Proc. Indian Acad. Sci. (Math. Sci.), Vol. 96, No, 2, November 1987, pp. 75-85. �9 Printed in India.

Nonlinear random equations involving completely closed operators

MOHAN JOSHI Department of Mathematics, Indian Institute of Technology, Bombay 400076, India

MS received 10 May t986; revised 26 February 1987

Abstract, We introduce the concept of completely closed operators in Banach spaces and then obtain the existence of random solutions of operator equations involving such operators. As simple corollaries we obtain the existence theorems for random operator equations involving monotone operators as well as operators of type (M).

Keywortls. Nonlinear random equations; closed operators.

1. Introduction

The study of random operator equations was first initiated by Spacek [21] and Hans [9] with a systematic development of the theory of random fixed points. Subsequently these results were extended and applied to various problems by Hans [10, 11] and Bharucha-Reid [2, 31. Now there is a wide range of literature in this field, for reference see Bharucha-Reid [4, 5].

In this paper we first introduce the concept of completely closed operators in Banach spaces and then obtain random solvability results for operator equations involving such operators. In studying such a class of operators we cover some important operators such as maximal monotone operators, operators of type (M), completely continuous operators and compact perturbations of the identity operator. As simple corollaries we obtain the measurability inverse theorems of Hans [10], Joshi [13], Nashed and Salehi [18] and random analogs of fixed point theorems of Banach and Schauder.

In w 2 we introduce the necessary basic concepts of the theory of nonlinear functional analysis and theory of random operators. Section 3 deals with existence results. We conclude the paper in w by giving the application of our theory to random Hammerstein operator equation including a concrete example.

2. Preliminaries

Throughout this paper (fl,/~,/a) denotes a complete probability measure space and X a separable real Banach space with dual X*, (. , .) denoting the duality pairing between X and X*. If X is a Hilbert space then (. , .) indicates the inner product in X. For any subset D of X, 2 ~ denotes the family of all subsets of D.

A mapping T:t'l ~ 2 x is said to be measurable, if for any closed subset C of X, T- ~ (C) = {coef~:T(o~)nC#~b} is in ~. A measurable mapping g : f l ~ X is said to be a

75

76 Mohan Joshi

measurable selector of a measurable mapping T if for any toef~ 0(to)e T(to). Any X- valued measurable mapping is called a random variable. T:f~ x D ~ X is called a random operator if for any xeD, T(to)x is a random variable. If T:f~ x D ~ X is a random operator which is 1-1 and onto for each to e f~ then T - 1: f~ x X --* D is defined and is said to be random if T-~(to)y is a random variable for any y~X. A random operator T:f~ x D --, X is called separable ifthere exists a countable dense set S in D and Neff with/~(N) = 0 such that for every closed subset K of X and open subset O of D we have

(toefl: T(to)(O n S) c K} A ( t o ~ : T(to)O c K} c N.

Remark. Separability of the operator T:f~ x D ~ X does not imply the joint measura- bility of T as a function of (to, x) with respect to the product a-field on t~ x D as we see by the following example.

Example. Let {X,, t/> 0} be a stochastic process on (t), f , #) such that for each t, X, has the Gaussian distribution with 0 mean and unit variance and E X , X , = 0, if s # t. One can look into Xz as a random operator T(to) from D = {t >/0, t~R} c R to R. Example 3.8.5 (page 534) of Billingsley [5a] shows that Xt is a separable process and hence T(to) is a separable operator. However example 1.2.5 of Kalianpur [14a] shows that this process is not measurable, i.e. T is not a measurable mapping.

An X-valued random variable x(to) is said to be a random solution of the random operator equation T(to)x(to) = y(to), where y(to) is a given random variable, if it satisfies the relation/J{(to: T(to)x(to)= y(to)} = 1. A random operator T:t2 x D-- .X is said to be a random contraction if there exists a random constant k(to)< 1 such that [[ T(to}x - T(to)y II ~< k(to)II x - y II for all x, y~X.

A mapping T :X ~ Y (Y is another real Banach space) is said to be bounded if it maps bounded subsets of X into bounded subsets of Y. T is called compact if it maps bounded sets into precompact sets. T is demicontinuous ifx, ~ x in X implies that Tx, converges weakly to Tx (denoted by Tx, --, Tx) in Y. T is hemicontinuous ifx~ ~ x along the lines in X implies that Tx, --" Tx. T is weakly continuous ifx~.~ x implies that Tx~ ~ Tx, T is completely continuous if x, ~ x implies that Tx, --, Tx.

T : X ~ X* is called monotone if ( T x - T y, x - y) >>. 0 for all x, y ~ D( T). T is called maximal monotone if it has no proper monotone extensions, that is, if for [x,y]~ X x X* we have ( y - Tu, x -u )>~O for all ueD(T), then xel ) (T) and y = Tx. Every demicontinuous monotone operator with dense domain is maximal monotone, refer [6]. A bounded linear monotone operator A: X--, X* is said to be angle-bounded with constant c >/0 if

I(Ax, y ) - (Ay, x)l <~ 2c[(Ax, x)(Ay, y)] 1/2 for all x, y e X .

It is clear that every symmetric bounded monotone operator is angle-bounded with constant c = 0.

T : X --,X* is said to be of type (M) provided the following two conditions hold:

(i) If x, ~ x in X, Tx , ~ y in X* and lira sup (Tx,, x,) <~ (y, x), then Tx = y. N

(ii) T is continuous from finite dimensional subspaces of X into X* equipped with weak* topology.

Random equations involving closed operators 77

T is called coercive if there exists a function c: R + ~ R such that (Tx, x) >1 c( II x II ) II x II and c(r) --, oo as r ~ oo.

A mapping J : X ~ X* is called duality mapping if (Jx, x) = II x II II Jx II and II x II = II Jx II for all xEX. The duality mapping is a demicontinuous monotone everywhere defined operator (refer Petryshyn [19]).

A random operator T:f l • X--. Y is called hemicontinuous (demicontinuous, bounded, etc.) if for each coEfl, T(o) is hemicontinuous (demicontinuous, bounded, etc.). Similarly we define random monotone and M-type operator if T(co) maps X into X* for ~o~f~.

With these definitions we have

THEOREM 2.1.[16]. Let X and Y be separable Banach spaces and let T:f2 x X---, Y be a demicontinuous random operator. Then for any X-valued random variable x, the function T(o~, x(co)) is a Y-valued random variable.

THEOREM 2.2.[17]. Let X be a separable complete metric space and let T:f~ ~ 2 x be a closed valued measurable multifunction. Then there exists at least one random selector of 7".

THEOREM 2.i.[4]. Let X and Y be separable Banach spaces and T:f~ x X ~ Y a random operator and S a countable dense subset of X. Then T is separable iff there exists NEff with #(N) = 0 such that for all ogr and x~X, there exists {xn} c S such that xn ~ x and T(~o)x~ ~ T(co)x.

THEOREM 2.4.[6]. Let T:X ~ X* be a demicontinuous coercive monotone operator. Then R(T) = X*.

3. Existence theorems

In the following, unless otherwise stated, X denotes a real separable reflexive Banach space with dual X*, Y another real separable Banach space and T:X ~ Y is a operator (not necessarily linear) with domain D(T).

DEFINITION 3.1. T:D(T) c X--* Y is said to be completely closed if whenever there exists a sequence {xn}cD(T) with x , ~ x in X and T x ~ o y in Y, then xED(T) and y = Tx.

Remark I. It is clear that if T is invertible with T-1 demicontinuous then T is completely closed. Also every weakly continuous operator is completely closed and so is negative of a completely closed operator.

Example 3.1. Let X = L2[0, 1] and T : X ~ X be the differential operator d/dt with domain D(T) defined as

D(T) = {x~L2[0 , 1]: .x; is absolutely continuous with

x'~L2 and x(O) = 0}.

78 Mohan Joshi

t " Then T - i is the integral operator [ T - lx] (t) = J , x(s)ds. T - ~ is continuous and hence

T is completely closed. If T is a linear operator which is everywhere defined on X, then complete closedness

of T is equivalent to boundedness.

PROPOSITION 3.1. Let T: X ~ Y be a linear operator with D(T) = X. Then T is bounded iff T is completely closed.

Proof. Let T be bounded. Since every bounded operator is weakly continuous, it follows that T is completely closed.

Conversely let T be completely closed and let (x,,, Tx.) ~ (x, y) in X x Y. This implies that x. ~ x and Tx . ~ y and hence complete closedness of T gives y = Tx. Thus the graph of T is closed and hence by closed graph theorem it follows that T is bounded.

Let Tbe a map from X into X*, then monotonicity of T is related to complete closedness in the following way.

PROPOSITION 3.2. Let T:D(T) c X ~ X * be a maximal monotone operator. Then T is completely closed.

Proof. Let {x,} be a sequence in D(T) with x , ~ x and T x , ~ y in X*. Then monotonicity of T gives

( T x , - Txo, X,-Xo)>_.O for all xoeD(T).

Taking limit in the above inequality as x, ~ x and Tx,--* y we get

( y - Txo, X - Xo)>~O for all xo~D(T).

Now maximal monotonicity implies that xeD(T) and y = Tx.

Remark 2. Every demicontinuous monotone operator with dense domain is com- pletely closed. So the duality mapping J:X ~ X * is a completely closed operator.

The converse of the above proposition is false. For, if T is maximal monotone then ( - T) is not maximal monotone, whereas it is completely closed.

Remark 3. It can be easily seen that every operator of type (M) is completely closed but the converse is not true as we will see in example 3.3. Also every completely continuo/us operator is completely closed but this is not so for compact operators as we see in the following example.

Example 3.2. Let X = 12. Define T:X ~ X as

T x = ( I - I lx l r ,0 ,0 . . . . ).

Since T is a finite rank operator, it is compact. But it is not completely closed. For this, consider the sequence {x,,} of elements in l: as

x , = ( � 8 9 . . . . ), x ~ = ( o , 3 , , o . . . . ) . . . . .

Then x , ~ 0 and Tx, --, (�89 0. 0 . . . . ). but Tor189 .. . . ).

Random equations involving closed operators 79

The sum of two completely closed operators need not be completely closed. However, a completely continuous perturbation of a completely closed operator is completely closed.

I~OPOSmON 3.3. Let T:D(T) c X ~ Y be a completely closed operator and S :X ~ Y a completely continuous operator with D(S) = X. Then T + S is completely closed with domain D(T).

Compact and continuous perturbation of a completely dosed operator with continuous inverse is closed.

l~oeosmoN 3.4. Let T:D(T) ~ X ~ Y be a completely closed operator with T -1 continuous and S:X --* Y be a compact and continuous operator with D(S) = X. Then T + S is completely closed with domain D(T).

Proof. Let x.eD(Sc~ T) = D(T) be such that x . ~ x and Tx, + S x . ~ z . Since {x.} is a bounded sequence and S is compact it follows that {Sx~} has a subsequence { S x J converging to y, which in turn implies that { T x J converges to z - y. Since T-1 is continuous, it follows that x~ --, T - l(z - y)eD(T). Uniqueness of limit implies that x = T - l ( z - y ) or Tx = z - y . So x~k~x and hence Sxn~--*Sx which together with Tx,~ --, z - y = Tx implies that Tx.~ + Sx,~ -~ T.x + Sx = z. This proves the assertion.

If both X and X* are strictly convex then the duality mapping J : X -* X* is maximal monotone with J - t continuous (refer [19]) and hence as a corollary of the above proposition we obtain a very useful result for compact perturbation of the duality mapping.

COROLLARY 3.1. Let X be a strictly convex real reflexive Banach space with strictly convex dual X*. Let T:X ~ X* be of the type T ~ J + C where C is compact and continuous with D(C) = X. Then T is completely closed.

If we consider the compact and continuous perturbation of the identity mapping we need not assume the strict convexity of X and X*.

COROLLARY 3.2. Let T : X ~ X be of the type I + C where C : X ~ X is compact and continuous, then T is completely closed.

Remark 4. The above two corollaries remain valid if, instead of T being defined on the whole space, we assume that it is defined on a closed subset of X.

We are now in a position to give the example of a map which is completely closed but not of type (M).

Example3.3. Let X = L 2 [-- 7t,/r]. Define T : X ~ X as

[Tx]( t )= - x(t)+ f ~ . I x ( t ) - 1 ] 3 dr.

Then T is of the type - [1 + 6"] where C is the compact operator

80 M o h a n Joshi

By corollary 3.2 it follows that T is completely closed. However, T is not of type (M) as we see below:

Take x~(t) = sinnt. Then [ T x j ( t ) = - [n + sin nt]. x . ~ O = x, Tx~--" - n = y and i i m s u p ( T x ~ , x . ) <<. (y ,x) . But T x = - 2~ # - n = y.

11

We now proceed to give the main existence theorem regarding the random solvability of operator equation of the form

T(og)x(og) = .v(co), (1)

where T:f l x X ~ Y is a random operator with domain

O = D ( T ) = ~ O(T(og)), to, e l l

t'l is a complete probability measure space.

THEOREM 3.1. Let T:f~ x D--, Y be a completely closed separable random operator such that {x: T(og)x = y(og)} ~: ~ for each ~oe~. Then (1) has a random solution. We presume that each D(T(co)) is a closed subset of X so that D is closed in X.

Proof. Letf(m) = { x e D : T(co)x = y(~o)}. T h e n f : f l ~ 2 D is a multivalued mapping. We show that this is measurable.

Let C = S n X , where S is a closed ball in X. We need to show that f - l (C)efl . Let {xn} c C be a countable dense subset of C. We claim that

f - l ( C ) = U ogef~:ll T(~o)xi-y(og)[r < n " (2) n = 1 x l e S

Let a~ef- I(C). This implies that there exists x e D such that T ( @ x = y(o~). Since {x.} is dense in C, separability of T implies that for every integer n there exists x,.)~ {x.} such that

1 1 II x,n~ - x II < - and I[ T(og)x,n) - y (@ II < -.

n n

This proves that ~ belongs to RHS of (2). Conversely, now assume that ~o belongs to RHS of (2). Ttiis implies that we have a

sequence {x,.)} c C such that T(~o)x,.) --. y(og). As {x,.)} is a bounded sequence in a reflexive Banach space it has a subsequence (which we do not differentiate) converging weakly to x. Thus x , . ) ~ x and T(og)xi(.) ~ y(og) and T(@ is completely closed, so we get that x e D and T(og)x = y(~o). This proves (2) and hence the measurability of f.

Also as T(a 0 is completely closed it follows that f has closed values. Thus f is a measurable multivalued mapping with closed values and hence by theorem 2.2 it has a measurable selector x(@ which is a random solution of(l). This completes the proof of the theorem.

As simple corollaries of this theorem, we obtain the following known results for random solvability of equation of the type (1) and also of random fixed point equation

T(~) x(@ = x(@. (3)

Random equations involving closed operators 81

COROLLARY 3.3. (Joshi [14]). Let T:I') x X--* X* be a separable random operator of type (M). Further, let condition (c) be satisfied:

(c) If {xn} is a bounded sequence in X such that for every toef~, {(T(to)xn, xn)} is also bounded, then {T(to)xn} is bounded for all toel/.

Further, assume that there exists an r > 0 such that (T(to)x, x) >1 0 for II x II = r and all toe~. Then there exists a random solution of the operator equation

T(to) x(to) = 0. (4)

Proof. For each toeD,., T(to): X --, X* is an operator of type (M) and hence by remark 3 it is completely closed. Also, each T(to) satisfies all assumptions of proposition 1.2 of [20] and hence it follows that {x:T(w)x = 0} # q~. Thus T: t ' )x X ~ X * satisfies all hypotheses of theorem 3.1 and so it follows that (4) has a random solution.

COROLLARY 3.4 (Itoh [12]). Let T:I) x X --* X* be a hemicontinuous coercive mono- tone separable random operator. Then (1) has a random solution for every random variable y(w).

Proof. As hemicontinuous monotone operator defined on the whole space is maximal monotone, it follows by proposition 3.2 that T(to):X ~ X * is completely closed for each toet~. Also by theorem 2.4 we have {x: T(to)x = y(to)} :# q~ for each toe~. Our result now follows by the application of the main theorem.

COaOLLAR't" 3.5 (Itoh [12]). Let U:~q x X - , X * be a random operator of the form U = T + S where T:FI x X ~ X* is a demicontinuous monotone separable random operator and S:~ x X-o X* is a completely continuous random operator. Assume further that U is coercive, then Ux(to) = y(to) has a random solution for every random variable y(to).

Proof. This follows from the Browder's result for deterministic operator equation (refer [7]) and proposition 3.3.

COROLLARY 3.6 (Hans [9]). Let T:D x X---* X* be a random contraction. Then there exists a unique random solution of the fixed point equation (3).

Proof. For each w~fl, T{to):X-~ X* is a contraction mapping and hence by Banach contraction mapping theorem it follows that { x e X : [ l - T(to)]x = 0} :/: ~b. Also, T(to) being a contraction, we get that [I - T(w)] is a monotone continuous operator defined on the whole space and hence maximal monotone. So [I - T(to)] is a completely closed separable random operator satisfying all hypotheses of theorem 3.1. Hence there exists a random solution x(w) of [ I - T(co)]x(og)= 0, which in turn implies that x(to) is a random fixed point of T.

COROLLARY 3.7 (Bharucha Reid [5]). Let E be a closed convex bounded subset of X and let T: f l x E--*E be a compact and continuous random operator. Then T has a random fixed point.

82 Mohan doshi

Proof. We want random solvability of the operator equation x(co) - T(co)x(to) = 0. By Sehauder's fixed point theorem for deterministic operators we have { x : x - T(co)x=0}~r for each coef~. Further, by corollary 3.2 and remark 4 [ I - T(co)] is completely closed and hence theorem 3.1 gives the required result.

We now assume that T:f~ x X ~ Y is an invertible random operator. Then the random solvability of the random operator equation T(co)x(co) = y for every ye Y is equivalent to the measurability of the inverse operator T - x:f~ x Y--, X. Theorem 3.1 immediately gives us the following result regarding the measurability of T-~.

THEOREM 3.2. Let T:f l x X ~ Y be a separable invertible random operator which is completely closed. Then T-~ : f l x Y :--, X is also a random operator.

From this theorem we derive some known measurability inverse theorems.

COROLLARY 3.8 (Nashed and Salehi [18]). Let T :~ x X -~ Y be a continuous random operator such that T(co) is invertible with T-~(co) continuous. Then T-~ is also a random operator.

COROLLARY 3.9 (Joshi [13]). Let T:f~ x X ~ X * be a separable demicontinuous monotone random operator such that T(c~) is invertible. Then T - t is also a random operator.

COROLLARY 3.10 (Hans [10]). Let T: f l x X ~ Y be a bounded invertible random operator. Then T-1 is also a random operator.

4. Random Hammerstein equation

In this section we discuss the solvability of random Hammerstein operator equation

x(co) + K(co)N(co)x(CO) = O, (5)

where N(co):X* ~ X is a bounded, continuous random operator and K(co):X ~ X* is a suitable bounded linear random operator. The class of such operator equations includes a random nonlinear integral equation of the form

x(s; 09) + f z K(s, t; co)f (t, x(t; co)) dt = O.

In the deterministic setting, such a type of equation has been extensively studied ([8]). In the random setting its study is limited to a few which includes Kannan [15], Kannan and Salehi [16]. ltoh [12] and Joshi [14]. Here we mainly give the random analogs of the results of Amann [1] for (5).

We conclude this section by giving a concrete example.

TI-IEOREM4.1. Let K: f l x X - * X * be a compact monotone random operator with range contained in a closed subspace Y of X*. Let N:f~ x Y--*X be a continuous bounded random operator. Assume that there exists a constant p > 0 such that for all

Random equations involvin9 closed operators 83

x ~ Y satisfying Ilull > P, we have

(x, N(to)x) >~ 0 for ~eO.

Then (5) has a random solution x(w) satisfying lix(o~)II ~< P.

Proof. By theorem 1 of Amann [1], {x:x + K(og)N(o)x --- 0} # ~b. As K(o) is bounded and random and N(co) is random, it follows that K(tn)N(w) is a random operator. Also continuity of both of these operators gives the separability of K(to)N(to). Thus for each og~O., T(o) = [1 - K(to)N(co)] is of the type I + C(o~), where C(o9) is a compact and continuous operator and hence by corollary 3.2 it follows that T(og) is completely closed. Theorem 3.1 gives a random solution x(tn) of (5) with IIx(oJ)ll ~< p.

If we assume that K(o):X--.X* is angle-bounded, we can considerably relax the condition imposed on N(tn). We first state without proof the following proposition which is a random analog of the splitting lemma for compact angle-bounded linear operators K:X-- . X* (refer Browder [8]7.

PROPOSITION 4.1. Let X be a real separable Banach space and let K:f l • X --.X* be a compact and monotone angle-bounded operator with IIK(o~)II ~< go and with a fixed constant c of angle-boundedness for all toef~. Then there exists a separable real Hiibert space H (independent ofoJ) and a compact linear random operator S:f~ x X --* H and a skew symmetric bounded linear operator B:f~ x H ~ H such that for each oJef~

and

and

K(o) = S*(r + B(o)))S(o),

[IS(o)l[ 2 ~< Ko, IlB(~o)ll ~ c

[[l + B(oJ)]-~h,h]n>~(l +c2) -~ ilhll~ for all h~H.

THEOREM 4.2. Let X and K be as in proposition 4. I with R(K(~o)) contained in a closed subspace Y of X* for all ~oe~. Let N : f / x Y ~ X be a continuous and bounded random operator and assume that there exists a function ~b:R + ~ R + satisfying 4~(P) = O(p2~ as p ---, oo such that for all xe Y

(x, N(o~)x) >1 - 4)( II x II) for all o~efl.

Then (5) has a random solution in Y.

Proof. The solvability of equation (5) is equivalent to the solvability of

T(oo)x(oJ) = 0 (67

in the Hilbert space H where T(o~) = [I + B(eo)] - 1 + S*(w)N(og)S(o~). First observe that {x: T(~) = 0} # q~ for each 0~r (Theorem 3 of Arnann [1]).

Further, as S(o) is a compact linear random operator and N(o~) is a bounded continuous random operatol, it follows that S*(o~)N(~)S(og) is a compact and continuous random operator. Also [I + B(to)] -? 1 is a bounded linear random operator (corollary 3.10 of 1-14]). Hence it follows by proposition 3.4 that T(~) is completely dosed and consequently theorem 3.1 gives a random solution of (6).

84 Mohan Joshi

Example 4.1. Consider a nonlinear random Sturm-Liouville's problem

d ~(p(t)x(t) ) + q(to; t)x(t) + f(to; t, x( t) ) = 0

x(0) = 0 = x(1). (7)

We assume that q is a random function. Likewise f is also a random function defined on [0, 1-1 x R satisfying Caratheodory conditions and a growth condition of the type

I f (to; t, x)l <~ a(t) + b(to)lx[,

where b(to) > 0 is a random constant and aeL2[O, 1-1. We further assume that there exists a constant p > 0 such that

x f (to; t, x) >>. O for Ixl>p for all toef~.

Assume that the Green's function K(to; s, t) of the system (7) exists and is of Hilbert- Schmidt type for each toeD. Also assume that K(to; s, t) is symmetric and monotone. Then (7) is equivalent to the Hammerstein integral equation

x(to; s) + f ~ K(to; s, 0f(to; t, x(to; t)) d t = 0. (8)

Define random operators K(to), N(to) on L2[0, 1] as

[K(to)]x(s) = f ~ K(to; s, t)x(t) dt

IN(to)Ix(s) = f(to; s, x(s) ).

Then the solvability of (6) is equivalent to the solvability of (8). To apply theorem 4.2 we set Y = X = X* = L2[0, 1].

As in Amann [1] it is easy to check the conditions of the above theorem and hence we get a random solution of (7), with relatively mild conditions on the nonlinear function f.

Acknowledgements

We would like to acknowledge the valuable suggestions of the referee and author's discussion with Dr A. Subramanian which led to the improvement of the paper. We also acknowledge CSIR for financial support for carrying out this research.

References

[1] Amann H, Existence theorems for equations of Hammerstein type, Appl. Anal. 2 11972) 385-397 [2] Bharucha-Reid A T, On random solutions of integral equations in Banach space, in: Trans. 11 Prague

conference on information theory, statistical decision function, random processes (Prague: Academe of Science) 27-48 (1960)

R a n d o m equations involving closed operators 85

[3]

[4] [5]

[Sa] [6]

[7] [8]

[9]

[1o]

[11]

[12]

[13]

[]4]

[14a] [IS]

[16]

[17]

[18]

[19]

[20]

[21]

Bharucha-Reid A T, On theory of random equations, in: Proc. symposia in applied mathematics (Rhode Island: Am. Math. Soc.) 40-69 (1964) Bharucha-Reid A T, Random integral equations (New York: Academic Press) (1972) Bharucha-Reid A T, Fixed point theorems in probabilistic functional analysis, Bull. Am. Math. Soc. 1112 (1976) 641-657 Billingsley P, Probability and measures (New York: John Wiley) (1986) Browder F E, Remarks on nonlinear functional equations, Proc. Natl. Acad. Sci. (USA) 51 (1964) 985-989 Browder F E, Problemes nonlineares (Montreal: Les Presses) (1966) Browder F E, Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type, in: Contributions to nonlinear functional analysis (ed) E Zarantonello (New York: Academic Press) (1971) Hans O, Random fixed point theorems, in: Trans. I Prague conference on information theory, statistical decision theory, random processes (Prague: Academe of Science) (1957) Hans O, Inverse and adjoint transforms of linear bounded random transformation, in: Trans. I Prague conference on information theory, statistical decision functions, random processes (Prague: Academe of Science) 127-133 (1957) Hans O, Random operator equations, in: Fourth Berkeley Symposium on mathematical statistics and probability (Berkeley: Univerisity Press) Vol. 2. 185-202 (1961) Itoh S, Nonlinear random equations with monotone operators in Banach spaces, Math. Ann. 236 (1978) 133-146 Joshi M, Measurability of inverses of random operators and existence theorems, Proc. Indian Acad. Sci. (Math. Sci.) 89 (1980) 95-100 Joshi M, Nonlinear random equations involving operators of type (M), d. Math. Anal. Appl. 94 (1983) 460-469 Kalianpur G, Stochastic filtering theory (New York: Springer Verlag) (1980) Kannan R, Random operator equations, in: Dynamical systems, Proceedings of the University of Florida international symposium (New York: Academic Press) (1977) Kannan R and Salehi H, Random nonlinear equation and monotone nonlinearities, J. Math. Anal. Appl. 57 (1977) 234-256 Kuratowski K and Ryll-Nardzewski C, A general theorem on selectors, Bull. Acad. Pol. Sci. Ser. Math. Sci. Astronom. Phys. 13 (1965) 397-403 Nashed M S and Salehi H, Measurability of generalized inverses of random linear operators, SIAM J. Appl. Math. 25 (1973) 681-692 Petryshyn W V, A characterization of strict convexity of Banach spaces and other uses of duality mapping, J. Func. Anal. 6 (1970) 282-291 Petryshyn W V and Fitzpatrick P M, New existence theorems for nonlinear equations of Hammerstein type, Trans. Am. Math. Soc. 160 (1971) 39-63. Specek A, Zufallige gleichungen, Czechoslovak. Math. J. 5 (1955) 462-466