Internat. J. Math. & Math. Sci.
VOL. ii NO. 3 (1988) 551-560
551
RANDOM FIXED POINTS AND RANDOM DIFFERENTIAL INCLUSIONS
NIKOLAOS S. PAPAGEORGIOU
Unlverslty of Callfornla
1015 Department of Mathematlcs
Davls, California 95616
(Received July 28, 1987
ABSTRACT.
using
and in revised form October 19, 1987)
In this paper, first, we study random best approximations to random sets,
point techniques, obtaining this way stochastic analogues of earlier
fixed
deterministic results by Broder-Petryshyn, KyFan and Reich.
Then we prove two fixed
point theorems for random mmltifunctlons with stochastic domain that satisfy certain
tangential conditions.
Finally we consider a random differential inclusion wth upper
semlcontinuous orlentor field and establish the existence of random solutions.
KEY WORDS AND PHRASES.
Random flxed point,
Random solutions,
Random Differential
Incluslon and Abstract Differential Equations.
1980 AMS CLASSIFICATION CODE.
I.
60H25, 47HI0.
INTRODUCTION.
Random fixed point theorems
are stochastic generalizations of
point theorems and are needed in the study of random equations.
classical fixed
Their study was
initiated by the Prague school of probabillsts, with the works of Hans [I] and Spacek
[2].
Recently the interest in these problems was revived by the survey article of
Bharucha-Reld [3].
Since then,
there has been a lot of activity in this area and
several interesting results have appeared.
In this paper,
approximations
and
we will study
will
derive
random fixed points in connection with random
stochastic
analogues of
some
results
by
Browder-
Petryshyn [4], KyFan [5] and Reich [6].
proved by Engl
We also extend a random fixed point theorem
[7] and finally we prove the existence of a solution for a random
differential inclusion with an upper semlcontlnuous orlentor field, extending this way
an earlier result of the author
[8] (theorem 5.1).
For the corresponding deterministic theory, we refer to the recent books of
Goebel-Relch [9] for fixed points (in connection with the study of the geometry of the
underlying space) and of Aubln-Cellna [I0] for differential ncluslons.
Another nice
wDrk, bringing together the two main mathematical branches considered in this note,
namely fixed point theory and differential equations,
where
an
interesting
approach
to
fixed
point
theory
existence theory of abstract differential equations.
is the paper of Reich
is
presented,
through
[II],
the
N.S. PAPAGEORGIOU
552
PRELIMINARIES
2.
(R,E) be a measurable space and X a separable Banach space.
Let
Throughout this
work, we will be using the following notations:
{Ac_X:
Pf(c)(X)
(convex)}
nonempty, closed
and
nonempty, compact, convex}
Pkc(X) --{At_X:
Pf(X)
K:R
Let
be
UcX open, we have that
_a
We say that K(.) is measur.able, if for all
multlfunction.
[mE:K(m)
K (U)
0
U}E.
It can be shown
see Himmelberg
that the above definition of measurability of K(.) is equivalent to saying that
[I0]
zEX,
any
for
map
the
d(z,K(m))
0
Furthermore, if there exists a complete
inf{Iz-xH :xK(m)}
measurable.
is
E, then the
{(re,x) ExX:xEK(m)}ExB(X),
above two
o-finite measure on
properties are equivalent to saying that GrK
where
Schal [12] and Engl [7], we will say
Following
B(X) is the Borel o-field of X.
and there exists a countable set
is
it
measurable
if
is
that K:
separable,
Pf(X)
It is not difficult to show that if K(.) is measurable
DcX s.t. cl(DO K(m))--K(m).
for all mER,
then K(.) is a
cl(intK(m))
with nonempty, closed values and K()
separable multifunctlon. This is the case for example, when K(.) has closed, convex,
solid values.
Y,Z
Let
be
topological
Hausdorff
spaces
+
G (U)
Pf(X). Recall that (Pf(X),h)
We
Pf(X) and let F:GrK Pf(X).
Hausdorff metric on
Let
and
K:
multlfunction with stochastic domain
K(.), if K(.)
say
that
F(.,.)
compact e.t.c.) on
fixed
point
(.)
Finally, if
will
denote
by
of
is a
S
3.
+
is
a
measurable
o-finite measure on
G
the
{gLI(X):g()G()-a.e.}.
and only if
xeX
and
is said to be u.s.c.
(continuous,
u.s.c.
Maps with stochastic domain were introduced by Engl [7].
K(m).
F(.,.)
mE, x(m)K(m) and x(0)EF(,x(m)).
we
a
random
a
is
is measurable and for all
Such an F(.,.)
{m:xK(m), F(,x)0U}E.
F(m,.) is
(continuous, compact e.t.c.), if for all ER,
random
be
is a complete metric space.
UcX open, we have
A
2Z{}
G:Y
let
We say that G(.) is upper semlcontlnuous (u.s.c.), if for all Uc._Z
Also by h( ,.) we will denote the
{yY:G(y)c_U} is open in Y.
multlfunction.
open,
two
inf{llx :xEG()
set
of
l
x:
map
and
integrable
G:R
X
s.t.
Pf(X)
selectors
for
all
is measurable,
of
G(.)
i.e.
It is easy to check that this set is nonempty if
belongs in
L+
RANDOM APPROXIMATIONS AND RANDOM FIXED POINTS.
We will start with a random version of proposition 2.3 of Reich [6], which in
KyFan [5] (theorem 2).
turn was an extension of an earlier very interesting result of
In this section
(R,E,)
is a complete
o-finite measure space.
Also recall
It is
llf(x)-f(y)l
llx-yH for all x, yEX.
on
a
Goebel-Reich
the
[9])
that
metric
(see
projection
for
known
closed,
example
well
convex set in a Hilbert space, is nonexpanslve. That’s why in theorems 3.1, 3.2, 3.3
that a map f:X
X is nonexpansive, if
553
RANDOM FIXED POINTS AND RANDOM DIFFERENTIAL INCLUSIONS
and 3.4, that follow and involve the metric projection (either in their statement or
proof), we assume that the ambient space is a Hilbert space.
If X is a separable Hilbert space, K:
Pfc(X)
in their
is a separable
THEOREM 3.1.
X is a random, nonexpanslve map, with stochastic domain
f:GrK
multlfunction and
, s,
K(.) s.t. for all
f(,K())
Then there exists
is bounded.
x:
X
x()K() and llx()-f()ll- d(f(0,x()),K()).
From theorem 3.4 of [13], we know that there exists
s.t. for all
PROOF.
Caratheodory extension of
IGrK
continuous and
f(.,.)
f)"
Let
We have already mentioned that
(,x)
(i.e.
p():X
p()(.)
K(t)
is measurable,
measurable
:xX
X
(,x)
x
also easy
is
K().
de the metric projection on
is nonexpanslve and it is
a
to show
Let
p(t0)(z) is
measurable.
zsX,
C(t0) =conv D (pof)(,y),
where D is the
yeD
C() is a
Hence
countable set postulated from the separability of K(.).
is
and
C(t0)
(pof)(,.):C()
measurable
multlfunction. For
every
[14]),
(see
every
for
that
conv(pof)(,K(t)).
C()
Note
that
,
So from Broler [15], we know that it has a fixed point.
nonexpanslve.
multlfunctlon L:
Pf(X)
[xC(m):(pof) (,x)--x}
L(0)
{xc()
(po) (,x)--x}
{(m,x)exX: (po) (,x)
GtL
(pof) (,x)
(,x)
But
x} N GrC
is measurable in
Applying theorem 3 of Salnt-Beuve [16], we get
e.
0
Hence it is
and continuous in x.
Also since C(.) is measurable,
jointly measurable.
for all
Consider the
defined by:
x:
GrCEExB(X).
X
GrLEExB(X).
Thus
measurable s.t.
x()eL(m)
x()eK(m) and llx()-f(,x(m))l] --d(f(,x()),K(m))
If K(.) is bounded values, the assumption of the range of f(,.) can
Therefore we have:
REMARK I.
be dropped.
REMARK 2.
Another result in the direction of theorem 3.1 above with a different
set of hypotheses, can be found in
[17] (theorem 4).
We have a similar result for condensing maps. Recall that f:X X
if it is continuous and for all BcX nonempty,
Y-condenslng,
(B)>O, ((f(B))<Y(B), where "((.)
THEOREM 3.2.
,
f:GrK
is said to be
bounded
s.t.
is the Kuratowskl measure of noncompactness.
If X is separable Hilbert space,
Pfc(X)
K:l
is separable and
X is a random condensing map with stochastic domain K(.) s.t. for all
x: X measurable s.t. for all
Then there exists
f(,K()) is bounded.
+
d(f(.x()),K()).
Ix()-f(,x())
Is the same as in theorem 3.1, using this time the fixed point of Furl-
el x()sK() and
PROOF.
Vignoli [18].
Using theorem 3.1, we can have the following random version of a fixed point due
to Broder-Petryshyn
THEOREM 3.3.
If X is a separable Hilbert space,
bounded values and
K(.) s.t.
[4].
for every
f:GrK
X
xebdK()
K:R
/
Pfc(X)
is separable with
is a random, nonexpansive map with stochastic domain
for which
p(,f(,x))
x,
we have
f(,x)
x.
554
N.S. PAPAGEORGIOU
Then f(..) admits a random fixed point.
x:R
X
d(f(,x()), K()))= Uf(,x())-p(,f(,x))ll.
x()
p(,f(,x())).
If
x(w)ebdK(),
Since the best approximation is unique,
Otherwise we must have that
then by hypothesis x()
f(,x()).
p(,f(,x()))
x(), wen.
f(,x())
f(,x())eK()
REMARK. In the previous theorem, we can instead assume that for all eR f(,.)
Then in the proof we have to use theorem 3.2.
is condensing on K().
(see remark I), we know that there exists
From theorem 3.1
PROOF.
lx()-f(,x())U
measurable s.t.
Now we pass to multlfunctlons and prove the following random fixed point theorem.
THEOREM 3.4.
F:GrK
Pfc(X)
K:R
If X is a separable Hilbert space,
is
h-contlnuous,
an
stochastic domain K(.) s.t. for all
Pfc(X)
random
Y-condenslng,
multlfunctlon
xebdK(), F(,x)
ei% and for
is separable and
0 p
-I
F(,K()) is bounded. Then F(.,.) admits a random fixed point.
PROOF. Let G:RxX
Pfc(X) be the multlfunctlon defined by
From our hypotheses on F(.,.), we see that
F(,p(,x)).
G(,x)
G(,x) is h-contlnuous. Also we claim that G(,.)
measurable, while x
So let BcX be nonempty, bounded, with y(B)>O.
We have
.Y-condenslng.
c__{x}
with
(,x)
and
Y(GC,B))
YCFC,p(,B)))
<
yCPC,B))
C(00)
{x
p(,.)
Then
F(.,),
G(,.):C(0)
clearly
we have
C()
that
cOnvnU
L(m)
F(,xn)
{xeC()
Then consider the multifunction defined by
measurable.
of
theorem
men,
is nonexpanslve.
Note that if
C().
is the countable set postulated from the separability of K(.) and exploiting
convF(,K()).
n n.>l
the h-continuity of
From
is
YCB)
the last inequality being a consequence of the fact that
Let
G(,x)
is
Himmelberg-Port e r-Van
19 ],
Vle ck
we
know
C()
is
xeG(,x)}.
that
for
al 1
{(m,x)exX:d(x,G(,x))
0}0 GrCeZxB(X).
Again
theorem 3 of Saint-Beuve [16], produces a measurable map x: + X s.t. x(m)eL(),
for all R.
Let ()=p(,x()).
Clearly (.) is measurable and ()bdK().
-I
Then x()
F(m,()) -x(m)ep
p (,(m)) and x()eG(,x())
N
(m) x(m)
x(m)eK(m) and x()eF(m,x(m)) i.e. x(.) is the desired
F(,())
L(m)0.
Also note that GrL
-l(,x(m))
random fixed point.
REMARK.
If
there
m dependence
no
is
of
the
data
in
the
previous
theorem
(deterministic case), then we can relax the hypotheses on F(.) and simply assume that
F(.) is closed, Y-condensing and with bounded range. Also in the deterministic case,
the
theorem
can
be
proved
for
general
Banach
spaces,
if
we
assume
approximatively w-compact and F(.) is w-u.s.c., with w-compact range.
analogous
to
the
random
mult ifunction
u.s.c,
theorem.
case
and
in
the
general
[6], which tells us that the
proposition 2.1 of Pelch
and
eventually
apply
the
Banach
space,
we
that
K
is
The proof is
have
to
use
metric projection on K is a w-
Kakutani-KyFa n
Both those deterministic versions of theorem 3.4,
fixed
point
extend theorem 3.3 of
[6].
Note that the second deterministic result that was stated in general
Banach spaces, can not be extended to the random case, since as it was illustrated
Reich
with a counter example in
u.s.c, in
[20],
a multifunction
G(m,x)
x, is not in general jointly measurable.
which is measurable in
,
w-
RANDOM FIXED POINTS AND RANDOM DIFFERENTIAL INCLUSIONS
2Xx {} is
F:K
F(K)
compact, if
l(x,K)={zX:z=x+(y-x) for some
translation
cone"
tangent
the
to
of
the
Recall that a ,altifunction
Also if KcX is convex
is compact.
yX and
>0}.
xK, we define
So l(x,K) is nothing else, but the
"Bouligand
the well known from nonsmooth analysis,
(see Aubin-Ekeland [21]).
to K at x
nonexpansiveness
x of
point
[7].
theorem 8 of Engl
result extends
The next
555
metric
the
projection,
Since we do not need any more the
result
can be
stated
for
general
separable Banach spaces.
K:
THEOREM 3.5.
If X is separable Banach space,
Pfc(X) is separable and
F:GrK
is a compact, u.s.c., random multifunction with stochastic domain
Pfc(X)
PROOF.
H:GrK
.
F(.,.) a random fixed point.
From proposition 5 of Engl [7], we know that there exists
F(,x)
K(.) s.t.
Pfc(X)
l(x,K())
c
a multifunctlon
s.t.
(i)
(m,x)gGrK, H(,x)cF(,x)
for each
(il)
for every
(iii)
H(.,.)
Then let
Then
L:
2
X
m,
H(m.,)
is u.s.c, on
K(m)
is jointly measurable
be defined by
{xeK(m):xeH(,x)}
L(m)
From theorem 3.1 of Reich [6], we know that for all
me,
L()0.
Observe that:
{(m,x)eRxX:xeK(m), xeH(,x)}
CrL
((RxD) N GrH)
proJ
xX
where
D=-{(x,y)eXxX:x=y}. But
note that GrHer.xB(X)xB(X). So
Then using the theorem in section 39.1V of Kuratowski
(xD) N GrHelxB(D).
get that
prOJRxx(RXD)
GrFeExB(X).
Hence
GrLeZxB(X).
x:R X measurable
x(o)sH(to,x(o))cF(o,x(o)).
3 Saint-Beuve [16], we get
x(to)eL(o), osn
4.
[22], we
Once again, through theorem
s.t.
RANDOM DIFFERENTIAL INCLUSIONS.
Let
interval
(,-,)
in
R+,
be a complete probability space,
T=[0,b]
X a finite dimensional Banach space and
a nonempty, closed, bounded
x
o
:
X measurable.
We
consider the following random differential inclusion:
x(m,t)eF(,t,x(,t)) a.e.
x(,0)
for all
wen
x ()
o
By random solution of (*), we understand a stochastic process
absolutely continuous realizations, satisfying (*) a.e. in t, for all
x:xT
X,
with
e.
In this section we present a theorem on the existence of random solutions of (*),
N.S. PAPAGEORGIOU
556
that generalizes theorem 5.1 of [8].
If F :xTxX
Pf (X) is a multifunction s.t.
measurable
is
F(m,t,x)
(m,t,x)
F(,t,x) is u.s.c.
for all (,t)eRxT, x
a(m,t)+b(m,t) xUa.e. for all JER, with a( ,.) ,b( ,.) measurable
Then (*) admits a random solution.
a(,.) ,b(
THEOREM 4. I.
(2)
(3)
and
IF(,t,x) l
.)L
We will by determining an a priori bound for the random solutions of
PROOF.
(*)
So let x(.,.) be a random solution.
meR,
Fixing
we have:
Applying Gronwall’s inequality, we get that:
(11 x ()11 +ila(,.)II
o
IIx(,tll
l)
exp lib(w,.)II
F(m,t,x)
Set
(m,t,x)
(.,.,.)
It is easy to check that
IIxII
if
(.,.)
random orientor field
(,.)L+I.
measurable and
(
>
M()
has the same measurability and semicontinuity
properties as F(.,.,.) and furthermore we have that
Let
M()
if Ilxll
M()x)
F(m,t,-
--(,t), with
M()
l(,t,x) la(m,t)+b(m,t)M(m)
We will consider (*) with the
).
W(m)cC(T,X) be defined by:
t
W(m)
Define
g(s)ds, tT, llg(t)ll (m,t) a.e.}
x () +
{xC(T,X):x(t)
o
:xC(T,X)xLI(x)
0
C(T,X) by:
t
(,x,g)(t)
Clearly
measurable.
g(s)ds-x(t)
x ()+
o
0
(.,.,.)
is measurable
Also if
(m)=B(O, (,.)III) is the closed ball
ll(m,.)lll, then (.) is measurable and
the origin with radius
GrW
((,x)exC(T,X):
GRWgr-xB(C(T,X)),
in
and continuous in (x,g).
ll(m,x,g)ll
0,
in
d(g,())=0}
(since g=x, see Kuratowski [22], 39.1V).
So is jointly
LI(X),
centered at
557
RANDOM FIXED POINTS AND RANDOM DIFFERENTIAL INCLUSIONS
Furthermore, a simple application of the Arzela-Ascoli theorem, tells us that for
me,
every
W(m) is a compact subset of C(T,X).
Next let
(m,x)
Define
:GrW
2
C(T’X)xLI(X)
[.}
be the multlfunctlon defined by:
{(y,f)eC(T,X)xLl(X):y(t)--x o (m)+
:xC(r,X)xLl(X) //R+ by:
(,x, f)
d(f,S
t
I^F(,
f(s)ds, teT, feS
0
,x(
(m,. ,x(.)
Note that:
d(f,s^
F(,. ,x(.))
inf{ [If-h
heS
F(,.,x(.))
b
inf{
llf(t)-h(t)ll dt:heS
(,. ,x(.))
:ze(0,t,x(t))}dt
0
b
inf{ llf(t)-z
0
b
d(f(t),(m,t,x(t))dt
0
d(z,(m,t,y)) a
(m,t,y)
measurable
and
z
d(z,(,t,y)) is continuous. Hence (w,t,y,z) d(z,(m,t,y)) is measurable.
A/so the evaluation map
e t(x(.))=x(t), is continuous from TxC(T,X) into
(t,x(.))
X.
Hence we deduce that
(m,t,x(.))
d(f(t),F(m,t,x(t))) is measurable from
xTxC(T,X) into
Rewrite (.,.) as follows:
But by hypothesis
(I),
IR+
(,x)
{(y,f)eC(T,X)xLI(x) [(,y,f) ll
P_cC(T,X)xLI(x)
Let
be defined
O,
by P={(y,f):=f}.
first variable is one-to-one, continuous.
So
if
6(,x,f)
for fixed
R(,x)
(m,x),
proj
, R(m,.):W() C(T,X)
W(m).
W(m)c__C( T, X) is compact,
C(T,X)xC(T,X). So let (x
n
it
0}
Then the projection to the
P) N
EZxB(C(T,X))xB(C(T,X)).
GrRer-xB(C(T,X))xB(C(T,X)).
then
Note that
We claim that it is u.s.c.
suffices
,yn)GrR(m
O}
Thus by Kuratowskl [22] (39.1V):
prOJxC(T,X)xC(T,rR=proJxC(T,X)xC(T,X)( (GrW
{(m,x,y,f):y(0)--x (),
o
(,x,f)
.)
to
show
n)l s.t.
that
(Xn,Yn)
To show this, since
GrR(m, .) is closed in
(x,y)
By definition:
t
Yn(t)--Xo()+
of
f (s)ds
tT
f eS
F(w,.
,Xn(.)
From the Dunford-Pettis compactness criterion, we deduce that
sequentially w-compact in
LI(x).
{f
w
n n)l
is
So by passing to a subsequence if necessary, we
N.S. PAPAGEORGIOU
558
may assne that
f
f in L (X).
n
f(t)econv
llmffn(t)} n) I=c
The
inclusion
last
cony
Using theorem 3.1 of [23], we have that:
llm
being a
(.,.,.).
convexity of the values of
F(m,t,Xn(m))
consequence of
So feS
a.e.
_c F(m,t,x(t))
the
upper semlcontinuity and the
a.e.
Also
F(m,. ,x(.))
t
y(t)
x (m)+
f
f(s)ds,
teT
0
(x,y)eGrR(,.)
R(m .) is
Let
L:R
2
L(m)
Since
for
C(T’x)
from W(m) into W(m).
u.s.c,
be defined by:
{xeC(T,X):xeR(m,x), xeW(m)}
fixed
wen,
R(m,.):W(m)/ W(m)
L(m)@,
fixed point theorem, we have that
is
u.s.c.,
for every
wen.
from the Kakutani-KyFan
Then as in the proof of
theorem 3.5, we have:
prOJnxC(T,X)((nxD)
GrL
0GrR)eExB(C(T,X))
Apply theorem 3 of Salnt-Beuve [16], to get
D
{(x,y)eC(T,X)xC(T,X):xffiy}.
wen, r(m)eL(m). Set x(m,t)fr(m) (t).
for all
s.t.
C(T,X)
measurable,
Clearly this is a random solution of (*) with orlentor field F.
But from the
definition of F, we see that
a(m,t)+b(m,t) Ax|a.e., for all wen and as
where
r:n
/
IF(m,t,x) l,
-
in the beginning of the proof, through Gronwall’s inequality, we get that
,M(m)
solution of (*).
(m,t,x(m,t))
AOINOWLEDGEMENT.
The author would llke to express his gratitude to the referee for
|x(m,t)A
F(m,t,x(m,t))
x(.,.) is
the
desired
random
his helpful suggestions and remarks.
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