RANDOM FIXED POINT THEOREMS ON PRODUCT SPACES

The existence of random fixed point of a locally contractive random operator in first variable on product spaces is proved. The concept "continuous random height-selection" is discussed. Some random fixed point theorems for nonexpansive self and nonself maps are also obtained in product spaces.


I. INTRODUCTION
The study of the fixed points of random operators of various types is a lively and fascinating field of research lying at the intersection of nonlinear analysis and probability theory.A wide spread interest in the domain and a vast amount of mathematical activity have led to many remarkable new results and viewpoints yielding insight even into traditional question.
Random fixed point theorems are stochastic generalizations of classical fixed point theorems.In Polish spaces, random fixed point theorems for contraction mappings were proved by Spacek [24] and Hans [7, 8].For a complete survey, we refer to Bharucha-Reid [2,  3].Itoh [10, 11, 12] gave several random fixed point theorems for various single and multivalued random operators, that is, c-condensing or nonexpansive random operators.He also gave several common random fixed point theorems for commuting random operators.
[1] studied the structure of common random fixed points and random coincidence points of a pair of compatible random multivalued operators in Polish spaces.They also proved a random fixed point theorem for contractive random operators in e-chainable Polish spaces.An interesting application to random approximation is also given.The aim of this paper is to prove several random fixed point theorems for various self and nonself random operators, that is, contractive and nonexpansive random operators in product spaces.The concept "continuous random height-selection" is discussed and its relation to the existence of random fixed points for a function is shown.Section 2 contains definitions and preliminary material.
In section 3, the existence of random fixed point of a contractive random operator in first variable on product spaces is proved.Other results are proved concerning the random fixed point theorem for product spaces.Section 4 contains some random fixed point theorems for nonexpansive self or nonself random operators.

PRELIMINARIES
Throughout this paper, let (X,d) be a Polish space, that is, a separable complete metric space, and (f2,t) be a measurable space.Let 2 X be the family of all subsets of X and WK(X) the family of all weakly compact subsets of X.A mapping T:2---,2 x is called measurable if for any open subset C of X, T-1(C) = {w E f2: T(w) N C } e A. This type of measurability is usually called weak measurability (cf.Himmelberg [9]), but in this paper since we only use this type of measurability, thus we omit the term "weak" for simplicity.A mapping (: fX is said to be the measurable seleclor of a measurable mapping T: f---,2 X if is measurable and for any w f2, ((w) T(w).A mapping f:xX--.,X is called a random operator if for any x X, f(.,x) is measurable.A measurable mapping (:f2---,X is called random fixed point of a random operator f: f2 x XX if for every 0 f2, f(w, ((w)) = ((w).A random operator f: f2 x X---,X is called continuous if for each w E f, f(w,.) is continuous.A random operator T:fxXX is said to be nonexpansive if for any [[ T(w, u) T(w, v) [[ < [[u-vii for allu, veX.A separable metric space Y has a random fixed point property for nonexpansive (continuous) random operators if every nonexpansive (continuous) random operator T: f2 x YY must have a random fixed point.In what follows, PI: X x Y---,X will denote the first projection mapping defined by Pl(z,y)= z, while P2:X x YY will denote the second projection mapping defined by P2(z, y) = y.

ILkNDOM FIXED POINT THEOREMS AND CONTINUOUS
ILkNDOM HEIGHT-SELECTION In 1968, Nadler [21] established some fixed point theorems for uniformly continuous functions on product metric spaces.Subsequently, Fora [6] further generalized these results.lecently, Kuczumow [18] proved a fixed point theorem in product spaces using the generic fixed point property for nonexpansive mapping.The aim of this section is to prove some random fixed point theorems using a recent result of Beg and Shahzad [1].
Let X be a Polish space and Y De any space.
A random operator f:fl x (X x Y)---.(Xx Y) is said to be a locally contractive random operator in the first variable if and only if for any x.E X there exists e > 0 and a measurable map A:f---,(0, 1) such that P,q 6 S(z,, e)= {z 6 X:d(z,,z) < e} and for any w 6 f d(PlOf(w p, y), PlOf(w, q, y)) <_ A(w) d(p, q) for any y Y.
A random operator f: f2 x (X x Y).--(X x Y) is called a contraction mapping in the first variable if and only if there exists a measurable map A: f---,(0, 1) such that for any y Y, we have for any w E f2 d(Pof(w,x,y), PlOf(w,x.,y))<_ A(w) d(x,x.)for all x, x, fi X.
A metric space (X, d) is said to be e-chainable if and only if given x, y in X, there is an e-chain from x to y (i.e., a finite set of points x Zo, Zl,...,z n y such that d(z.i_ 1,zj) < e for j 1,2,...,n).
Theorem 3.1: Let X be an e-chainable Polish space, let Y be a separable metric space with the random fixed point property and let f:fx(X x Y)---X x Y be a continuous random operator.If f is a locally contractive operator in the first variable, then f has a random fixed point.
Corollary 3.2: Let X be a Polish space, let Y be a separable metric space with random fixed point property and let f'f2 x (X x Y)---,(X x Y) be a continuous random operator.If f is a contraction random operator in the first variable, then f has a random fixed point.
Let f: f2 x (X x Y)(X x Y) be a continuous random operator.A measurable mapping rlz:flY for which P2of(w,z,,lz(w))= fix(w) for each w fi fl is called a random fixed f-height of z.The set {r/z: r/z: f2Y is a random f-height of z} is called the random fixed f-height of z and is denoted by F(f,z).The set U{F(f,z):z X} is called the random fixed f-height of X and is denoted by F(f, X).A continuous random height-selection of f is a continuous random operator g: flx XF(f, X) such that g(., z): fY is a random f-height of z.
Theorem 3.3: Let X and Y be separable metric spaces with the random fixed point property, and let f:f2 x (X x Y)---.(Xx Y) be a continuous random operator.If f has a continuous random height-selection, then f has a random fixed point.
Using the proof of Theorem 3.1 and the technique used in the proof of Theorem 3.2, one can obtain the following theorem.Theorem 3.4: Let X be an e.chainable Polish space, let Y be a separable metric space with the random fixed point property and let f:fx (Y x X)---(Y x X) be a continuous random operator.If f is a locally contractive random operator in the second variable, then f has a continuous random height.selectionand hence f has a random fixed point.

ILNDOM FIXED POINT THEOREMS OF NONEXPANSIVE ILkNDOM
OPEILkTORS IN PRODU SPACES Let E and F be two Banach spaces with XCE and YCF.For It was shown in Kirk and Sternfeld [17] that if E is uniformly convex (or reflexive with the B-G property), if X is bounded closed convex, and if Y is bounded closed and separable, then the assumption that Y has the fixed point property for nonexpansive mappings assures the same is true of (X Y).Subsequently, various results on fixed point theorems in product spaces were given by many authors (cf.Khamsi [13], Kirk [14, 15], Kirk and Yanez [16] and their references).
Pecently Tan and Xu [25] proved some fixed point theorems for nonexpansive self and nonself mappings in product spaces.They also generalized and improved some results of Kirk [15] and Kirk and Sternfeld [17].In this section we give a stochastic version of the results of [14,16,17,25].
A subset K of a Banach space E has the Browder-GShde (B-G) property [14] if for each nonexpansive mapping T:KE, the mapping (I-T) is demiclosed on K (i.e., for each sequence {uj) in K, the condition u.i-.-,u weakly and uj-T(uj)--.,pstrongly implies u K and u-T(u) = p).It is known (Browder [4]) that all closed convex subsets of uniformly convex Banach spaces have this property.Let E and F be two separable Banach spaces with X C E and Y C F. Suppose that X is weakly compact, convex, and has the B-G property.Suppose also that Y has the random fixed point property for nonezpansive random operaors.Then (X Y)oo has the random fixed point property for nonezpansive random operators. Proof: Let P1 and P2 be the natural projections of (EOF) onto E and F, respectively.For each y in Y, we define T:xX--X by Tu(w,x)= PoT(w,x,y),x.X.Then Tu is a nonexpansive random operator.Let Sy-(I + Tu)/2 (I denotes the identity operator on E).Let o: fl--,X be a fixed measurable mapping.We have (1) For each n, define Fu, n(w) = w ct{u, (w)" >_ n }, where w-el(c) is the weak closure of C.
Let ru'fb--.WK(X be a mapping defined by ru(w = ru,n(W ).Then, since the n=l weak-topology is a metric topology F is w-measurable by [9, Theorem 4.1].Thus there is a w-measurable selector of F u [19].For any x*E E* (the dual space of E), x*(,u(.))ismeasurable as a numerically-valued fimction on f.Since E is separable, is measurable.Fix w E a arbitrarily.Then some subsequences {,m(W)} of {u,n(w)} converges weakly to y(w).Then by the B-G property of X and (1), it follows that y(w) is a random fixed point of Sy and hence of Tu, that is, PlOT(, u(w), y) = u().Also, Now let f: x Y-,Y defined by f(w, y) = P2oT(w, u(w), y), y Y. Then for u, v in Y, we have for any w f II f(w, u) f(w, v) I I F !1 P2oT( w, u(w), u) P2oT(w, v(W), v) I I F <_ I[ T(w, u(w), u) T(w, v(w), v) II Therefore f is nonexpansive on Y and thus has a random fixed point /: l'Y.It follows that T(w, o()(v), r/(w)) = (o()(w), /()) for each w .
For a subset K of a separable Banach space, a random operator T:fl x K.--,K is said to be strictly contractive if Theorem 4.2: Let E and F be two Banach spaces with X C E and Y C F separables.Suppose that X is a closed bounded convex subset of E. Suppose also that Y has the random fixed point property for strictly contractive random operators.Then for each 1 _< P _< cx, (X 9 Y)p has the random fixed point property for strictly contractive random operators.
Proofi For a fixed p, 1 g p _< cx, suppose T: f2 (X 9 Y)p--(X 9 Y)p is strictly contractive.As before, for each y (5 Y, we define Ty: f2 x X---,X by Tu(w,x PloT(w,x,y), e f, E X.
Then it is easily checked that in any case of P, T u is strictly contractive and hence has a random fixed point ,u:w.-,X [11,Theorem 2.1].Now define f:fx Y--,Y be f(w,x) = P2oT(w, u(w), y), w a, 9 e Y.
As in [25, Theorem 2.2], for any case of p, f: f2 x YY is strictly contractive.Thus f has a random fixed point rl:fY.It follows that, for each = Finally, we prove random fixed point theorems for nonself random operators in product spaces.Recall that for a closed subset C of a separable Banach space E, the inward set of C at a point z in C, It(z), is defined by tc( ) = + e c, > 0}.
A random operator f:gtxC.-..E is said to be weakly inward if for each chEFt, f(w,z) cl(Ic(z)) for z C. We will say that C has the random fixed point property for a nonexpansive (continuous) weakly inward random operator if every nonexpansive (continuous) weakly inward random operator T: fl x C---,E has a random fixed point.
Theorem 4.3: Let E and F be two separable Banach spaces with X C E and Y C F. Let E F be a product space with an L P norm, l <_P < c.Suppose that both X and Y have the random fixed point property for nonezpansive weakly inward random operators.Then (X Y)p also has the fixed point property for nonezpansive weakly inward random operators. Proof: Let T: l't x (X Y)p---,(E F)p be a nonexpansive and weakly inward random operator.For a fixed y in Y, we define Tu: fl x XE by Ty(ca, x) = PloT(ca, x, y), ca gt, x X.
Then Ty is a nonexpansive random operator.each ca ft.Since It is also weakly inward.Indeed, for T(ca, z,y) cl(I X y(z,y)) for (z,y) X x Y, we have Ty(ca, x) ": P1 (cl Ix 9 y(X, y)) C_ cl Ix(x).
Hence there exists a measurable map u(ca)) = y(ca)" Now define f: x Y-.-,F by such that for each f(ca, x) P2oT(ca, y(ca), y), ca , y Y.
Then it is easy to see that f is nonexpansive.Also, for each ca E fl, f(ca, y) P2(cl Ix$y((w),y)) C_ cl I(y) for y Y. Therefore, f has a random fixed point r/:fl--,Y and hence T(ca, o(w)(ca)' 'l(ca)) = (o(w)(ca), 'l(ca)) for ca . in the next theorem, we assume that X has the net B-G property, that is, if T: X--.,E is nonexpansive and if {a:c}a e A (we also assume that A is countable) is a net in X for which a:a---,x weakly and a: a -T(a:a)p strongly, then : E X and a:-T(a:) = p.Theorem 4.4: Let E and F be two separable Banach spaces with X C E and Y C F. Suppose that X is weakly compact, convex, and has the net B-G property.Suppose also that Y has the random fixed point properly for nonexpansive weakly inward random operators.Then (X q Y)oo has the random fixed point properly for nonexpansive weakly inward random operators. Proof: Let T:fl x (X Y)oo(E q F)o o be a nonexpansive and weakly inward random operator, then for each y E Y, the operator Tv:fxX---,E defined by Tv(w,x)= PloT(w, z, y), (w f2,x X) is nonexpansive and weakly inward.For a fixed z X and t E (0,1), the contraction operator (1-t)z + tTv is weakly inward and has a unique random fixed point [Han's, 7] which we denote by v,t" Thus for ,(v,t(w) = (1 t)z -I" Tv(w,v,t(w)).Now let {ta:a A} be a universal subnet of the net {t:0 <t< 1) in [0,1].It follows that {v, ta(w)) is a universal net in X (v, ta'fl--,X is a measurable map).Also, since t---,1, and for each w I I ,.t()-Tv(w,v, ta(w))II E (1 t)II z--Tv(w,v, ta(w))II E 0. (*) For each a, define Fv, a'fl-.-.WK(X) by F,() = w-t{,t .().,* >_ }.
Define Fu:fl--WK(X by Fv(w = Fu,(w ).Then, as in the proof of Theorem 4.1, F is w-measurable and has a measurable selector .urthermore, since X is weakly compact, {,tc(w)} converges weakly, say o (u(w).Combining this with (,), the B-G property implies Tu(w, u(w)) = u(w) for w e f2.Now let u,v Y, w E f2 I I C,.t() o. ,(,) II E I I Tu(w, [u, t(w)) To(w, v, t(w)) I I E !1 T(w, [u, t(w), u) T(cz, v,t(w), v) [[ _< ll (,, t(), u) --(o, t(,), v)II = max.{I I , , , t()-,, t()II E, I I ' v II F} l_<P<c and (x, y) E F, set il (,y)II p (11 I I z + il y I I and fo e , I I (,Y)II ma{ I I I I E, I I Y I I F}" for u,v in Y and w .It further implies I I o( )II E <ldrncosuplinrtsup I I < I I = v II F"