Random fixed point theory with applications

Nonlinear

Analysis,

Theory,

Methods

& Applicarionr, Vol. 30. No. 6, pp. 3295-3299, 1997 Proc. 2nd World Congress of Nonlinear Analysts 8 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00

PII: SO362-546X(%)00158-7

RANDOM FIXED POINT THEORY WITH APPLICATIONS DONALO'REGAN Department

Of Mathematics,

University College Galway, Galway, Ireland

Keywords and phrases: Random operators, fixed point 1. INTRODUCTION This paper presents new fixed point theory for random operators. Our results were motivated by ideas in [l, 2, 31. We then apply our theory to obtain a general existence principle for the random operator equation y(t,w) = N(t,y(t,w)) on [O,T]. We now introduce some concepts which will be used throughout this paper. Let (9,A) denote a measurable space. For a metric space (X, d) we denote by CD(X) all nonempty closed subsets of X, CB(X) all nonempty closed bounded subsets of X, K(X) all nonempty compact subsets of X. A multivalued mapping F : R --+ X is called measurable if for every open subset B of X, F-‘(B) = {w E R : F(w) n B # 0} E A (this type of measurability is usually called weakly measurable in the literature [4]). Notice that when F(w) E K(X) for all w E 0 then F is measurable iff F-‘(C) E A for every closed set C of X [4]. A measurable mapping [ : R -+ X is called a measurable selector of a measurable mapping F : fi + CD(X) if t(w) E F(w) for each u) E 0. Let Z be a nonempty closed subset of X. Then a mapping F : 51 x Z -+ X is called a random operator if for every z E Z, the map F( . ,x) : fl + X is measurable . A measurable map [:i2X is called a random fized point of a random operator F : R x Z + X if for every w E 0 we have F(w,e(w)) = t(w). We A single valued mapping F : Z C X + X is called a compact map if F(Z) is precompact. call F a a-Lipschitzian map if there is a constant Ic > 0 with cy(F(Y)) 5 ko(Y) for all bounded sets Y & Z; here o(Y) is the measure of noncompactness of Y i.e. o(Y)

= inf {t > 0 : Y can be covered by a finite number

F is called a condensing map if bounded sets Y C Z with o(Y) (condensing etc.) if for each w E a Banach space, then let CK(Z)

F # 0, be

of sets of diameter

5 c} .

is a-Lipschitzian with E = 1 and a(F(Y)) < o(Y) for all 0. A random operator F : R x Z -+ X is called continuous etc.). If Z is any subset of F(u), . ) is continuous (condensing the family of all nonempty compact convex subsets of Z. 2. FIXED

POINT

THEORY

We present our general fixed point result immediately.

THEOREM 2.1. Suppose E is a separable Banach space (let D be a countable dense subset of E), T : R + R measurable with r(w) > 0 for each w E R, and Q,(,) = {z E E : ]z] 5 T(W)}. Assume

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the random operator F : R x E --f E is such that F(w, .) : Q,(,) every w E R. For each w E R let R, : E + Q,+,,, be the continuous

and suppose

the following = (2 E E :

H(w)

each sequence

1

conditions F(w,

map for

are satisfied:

R,(z))

C D with

{y,}

+ E is a continuous retraction given by

= z)

is nonempty

F(w,R,(y,))

if there exists a measurable for each w E R then d(w)

and precompact

for each

- yn + 0 has a convergent

2u E s2

(2.1)

subsequence

(2.2)

,$ : R -+ E such that 4(u)) = F(w, R,(qG(w))) E Q,(,J for each w E R.

Then F has a random fixed point (i.e. there exists a measurable and F(w,d(w)) = d(w) on 0).

4 : R + E such that

4(w)

(2.3) E QT(,,

Proof. Let G(w,z) = F(w,R,(z)). Notice G(w, .) : E + E is continuous. Fix w E R. From (2.1), H(w) # 0 and H( w ) IS c1ose d since G(w, ) is continuous. Now since H(w) is precompact we have that H(w) is compact valued. We now show G( . , z) is measurable for every 3: E E; to see this notice for any B E L: (Bore1 a-algebra of E) we have {W

: G(w,s)

E B}

=

[(F(.

,x))-‘(B)

U [{w

ft T-‘[lzl,ffi)]

:

F(w.T(w) 6)

E B} fl

‘-I(@

IT/)].

Now since T( . ) is a pointwise limit of step functions we have that G( . , z) is measurable for every z E E. Next we clainl H is measurable. Since H is compact valued it suffices to show [4] that H-‘(A) is measurable for any closed subset A of E. Let D = {Zig}? and look at L(A)

= fi u { w E n : IG(w,z,) 71=1 Z,EA”

- z;I < 2 n >

where

A,=

i

a~E:d(s,A)<-!

111

and

d(z,A) = inf{lr

- y[ : y E A}

A standard arguement (see [2] f or example) yields H-‘(A) = L(A) so H is measurable; for COIIIpleteness we supply the arguement. Suppose w E L(A). Then for each n there exists z2(71) such by (2.2) there is a subsequence that d(z,(,): A) < $ and /G(uI.I,(,,)) - zL(lL)I < $. (‘onsequently of {z2(71) : n = 1,2, ...} converging to some yo E E. Thus yo E A and G(u), yo) = ~0. Hence ‘[u E N-‘(‘4) so L(A) C H-‘(A). On theotherhand suppose w E H-‘(A). Then thereexists x E A with or each n. For fixed R there .c = G(w,z). We will show w E (Jz,EA, { w E R : JG(w,z; )-r;l
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The next result replaces THEOREM

2.2. Assume

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condition

(2.3) in theorem

the conditions

of theorem

if y : 0 + E is measurable { we have y(w) # xF(w,y(w)) Then

I;’ has a random

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2.1 with a condition

of Leray-Schauder

2.1 except (2.3) is replaced

then for any w E R with for every X E (0,l).

type.

by

y(w) E aQ,(,l

(2.3)*

fixed point.

Proof. The result follows from theorem 2.1 once we show (2.3) is true. Suppose there exists a measurable C$: R -+ E such that d(w) = F(w,+(w)) for each w E R. If (2.3) is not true then there exists w1 E R with I~$(wr)l > ~(201) and ~$(wi) = F(w~,R,,(~(w~))). Let U(W) = R,, 04(w). Now 11: R - E is measurable, ~(20,) = X F(wi,v(wr))

with

~(2~1) =

+I

)@(w

)

M(Wl)l This contradicts

(2.3)*.

E aQ,(,,,,,

and

X= w

Thus (2.3) is true so the result follows from theorem

We now discuss conditions

E (O-1)

Ib(w)l

(2.1) and (2.2) for operators

usually

2.1

encountered

in applications.

THEOREM 2.3. Let E be a separable Banach space (let D be a countable dense subset of E), T-:0 -, R measurable with T(W) > 0 for each w E 0, and Q,(,) = {z E E : (z( < r(w)}. Assume the random operator F : R x E + E is such that F(w. .) : Q,(,,l + E is a continuous. bounded. condensing map for every w E R. Let R,, be as in the statement of theorem 2.1 and assume (,2.3) holds. Then F has a random fixed point. Remark.

Condition

(2.3) can be replaced

by condition

(2.3)” in theorem

2.3.

Proof. The result follows from theorem 2.1 once we show (2.1) and (2.2) are satisfied. Fix ~1 E R. Firstly H(w) # 0, see [fi, i’]. Next we show G(w, . ) = F(w, R,,,( .)) is a bounded condensing map. To see this let Z be a bounded subset of E with o(Z) > 0. Then if n(R,(Z)) > 0 we have 4G’(w, since R,( 2) C co(Z U {0}),

whereas

a(~(w,Z)) Thus

G(w, .) is a bounded

2)) = a(F(w,

if o( R,(Z))

= a(F(u!,R,(Z)))

condensing

< 4&(Z))

i a(z)

= 0 WP have 5 4&(Z))

= 0 < a(z).

map. Also since H(w)

we have if cu( H(w))

R,(Z)))

c G(w,H(w)),

# 0, a( H(w))

I cx(G(w, H(w)))

< a(H(w)),

Consequently a contradiction. Thus o( H(w)) = 0 so H(w) is precompact. b e a sequence in D with F(w, To show (2.2) is true fix w E R. Let {yll}y must show {y,} has a convergent subsequence. The proof follows the ideas If we show A is precompact then (2.2) is true. Let un = a(F(w, R,(A))). < ae + 5 such c > 0 there exist sets A,, i = 1>2,..., n of E with diarn(A,) F(w, R,(A))

C_ uyE1 A;.

(2.1) is true. Rw(yn))-yn + 0. We in [6]. Let A = {yn}T. By definition? for any that

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For i = 1, .. .. n let B; be an 5 neighborhood

of Nonlinear

of Ai and choose N E N such that

Iyn - F(w,&,,(y,))l (this is possible since F(w,Rw(yn))-yn for i = 1, .. .. n. Since c is arbitrary a(A) Suppose

a(A)

# 0; then if a(R,,,(A)) 4.4)

whereas

if @(R,(A))

< i

-+ 0). Thus we have

= cr({y,

Analysts

: n 2 N})

for all n > N

{yn : TL 2 N} c

2 a0 = a(F(w,

B; and diam(B;)

Uy==,

< ao+c

R,(A))).

> 0 we have

I 4F(us

&(A)))

< 4&(A))

I a(A)

= 0 then (Y(A) 5 a(F(w,

The above contradiction

implies

a(A)

R,(A)))

5 4&(A))

= 0 < a(A).

= 0 so A is precompact

and (2.2) is true.

3. APPLICATION

In this section we use theorem

2.3 to establish

an existence

principle

for

y(t, w) = N(w, ~(6~1) on 10,Tl.

(3.1)

By a solution to (3.1) we mean a process y : [O,T] x R + Rn (i.e. y is measurable in w for each t E [0, T]) such that y( . ?w) is continuous on [0, T] for each w E R and (3.1) is satisfied on [0, T] x R. We assume the following conditions are satisfied throughout this section: there exists a measurable T : R -+ R with r(w) > 0 for each w E R, also suppose for each fixed u’ E R that ]y(u))]o = ~up~el~,~l (y(t,w)] # T(W) for any solution y(t, UJ) (u’ fixed) to y(t, w) = X N(tu, y(t, w)) on [O,T] for each X E (0,l); also assume T(W) is independent of X 1 for each w E a, assume N(w, . ) : Q,(,) + C[O, T] is a continuous compact map; here Q,.(,,,) = {y E CIO, T] : ]y]o F r(w)} { for each y E C[O, ?‘I assume N( . , y) is measurable.

(3.2)

(3.3) (3.4)

Remark. For fixed w E R, by a solution to y(t, w) = X N(zu,y(t, 20)) on [0, T] we mean a function for t E [O,T]. Y(t) (= Y(h WI) E CIO,Tl with y(t) = X h’(w,y(t)) THEOREM above).

3.1. Suppose

(3.2), (3.3) and (3.4) are satisfied.

Then

(3.1) has a solution

(as described

Proof. We wish to apply theorem 2.3. To do this we need to show (2.3)* is satisfied. If (2.3)* is not true then there exists a measurable y : Sz -+ C[O,T] with y(w) = AN(w,y(w)) for some i.e. for t E [O,T], y(w)(t) = X N(w, y(w)(t)) for w E Kl and some X E (0,l) with y(w) E aQ,(,) some w E fi and some X E (0,l) with r(w) = (y(w)]0 = suplo,Tl ]y(w)(t)]. For the above w, y(w) on [O,T] (for the above fixed w). This is a contradiction. is a solution of z(t,~) = X N(w,z(~,w)) Consequently (2.3)” is true so theorem 2.3 implies that there exists a measurable 4 : IR + C[O,T] on fi (i.e. for t E [O,T] we have $(w)(t) = N(w,~(w)(~))). Define with I = N(w,$(w)) c#* : [o,T] x 52 + C[O,T] by +*(u~u, t) = ai( Now +*( . ,t) is measurable for every t E [O,T]

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since for t fixed we have d*(w, t) = j, o d(u) where j, : C[O,T] ---f R” is the continuous mapping given by jt(u) = u(t). Also +*(w, .) is continuous for every ‘u) E a. In addition for each w E R and each t E [O,T] we have dJ*(%t) Thus

#~*(w,t)

is a solution

= @J(w)(t) = N(w, 4(w)(t))

= N(%4**(%t)).

of (3.1) REFERENCES

[l]. DEIMLING K., LADDE problems, Stochastic Anal.

G.S. & LAKSHMIKANTHAM and Appl., 3, 153-162(1985).

[2]. ITOH S., Random fixed point theorems with spaces, Jour. Math. Anal. Appl., 67, 261-273(1979).

V., Sample

an application

to random

[3]. SEHGAL V.M. & WALTERS C., S ome random fixed point linear Fvnclronal Analysis (edited by S.P. SINGH, S. THOMEIER Mathematical Society, Vol. 21, Providence (1983). [4]. HIMMELBERG [5]. KURATOWSKI Ser. Sci. Math.

C.J.,

[6]. FUR1 M. & VIGNOLI Rend. CI. Sci. Fis. Mat. [7]. TARAFDAR convex topological

Measurable

relations,

K. & RYLL-NARDZEWSKI Astronom. Phys., 13, 397403( A., Natur.,

On a-nonexpansive 48, 195-198(1970).

Fundamenta C., A g eneral 1965). mappings

solutions

of stochastic

differential

boundary

equations

value

in Banach

theorems, in Topological Methods zn Nonand B. Watson), pp. 215-218, American

Math.,

8’7, 53-72(1975).

theorem and

fixed

E. & VYBORNY, Fixed point theorems for condensing space, Bull. Austral. Math. Sot., 12, 161-170(1975).

on selectors, points, multivalued

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Acad.

Accad. mappings

Polon. Naz.

Sci. Ljncei

on a locally