693
Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 693-696
A VARIANT OF A FIXED POINT THEOREM OF
BROWDER-FAN AND REICH
V. M. SEHGAL
Department of Iathematics
University of Wyoming
Laramie, Wyoming 82071 U.S.A.
(Received January 8, 1985)
Let
ABSTRACT.
be a convex, weakly compact subset of a locally convex Hausdorff
S
p(A, B)
A
be a continuous seminorm on (E, ) and for subsets
p
let
be a continuous multifunction from its weak topology
E
space (E, ) and f: S
y): x
inf:p(x
B}
A, y
veloped for the existence of an
m to
let
In this paper, sufficient conditions are deS
x
of E
B
satisfying
p(x, fx)
p(fx, S)
The result
is then used to prove several fixed point theorems.
KEY WORDS AND
I.
Multifunction,, convex topology, fixed point,:.
47H10, 60H25.
PII.q’AL’/’,’,.
AMS SUBJECT CLASSIFICATION CODES.
lqO
INTRODUCTION
(E, z)
Let
and
logy
of
be a locally convex Hausdorff topological vector space with topology
E*
(E, )*
E
Let
be its topological dual.
P
the topologies
m(E,
m
E*)
be the weak topo-
denote the family of continuous semi-norms generating
For sets
respectively.
m
y): x
inf{p(x
let p(A, B)
Q
and
and
Let
B}
A, y
A
B
and
of
E
P,
p
and a
In this paper, we prove the following
result.
THEOREM Io
(E, r)
each
x
Let
S
be a nonempty co,vex, m-compact subset of
be a continuous multifunction such that
S
Then f,r each
P
p
there exists a
p(x, fx)
p(x, fx)
F.ther if
0
then
a(S, )
It may be remarked the since
f.nction from
each
q
Q
(S
....
(E. ,,)
S
x
E
and
is convex and
f:
f
(S, m)
-compact for
satisfying
p(fx. S)
a(S, ,)
x
f(x)
(1.1)
where
in Theorem
denotes the boundary.
is also a continuous multi-
Consequently it follows by Reich (Lemma 1.6 [1]) that
satisfies (1.1) for some
[1] is not applicable for arbitrary
S
x
p
P
However, since
In fact, Theorem
Q
p
the lemma in
contains the above
1emma [1] (see Corollary 2) and it provides a generalization of a well-known result of
Ky Fan [2] for single valued mappings.
2. PRELIMINARY RESULTS.
X
Recall that if
(fx *
for each
x)
Y
are topological spaces then a multifunction
f: X
is upper (lower) semicontinuous iff for each closed
Y
(open) sub-
V.M. SEHGAL
694
A
set
,x
f-1 (A)
y
of
X: f(x)
x
and
X
in
U
fz
then
for each
then
fx
ya
with
fX
values then
xa
X
in
x
and
Further,
Ya
in
Y
is u.s.c, with compact
f
is compact and
X
of
Y
of
is called
mu]tifunction which is both u.s,c, and 1.s,c.
A
is compact.
X
(ii) if
fx;
v
V
in some neighborhood
z
U
for some open set
It
X
(open) subset of
is a closed
is u.s.c, and a net
it is well-known (i) that if
Y
}
A
0
is 1.s.c. iff fx n U
follows by definition that
cntinuous.
Let A, B
LEIA 1.
p(x
y)
for some
PROOF.
,-compact sets of
be
{Xn
sequences
Cho,.e
We may assume that
x
{yn _c
_c A,
weakly for some
x
P
E
and p
B
such that
A
x
and
p(x
y)
-< p(u)
for each
p(x
y)
LEtA 2.
Let
p(x, B)
x*(x
p(A, B)
x*
E*
x
B.
y
x*(x
with
Consequently, since
S
each
P
p
(x
U
y)
Yn
n
fx
n U
S) -<
p(fx,
S) <-
P(Ya’
U
Then
For such
eventually.
x
let
c
(E, )
(S, m)
be a l.s.c.
then for
S
weakly in
x
eventually.
y
fx
y
n U.
fx
p(fx, S)
with
fx n U.
y
-open and
is
p(A, B) -< p(x, B)
Yn
n
f:
and
If a net
that there is a
e}
y)
E: p(x
im p(x
S) -< p(fx, S) +
p(fx
It follows by Lemma
PROOF.
Let
-<
E
-compact subset of
be
O,
and
yn)l
lim[x*(x n
y)
multifunction with weakly compact values.
p(y, S)
Ilence by l.s.c.,
Then eventually,
y) + P(Y’ S) -< p(fx, S)+ e.
P(Ya-
MAIN RESULTS.
3.
PROOF OF THEOREM I.
Let
P.
p
Then by Lemma 1, g(x) z
Let
x
We may
that
(E, m)
that
0
z
E*
weakly for some
za fxa
weakly for some
x*(y
such that
Choose
quently, for
by
-closed and
is
m-compact subset of
g(x) n C
with
being u.s.c., it follows that
z
x*
that is
y
Ycz
S), there exists
p(fxa,
S
aO
0
p(y, fx)
A
_
S
We show
be a weakly closed (hence weakly compact) aubset of
C
g-I (C),
We show that
asgu..e
(S, m)
(S, m)
<- p(y- z).
p(z, fx)
fx)
-closed convex and hence a
is
g is u.s.c.
that
Further, since
and is clearly convex.
IP(Y,
g(x)
It follows
g:
p(fx, S)}.
S: p(y, fx)
g(x), the triangular inequality implies
y, z
fr any
Define a multifunction
{y
g(x)
a
E
u
yn
n
weakly,
y
x
x*(u)
and
p(x, B)
weakly for some
y
Yn
By Hahn Banach Theorem (see [3], Cot. 2, p. 29) there exists a
l(x
p(A, B)
Then
B
A, y
x
be
E
a nonempty subset of
S
as stated in the beginning and
E
Throughout, let
We prove two lemmas that simplify the proof of Theorem 1.
x*(y
fS
fx.
Thus
z)
p(y
z)
z)[
p(y- z)
C.
y
p(ya
z
such that
Choose for each
p(fxa,
-< p(fx
limlx*(y
Also
za)
since
p(fXc
S)
S
Ya gxa
P(Ya’ fx)
C
a
Further
f: (S,
is weakly compact and hence we may assume
ya
and
z
y
[x*(u)
z weakly.
-< p(u)
S) -< p(fx, S) + e for
S)
p(fx
z)
S) +
Choose as before
for each
E.
-> a
O.
Conse-
a
and hence
<- p(fx, S) + c.
Let
695
FIXED POINT THEOREM OF BROWDER-FAN AND REICH
Since
g(x) n C.
y
is
g(x)
with x
p(fx, S)
p(fx, S) -< p(y, fx), we have p(y, fx)
Hence by Glicksberg [4] there exists a x
is u.s.c.
Thus
g
Now, if
p(x
(x, y) o S
z
y
a
fx
int(S, t), then since
0
p(fx, S) -<
int(S, m) or
x
p(fx, S), a contradiction.
y)
I,
Choose by Lemma
fx n S
Then
being weakly closed and convex, there is a
z)
S
p(fx, S).
p(x, fx)
This implies
0
Now, suppose p(x, fx)
y)
p(x, fx).
satisfying p(x
p(y
that
is arbitrary and
0
e
S
with
This proves the result.
As a consequence of Theorem I, we have
COROLLARY I.
(E, )
Let
S
has a fixed point or there exists a
f
E
be a convex and weakly compact set in
P
p
and
f: (S, m)
and
u-compact values.
be a continuous multifunction with convex and
p(x, fx)
satisfying 0
S
x
Then either
p(x, s)
For each
PROOF.
each
A
P
p
P
p(x, fx)
satisfying (I).
S
P
{x
0}
S: p(x, fx)
P
0
{x*(x
x
(S,m)
for
is continuous, it
Consequently, there exists
S
x
with
fx
is
t-closed and
E*
such that
P
p
Now, if
fx, then since
x
x
Ix*I.
y): y
0
fx, there exists (see [3], Cot. I, p. 30) a x*
fx}. Let p
P and p(x, fx)
Then p
for each
0
(S,m)
f:
p(Xp, fXp)
If
is nonempty, weakly compact and the family
has finite intersection property.
convex and
0
x
P, then using the implication that
p
follows that
{A
P, let
p
0
a contra-
diction.
The following corollaries result from Theorem 1.
COROLLARY 2.
(E, )
(S, )
either
f: (S, )
(Waters [5]).
(E, )
P
p
and
S
x
satisfying
f:
Then
0
Ten
E
Let
S
(E, t)
be a compact and convex subset of
be a continuous multifunction with convex and weakly compact
P
p
f: (S, m)
follows that
set in
and
there exists a
satisfying (I.I).
S
x
It suffices to show that the hypotheses in Corollary 2 and Corollary 3
imply that
E
(E, t)
be a compact and convex in
be a continuous mutifunction with convex and compact values.
Then for each
PROOF.
n
S
p(fx, S).
COROLLARY 3.
and
Let
has a fixed point or there exists a
f
p(x, fx)
values.
(Reich [I]).
f-I (A)
f-I (A)
then
(E, T)
is
is a continuous multifunction.
T-compact subset of
is weakly closed.
f-I(E\A)
S\f-I (A)
is
S.
S
Since
is u.s.c.
Thus
m-closed and hence
Let
A
r-closed
be
is weakly closed, it
Similarly if
f-I (A)
A
is t-open
is u-open.
Thus
is l.s.c.
In the setting of semi-reflexive locally convex spaces, we have
COROLLARY 4.
Let
locally convex space
S
E.
be a closed, bounded and convex subset of a semi-reflexive
If
f:
(S, 0)
(E, )
closed, bounded and convex values then for each
is continuous multifunction with
p
P
there exists
x
S
satis-
fying (I.I).
*Theorem
|983),
of this paper was presented at the summer meeting of the Amer. Math. Society,
Albany, New York.
V.M. SEHGAL
696
REFERENCES
i.
REICH, S.
Fixed point theorems in locally convex spaces, Math. Z. 125
(1972),
pp. 17-31.
2.
KY FAN
3.
(1969), pp. 234-240.
ROBERTSON, A. P., and W. J.
(1966).
4.
GLICKSBERG, I. L.
5.
WATERS, C.
Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112
Topological Vector Spaces, Cambridge, University Press
A further generalization of Kakutani’s fixed point theorem with
application to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952),
pp. 170-174.
Ph.D. Thesis (1984), University of Wyoming.