RANDOM FIXED POINTS OF RANDOM ... - ScienceDirect.com

Nonlinear Analym, Theory, Methods Printed in Great Britain.

RANDOM

& Applicafrons.

Vol. 20, No. 7, pp. X35-847, 1993.

0362-546X/93 $6.00+ .Oil C 1993 Pergamon Press Ltd

FIXED POINTS OF RANDOM MULTIVALUED OPERATORS ON POLISH SPACES? ISMAT BEG and NASEER SHAHZAD

Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan (Received 25 October 1991; received in revised form 17 December 1991; received for publication 2 June 1992) Key words and phrases: Polish space, random multivalued operator, coincidence point.

random fixed point, random

1. INTRODUCTION RANDOM coincidence point theorems and random fixed point theorems are stochastic generalizations of classical coincidence point theorems and classical fixed point theorems. Random fixed point theorems for contraction mappings in Polish spaces were proved by Spacek [l] and Hans (2, 31. For a complete survey, we refer to Bharucha-Reid [4]. Itoh [5] proved several random fixed point theorems and gave their applications to random differential equations in Banach spaces. Recently, Sehgal and Singh [7], Papageorgiou [8] and Lin [9] have proved different stochastic versions of the well-known Schauder’s fixed point theorem. The aim of this paper is to prove various stochastic versions of Banach type fixed point theorems for multivalued operators. Section 2 is aimed at clarifying the terminology to be used and recalling basic definitions and facts. Section 3 deals with random coincidence point theorems for a pair of compatible random multivalued operators. The structure of common random fixed points of these operators is also studied. In Section 4, the existence of a common random fixed point of two random multivalued operators satisfying the Meir-Keeler type condition in Polish spaces is proved. Section 5 contains a random fixed point theorem for a pair of locally contractive random multivalued operators in .s-chainable Polish spaces. As an application, a theorem on random approximation is also obtained. 2. PRELIMINARIES

Throughout this paper, let (X, d) be a Polish space, that is, a separable complete metric space, and (Q, a) be a measurable space. Let 2’ be the family of all subsets of X, K(X), a family of all compact subsets of X and CB(X) denote the family of all nonempty bounded closed subsets of X. Let T: X + CB(X) be a mapping and (x,1 a sequence in X. The sequence lx,) is said to be asymptotically T-regular if d(x,, , TX,) -+ 0. Let f: X + X be a mapping such that T(X) Gf(X). The sequence lx,) is called asymptotically T-regular with respect to f if d(fx,, TX,) --f 0 (cf. [IO]). A mapping T: L2 --f 2’ is called measurable if for any open subset C of X, T-‘(C) = {o E !A: T(o) fl C # 4) E a. This type of measurability is usually called weakly measurable (cf. Himmelberg [l l]), but in this paper since we only use this type of measurability, we therefore omit the term “weakly” for simplicity. A mapping [: Q --t X is said to be a measurable selector of a measurable mapping T: R + 2” if c is measurable and t This work was supported by NSRDB grant No. M.Sc.Sc.(S)/QAU/90. 835

I. BEG and

836

N. SHAHZAD

for any o E Q c(o) E T(o). A mappingf: Q x X --t X is called a random operator if for any x E X,f( *, x) is measurable. A mapping T: Sz x X -+ CB(X) is a random multivalued operator if for every x E X, T(. , x) is measurable. A measurable mapping c: Sz -+ X is called a random fixed point of a random multivalued operator T: Cl x X 4 CB(X) (f: Szx X + X) if for every o E Sz, c(w) E T(o, C(o)) (f(o,[(co)) = C(w)). A measurable mapping [: Sz + X is a random coincidence point of T: &I x X -+ CB(X) and f:C2x X + X if for every w E Sz,

f(w,C(o)) E T(o,C(o)).

Mappings T: X -+ CB(X), f:X + X are compatible if, whenever there is a sequence (x,) in X satisfying lim fx, E lim TX,, (provided lim fx,, exists in X and lim TX, exists in CB(X)), n-m n-4, n-m n-05 then lim H(fTx,, TfxJ = 0, where H is a Hausdorff metric on CB(X) induced by the metric n-c= d. For details we refer to Jungck [12] and Beg and Azam [ 131. Random operators f: &2x X + X and T: Cl x X -+ CB(X) are compatible if f (0, *) and T(o, -) are compatible for each w E Q. A random operator f: Sz x X -+ X is called continuous if for each w E Q, f(cu, a) is continuous. For every B > 0, the measurable mapping k(* , p): Q -+ (0, 1) is said to have property (P): if for t > 0, there exists measurable maps a(*, t): L2 --t (0, oo), S(*, t): Cl + (0, 1) such that for every w E $2, 0 I r - t < &co, t) implies k(o, r) I S(o, t). We shall require the following well known facts. LEMMA 2.1. (Nadler

[14].) Let (A,] be a sequence in CB(X) and lim H&4,, A) = 0 for n-m A E CB(x). If x, E A, and lim d(x, , x) = 0 then x E A. n-m

LEMMA2.2. (Itoh [5].) Let F, G: Sz + CB(X) be measurable mappings, [: Sz + X be a measurable selector of F, r: Sz + (0,~) a measurable function, then there exists a measurable selector II: &2-+ X of G such that for any cc)E Q d(t;(o), q(o)) I H(F(o), G(o)) + r(u). The proof of the following lemma follows immediately from lemma 2.2. LEMMA2.3. Let F, G: 0 --f CB(X) be two measurable mappings, 4’:Q -+ X a measurable selector of F with H(F(w), G(o)) < r*(o), where r *: CJ + (0, 1) a measurable function, then there exists a measurable selector q: Q -, X of G such that d(c(u), v(o)) 5 r*(o). 3. RANDOM COINCIDENCE POINTS OF COMPATIBLE RANDOM MULTIVALUED OPERATORS Jungck [12] gave the notion of compatible single valued mappings. Subsequently, Beg and Azam [13] defined the compatible multivalued mappings and studied many fixed point and coincidence point theorems for generalized contractive type multivalued mappings. In this section we give a stochastic version of results of Beg and Azam [ 131. THEOREM 3.1. Let T: Sz x X -+ CB(X) be a random multivalued operator and let f: fi x X + X be a continuous random operator such that T(w, X) C f(o, X) for every

Random fixed points

o E a. If f and T are compatible H(T(w,

837

and for all x, y E X and o E Q

x), T(w, Y)) < @a, d(f(w,

x),f(o,

_v))Mf(~

x),f(w,

Y))

(1)

where for every r > 0, the measurable function k(. , r): L2 + (0, 1) has property (P). Then there exists a sequence (in(~)) of measurable mappings which is asymptotically T(w, *)-regular with respect tof(o, *) andf(a, &(o)) converges to a random coincidence point off and T.

Proof. Let co: 0 + X be an arbitrary measurable mapping and yO(w> = f(w, i,,(w)>. Let c, : Q -+ X be a measurable mapping such that yr : L2 -+ X defined by y,(o) = f(o, T,(o)) E T(cu, &(o)). Indeed, because of inequality (l), for every u E X, the map (cu, x) --t d(u, T(o, x)) is a Caratheodory function (that is, measurable in o E Q, continuous in x E X). Thus it is jointly measurable. Hence since co : C2+ X is measurable w -+ d(v, T(o, &(a))) is measurable, therefore, cu + T(o, &,(cu)) is measurable (see Himmelberg [ll]). By the KuratowskiRyll-Nardzewski selection theorem [15], there exists a measurable map yr : Q + X such that Y,(O)

E Vu,

h,(o)).

Now inequality W(T(w,

MO)),

(1) implies

that

T(w, i,(o)))

< k(o,

cW-(o,

io(o)),f(w,

c,(w))))W(o,

0, there exists a natural number N such that whenever n > N. Hence k(u, d(y,(u),y,+,(u))) I 0 5 d(Y,,(w),Y,,+,(w)) - t < 4u, I), S(u, t), whenever n > N. Let o(u)

= maxMu,

~(Y,(u),Y,(u))),

Mu,

d(y,(u),~Au))),

. . . , Mu,

d(y,,-

,(u),Y,~(u))),

S(u,

[)I,

838

I.BEG and N. SHAHZAD

for each w E Sz. For n = 1,2, . . . &Y,(o),

Y,,lkJ))

+Oasn+co which contradicts implies

the assumption

that t > 0. Consequently

lim d(y,(o),y,+,(o)) = 0, which n+m = 0. Therefore the sequence l 0, there exist measurable mappings 6( *, t*): L2 -+ (0, co), S(*, t*): L2 + (0, 1) such that 0 i r - t* < 6(0, t*) implies k(o, r) 5 S(o, t*). For 6(w, t*) > 0, there exists a natural number N, such that i 1 N, implies 0 5 d(y,&o),y,(,,(w)) - t* < &co, t*). Hence k(w, d(Y,(i,(o), Y,(i,(W))) 5 S(W, t*) for i 2 No. Thus for each CC)E Q d(Y,(;j(w)3 Ym(i)C”)) i d(Yn(i)(“),

Yn(i)+l(O))

+ d(Yn(i)+l(“)9 Ym(i)+l(“))

+ d(Ym(i)+l(“)9 Ym(i)CO)) 5 d(Yn(i)(“)7 Yn(i)+ l(W)) + ho, x d(Yn(i)(“),

d(Yn(i)(“)9 Ym(i)Cw)))

Ym(i)(W)) + d(Ym(i)+l(w)9 Y*(i)(w))

s d(Yn(i)(“)7 Y,(i)+ lCw)) + s(03 t *V(Yn(i)Ccu), Ym(i)CW)) + d(Ym(i)+l(w),

Ym(i)Cw)).

Letting i --f 00, we get t* 5 S(w, t*)t* < t*, a contradiction. Thus lf(w, [,(o))] is a Cauchy sequence. By completeness of the space, there exists y(o) E X such that for each w E a d(Y,(o), y(w)) -+ 0 as n + 43. Continuity off implies that d(S(o, y,(w)),f(o, y(o))) + 0. It further implies that H(T(w,

Y,(W)), V%

Y(O))) < Mm, a_eo, < d(f(w,

Yn(o)),f(w

Yn(o)),.l-(o,

Inequality (1) and the fact that (f(o, &(w))) A(o) E CB(x) such that T(w, c,(w)) * A(o).

?4~))))w-(~,

Yn(~)),f(w,

Y(W)))

Y(O))) -+ 0.

is a Cauchy sequence imply that there exists (By Itoh [5, proposition 11, A is measurable.)

839

Random fixed points

Furthermore

for each o E Q W(o),

A(o)) 5 lim WW, n+m

i,-i(a)),

VW Ma)))

= 0.

Now W-(a,

Y,+i(W)), T(o, Y,(O))) 5 H(f(o,

T(a, r,(w))), T(%f(%

MO)))).

Letting n + 00, we obtain d(f(w, y(o)), T(o,y(o))) = 0. Hence f(~, y(o)) E T(o, y(o)) for each cc)E a. (The mapping y: Q -+ X is the pointwise limit of measurable mappings (u,), hence measurable.) THEOREM 3.2. If in addition to the hypothesis f: Q x X -+ X satisfies, for all x, y E X

W-(w, x),f(o,

of theorem

3.1, the random

operator

u))

5 ~4~1 max 4-c ~1, G,f(w

41, KYJ(w

Y)>,

d(Xf(W

Y)) + W,f(W 2

4)

,

(2)

1

( where ,u: Q --f (0, 1) is a measurable mapping, then there exists a common random fixed point off and T. Proof. Let R: Sz + 2” be the multifunction defined by R(o) = (x E X: x = f(w, x), x E T(o, x)]. From the deterministic theory (see [13, corollary 5.3]), it follows that for every w E Q, R(o) # q5 and because of the continuity properties of f(o, *) and T(o, *), R( *) is closed valued. Set R,(o) = {x E X: f(w, x) = x) and R2(u) = (x E X: x E T(o, x)). Then R,(e) and R2( 0) of theorem in Itoh [6]). f and T have a common random

fixed point. Remark 3.3. If all the hypothesis of theorem 3.1 are satisfied then as in the proof of theorem 3.1 there is a random coincidence point t,(w) off and T. Now, consider a constant sequence of measurable maps v,, : Q + X as follows VJW) = to(w)

for each w E a, then lim f(o,

rln(o)) = f(o,

n-m

By compatibility HWo,

off

&(o))

E T(w h(w))

= h T(w r,(w)). n+m

and T

T(w, b(w))), T&f

(0, t,(w))))

= lim Wf (a, T(w v,(cJJ))), T(o, f (w v,(a)))) n-m

Hence f (0, G,(o))~ = f (o,f

(0, b(o)))

E f (0, T(w t,(o)))

= T(o,f

(a, h(w))).

Choose another constant sequence of measurable mappings V,(o) = f(o, lim f(w, V,(w))* = f(w, b(~))*

n-m

E T(wf(w

&(a)))

t,(o)).

= lim T(o, n- m

and H(f(o,

T(o,f(o,

= J$_H(f

&do)))), T(w,f(o, (w, T(w, K(o))),

&Ad)*))

T(mu,f (0, Mco))))

= 0.

Then

I/n(o)),

= 0.

840

I.B~cand N.SHAHZAD

Thus for o E a, f(w9 to(o))3 =

f(o, f(o, to(W))2) E f(% T(w,f(o,

= VW,f(Ol Consequently,

MO))))

to(o))2).

we have

f(w 4h)r+ 1E T(w,f(o, b(~N”)

for any w E Q.

In theorem 3.1 our hypothesis that f is continuous implies that T is continuous. We used the continuity off and T in our proof. In the next theorem we show that if f(o, X) is complete then the continuity and compatibility off and T are not required. THEOREM 3.4. Let T: Q x X + CB(X) be a random multivalued f: Q x X -+ X be a random operator such that T(o, X) C f(o, X), f(o, each o E Q and the condition (1) is satisfied. Then: (i) there exists a sequence (i,,(o)) (where each [, : Sl --* X is a measurable asymptotically T(o, *)-regular with respect to f(w, *); and (ii) f and T have a random coincidence point.

Proof. Examining the proof of theorem f(o, X) allows us to obtain a measurable f(w, z(o)), for each o E Q. Now d(f(o,

z(o)),

operator and X) is complete mapping)

let for

which is

3.1 the only change is that the completeness of map z: Q -+ X such that f(w, [,,(a)) -+ y(o) =

T(w, z(w)))

5 d(f(o,

z(u)),f(a,

in+~(~)))

+ d(f(w,

in+,(~)),

s d(f(o,

z(o)),f(w,

inil(~

+ MT(w,

L(O)),

< d(f(w,

z(w)),f(o,

in+,(m)))

+ k(o, d(f(o,

T(o, z(w))) T(o, z(w)))

MW)),f(O,

z(0))))

x d(f (0, in(W)), f (0, z(w))) < d(f(QJ7 dw)),f(ot Letting

in+,(o)))

+ d(f(w,

iJO)),f(O,

z(o))).

n 4 co, we obtain d(f(a,

z(o)),

T(w, z(w))) 5 d(f(w,

Hence f(cu, z(w)) E T(o, z(o))

z(w)), Y(O)) + d(y(o),f(w,

z(o)))

= 0.

for o E a.

COROLLARY 3.5. Suppose that, in addition to the hypothesis of theorem 3.3, f satisfies (2) and f and T are compatible. Then (f(a,&(u))) converges to a random coincidence point (say y(o)) off and T, and (f (w, y(o))” ] converges to a common random fixed point off and T.

Proof. By theorem 3.4, there exists z: Q -+ X such that f(o, As in remark 3.3, compatibility off and T implies that f(w>f(o,

z(o)))

~f(a,

T(a, z(o)))

z(o))

= T(o>f(w,

E T(co, z(w)) for w E Q.

z(m))).

Random

fixed points

841

Sincef(w, c,(w)) -+ f(w, z(o)) (see theorem 3.4). Thereforef(w, c,(o)) converges to a random coincidence point off and T. Now inequality (2) implies that (f(o, z(o))“] is a Cauchy sequence. Letf(o, z(o))” -+ c*(o). Since (as in remark 3.3) f(o, z(o))“+’ E T(o,f(o, z(o))“). Therefore for any o E Q, d(c*(a),

Letting

T(w, c*(o)))

5 d(c*(o),f(o,

z(o~))“+‘) + WWA.W,

< d(r*(o),f(o,

z(o))““)

+ k(w, W(W

< d(r*(o),f(oA

z(o))“+‘)

+ d(f(w,

z(w))“), T(o, z(o))“,

z(o))“,

c*(w)))

r*(o)))d(f(w,

z(o))“,

r*(w))

r*(o)).

n 4 00, we obtain d(c*(o),

Moreover,

i.e. c*(w) E T(o,

T*(w))) = 0

c*(o)).

for w E Q

ac*(u),f(u,

TX(u)))

5

ar*(u),f(u,

5

ai* (0) +

T(w

I

z(o))“+l)+ d(f(w, z(w))“+‘,f(w,(-*(co)))

f(o , z(w))“+9

Au) max W(o, z(o))“, C*(o)), d(f(~, z(u))",~(u,

z(u))"+~), d(~*(~~),f(~,

a-(% Z(~)Y,fh

i*(o))) + d(f(o, z(o))““, c*(w)) 2

Letting

pyw))),

i

1.

n + co, we get

Hence c*(o)

= f(o,

i*(o))

4. COMMON

for each o E s2. RANDOM

FIXED RANDOM

POINTS

OF

MEIR-KEELER

TYPE

OPERATORS

In 1969 Meir and Keeler [16] established a fixed point theorem T: X -+ X that satisfies the following condition.

for a single valued

Given E > 0, there exists a 6 > 0 such that E I d(x, y) < E + 6 implies

mapping

d(Tx, Ty) < E.

In 1981 Park and Bae [17] extended it to a pair of commuting single valued mappings. In this section we prove a random fixed point theorem for a hybrid of multivalued mappings satisfying Meir-Keeler type contractive definition. THEOREM

operator

4.1. Let X be a Polish space and f: ~2 x X + X a continuous

and T: L? x X --t CB(X) a random multivalued random operator such that T(o, X) E f(w, X)

842

I. BEG and N. SHAHZAD

for every o E Q. If f and T are compatible

and the following

condition

is satisfied;

for E > 0 there exists a 6 > 0 such that &scl(f(~, x),f(~,y)) < E + 6 implies d(u, u) < E, u E T(o, x), u E T(o, Y) and T(w, x) = T(co, Y) when f(w, x) = f(o, y) for each o E Sz; then there is an asymptotically T(o, have a common random fixed point.

*)-regular

sequence

with respect

to f(o,

a) and T and f

Proof. Let &, : Q -+ X be an arbitrary measurable mapping. Consider the following of measurable mappings in: fi + X, y,: Q -+ X and A,: Q + CB(X) such that y,(o)

= f(o,

in(~))

E T(o,

L+,(o))

(3)

sequence

= A,-,(o)

for any o E fi and n > 0. Then for each E > 0 there exists a 6 > 0 such that F < d(f(o, L(c0)),f(u, L(W))) < E + 6 implies d(f(QJ7 im+l(~)),f(w r,+,(~))) < E. It follows that d(y,(o),y,+,(o)) < d(y,_,(o),y,(w)) for each o E 0. Thus the sequence [d(y,(o), y,+t(o))l is decreasing and converges to the greatest lower bound which we denote by r. Now r L 0, in fact r = 0. If otherwise r > 0. Pick N so that n 2 N implies r s d(y,(o), Y,+~(o)) -=cr + 6. It further implies that d(y,+,(o), ~~+~(a)) < r which is a contradiction to the fact that r = inf d(y,(o), y,+,(o)). Thus for each w E Q n d(f(w,

in(~)), VW, MO)))

5 d(f(o,

i,@)),f(w

in+,(~)))

+ 0

as n + 00. Therefore 1(,(w)] is asymptotically T(w, *)-regular with respect to f(w, 0). To show that (y,(o)) is a Cauchy sequence. Suppose that d(y,(o), y,, ,(w)) = 0 for some n > 0. Then d(Y,,@),

for all m > n.

Ynzc1(m)) = 0

Otherwise d(y,(o), y,+,(w)) = 0 < d(y,+,(w),y,+,(~)), a contradiction. Hence (y,(o)) is a Cauchy sequence. In the case that d(y,(o),y,+,(cu)) # 0 for each n. Define E’ = 2~ and choose 6, 0 < 6 < E 4 0, there exists an integer N such that such that (3) is satisfied. Since d(y,(w), y,+,(o)) d(yj(w),yi+,(o)) < 6/6 for i 2 N. Let q > p > N then d(y,(o), y,(w)) 5 E’. Indeed, assume that d(Yp(o),Y,(w))

We first show that there exists an integer

> 2E = E’.

m > p such that,

s + 4 < aYp(o),Y,,,(P)) with p and m of opposite

parity.

(4)

Let k be the smallest

< E+ d

integer

greater

(5) than p such that

d(Yp(o),Y,W))

> & + ;.

(6)

d(Yp(w),Y,(o))

< E + 7.

(7)

Moreover,

Random fixed points

843

For otherwise, e + 7

5 d(Y#),Y,-,(a))

Since k - 1 2 p 2 N, therefore

which is a contradiction

+ d(Yk-l(O),

Y/JO)).

d(Yk_,(~),Yk(m))

< 6/6, it implies

d(Y,(o),Y,-,(a))

> e + ; 3

to the fact that k is the smallest & +

;

d(y,(o),


0 and il: 0 + (0, 1) is a measurable map) if x,y E X and d(x, Y) < E, then HUW, 4, T(o, Y)) 5 4o)dk Y).

Random fixed points

845

A metric space (X, d) is said to be c-chainable if and only if given x, y in X, there is an a-chain from x to y (i.e. a finite set of points x = zO, zl, . . . , Z, = JJ such that d(Zj_l , Zj) < E for j= 1,2,3 ,..., n). THEOREM 5.1.Let X be a c-chainable Polish space and Ti : Q x X -+ CB(X), Tz : Cl x X --+CB(X)

be two random multivalued operators that satisfy the following condition: for each x, y E X and o E 52 0 < d(x, y) < E

implies H(T(w, x), q(o4 v)) < k(o, 0,

y))d(x, Y),

(10)

for i, j = 1,2 and where for every 0 c /3 < E, k( *, /3): Sz + (0, 1) is a measurable mapping satisfying property (P). Then there exists a common random fixed point of q and T,. Proof. Let R: a + 2” be the multifunction

defined by

R(w) = (x E X: x E T&D, x), x E T&J, x)), then by Beg and Azam [19], R(o) is nonempty for any w E Q. Because of the continuity properties of q(o, *) and G(u, -), R(e) is closed valued. Set R,(o) = fx E X: x E T,(w, x)) and R,(w) = (x E X: x E T2(w, x)1. Then RI(-) and R2(*) are measurable (see the proof of theorem 3.1 in Itoh [6]). And hence R(e) is measurable. The Kuratowski-Ryll-Nardzewski selection theorem [15] further implies that T, and T, have a common random fixed point. COROLLARY 5.2. Let X be a Polish space. Suppose Ti : &I x X -+ CB(X), T, : Q x X -+ CB(X)

are two random multivalued operators such that for all x, y in X and u E Q, H(T(:(o, x), qo,

Y)) c Ho, d(x, _Y)V(x,Y)

for i,j = 1,2,

where for each 0 < /I < 00, k(*, p): CJ --) (0, 1) is a measurable mapping having property (P). Then Tl and T, have a common random fixed point. Remark 5.1 and affecting Let X define

5.3. The condition that H(T(o,x), Tj(w,r)) < k(u, d(x,y))d(x,y) as stated in theorem corollary 5.2 can be replaced by H(T(o, x), T(w,y)) < k(w, d(x,y))d(x,y) without the validity of theorem 5.1. be a real normed space and G a nonempty subset of X. For a bounded set A c X,

rado(A) = ii:

=p~ 11~- gll,

and

Cent,(A)

= (go E G: ;tii ][a - g,(/ = rado(A)).

The number rad,(A) is called the Chebyshev radius of A with respect to G and an element g, E Cent,(A) is called a best simultaneous approximation of A with respect to G. When A is a singleton, say A = (xl, x E X, then rad,(A) is the distance of x to G and Cent,(A) is the set of all best approximation of x out of G. For further details we refer to Milman [20]. THEOREM 5.4. Let X be a separable Banach space and T, S: Q x X + CB(X) be two random multivalued operators. Let G E CB(X). For any bounded set A of X, if Cent,(A) is nonempty, compact, star-shaped and: (i) T(w, x) C CentG(A) and S(w, x) c Cent,(A), whenever x E Cento(A) and o E Q; (ii) T, S are continuous on Cent,(A); and

I. BEG and N. SHAHZAD

846

(iii) 11x- y\( < H(A, G) implies H(T(w, w E Q, then Cent,(A) contains a common Proof. Let p be the starcentre measurable mappings and k,(o)

x), T(w, u)) < 11x- y(] for all x, y in Cent,(A) random fixed point of S and T.

of Centc(A). -+ 1. Define

S, , T, : L2 x Cent,

Let (k,), where k,: Q + (0, l), be a sequence random operators (A) + CB(Cent,

and

of

(A))

as follows T,(o,x)

= (1 - k,(w))p

+ k,,(o)T(w,x)

and S,,(O, x) = (1 - k,(w))p

+ k,(o)S(o,

x)

for each w E 0

and x E Cent,(A). Obviously H(T,(o, x), &(w, y)) < k,(w)Ilx - yll for all 1(x - yll < H(A, G). It follows by theorem 5.1 that T, and S, have a common random fixed point in(a). For each n, define F,: Q + K(Cent,(A)) by F,(w) = Cl(&(o): i L n) (where Cl denotes the closure). Define G: Q + K(Cent,(A)) by G(w) = n;= 1F,(o), then G is measurable by Himmelberg [l 11. Thus there exists a measurable selector c of G. This selector [ is a desired common fixed point of T and S. Indeed, assume that {[,(o>] converges to l(o). For each n, there exist yn E T(w, [JO)) and z, E S(co,[,(w)) such that i,(w) = (1 - k,(o))p + K,(o)y, and in(w) = (1 - k,(o))p + K,(o)z,. It implies that (z,) and (yn] both converges to c(o) and since S and T are continuous, it follows that c(o) E T(w, c(o)) and i(w) E S(w, c(w)).

6. CONCLUDING

REMARKS

If we assume that Q. is a Souslin family (this is the case if for example, there is a a-finite measure defined on @, with respect to which Q. is complete). Then the existing deterministic results, together with a selection argument can give results presented in this paper. This point is further illustrated by the following observation: if F,(co, x), F2(co, x) are continuous random multifunctions which for every o E a, have a common fixed point, then F, and F, have a common random fixed point. Indeed, let R: f2 + 2” be the multifunction defined by R(w)

Because Note that

of the deterministic

= ix E X: x E F,(m, x), x E F2(o, x)).

solvability

hypothesis,

GrR = ((co, x) E Q x X: d(x, F,(o,

we have that for all w E 0, R(w) # 4.

x)) = 0, d(x, F,(o,

x)) = 0).

But (0, x) -+ d(x, F,(o, x)), d(x, F2(w, x)) are Caratheodory functions, thus jointly measurable. Hence GrR E C? x B(X) (B(X) = Bore1 o-field of X). Aumann’s selection theorem (see theorem 5.10 of Wagner [21]) implies that FL and F2 have a common random fixed point.

Acknowledgement-The authors are grateful to the referee for a thorough perusal of the paper and criticism of some details of its presentation. Many of the suggestions have been followed, which led to the straightening of some arguments and the elaboration of others.

Random

fixed points

847

REFERENCES 1. SPACEK A., Zufsllige Gleichungen, Czech. Math. J. 5, 462-466 (1955). 2. HANS O., Reduzierende Zuflllige Transformationen, Czech. Math. J. 7, 154-158 (1957). 3. HANS O., Random operator equations. Proc. 4th Berkeley Symp. Mathematical Statistics and Probability, Vol. II, Part I, pp. 185-202. University of California Press, Berkeley (1961). 4. BHARUCHA-REID A. T., Random Integral Equations, Mathematics in Science and Engineering, Vol. 96. Academic Press, New York (1972). 5. ITOH S., A random fixed point theorem for a multivalued contraction mapping, Pacif. J. math. 68, 85-90 (1977). 6. ITOH S., Random fixed point theorems with an application to random differential equations in Banach spaces, J. math. Analysis Applic. 67, 261-273 (1979). 7. SEHGAL V. M. & SINGH S. P., On random approximations and a random fixed point theorem for set-valued mappings, Proc. Am. math. Sot. 95, 91-94 (1985). 8. PAPAGEORGIOU N. S., Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Am. marh. Sot. 97, 507-514 (1986). 9. LIN T. C., Random approximations and random fixed point theorems for non-self-maps, Proc. Am. math. Sot. 103, 1129-1135 (1988). 10. RHOADES B. E., SESSA S., KHAN M. S. & SWALEH M., On fixed points of asymptotically regular mappings, J. Austral. math. Sot. (Ser. A) 43, 328-346 (1987). Il. HIMMELBERCC. J., Measurable relations, Fund. Math. 87, 53-72 (1975). 12. JUNGCK G., Common fixed points for commuting and compatible maps on compacta, Proc. Am. math. Sot. 103(3), 977-983 (1988). 13. BEG I. & AZAM A., Fixed points of asymptotically regular multivalued mappings, J. Austral. math. Sot. (Ser. A) 53(3) (1992). 14. NADLER S. B. JR., Multivalued contraction mappings, Pucif. J. math. 30, 475-480 (1969). 15. KURATOWSKI K. & RYLL-NARDZEWSKI C., A general theorem on selectors, Bull. Acad. polon. Sci. S&r. Sci. math. ustro. phys. 13, 379-403 (1965). 16. MEIR A. & KEELER E., A theorem on contraction mappings, J. math. Analysis Applic. 2, 526-529 (1969). 17. PARK S. & BAE J. S., Extensions of a fixed point theorem of Meir and Keeler, Ark. Math. 19, 223-228 (1981). 18. EDELSTEIN M., An extension of Banach’s contraction principle, Proc. Am. math. Sot. 12, 7-10 (1961). 19. BEG I. & AZAM A., Fixed points of multivalued locally contractive mappings, Boll. C/n. mat. Ital. 4-A(7), 227-233 (1990). 20. MILMAN P. D., On best simultaneous approximation in normed linear spaces, J. Approx. Theory 20, 223-238 (1977). 21. WAGNER D. H., Survey of measurable selection theorems, SIAM J. Control Optim. 15, 859-903 (1977).