measurability of inverses of random operators and existence theorems

Prec. Indian Acad. Sei. (Math. Sci.), Vet. 89, Number 2, May 1980, pp. 95-100. (~) Printed in India.

Measurability of inverses of random operators and existence theorems

M O H A N J O S H I Department of Mathematics, Birla Institute of Technology and Science, Pilani 333 031, India

MS received 29 December 1978; revised 9 June 1979

Abstract. Let (~Q, q~, p) be ameasure space and X a separable Hilbert space. Let T be a random operator from ~ >< X into X. In this paper we investigate the measurability of T -1. In our main theorems we show that if T is a separable random operator with T(o~) almost sure invertible and monotone and demi- continuous then T -1 is also a random operator. As an application of this we give an existence theorem for random Hammerstein operator equation.

Keywords. Separable random operator; monotone operator; Hammerstein operator; existence theorems.

1. Introduction

N a s h e d and Salehi [5] ob t a ined the fo l lowing t he o re m on the m e a s u r a b i l i t y o f the inverse of r a n d o m non - l i nea r o p e r a t o r .

Theorem 1.1

Le t (f2, q~,/t) be a c o m p l e t e p r o b a b i l i t y space , X be a s epa rab le met r ic space and Y be a metr ic space. Le t T be a sepa rab le r a n d o m o p e r a t o r f r o m I2 • X on to Y such t ha t a l m o s t sure T( to) is inver t ib le a n d i ts inverse T -1 (to) is con t inuous . T h e n T - t is a lso a r a n d o m o p e r a t o r f rom t'2 • Y into X.

In our mah l t h e o r e m in this p a p e r w e o b t a i n a s imi la r resu l t on a H i lbe r t space J~ wi thou t the c o n d i t i o n o f con t i nu i t y on the inverse o p e r a t o r T -1 (09). Ins tead , we impose the monoton ic~ ty c o n d i t i o n o n t h e o p e r a t o r T(to) . The s t a t emen t o f

o u r ma in t h e o r e m reads as fol lows.

Theorem 1.2

Le t (,(2, q] , / t ) be a c o m p l e t e p r o b a b i l i t y space a n d X be a s epa rab l e H i l b e r t space. L e t T be a s e p a r a b l e r a n d o m o p e r a t o r f r o m K2 • Af on to X such t h a t a l m o s t sure T(og) is invert~ble a n d m o n o t o n e and d e m i - c o n t i n u o v s . Then T -1 is also a r a n d o m

o p e r a t o r f rom t2 • X in to g,. Th is t heo rem is then fo l lowed b y an i m p o r t a n t resul t r e g a r d i n g the so lvab i l i ty

o f r a n d o m H a m m e r s t e i n equa t ion wi th an a p p l i c a t i o n to conc re t e r a n d o m n o n -

l i nea r in teg ra l equa t i on . 95

P .(A)--3

96 Mohan Joshi

2. Preliminaries

Let (s q~,H) be a probabili ty space with a probabili ty measure p ; that i.s, K2 is a nonempty set, q~ is the o'-algebra of subsets of 12 a n d / t is a probability measure. We say that the probabil i ty space is complete if B ~ q~,/z (B) = 0 and B0 ___ B mplies that B0 ~ q~.

A function g from I2 into a normed space Y is Y-valued random variable if the inverse image, under the function g, of each Borel set B ~ q~'t belongs to q~ where q~, is the o'=algebra generated by closed subsets o f Y.

The mapping T from f2 • F into Y, where F an arbitrary set is called a random operator if for each 7 ~ F, the function T ( . ) 7 is a random variable.

A random operator T(~o) : ~r ~ y said to he continuous at x0 ~ X if x , ~ Xo implies that T(co)x,--* T(co)xo almost surely. I t is called demi-continuous i f convergence o f T(~o)x~ to T(og)x0 is weak.

Theorem 2 . 1

Let g be a random variable with values in a separable Banach space X and let T be a cont inuous random operator of the space s • X into metric space Z. Then the mapping W of t2 into Z defined by, for every ~o~s lJ z (o~)----- T(oa)g(co) is a random variable with values in Z.

A random operator T f rom s • X into Y, where f2 is a complete probabil i ty space, .E a separable metric space and Y a metric space, is said to be separable if there exists a countable set S C X and negligihle set N e q~,/z (N) = 0, such that

x rn s) A{o : T ( m ) x e K , x e F ) C N

for every closed set K in q~.~ and every F in q~x- For a further study of separable random operators we refer to [2]. I t is easy

to see that the above definition of separability is equivalent to the following : there exists a negligible set N e q~ and a countable set SC X such that for 09 r N and each x e ~ there exists a sequence {x~} e S such that x i ~ ~ and T (co) x~ ~ T (to) x. We can now state the following result [1].

Theorem 2 . 2

Let X be a separable Banach space and T : t? • X ~ X be a continuous random operator. Then T is separable.

Let T be a random operator f rom s • X into Y. An equation of the type T ( ' ) x (.)-----y ( ' ) w h e r e y is a given random variable with values in I z is called a random operator equation. Any X-valued random variable x (a~) which satisfies

i z{~o:T(og)x(og) = y (o9)) = 1

is said to he random solution of the above equation. We now give few important definitions and theorems regarding mono tone

operators. In what follows X is a Banach space (,) a bilinear form on X • X* and T a nonlinear operator f rom X into X r*.

T is called monotone if (Tx~ -- Tx 2, xl -- xz) >~ 0 for all xl, xz e X. T is called strictly mon.otone if the above inequality is strict for xl ~ x2. T is strongly mono-

Random operators and existence theorems 97

tone if there exists c > 0 such that (Txl -- Tx2, xl -- x~) >i c [1 xl -- x~ I1 ~ for all Xl, xz in X.

We have the following theorems regarding monotone operators. For reference see [31.

Theorem 2 .3

Let T be a demi-continuous monotone operator from a Banach space X to its dual J(* such that

(y -- Tx, xo ~ x) >/ 0 for all xeRr , then y = Txo.

Theorem 2 .4

Let T be a demi-continuous strongly monotone operator from X into X*. Then T is 1-1 and on to with T -1 continuous.

We say that T : X ~ X* is coercive if (Tx, x ) / l l x II - " ~o a s 11 x II --" ~o.

Theorem 2 .5

Let T : j r - ~ X* be a dotal-continuous, monotone and coercive operator. Then R ( T ) = Jr*.

Finally we give a definition, called angle-boundedness, for the bounded mono- tone linear operator K from Jr to X*. A bounded monotone linear operator K f rom I f to J(* is called angle bounded if there exists a constant a >~ 0 such that

I (Kx, y) -- (Ky, x) I ~< 2a [(Kx, x) (Ky, y)l 1/2

for x, y in Jr. I t is clear that every symmetric, monotone linear operator is angle-bounded with

constant zero.

3. Existence theorems

Let J~ be a separable Hilbert space and (12, q~,p) a complete probabili ty measure space. T : 12 • X ~ X is a random operator. Following is the main theorem of this paper.

Theorem 3.1

Let T be a separable random operator from Y2 7,, X o n t o X such that almost sure T(og) is invertible and monotone and demi-continuous. Then T -x is also a random operator from 12 • X into X.

Proof

Let superscript e denote the complementation, S (., r) and S (.,r) denote the open and closed ball of radius r around respectively. To prove the mea~rabi l i ty of T -x it suffices to show that for an arbitrary y ~ X and a closed ball ~ (x, r) the event {e) : T -1 (o9) y e S (x', r)} is in B. We have

{o~: r - 1 (co) y e ~ (x' , r)} = u {~o : T(o)) x = y}

98 Mohan Joshi

We claim that

O {09 : T ( ~ o ) x = y } = f3 U {co: T ( o ~ ) x ~ S ( y , 1/n)}. (3.1) x ~ ~ - {a , t , r ) n : l : e S { x ' . r + l l n )

It is enough to show that the right side is contained in the left side. Let

oo

~o ~ n u {co : F ( o g ) x e S ( y , l/n)}. n = l ee~(m', r-l-lln)

Then for eazh n there exists x , ~ S ( x ' , r + 1 / n ) s u c h that T ( w o ) x , ~ S ( y , 1/n). It follows that

lim T(ogo) x, = y.

Since the sequence {x,} is a bounded sequence in a Hilbert space, it follows that there exists a subsequence

{x.~}, X, k e S x ' , r + -~ ,

and xo such that x.k ~ Xo weakly. We claim that xo ~ S (x ' , r). This follows easily from the relation

( x o - x ' , Xo - x ' ) = ( x 0 - x . ~ , x o - x ' ) + ( x . ~ - x ' , x ~ - x ' )

< ~ l ( x o - x . ~ , x o - x ' ) l + l lx.~-x ' l l l lx0-x ' l l ,

( using the fact that x,~ ~ Xo weakly and x,~ ~ S x ' , r + ~ j .

Moreover, we have

{T(coo) x,~ - - T(o)o) x , x,~ - - x) >1 0 for all x ~ .5(.

Now, since

lim T(oJo) x.~ = y and x.~ ~ Xo k,..~ oo

weakly; passing over to the limit we get

( y - - T(O9o) x, xo - - x) ~ 0 for all x ~ X.

Since T(~oo) is monotone and demi-continuous, it follows by theorem 2 .3 that y = T(co0) x0. This together with the fact that

Xo ~ g' (x', r) implies that COo ~ U {co : T(co) x = y}.

But

[.~ u {~: r(o))xeS(y, ~))] ~ ~(~,,+ ~)

= U A co :T (co) x c S ~ y, (3 .2)

Random operators and existence theorems 99

B ~ a u s e of separabili ty of T,

is measurable. This together with (3.1) and (3.2) gives the result. As a corollary we can get existence artd uniqueness of a r andom solution x (09)

of the operator equat ion T(og) x = y (co).

Corollary 3.1

Let T : I 2 • r ~ X be a continuous r andom operator such that a lmost surely T(09) is strictly monotone znd coercive. Then there exists a unique random solution x (39) of the opera tor equation T(09) x (o9) = y (e~).

Proof. Since T(09) is strictly monotone and coercive it follows f rom the theorem 2 .5 that T(og) is 1-I and onto and hence almost sure T(~o) is invertible. Far ther , since X is separable and T is continuous, it follows by theorem 2 .2 tha t T is separable. Thus T satisfies all the condit ions of the above theorem and hence T -1 is also a r andom opera tor f rom s • X into X. That is, there exists a random variable x (09) such that T(09) x (o9) = y (09). Uniqueness of x (09) follows f rom 1-1 proper ty of T(09).

As aa applicat ion we now state an existence result for r andom Hammers te in opera tor equat ion

x (09) q- K N x (09) = y (09)

on a separable Banach space X. Here K : O • X ~ X* it a random linear operator and N : s • X ~* --, X is a random nonlinear operator.

Theorem 3.2

Let Rr he a separable Banach space and let

(i) K(09) : .~ ~ X* 1Je a continuous r andom monotone opera tor with l[ K(09) 1[ K0 and with a fixed constant o f angle boundedness a.

(it) N(09) : X* --* ~ be a continuous r andom operator such tha t <x - - y, N (09) x - - N (09) y> >~ - - k (09) [l x - - y [12 for all x, y ~ X* (3.3) and for a lmost all o9 ~ s

Suppose that k (09) Ko (1 -b a 2) <: 1 for a lmost all o9 ~ g2, then there exists a unique random solution x (co) in X* such tha t

x (o9) + K N x (09) = y (o9). (3.4)

Tttis theorem geaeral~ses the Browder and Gup ta ' s theorem [3] to random I-Iammerstein equations. The proof is similar to that of Browder [3].

4. ~xample

We now give an example which depicts the application of theorem 3 .2 to a con,:rete nonlinear integral equation. We consider an eqt:ation of the type

x (s;09) -b J" K(s , t ; o ~ ) f ( t , x ( t ) ; 0 9 ) dt = y(s ;o9) (4.1)

100 Mohan Joshi

where (i) 09 ~ f2, K2 is the supporting set of the probabili ty space (t2, q~,/t),

(ii) 27 is a a-finite measure space,

(iii) K(s, t ; co) is a random kernel defined on 27 • Z',

(iv) f ( t , x, 09) is a nonlinear random function defined on 27 x R" wit h values in R",

(v) y (s; 09) is a known and x (s; co)~is an unknown n-dimensional valued random variable defined for s~27.

In order to consider this equat ion for the existence o f a random solution we transfer it into random Hammerstein operator equation. We define the random linear operator K and the random nonlinear operator N as

K (og) x ( s ) = I g ( s , t ;09) x ( t )d t

N (09) x (t) = f ( t , x (t) ; 09).

(4.1) is then equivalent to the random operator equat ion

x (09) q- KNx (o9) = y (09). (4 .2)

We assume that the function f ( t , u ; to) satisfy the conditions

(i) I f ( t , x ;09) l<.~b(09)[g( t )+blxl] , g~LZ(Z,), 6 ( 0 9 ) > 0 .

(ii) ( f ( t , x ; 0 9 ) - - f ( t , y ; 0 9 ) ) ( x - - y ) > / O for almost all 0 9 e l l

The random kernel K(s, t ; 09) is assumed to belong to L 2 (Z • 27) with I] K(s, t ; 09) I I ~/Co, for every to ~ t2. Fur ther we assume that it is symmetric and mono- tone for almost all co e ft. Under these conditions it can be easily seen that the operator K(09) is a continuous random linear angle bounded operator with con- stunt a = 0. The nonlinear operator N(co) is also continuous and monotone. Hence it follows by theorem 3 .2 that there exists a unique random solution x (09) o f (4.1).

References

[1] Bharucha-Reid A T 1972 Random integral equations (New York :Academic Press)

[2l Bharucha-Reid A T and Mukherjea A 1969 Rev. Roumaine Math. Pures AppL 14 1553

[3] Browder F E 1971 Contributions to nonlinear functional analysis ed. E ZarantoneUo (Now Y o r k : A c a d e m i c Press) p. 99

[4] Minty G J 1963 Prec. Natl. Acad. Sci. USA 50 1038

[5] Nashed M Z and Salehi H 1973 SIAM J. AppL Math. 25 681