measurability of inverses of random operators and existence theorems

6
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 MOHAN JOSHI 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 Nashed and Salehi [5] obtained the following theorem on the measurability of the inverse of random non-linear operator. Theorem 1.1 Let (f2, q~,/t) be a complete probability space, X be a separable metric space and Y be a metric space. Let T be a separable random operator from I2 • X onto Y such that almost sure T(to) is invertible and its inverse T -1 (to) is continuous. Then T -t is also a random operator from t'2 • Y into X. In our mahl theorem in this paper we obtain a similar result on a Hilbert space J~ without the condition of continuity on the inverse operator T -1 (09). Instead, we impose the monotonic~ty condition onthe operator T(to). The statement of our main theorem reads as follows. Theorem 1.2 Let (,(2, q],/t) be a complete probability space and X be a separable Hilbert space. Let T be a separable random operator from K2 • Af onto X such that almost sure T(og) is invert~ble and monotone and demi-continuovs. Then T -1 is also a random operator from t2 • X into g,. This theorem is then followed by an important result regarding the solvability of random Hammerstein equation with an application to concrete random non- linear integral equation. 95 P .(A)--3

Upload: mohan-joshi

Post on 16-Aug-2016

213 views

Category:

Documents


1 download

TRANSCRIPT

Page 1: 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

Page 2: Measurability of inverses of random operators and existence theorems

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-

Page 3: Measurability of inverses of random operators and existence theorems

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}

Page 4: Measurability of inverses of random operators and existence theorems

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)

Page 5: Measurability of inverses of random operators and existence theorems

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)

Page 6: Measurability of inverses of random operators and existence theorems

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