Journal of Applied Mathematics and Stochastic Analysis
8, Number 3, 1995, 261-264
RANDOM FIXED POINTS OF WEAKLY INWARD
OPERATORS IN CONICAL SHELLS
ISMAT BEG
Kuwait University
Department of Mathematics
P.O. Box 5969
Safat 13050,
Kuwait
NASEER SHAHZAD
Quaid-i-Azam University
Department of Mathematics
Islamabad, Pakistan
(Received February, 1995; Revised March, 1995)
ABSTRACT
Conditions for random fixed points of condensing random operators are
obtained and subsequently used to prove random fixed point theorems for weakly
inward operators in conical she]Is.
Key words: Banach Space, Inward Operator, Random Fixed Point.
AMS (MOS)subject dassifications:47H10, 60H25, 47H40, 47H04.
1. Introduction
The fixed point theorems play a very important role in the questions of existence, uniqueness
and successive approximations for various type of equations. The random fixed point theorems
are useful tools for solving various problems in the theory of random equations, which form a part
of random functional analysis. In recent years, there has been exciting interaction between
analysis and probability theory, furnishing a rich source of problems for analysis. Bharucha-Reid
[2] proved the random version of Schauder’s Fixed Pint Theorem. Random fixed point theory has
further developed rapidly in recent years; see e.g. Itoh [4], Sehgal and Waters [8], Sehgal and
Singh [7], Papageorgiou [6], Lin [5], Xu [10], Tan and Yuan [9] and Beg and Shahzad [1]. This
paper is a continuation of these investigations. We have obtained random fixed points of weakly
inward random operators in the shell, a stochastic analogue of the results of Deimling [3].
2. Preliminaries
Let (D, at) be a measurable space with at as a sigma-algebra of subsets of D. Let X be a real
Banach space; a map q:--X is called measurable if for each open subset G of X, q-I(G) E at.
Let S be a nonempty subset of X; a map f: 12 x S-X is called a random operator if for each fixed
x E S, the map f(. ,x): D--+X is measurable. A measurable map :gt--+S is a random fixed point
of the random operator f if f(w, {(w)) (w) for each w e gt.
Let K C X be a cone; that is, K is closed and convex such that AK C K for all A > 0 and
KN(-K)-{0}. Denote {xK: Ilxll <r} by g r and {xK’p< Ilxll <r} by
r for
gp,
Printed in the U.S.A.
()1995 by North Atlantic
Science Publishing Company
261
ISMAT BEG and NASEER SHAHZAD
262
Kr+X
is a-condensing if f is continuous and bounded and a(f(B)) <
< p < r. A map f:
g
for
with
all
B
0,
C r
a(B)
a(B) > where a(B)- inf{d > 0: B can be covered by a finite number
of sets of diameter _< d}. A random operator f:f.x Kr---*X is a-condensing if for each w
f(w,. is a-condensing. An operator f:f x C--,X is said to be weakly inward on a closed convex
subset C of X if for each w f, f(w, x) I c(x) for all x e C, where I c(x denotes the closure of
{x + A(y x)" A _> 0, y C}. In the case C K, it simply becomes x Oh’. The latter, x* E K*
(the dual cone), and x*(x) 0 imply that
some 0
x*(f(w, x)) > 0
(1)
for each w f, where X* is the dual space, K* = {x* e X*:x*(x) >_ 0 on K} and OK is the boundary of g. Suppose f:f x grx satisfies x OK, [[ x [[ _< r, x* K* and x*(x)= 0 imply that
x*(f(w,x)) >_ 0 for each w E f. Let Pr:ggr be the radial projection; that is, Pr(x)= x for
for 1[ z 11 > r. Then for each w e f, f(w,. )o Pr satisfies (1). Since
I[ x 1[ <_ r and Pr(x)
IlrX
II
c(Pr(B)) _< c(B) for all bounded B C K, f(w,.
3. Main Results
o
_
Pr is a-condensing if f is.
-
Theorem 1: Let X be a separable Banach space, K C_ X a cone and f’f Kr---*X an a-condensing random operator, such that
and
x e OK, Il x ll
r,x* e g* and x*(x) O imply that x*(f(w,x)) O for all w
(i)
on
and
and
all
are
f
w
Ax
r
A
E
> l,
(ii) f(w,x)
[[x[[
for
satisfied.
Then f has a random fixed point.
,
Proof: Define a random operator g:f x gr--X by g(w,x)- (f(w,.)o Pr)x. It is clear from
Deimlig [3, proof of Theorem 1] that for any w f, g(w,. is a-condensing and weakly inward
on K r for some r > 0. Further application of Xu [10, Theorem 2] or Tan and Yuan [9, Corollary
[3
2.6] gives the desired result.
Theorem 2: Let X be a separable Banach space, K C X a cone,
ing random operators, such that (i), (ii) (from Theorem 1) and
(iii) there exists p G (0, r) and e K\{O} such that for any
for
are
Then
f
I I
f: x Kr---*X
an a-condens-
P and A > 0
satisfied.
has a random fixed point in K
Proof: Let
Pr be as before and for all w
C(w)
sup{ I f(w, x)I1" e Kr}.
I
We
use (ii) to get a "barrier" at
x l]
P as follows. Let Cn: x [0, r]--,[0, cx) be a continuous
random map such that ,(w, t) 0 for t > p and ,(w, t) 5(w) for t < p- and large n, with a
measurable map 5:(0,)such that 5(w) llll > p/C() for each w. Let f,(w,x)=
/
Evidently, fr, is a-condensing and weakly inward random operator on K. By Theorem 1, there exists a measurable map ,:K such that (,(w)=
Fix w
arbitrarily. We cannot have
Hence,
f(w,,(w)) for each w E
>
(n(w) f(w, (n(w))+ Ca(w, 11 (n(w)II )" By the choice of we cannot have
and we are done if [[(n(w)[[ >p. So assume that
II(n( w)[[ <P for all large n. Since
{Ca( w, ((w)[] )} is bounded and f is a-condensing, we have without loss of generality (n(w)-
.
I I
I
,
p-<
I
I
.
Random Fixed
_Points
of Weakly Inward Operators
f(w,(n(w))-Ae and tn(w)o(W) with I 0(w) II-P
f(w,(o(W)) + Ae and therefore A- 0 by (iii).
in Conical Shells
where A depends on w. Hence,
263
(o(W)E]
.
Theorem 3: Let X be a separable Banach space, K C X a cone such that K 1 is not compact,
random operator satisfying (i) and (ii) of Theorem 1 and
(o.) uc tat f(..)
I111- ad e(0.1) and
f’ KrX a compact
t
()
Then
f
has a random
fixed point
in K p,r"
Proof: Since for any wgt inf{llf(w,x)
not compact, we can find e A such that -Ae
Consider a random operator
o
-P} >0, and A-{xK: ]lxll -1}
f(w, pA) for all A _> 0.
ll’llxll
fn: x KX defined by
(f(w, .)o Pr)x
f n(w,x)
lii
fr
+(’
I
< po
for
fn is compact and a weakly inward random operator, it has a random fixed point n"
We cannot have Po<- IIn( w) ll <P or IIn( w) ll -<P0(1-) by (iv) and
the
I
tn(W, p0)- 0
and
p) for each
as before.
choice of Ca" If, for any w
such that Po-1 < n(w)
I < Po for all large n.
I
I ()p I ()_ I
with
Yo
I I
tn(w,t)- 5(w)
5(w) > Po -4-sup{ I f(, )I1"11
Kr
>_,o
0<Po-< IIII <
for
where tn:f [0, P0]---*[0, cx is a continuous random map,
t <_ P0(1- ), with a measurable map 5: fl(0, o) such that
Since
IIII
for
z)
0
is
p
’() II- p,
I /,,(w)II- I II ()II
pA and 0 < A < 5(w) depending
contradicting the choice of e.
on
Then
By letting
noo we
get
Yo-f(w, Yo)+ Se
with
e
f(w, pA)
> 0,
, hence po0 yields o
for some $o
Pemark 4: Theorem 3 does not hold if K 1 is compact. For a counterexample, see [3].
Acknowledgement
This work is partially supported by Kuwait University Research Grant No. SM-108.
References
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264
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IS]
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ISMAT BEG and NASEER SHAHZAD
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