Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
SOME
TWO-FUNCTION
TOPOLOGICAL
MINIMAX
THEOREMS
∗
Ornella
Naselli
†
Dedicated
to
Professor
Constantin
Fetecau
on
the
occasion
of
his
80th
anniversary
DOI 10.56082/annalsarscimath.2026.4.11
Abstract
In
this
note,
we
extend
some
topological
minimax
theorems
of
Ric-
ceri
(see
[
3
])
to
the
case
of
two
functions.
Keywords:
minimax,
semicontinuity,
connectedness.
MSC:
90C47,
49K35.
1
Introduction
and
statements
of
the
main
results
Let
X,
Y
be
two
non-empty
sets
and
let
ϕ
be
a
real-valued
function
on
X
×
Y
.
Set
ϕ
∗
=
sup
y
∈
Y
inf
x
∈
X
ϕ
(
x,
y
)
and
ϕ
∗
=
inf
x
∈
X
sup
y
∈
Y
ϕ
(
x,
y
)
.
It
is
clear
that
ϕ
∗
≤
ϕ
∗
.
∗
Accepted
for
publication
on
May
18,
2026
†
ornella.naselli@unict.it
,
Department
of
Mathematics
and
Computer
Science,
University
of
Catania,
Italy
11
Some
two-function
topological
minimax
theorems
12
This
is
called
the
trivial
minimax
inequality.
The
opposite
inequality
ϕ
∗
≤
ϕ
∗
is
called
non-trivial
minimax
inequality
and
of
course
it
is
equivalent
to
the
minimax
equality
ϕ
∗
=
ϕ
∗
.
(1)
Starting
from
the
pioneristic
work
of
von
Neumann
(see
[
10
]),
many
results
ensuring
(see
1
)
were
established.
For
an
introductory
bibliography
see,
for
example,
the
classical
survey
of
Simons
(see
[
8
]).
In
particular,
in
the
last
century
many
minimax
theorems
were
established
in
which
the
function
is
defined
in
topological
vector
spaces
with
both
vectorial
and
topological
hypotheses.
In
other
theorems
the
function
is
defined
in
topological
spaces
and
the
vectorial
hypotheses
are
replaced
by
topological
hypotheses:
they
are
the
so-called
topological
minimax
theorems.
Many
topological
minimax
theorems
have
been
established
by
Ricceri,
(see,
for
example,
[
3
],
[
5
]
and
[
6
]).
In
particular,
in
[
3
],
assuming
that
Y
is
a
real
interval,
he
proved
some
minimax
theorems
where
the
quasi-convexity
of
f
in
X
(present
in
each
of
the
most
classical
results)
was
replaced
by
inf-connectedness.
This
allowed
one
to
obtain
several
consequences
and
applications
in
non-convex
settings
(see
[
4
]).
Now,
let
f,
g
:
X
×
Y
→
R
,
with
f
(
x,
y
)
≤
g
(
x,
y
)
for
every
x
∈
X,
y
∈
Y
.
We
call
non-trivial
minimax
inequality
involving
f,
g
the
following
f
∗
≤
g
∗
.
(2)
So,
if
f
=
g
,
(
2
)
is
equivalent
to
(
1
).
For
a
given
minimax
theorem
for
one
function
ϕ
,
it
is
an
usual
fact
to
see
whether
it
is
possible
to
find
a
two-function
version
of
it.
The
most
natural
way
to
get
this
is
to
split
the
hypotheses
on
ϕ
to
f
and
g
.
For
example,
the
two-function
version
of
the
most
classical
Fan-Sion’s
theorem
(
[
9
])
has
been
obtained
by
Simons
(
[
7
]).
In
[
1
]
and
[
2
]
the
two-function
versions
of
the
results
of
[
5
]
and
[
6
]
have
been
established.
The
aim
of
the
present
paper
is
to
extend
to
two
functions
the
above
mentioned
classical
Ricceri’s
results
of
[
3
].
In
particular,
we
will
establish
the
Theorems
1,
2
and
3
below.
To
this
end,
if
X
is
a
topological
space,
I
is
a
real
interval
and
f,
g
are
two
real
functions
defined
on
X
×
I
,
we
will
consider
the
following
conditions:
H1)
for
every
(
x,
λ
)
∈
X
×
I
,
one
has
f
(
x,
λ
)
≤
g
(
x,
λ
).
O.
Naselli
13
H2)
for
every
λ
∈
I
,
f
(
·
,
λ
)
is
lower
semicontinuous.
H3)
the
functions
f
is
upper
semicontinuous
in
X
×
I
.
H4)
for
every
x
∈
X
and
for
every
α
∈
R
the
set
{
λ
∈
I
:
f
(
x,
λ
)
>
α
}
is
connected.
H5)
for
every
x
∈
X
,
g
(
x,
·
)
is
upper
semicontinuous.
H6)
the
functions
g
is
lower
semicontinuous
in
X
×
I
.
H7)
there
exists
a
dense
subset
D
⊆
I
such
that,
for
every
λ
∈
D
,
and
for
every
α
∈
R
,
the
set
{
x
∈
X
:
g
(
x,
λ
)
<
α
}
is
connected.
H8)
I
is
compact
or
there
exists
λ
0
∈
I
such
that,
for
every
α
∈
R
,
the
set
{
x
∈
X
:
f
(
x,
λ
0
)
≤
α
}
is
compact.
H9)
for
every
λ
∈
I
and
for
every
α
∈
R
the
set
{
x
∈
X
:
g
(
x,
λ
)
≤
α
}
is
connected.
Theorem
1.
Let
X
be
a
topological
space,
I
⊆
R
an
interval
and
f,
g
:
X
×
I
→
R
two
functions
satisfying
conditions
H1),
H2,
H4),
H5),
H7)
and
H8).
Then,
the
relation
(
2
)
holds.
Theorem
2.
Let
X
be
a
topological
space,
I
⊆
R
a
compact
interval
and
f,
g
:
X
×
I
→
R
two
functions
satisfying
conditions
H1),
H2),
H3),
H4),
H5)
and
H7).
Then,
the
relation
(
2
)
holds.
Theorem
3.
Let
X
be
a
compact
topological
space,
I
⊆
R
an
interval
and
f,
g
:
X
×
I
→
R
two
functions
satisfying
conditions
H1),
H2),
H4),
H6)
and
H9).
Then,
the
relation
(
2
)
holds.
Observe
that,
if
f
=
g
,
Theorems
1,
2
and
3
coincide
with
Theorem
1.1
of
[
3
].
2
Proofs
of
the
main
results
The
proofs
of
the
Theorems
1,
2
and
3
are
based
on
the
Lemma
1
below
(see
[
3
],
Theorem
2.5).
Before
stating
it,
we
recall
that,
if
X,
Y
are
two
non-empty
sets
and
V
is
a
subset
of
X
×
Y
,
one
has
the
following
notations:
∗
p
X
(
V
)
=
{
x
∈
X
:
∃
y
∈
Y
:
(
x,
y
)
∈
V
}
;
∗
p
Y
(
V
)
=
{
y
∈
Y
:
∃
x
∈
X
:
(
x,
y
)
∈
V
}
;
∗
if
x
∈
X
,
V
x
=
{
y
∈
Y
:
(
x,
y
)
∈
V
}
;
∗
if
y
∈
Y
,
V
y
:
{
x
∈
X
:
(
x,
y
)
∈
V
}
.
Lemma
1.
Let
X
be
a
topological
space,
I
a
compact
real
interval,
and
let
S,
T
two
subsets
of
X
×
I
.
Assume
that
at
least
one
of
the
following
sets
of
conditions
are
verified:
Some
two-function
topological
minimax
theorems
14
(
δ
1
)
p
I
(
S
)
is
connected,
S
x
is
open
for
every
x
∈
X
,
the
set
{
λ
∈
p
I
(
S
)
:
S
λ
is
connected
}
is
dense
in
p
I
(
S
)
,
T
x
is
non-empty
and
connected
for
every
x
∈
X
,
T
λ
is
open
for
every
λ
∈
I
.
(
δ
2
)
p
I
(
S
)
is
connected,
S
x
is
open
for
every
x
∈
X
,
the
set
{
λ
∈
p
I
(
S
)
:
S
λ
is
connected
}
is
dense
in
p
I
(
S
)
,
T
is
closed,
T
x
is
non-empty
and
con-
nected
for
every
x
∈
X
.
(
δ
3
)
p
I
(
S
)
is
connected,
X
is
compact,
S
is
closed,
S
λ
is
connected
for
every
λ
∈
I
,
T
is
closed,
T
x
is
non-empty
and
connected
for
every
x
∈
X
,
T
λ
is
open
for
every
λ
∈
I
.
Then,
at
least
one
the
following
two
assertions
does
hold:
i)
p
I
(
S
)
=
I
;
ii)
S
∩
T
=
∅
.
We
will
also
use
the
following
Proposition
1.
Let
X
be
a
topological
space,
Y
a
non-empty
set,
f,
g
:
X
×
Y
→
R
two
functions
satisfying
the
following
conditions:
h1)
for
every
(
x,
y
)
∈
X
×
Y
,
one
has
f
(
x,
y
)
≤
g
(
x,
y
)
;
h2)
for
every
y
∈
Y
,
f
(
·
,
y
)
is
lower
semicontinuous;
h3)
there
exists
y
0
∈
Y
such
that,
for
every
α
∈
R
,
the
set
{
x
∈
X
:
f
(
x,
y
0
)
≤
α
}
is
compact;
h4)
there
exists
an
increasing
sequence
{
Y
n
}
of
subsets
of
Y
such
that
∪
n
∈
N
Y
n
=
Y
and
f
∗
(
n
)
≤
g
∗
(
n
)
for
every
n
∈
N
,
where
f
∗
(
n
)
=
inf
x
∈
X
sup
y
∈
Y
n
f
(
x,
y
)
and
g
∗
(
n
)
=
sup
y
∈
Y
n
inf
x
∈
X
g
(
x,
y
)
.
Under
such
hypotheses,
the
relation
(
2
)
holds.
Proof.
It
is
not
restrictive
to
assume
that
y
0
∈
Y
1
.
Arguing
by
contra-
diction,
assume
that
g
∗
<
f
∗
and
fix
h
∈
R
satisfying
g
∗
<
h
<
f
∗
.
For
every
n
∈
N
,
put
C
n
=
{
x
∈
X
:
sup
y
∈
Y
n
f
(
x,
y
)
≤
h
}
.
Each
set
C
n
is
non-empty.
Otherwise,
one
would
have,
taking
into
ac-
count
h4)
h
≤
inf
x
∈
X
sup
y
∈
Y
n
f
(
x,
y
)
=
f
∗
(
n
)
≤
g
∗
(
n
)
=
sup
y
∈
Y
n
inf
x
∈
X
g
(
x,
y
)
≤
g
∗
<
h.
O.
Naselli
15
Since
the
sequence
{
Y
n
}
is
increasing,
the
sequence
{
C
n
}
is
decreasing.
So,
thanks
to
h2),
{
C
n
}
is
a
decreasing
sequence
of
non-empty
closed
and
compact
sets.
So,
there
exists
x
∗
∈
∩
n
∈
N
C
n
.
Then,
for
every
n
∈
N
,
one
has
sup
y
∈
Y
n
f
(
x
∗
,
y
)
≤
h
and
so
sup
y
∈
Y
f
(
x
∗
,
y
)
=
sup
n
∈
N
sup
y
∈
Y
n
f
(
x
∗
,
y
)
≤
h.
Hence
f
∗
=
inf
x
∈
X
sup
y
∈
Y
f
(
x,
y
)
≤
h
,
absurd.
Proof
of
Theorem
1.
Thanks
to
Proposition
1
and
H2),
we
can
limit
ourselves
to
the
case
in
which
I
is
compact.
Arguing
by
contradiction,
suppose
that
g
∗
<
f
∗
.
Fix
h
∈
R
satisfying
g
∗
<
h
<
f
∗
.
Let
us
put
S
=
{
(
x,
λ
)
∈
X
×
I
:
g
(
x,
λ
)
<
h
}
,
T
=
{
(
x,
λ
)
∈
X
×
I
:
f
(
x,
λ
)
>
h
}
.
Consider
the
set
S
.
First
of
all,
observe
that
p
I
(
S
)
=
I
.
In
fact,
if
λ
0
∈
I
,
one
has
h
>
g
∗
≥
inf
x
∈
X
g
(
x,
λ
0
),
hence
there
exists
x
∈
X
such
that
g
(
x,
λ
0
)
>
h
.
Moreover,
if
x
0
∈
X
,
the
set
S
x
0
=
{
λ
∈
I
:
g
(
x
0
,
λ
)
<
h
}
is
open
thanks
to
H5).
Finally,
from
H7),
the
set
{
λ
∈
P
I
(
S
)
:
S
λ
is
connected
}
,
since,
as
we
have
seen,
P
I
(
S
)
=
I
,
contains
D
,
so
it
is
dense
in
p
I
(
S
).
Now,
consider
the
set
T
.
Observe
that,
if
x
0
∈
X
,
the
set
T
x
0
=
{
λ
∈
I
:
f
(
x
0
,
λ
)
>
h
}
is
non-empty
because,
since
h
<
f
∗
≤
sup
λ
∈
I
f
(
x
0
,
λ
),
there
exists
λ
∈
I
such
that
f
(
x
0
,
λ
)
>
h
,
so,
T
x
0
is
non-empty.
Moreover,
it
is
connected
thanks
to
H4).
Finally,
if
λ
0
∈
I
,
the
set
T
λ
0
=
{
x
∈
X
:
f
(
x,
λ
0
)
>
h
}
is
open
thanks
to
H2).
So,
the
conditions
(
δ
1
)
of
Lemma
1
are
satisfied
but
this
is
absurd
because
both
the
assertions
i)
and
ii)
are
clearly
false.
Hence,
Theorem
1
is
proved.
Proof
of
Theorem
2.
Arguing
as
in
Theorem
1
by
contradiction,
let
us
fix
h
∈
R
satisfying
g
∗
<
h
<
f
∗
.
Let
us
put
S
=
{
(
x,
λ
)
∈
X
×
I
:
g
(
x,
λ
)
<
h
}
,
T
=
{
(
x,
λ
)
∈
X
×
I
:
f
(
x,
λ
)
≥
h
}
.
Some
two-function
topological
minimax
theorems
16
Consider
the
set
S
.
As
in
Theorem
1,
it
is
possible
to
see
that
p
I
(
S
)
=
I
.
Moreover,
if
x
0
∈
X
,
the
set
S
x
0
=
{
λ
∈
I
:
g
(
x
0
,
λ
)
<
h
}
is
open
thanks
to
H5).
Finally,
thanks
to
H7),
the
set
{
λ
∈
P
I
(
S
)
:
S
λ
is
connected
}
,
since,
as
we
have
seen,
P
I
(
S
)
=
I
,
contains
D
,
so
it
is
dense
in
p
I
(
S
).
Now,
consider
the
set
T
.
Observe
that
it
is
closed,
thanks
to
H3).
More-
over,
if
x
0
∈
X
,
the
set
T
x
0
=
{
λ
∈
I
:
f
(
x
0
,
λ
)
≥
h
}
is
non-empty
because
h
<
f
∗
≤
sup
λ
∈
I
f
(
x
0
,
λ
)
and
it
is
connected
thanks
to
H4).
So,
the
conditions
(
δ
2
)
of
Lemma
1
are
satisfied
but
this
is
absurd
because
both
the
assertions
i)
and
ii)
are
clearly
false.
Hence,
Theorem
2
is
proved.
Proof
of
Theorem
3
.
By
H2)
and
the
compactness
of
X
,
thanks
to
Proposition
1
we
can
limit
ourselves
to
the
case
in
which
I
is
compact.
Arguing
again
by
contradiction,
let
us
fix
h
∈
R
satisfying
g
∗
<
h
<
f
∗
.
Let
us
put
S
=
{
(
x,
λ
)
∈
X
×
I
:
g
(
x,
λ
)
≤
h
}
,
T
=
{
(
x,
λ
)
∈
X
×
I
:
f
(
x,
λ
)
>
h
}
.
Consider
the
set
S
.
As
in
Theorem
1,
it
is
possible
to
see
that
p
I
(
S
)
=
I
.
Moreover,
we
can
observe
that
S
is
closed
thanks
to
H2).
Finally,
if
λ
0
∈
I
,
the
set
S
λ
0
=
{
x
∈
X
:
g
(
x,
λ
0
)
≤
h
}
is
connected
thanks
to
H9).
Now,
consider
the
set
T
.
Observe
that,
if
x
0
∈
X
,
the
set
T
x
0
=
{
λ
∈
I
:
f
(
x
0
,
λ
)
>
h
}
is
non-empty
because
h
<
f
∗
≤
sup
λ
∈
I
f
(
x
0
,
λ
),
and
it
is
connected
thanks
to
H4).
Moreover,
if
λ
0
∈
I
,
the
set
T
λ
0
=
{
x
∈
X
:
f
(
x,
λ
0
)
>
h
}
is
open
thanks
to
H2).
So,
the
conditions
(
δ
3
)
of
Lemma
1
are
satisfied
but
this
is
absurd
because
both
the
assertions
i)
and
ii)
are
clearly
false.
Hence,
Theorem
3
is
also
proved.
Acknowledgements.
The
author
is
grateful
to
the
anonymous
referee
for
her/his
comments.
Paper
written
with
financial
support
of
Universit`
a
degli
Studi
di
Catania,
PIACERI
2024-2026,
Linea
di
intervento
1,
Progetto
”PAFA”.
References
[1]
O.
Naselli,
A
two-function
extension
of
a
minimax
theorem,
Le
Matem-
atiche
79
(2024),
441-447.
O.
Naselli
17
[2]
O.
Naselli,
Two-function
extensions
of
some
minimax
theorems
of
Ric-
ceri,
Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
18
(2026),
73-80.
[3]
B.
Ricceri,
Some
topological
mini-max
theorems
via
an
alternative
prin-
ciple
for
multifunctions,
Arch.
Math.
(Basel)
60
(1993),
367-377.
[4]
B.
Ricceri,
On
a
topological
minimax
theorem
and
its
applications
,
In:
Minimax
theory
and
applications
(Erice,
1996),
Nonconvex
Optim.
Appl.,
vol.
26,
Kluwer
Acad.
Publ.,
Dordrecht
(1998),
191-216.
[5]
B.
Ricceri,
Multiplicity
of
global
minima
for
parametrized
functions,
Rend.
Lincei
Mat.
Appl.
21
(2010),
47-57.
[6]
B.
Ricceri,
Minimax
theorems
in
a
fully
non-convex
setting,
J.
Nonlin-
ear
Var.
Anal.
3
(2019),
45-52.
[7]
S.
Simons,
Two-function
minimax
theorems
and
variational
inequalities
for
functions
on
compact
and
noncompact
sets,
with
some
comments
on
fixed
point
theorems,
Proceedings
of
Simposia
in
Pure
Mathematics
45
(1986),
377-391.
[8]
S.
Simons,
Minimax
theorems
and
their
proofs
,
In:
Minimax
and
Ap-
plications,
Nonconvex
Optim.
Appl.,
vol.
4,
Kluwer
Acad.
Publ.,
Dor-
drecht
(1995),
1-23.
[9]
M.
Sion,
On
general
minimax
theorems,
Pacific
J.
Math.
8
(1958),
171-
176.
[10]
J.
von
Neumann,
Zur
Theorie
der
Gesellschaftspiele,
Math.
Ann.
100
(1928),
295-320.