
every
V
∈
V
contains
α
n
for
all
but
nitely
many
n
.
We
assume
the
following:
(A)
∀
α
∈
K
′
,
{
β
∈
K
:
β
≤
α
}
⊂
K
′
.
(B)
∀
α,
β,
γ
∈
K
,
α
≤
β
⇒
α
+
γ
≤
β
+
γ
.
(C)
∀
V
∈
V
,
∀
α
∈
K
′
∩
V,
{
β
∈
K
:
0
≤
β
≤
α
}
⊂
V
.
(D)
∀
V
∈
V
,
∀
α
∈
K
′
,
∃
N
≥
1
,
∀
β
1
,
.
.
.
,
β
N
∈
K
′
+
\
V,
β
1
+
.
.
.
+
β
N
̸≤
α
.
(E)
∀
α
∈
K
,
∀
(
β
n
)
n
∈
N
⊂
K
+
,
(
α
≤
β
n
∀
n
and
β
n
→
0)
⇒
α
≤
0.
•
Let
P
be
a
nonempty
set
endowed
with
a
preorder
≤
and
a
map
δ
:
P
2
≤
=
{
(
x,
y
)
∈
P
×
P
:
x
≤
y
}
→
K
+
such
that
δ
(
x,
x
)
=
0
for
all
x
and
δ
(
x,
z
)
≤
δ
(
x,
y
)
+
δ
(
y,
z
)
whenever
x
≤
y
≤
z
.
•
In
this
context,
a
≤
-ascending
sequence
(
x
n
)
⊂
P
is
said
to
be
•
δ
-convergent
to
x
if
x
n
≤
x
for
all
n
and
δ
(
x
n
,
x
)
→
0;
•
δ
-Cauchy
if
for
every
V
∈
V
there
is
n
0
such
that
δ
(
x
m
,
x
n
)
∈
V
whenever
n
0
≤
m
≤
n
.
L.
Fresse,
V.V.
Motreanu
379
Finally
P
is
called
(
≤
,
δ
)
-complete
if
every
δ
-Cauchy
≤
-ascending
se-
quence
(
x
n
)
⊂
P
is
δ
-convergent
to
some
x
.
•
Given
a
map
f
:
P
→
K
,
•
we
write
dom(
f
)
=
f
−
1
(
K
′
);
•
we
say
that
f
is
(
≤
,
δ
)
-submonotone
if,
for
every
≤
-ascending
sequence
(
x
n
)
⊂
dom(
f
)
which
is
δ
-convergent
to
x
,
(
f
(
x
n
)
≤
f
(
x
m
)
∀
m,
n
with
m
≤
n
)
⇒
f
(
x
)
≤
f
(
x
m
)
∀
m
;
•
we
say
that
f
is
locally
bounded
from
below
if,
for
every
x
∈
dom(
f
),
there
is
µ
∈
K
′
such
that
∀
y
∈
P,
(
x
≤
y
and
f
(
y
)
≤
f
(
x
))
⇒
f
(
y
)
≥
µ.
(The
latter
notion
is
a
special
case
of
the
notion
of
“locally
nitely
bounded
from
below”
map
considered
in
[
3
].)
In
this
setting,
we
have
(see
[
3
,
Theorem
1]):
Theorem
3.
Let
K
,
P
be
as
above,
assume
that
P
is
(
≤
,
δ
)
-complete
and
let
f
:
P
→
K
be
(
≤
,
δ
)
-submonotone
and
locally
bounded
from
below.
Then,
for
every
x
∈
dom(
f
)
,
there
is
y
∈
dom(
f
)
such
that
(a)
x
≤
y
;
(b)
δ
(
x,
y
)
≤
f
(
x
)
−
f
(
y
)
;
(c)
whenever
z,
z
′
∈
dom(
f
)
satisfy
y
≤
z
≤
z
′
and
δ
(
z,
z
′
)
̸
=
0
,
we
have
δ
(
y,
z
)
̸≤
f
(
y
)
−
f
(
z
)
or
δ
(
z,
z
′
)
̸≤
f
(
z
)
−
f
(
z
′
)
.
Example
1.
(a)
The
typical
example
of
monoid
to
which
the
result
applies
is
K
=
R
∪
{
+
∞}
endowed
with
the
standard
addition
+
and
total
order
≤
,
and
with
the
countable
collection
of
subsets
V
:=
{
(
−
r,
r
)
}
r
∈
Q
∗
+
.
Then
conditions
(A)–(E)
are
easily
fullled.
Note
also
that
K
′
=
R
,
K
+
=
[0
,
+
∞
],
and
K
′
+
=
[0
,
+
∞
).
(b)
As
another
example,
consider
the
set
of
sequences
K
=
(
R
∪
{
+
∞}
)
N
=
{
(
α
n
)
n
∈
N
:
α
n
∈
R
∪
{
+
∞}}
On
Ekeland
variational
principle
for
preordered
uniform
space
380
which
is
a
commutative
monoid
for
the
standard
componentwise
addition,
the
neutral
element
is
the
zero
constant
sequence.
We
endow
K
with
the
partial
order
given
by
(
α
n
)
≤
(
β
n
)
if
α
n
≤
β
n
∀
n
∈
N
.
We
consider
the
countable
collection
of
subsets
V
=
{
V
I,r
}
parametrized
by
pairs
(
I,
r
)
formed
by
a
nonempty
nite
subset
I
⊂
N
and
a
map
r
:
I
→
Q
∗
+
,
i
→
r
i
,
and
with
V
I,r
:=
{
(
α
n
)
∈
K
:
α
i
∈
(
−
r
i
,
r
i
)
∀
i
∈
I
}
.
Here
we
have
K
′
=
{
(
α
n
)
:
α
n
<
+
∞
∀
n
}
=
R
N
,
K
+
=
[0
,
+
∞
]
N
,
and
K
′
+
=
[0
,
+
∞
)
N
.
Again
it
is
straightforward
to
check
that
(A)–(C)
hold
true.
For
showing
(D),
let
V
=
V
I,r
∈
V
,
with
I
⊂
N
nonempty
nite
and
r
:
I
→
Q
∗
+
.
Let
α
=
(
α
n
)
∈
K
′
=
R
N
.
For
every
i
∈
I
,
since
r
i
∈
Q
∗
+
and
α
i
∈
R
,
we
can
nd
N
i
∈
N
∗
such
that
N
i
>
α
i
r
i
.
We
set
N
0
=
max
{
N
i
:
i
∈
I
}
and
N
=
N
0
card(
I
).
Now
let
β
1
,
.
.
.
,
β
N
∈
K
′
+
\
V
.
For
m
∈
{
1
,
.
.
.
,
N
}
,
we
write
β
m
=
(
β
m,n
)
n
∈
N
.
For
every
m
∈
{
1
,
.
.
.
,
N
}
,
the
assumption
that
β
m
∈
K
′
+
\
V
yields
I
m
:=
{
i
∈
I
:
β
m,i
≥
r
i
}
̸
=
∅
.
Fix
any
i
m
∈
I
m
,
so
that
we
obtain
a
map
F
:
{
1
,
.
.
.
,
N
}
→
I,
m
→
i
m
.
Since
N
=
N
0
card(
I
),
there
is
i
∈
I
with
card(
F
−
1
(
i
))
≥
N
0
.
Since
β
m,i
≥
0
for
all
m
∈
{
1
,
.
.
.
,
N
}
(because
β
m
∈
K
′
+
),
we
obtain
N
m
=1
β
m,i
≥
m
∈
F
−
1
(
i
)
β
m,i
≥
N
0
r
i
>
α
i
.
Whence
β
1
+
.
.
.
+
β
N
̸≤
α
.
This
shows
(D).
For
showing
(E),
let
α
∈
K
and
(
β
n
)
n
∈
N
⊂
K
+
be
such
that
α
≤
β
n
for
all
n
and
β
n
→
0.
We
write
β
n
=
(
β
n,i
)
i
∈
N
.
For
every
i
∈
N
,
we
aim
to
show
that
α
i
≤
0.
To
this
end,
let
ε
∈
Q
∗
+
,
set
I
=
{
i
}
and
r
:
I
→
Q
∗
+
,
i
→
r
i
=
ε
.
Since
V
:=
V
I,r
∈
V
and
β
n
→
0,
there
is
n
0
∈
N
such
that
β
n
∈
V
for
all
n
≥
n
0
.
Using
in
addition
that
β
n
∈
K
+
,
this
yields
0
≤
β
n,i
<
ε
for
any
n
≥
n
0
and,
therefore,
α
i
≤
β
n,i
<
ε
.
Since
this
holds
for
any
ε
∈
Q
∗
+
,
we
conclude
that
α
i
≤
0
and
the
argument
is
complete.
L.
Fresse,
V.V.
Motreanu
381
5.2
Proof
of
Theorem
1
We
aim
to
apply
Theorem
3
to
the
monoid
K
=
R
∪
{
+
∞}
equipped
with
the
standard
addition
+
and
total
order
≤
,
as
in
Example
1
(a)
and
with
V
=
{
(
−
ε,
ε
)
}
ε
∈
Q
∗
+
.
As
explained
in
Example
1
(a),
conditions
(A)–(E)
are
fullled.
For
applying
Theorem
3
,
we
also
consider
our
preordered
set
(
P,
≤
)
endowed
with
the
map
δ
:
P
2
≤
→
[0
,
+
∞
]
,
(
x,
y
)
→
sup
d
∈D
d
(
x,
y
)
.
We
clearly
have
δ
(
x,
x
)
=
0
for
all
x
∈
P
,
moreover
it
is
straightforward
to
check
that
δ
(
x,
z
)
≤
δ
(
x,
y
)
+
δ
(
y,
z
)
whenever
x
≤
y
≤
z
.
For
checking
the
applicability
of
Theorem
3
to
the
considered
setting,
we
need
to
show
that
•
P
is
(
≤
,
δ
)-complete;
•
f
is
(
≤
,
δ
)-submonotone.
For
checking
the
rst
assertion,
let
(
x
n
)
⊂
P
be
a
≤
-ascending
sequence
which
is
δ
-Cauchy.
For
every
d
∈
D
,
the
fact
that
d
(
x,
y
)
≤
δ
(
x,
y
)
for
all
(
x,
y
)
∈
P
2
≤
easily
implies
that
(
x
n
)
is
d
-Cauchy.
Hence
(
x
n
)
is
D
-Cauchy.
Knowing
that
P
is
(
≤
,
D
)-complete,
we
nd
x
∈
P
such
that
x
n
D
→
x
,
in
particular
x
n
≤
x
for
all
n
(see
Remark
6
(c)).
We
claim
that
(
x
n
)
is
δ
-
convergent
to
x
.
To
see
this,
x
V
∈
V
,
V
=
(
−
ε,
ε
)
with
ε
∈
Q
∗
+
.
Using
that
(
x
n
)
is
δ
-Cauchy,
there
is
n
0
∈
N
such
that
δ
(
x
m
,
x
n
)
<
ε
2
whenever
n
0
≤
m
≤
n
.
We
claim
that
δ
(
x
m
,
x
)
<
ε
whenever
m
≥
n
0
.
(2)
Indeed,
let
m
≥
n
0
and
let
d
∈
D
.
Since
x
n
D
→
x
,
by
virtue
of
Lemma
5
we
have
also
x
n
d
→
x
,
so
there
is
n
′
0
∈
N
such
that
d
(
x
n
,
x
)
<
ε
2
whenever
n
≥
n
′
0
.
Now
we
set
n
=
max
{
m,
n
′
0
}
and
we
obtain
d
(
x
m
,
x
)
≤
d
(
x
m
,
x
n
)
+
d
(
x
n
,
x
)
≤
δ
(
x
m
,
x
n
)
+
d
(
x
n
,
x
)
<
ε
2
+
ε
2
=
ε.
Since
this
holds
for
all
d
∈
D
,
we
get
(
2
).
The
fact
that
(
x
n
)
is
δ
-convergent
to
x
is
now
justied,
and
we
have
established
that
P
is
(
≤
,
δ
)-complete.
For
showing
the
second
assertion,
let
(
x
n
)
⊂
dom(
f
)
be
a
≤
-ascending
sequence,
which
is
δ
-convergent
to
x
∈
P
(in
particular,
x
n
≤
x
for
all
n
)
and
such
that
f
(
x
n
)
≤
f
(
x
m
)
for
all
m,
n
with
m
≤
n
.
For
every
d
∈
D
,
On
Ekeland
variational
principle
for
preordered
uniform
space
382
since
d
(
x,
y
)
≤
δ
(
x,
y
)
for
all
(
x,
y
)
∈
P
2
≤
,
we
obtain
that
x
n
d
→
x
.
Whence
x
n
D
→
x
(see
Lemma
5
).
Using
that
f
is
(
≤
,
D
)-submonotone,
we
conclude
that
f
(
x
)
≤
f
(
x
m
)
for
all
m
,
and
the
fact
that
f
is
(
≤
,
δ
)-submonotone
is
established.
We
are
in
position
to
apply
Theorem
3
,
which,
given
x
∈
dom(
f
),
pro-
vides
us
with
an
element
y
∈
dom(
f
)
satisfying
conditions
(a)–(c)
therein.
Let
us
deduce
that
the
same
element
y
satises
conditions
(a)–(c)
of
Theo-
rem
1
.
Condition
(a)
of
Theorem
1
coincides
with
condition
(a)
of
Theorem
3
.
Condition
(b)
of
Theorem
3
tells
us
that
δ
(
x,
y
)
≤
f
(
x
)
−
f
(
y
).
By
denition
of
δ
,
this
means
that
d
(
x,
y
)
≤
f
(
x
)
−
f
(
y
)
for
all
d
∈
D
,
which
is
exactly
condition
(b)
of
Theorem
1
.
Finally,
let
z
∈
dom(
f
)
and
z
′
∈
P
be
such
that
y
≤
z
≤
z
′
and
assume
that
d
(
z,
z
′
)
>
0
for
some
d
∈
D
.
By
denition
of
δ
,
this
yields
δ
(
z,
z
′
)
̸
=
0.
If
f
(
z
′
)
=
+
∞
then
the
inequality
d
′
(
z,
z
′
)
>
f
(
z
)
−
f
(
z
′
)
is
immediate
for
all
d
′
∈
D
,
so
we
can
assume
that
z
′
∈
dom(
f
).
Condition
(c)
of
Theorem
3
now
implies
that
δ
(
y,
z
)
>
f
(
y
)
−
f
(
z
)
or
δ
(
z,
z
′
)
>
f
(
z
)
−
f
(
z
′
)
(recall
that
in
our
situation
the
monoid
K
=
R
∪
{
+
∞}
is
endowed
with
its
total
order
≤
).
Invoking
again
the
denition
of
δ
,
these
formulas
mean
that
there
is
d
∈
D
such
that
d
(
y,
z
)
>
f
(
y
)
−
f
(
z
)
or
there
is
d
∈
D
such
that
d
(
z,
z
′
)
>
f
(
z
)
−
f
(
z
′
),
which
matches
with
the
conclusion
of
part
(c)
of
Theorem
1
.
The
proof
of
Theorem
1
is
complete.
5.3
Proof
of
Theorem
2
We
write
D
=
{
d
n
}
n
∈
N
(that
is,
we
x
a
surjective
map
N
→
D
,
which
is
pos-
sible
since
D
is
at
most
countable).
Also
we
write
for
simplicity
f
n
=
f
d
n
for
all
n
∈
N
,
so
that
F
=
{
f
n
}
n
∈
N
.
We
consider
the
monoid
K
=
(
R
∪{
+
∞}
)
N
equipped
with
the
addition
+,
the
partial
order
≤
,
and
the
set
V
as
in
Exam-
ple
1
(b).
Furthermore,
let
δ
:
P
2
≤
→
K
+
=
[0
,
+
∞
]
N
,
(
x,
y
)
→
(
d
n
(
x,
y
))
n
∈
N
.
Clearly,
δ
(
x,
x
)
=
0
for
all
x
∈
P
,
where
0
denotes
the
neutral
element
in
K
(i.e.,
the
zero
constant
sequence).
Moreover,
whenever
x
≤
y
≤
z
,
we
have
d
n
(
x,
z
)
≤
d
n
(
x,
y
)+
d
n
(
y,
z
)
for
all
n
∈
N
,
whence
δ
(
x,
z
)
≤
δ
(
x,
y
)+
δ
(
y,
z
).
We
aim
to
apply
Theorem
3
with
K
and
δ
as
above
and
to
the
map
f
:
P
→
K
,
x
→
(
f
n
(
x
))
n
∈
N
.
We
need
to
check
that
•
P
is
(
≤
,
δ
)-complete;
L.
Fresse,
V.V.
Motreanu
383
•
f
is
(
≤
,
δ
)-submonotone;
•
f
is
locally
bounded
from
below.
Let
us
prove
the
rst
assertion.
Let
(
x
n
)
⊂
P
be
a
≤
-ascending
sequence
and
assume
that
(
x
n
)
is
δ
-Cauchy.
Let
us
show
that
(
x
n
)
is
D
-Cauchy.
To
do
this,
for
every
d
=
d
i
∈
D
with
i
∈
N
,
we
have
to
show
that
(
x
n
)
is
d
-Cauchy.
So
let
ε
∈
Q
∗
+
.
Set
I
=
{
i
}
and
r
:
I
→
Q
∗
+
,
i
→
r
i
=
ε
.
Considering
V
=
V
I,r
(see
Example
1
(b)),
we
nd
n
0
∈
N
such
that
whenever
n
0
≤
m
≤
n
,
we
have
δ
(
x
m
,
x
n
)
∈
V
.
The
latter
inclusion
means
that
d
(
x
m
,
x
n
)
<
ε
.
We
obtain
that
(
x
n
)
is
d
-Cauchy
for
all
d
∈
D
,
and
so
D
-Cauchy
as
claimed.
Knowing
that
P
is
(
≤
,
D
)-complete,
it
follows
that
there
is
x
∈
P
such
that
x
n
D
→
x
,
in
particular
x
n
≤
x
for
all
n
(see
Remark
6
(c)).
We
verify
that
(
x
n
)
is
δ
-convergent
to
x
and
this
will
conclude
the
argu-
ment
for
showing
that
P
is
(
≤
,
δ
)-complete.
Let
V
=
V
I,r
∈
V
with
I
⊂
N
nite
and
r
:
I
→
Q
∗
+
.
For
every
i
∈
I
,
we
know
that
x
n
d
i
→
x
(see
Lemma
5
).
Hence
there
is
n
i
∈
N
such
that
d
i
(
x
n
,
x
)
<
r
i
whenever
n
≥
n
i
.
Set
n
0
=
max
{
n
i
:
i
∈
I
}
.
For
every
n
≥
n
0
,
we
then
obtain
that
d
i
(
x
n
,
x
)
<
r
i
for
all
i
∈
I
,
whence
δ
(
x
n
,
x
)
∈
V
.
Therefore,
(
x
n
)
is
δ
-convergent
to
x
.
Next
we
show
that
f
is
(
≤
,
δ
)-submonotone.
To
do
this,
let
(
x
n
)
⊂
dom(
f
)
be
a
≤
-ascending
sequence
which
is
δ
-convergent
to
x
∈
P
(thus
x
n
≤
x
for
all
n
)
and
such
that
f
(
x
n
)
≤
f
(
x
m
)
for
all
m,
n
∈
N
with
m
≤
n
.
By
denition
of
f
,
the
latter
inequality
tells
us
that
f
i
(
x
n
)
≤
f
i
(
x
m
)
for
all
m,
n
∈
N
with
m
≤
n
and
all
i
∈
N
,
i.e.,
(
f
i
(
x
n
))
n
∈
N
is
nonincreasing
for
all
i
∈
N
.
Moreover,
note
that
x
n
D
→
x
.
Indeed,
by
Lemma
5
,
it
suces
to
see
that
x
n
d
i
→
x
for
every
i
∈
N
.
For
seeing
this,
let
ε
∈
Q
∗
+
.
Set
I
=
{
i
}
and
r
:
I
→
Q
∗
+
,
i
→
ε
.
Since
(
x
n
)
is
δ
-convergent
to
x
,
there
is
n
0
∈
N
such
that
δ
(
x
n
,
x
)
∈
V
I,r
whenever
n
≥
n
0
;
whence
d
i
(
x
n
,
x
)
<
ε
whenever
n
≥
n
0
.
Therefore,
x
n
D
→
x
.
Now
we
can
invoke
the
assumption
that
F
is
(
≤
,
D
)-submonotone,
from
which
we
obtain
that
f
i
(
x
)
≤
f
i
(
x
m
)
for
all
i
∈
N
and
all
m
∈
N
,
hence
f
(
x
)
≤
f
(
x
m
)
for
all
m
∈
N
.
The
argument
for
showing
that
f
is
(
≤
,
δ
)-
submonotone
is
complete.
Let
us
now
establish
the
third
assertion
saying
that
f
is
locally
bounded
from
below.
Let
x
∈
dom(
f
).
This
means
that
f
i
(
x
)
<
+
∞
for
all
i
∈
N
,
i.e.,
x
∈
i
∈
N
dom(
f
i
)
=
dom(
F
).
Using
that
F
is
≥
-locally
bounded
from
below,
there
is
a
family
{
µ
i
}
i
∈
N
⊂
R
such
that
µ
i
≤
f
i
(
y
)
whenever
y
∈
P
satises
x
≤
y
.
Set
µ
=
(
µ
i
)
i
∈
N
∈
R
N
=
K
′
.
We
obtain
that
for
every
y
∈
P
such
that
x
≤
y
,
we
have
µ
≤
f
(
y
).
The
argument
is
complete.
On
Ekeland
variational
principle
for
preordered
uniform
space
384
The
above
assertions
allow
us
to
apply
Theorem
3
.
Let
x
∈
dom(
F
)
=
dom(
f
).
There
exists
y
∈
dom(
f
)
=
dom(
F
)
fullling
(a)–(c)
of
Theorem
3
.
We
claim
that
the
same
element
y
also
fulls
(a)–(c)
of
Theorem
2
.
Condition
(a)
is
exactly
the
same
for
both
theorems
so
we
only
have
to
consider
conditions
(b)
and
(c).
Theorem
3
(b)
tells
us
that
the
relation
δ
(
x,
y
)
≤
f
(
x
)
−
f
(
y
)
holds
in
the
monoid
K
.
This
means
that
d
i
(
x,
y
)
≤
f
i
(
x
)
−
f
i
(
y
)
for
all
i
∈
N
,
in
other
words
d
(
x,
y
)
≤
f
d
(
x
)
−
f
d
(
y
)
for
all
d
∈
D
.
Condition
(b)
of
Theorem
2
is
established.
Finally,
let
z
∈
dom(
F
)
=
dom(
f
)
and
z
′
∈
P
be
such
that
y
≤
z
≤
z
′
and
d
(
z,
z
′
)
>
0
for
some
d
=
d
i
∈
D
.
Since
δ
(
z,
z
′
)
=
(
d
j
(
z,
z
′
))
j
∈
N
,
we
obtain
that
δ
(
z,
z
′
)
is
dierent
from
the
zero
constant
sequence
(the
neutral
element
of
K
).
If
z
′
/
∈
dom(
f
),
then
there
is
d
′
=
d
j
∈
D
such
that
f
d
′
(
z
′
)
=
f
j
(
z
′
)
=
+
∞
,
thus
d
′
(
z,
z
′
)
>
f
d
′
(
z
)
−
f
d
′
(
z
′
).
Now
assume
that
z
′
∈
dom(
f
).
By
Theorem
3
(c),
we
then
have
δ
(
y,
z
)
̸≤
f
(
y
)
−
f
(
z
)
or
δ
(
z,
z
′
)
̸≤
f
(
z
)
−
f
(
z
′
).
Due
to
the
denition
of
the
partial
order
≤
of
K
,
we
conclude
that
there
is
d
′
=
d
j
∈
D
such
that
d
′
(
y,
z
)
>
f
d
′
(
y
)
−
f
d
′
(
z
)
or
there
is
d
′
∈
D
such
that
d
′
(
z,
z
′
)
>
f
d
′
(
z
)
−
f
d
′
(
z
′
).
Thus
condition
(c)
of
Theorem
2
holds,
and
the
proof
of
the
theorem
is
complete.
References
[1]
N.
Bourbaki,
´
El´
ements
de
Math´
ematiques.
Topologie
G´
en´
erale.
Chapi-
tres
5
`
a
10
.
Hermann,
Paris,
1974.
[2]
L.
Fresse
and
V.V.
Motreanu,
Nonlinear
error
bounds
for
maps
on
pre-
ordered
pseudometric
spaces,
Ann.
Acad.
Rom.
Sci.,
Math.
Appl.
17
(2025),
169-191.
[3]
L.
Fresse
and
V.V.
Motreanu,
Ekeland
variational
principle
over
pre-
ordered
monoids,
Ann.
Acad.
Rom.
Sci.,
Math.
Appl.
18
(2026),
133-
151.