RANDOM FIXED POINT THEOREMS FOR NONEXPANSIVE AND CONTRACTIVE-TYPE RANDOM OPERATORS ON BANACH SPACES ISMAT

The existence of random fixed points for nonexpansive and pseudocontractive random multivalued operators defined on unbounded subsets of a Banach space is proved. A random coincidence point theorem for a pair of compatible random multivalued operators is established.


Introduction
Random fixed point theorems for contraction mappings in Polish spaces were proved by Spacek [20], Hans [7, 8] and many others.For complete survey, we refer to Bharucha-Reid [3].Itoh [10, 11] established several random fixed point theorems for various commuting single and multivalued random operators.Afterwards, Sehgal and Singh [18], Papageorgiou [17] and Lin  [15] proved different stochastic versions of well-known approximation result of Fan [4] and obtained some random fixed point theorems.Recently, Beg and Shahzad [2] studied the structure of common random fixed points and random coincidence points of a pair of compatible random multivalued operators in Polish spaces.The purpose of this paper is to prove some random fixed point theorems for random multivalued nonexpansive and pseudocontractive operators defined on closed convex unbounded subsets of a Banach space.Section 2, is aimed at clarifying the termino- logy to be used and recalling some necessary definitions, in Section 3, the existence of random fixed points for nonexpansive random multivalued operators defined on an unbounded closed convex subset of a Banach space is established.A random fixed point theorem for Lipschitzian pseudocontractive operators is also proved.Section 4 contains a random fixed point theorem for a (a)-firmly nonexpansive random operator in separable Banach spaces.Section 5 deals with random coincidence point theorems for a pair of compatible random multivalued operators satis- iResearch supported by the NSRDB Research Grant No. M. Sc. (5)/QAU/90.2Florida Institute of Technology, Program of Applied Mathematics, Melbourne, Florida 32901-6988, USA.

Preliminaries
Throughout this paper, (fl, A) denotes a measurable space.Let (X,d) be a metric space, (X) the family of all subsets of X, %(X) the family of all nonempty compact subsets of X, %(X). the family of all nonempty closed bounded subsets of X and %(X) the family of all nonempty closed bounded convex subsets of X.A mapping T:(X) is called measurable if for any open subset C of X, T-1(C) {w E f: T(w)gl C # } E A. A mapping : f--,X is said to be a measurable selector of a measurable mapping T:f--,(X) if is measurable and for any c0 f, (co) T(w).Let M be a subset of X.A mapping f:fxM--X is called a random operator if for any x M, f(.,x) is measurable.A mapping T:fxMe%(X) is a random multivalued operator if for any x M, T(. ,x) is measurable.A measurable mapping : f---M is called a random fixed point of a random multivalued (single-valued) operator T:f x M--,e%(X) (/: f x MX) if for every a E f, (o) T(o, (o))((o) f(0, (o))).A measurable mapping : f--,M is a random coincidence point of T: f x Me%(X) and f: f x M---,X if for every w f, f(w,((c0)) T(co,((o)).A mapping T: M---,e%(X) is upper (lower) semicontinuous if for any closed (open (open).A mapping T is called continuous if T is both upper and lower semicontinuous.A mapping T:M--,E%(X) is called Lipschitzian if H(Tx, Ty) < k d(x,y) for any x,y M, k > O, where H is the Hausdorff metric on %(X), induced by the metric d.When k < 1 (k 1) then T is called contractive (nonexpansive).A mapping T: M---,C%(X) is pseudocontractive if for any x, y M, u Tx, v Ty, r > 0, we have I Iy I I < II(1 + r)(x-y)-r(u-v)II- A random operator T: f x MC%(X) is pseudocontractive if T(w, is pseudocontractive for each 0 e a (in this case, r:a[0,oc)is a measurable mapping).Following Itoh [10], a measurable mapping L:a---,[0, oc) is called Lipschitz measurable mapping of T if, H(T(co, x),T(w,y)) <_ L(co)d(x,y).Let T:f x M--,E%(X) be a random operator and {,,} be a sequence of measurable mappings n:f--M.The sequence {n} is said to be asymptotically T-regular if d(n(co), T(co, n(co)))--0 for each a e a. Mapping T: M--,e%(X) is said to be demiclosed if the conditions that x n converges weakly to x,y n converges to y, and Yn Txn' imply that y Tx.For any A e %(X), we denote with CD(A) the closed convex hull of A.
3. Pudom Fixed Points for Random Multivalued Operators Defined on Unbound- ed Sets In 1978, Goebel and Kuczumov [5] proved that, if X is a closed convex subset of 2 and T: XX is nonexpansive for which there exists a point x X such that the set LS(x, Tx; X): {z e X: <z x, Tx x) >_.0) is bounded, then T has a fixed point in X. Kirk and Ray [13] have shown that if X is an un- bounded closed convex subset of a uniformly convex space and T: XX is a Lipschitzian pseudo- contractive mapping for which the set (, Tx; x): (z e X: I I z-Tx I I I I z-x II} is bounded for some x E X, then T has a fixed point in X.Subsequently, Marino [16] extended Random .Fixed Point Theorems for Nonexpansive 8J Contractive-Type Random Operators 571 these results to the multivalued case and improved some known results. Let X be a real Banach space and let K be a nonempty convex subset of X.We set (for any x, y E X) r(z,y): lim I I + tu I I I I I I tO + Following [16] we define, for z, y E X, e > 0, A C_ X, LS(x, y; K): -{zGK:v(x-z,y-x)<0} and LS(x,A;K)" -{zK'3aGA:v(x-z,a-x)<0} U LS(x,a;K) LS(x, A, ; K): -{zK'qaEA:r(x-z,a-x)<} [.J cEA The aim of this section is to establish random fixed point theorems for nonexpansive and pseudocontractive random operators defined on unbounded sets in Banach spaces.Theorem 3.1: Let X be a separable closed convex subset of a real Banach space, and let T: f2 x X---%(X) be a nonexpansive random operator.Suppose that for some bounded set W C_ X the set LS(W,T(w,W);X): LS(z,T(w,z);X) is bounded for each w f2.Then there .existszW a bounded sequence {n} of measurable mappings n:f2---X which is asymptotically T-regular.
Proof: Choose an element y X and a sequence icon} of measurable mappings a n" D(0, 1) such that cn(w)-l as n--< pointwise in w.For each n, define a contractive random operator Tn: f2xX---+%(X) by Tn(w,x)"-(1-(w))y+an(w)T(w,x).Then by Itoh [10], T, has a random fixed point n" Assume that the set {n(W)}n e N is unbounded.Then it is possible to choose k @ N such that for each w sup H({y},T(w,z)) < d(k(w),W (1) zW and sup IlPll < lick(w) ll" (2) p @ LS(W, T(w, W); X) We will prove that for any z E W and D, there exists x z T(,z) such that 7(z- z) < 0. Indeed, by .k() (1ak(w))y + ak()T(, k()), it follows that k(w)-(1-ak(w))y + alc(w),k(w), rlk(w for each w E f. (The existence of measurable maps r/k: f/-X is due to Kuratowski and Ryll-Nard- zewski [14].)From the nonexpansivity of T, there exists x z e T(w,z)such that for any w e a, I I ()-z I I _< I I ()-z I I and therefore r(z ,k(w), x z z) < 0 for each w a [16, see proof of Theorem 2]; that is, k(w) LS(W,T(w,W);X), contradicting (2).Thus, M(w)'- sup{ N)< o for all weft, (the mapping M'a---N + is measurable)and W C_ X and e > O, the set Suppose that, for some bounded set LS(W, T(w, W), ; X)" f' LS(z, T(w, z), ; X) zEW is relatively compact for each w E .Then T has a random fixed point.
Corollary 3.3: Let X be a separable closed convex subset of a reflexive (real) Banach space, T:Qx X---,%(X) be a nonexpansive random operator, and for each w , I-T(w,.) be demi- closed on X. Suppose that, for some bounded set W C_ X the set LS(W,T(w,W);X) is bounded for each w .Then, T has a random fixed point.
Proof: As in the proof of Theorem 3.1, there exists a bounded sequence of measurable mappings n: ---,X such that n(w) (1-on(w))y + an(w)rin(w with rln(w for each w a, {tin(w)} is also bounded, and I I n(w) fin(w)II (1 -an(w))II '()II-0 s nc.Fix w E .By reflexivity, there exists a subsequence {k(w)} of {n(w)} such that converges weakly to (w), where :--.X is a measurable mapping.Since n(w)-r/n(w (I- T(w,.)), both (n(w) and n(w)-n(w) converge to 0. Since I-T(w,. is a demiclosed mapping, Corollary 3.4: Let X,T,W be as in Theorem 3.1 and suppose that there exists > 0 such that LS(W,T(w,W),;X) is relatively compact for each w .If I-T(w,.) is demiclosed on X for every w , then T has a random fixed point.
Corollary 3.5: Let X, T, W, LS(W, T(w, W); X) be as in Theorem 3.1.If the Banach space is reflexive and satisfies Opial's condition (that is, if z n converges weakly to z and z v), liminfllzn-z]l <liminfllZn-Vll) then T has a random fixed point.
Corollary 3.6: Let X, T, W, LS(W, T(w, W); X) be as in Theorem 3.1.If Banach space is uni- formly bounded, then T has a random fixed point.
Random Fixed Point Theorems for Nonexpansive Contractive-Type Random Operators 573 Theorem 3.7: Let Y be a reflexive real Banach space which satisfies Opial's condition.Let X be a separable closed convex subset of Y and let T:f x X---,%(Y) be a Lipschitzian pseudocon- tractive random operator which satisfies the inwardness condition: for any x E X, T(w,x) C_ Ix(x), for each w G f2. Suppose that there exist x o G X and e > 0 such that LS(Xo, Co(T(W, Xo)), ; X) i8 boltnded for each co ft.If Z(co, X)(-1B r is compact for any B r {z G X: I I I I _< ) then T has a random fixed point.
Proof: Let L be a Lipschitz measurable mapping of T.
Then, for any y G X, define a contractive random operator Tu'f x X--,%(Y) by Tu(w,x -(1- a(co))y+c(co)T(co, x).The operator T u satisfies the inwardness condition" for any z e X Tu(co z) C_ Ix(x for each co e a. Consider the mapping G:fxX--.(X)defined by G(w,y)-{x:x G Tu(w,x)}.For any y e X, G(.,y):ft--2(X) is measurable [11, see proof of Theorem 3.1].Also G(w,y)is nonempty and closed for every y G X and co G ft.It is easy to verify that G(w, y) c_ (1-a(w))y + a(w)T(w, G(w, y)) (3) and (w) e G(co, (co)) iff (w)e T(co,(co)) for each co e a (where c: a---X is a measurable map).For u, v ( X (fixed), I I a-b I I I I u-v I I for any a e G(co, u) and b e G(co, v), for each co ft. (5) and so, from pseudocontractivity of T, from ( 6) and ( 7) and choosing r(co)< a(w)/(1-a(w)) for each co Eft, I I a b I I I1(1 + r(w))(a b) r(w)(r (,)II I I (), I I a-b I I + Therefore r(co)(1 --C(co))C-l(co)II a-b I I < r(co)(1 --C(co))C-l(co)l] tt--V il, proving (5).
It follows, in particular, that G(w,y) belongs to N(X) for any y E X and w E .Besides, T(w,G(w,y)) is bounded (since T is Lipschitzian), and by (3) we can conclude, under the hypothesis, for each w G, T(w,X) VB r is compact for any r, that, for any y G X, a(,y) e %(X).From H(G(w, x), G(w, y)) <_ sup{ I I a-b I I a e G(w, x), b e G(w, y)), and from (5)it follows that the random operator G:ax X--.%(X)is nonexpansive, and by (4), has the same random fixed point of T. The set LS(xo, G(w, Xo);X is bounded for each w [16, proof of Theorem 8], Corollary 3.5 implies that G has a random fixed point.
4. Random Fixed Points of Firmly Nonexpansive Random Operators Let C be a nonempty subset of a Banach space X, and let A ( (0,1).
T: C--,X is said to be A-firmly nonezpansive if Then, a mapping I I Tx-Ty I I <_ II( 1 A)(x-y)+ A(Tx-Ty) I I (8) for all x, y C. In particular, if (8) holds for every A (0, 1), then T is said to be a firmly nonex- passive mapping.It is clear that every A-firmly nonexpansive mapping is nonexpansive.Conversely, with each nonexpansive mapping T:C--+C ..one can associate a firmly nonexpansive mapping with the same fixed point set, whenever C is closed and convex [6].A random operator T:a x C-C is said to be A(w)-firmly nonexpansive for some measurable mapping A:f--,(0, 1), if T(w,.) is (w)-firmly nonexpansive for each w E 12.The aim of this section is to obtain a random fixed point theorem for a A(w)-firmly nonexpansive random operator which is a stochastic analogue of theresult by Smarzewski [19].
5. A Random Coincidence Point Theorem Jungck [12] gave the notion of compatible single valued mappings.Subsequently, Beg and Azam [1] introduced th notion of compatible multivalued mappings and proved various Banach type fixed point theorems for multivalued mappings.In this section we obtain a random coinci- dence point theorem and random fixed point theorem for random operators satisfying a contrac- tive type condition.
.eom compatible and for all x,y G X and H(T(w, x), T((0, y)) < A(w)d(f(w, x), f(w, y)) ( 13) Random Fixed Point Theorems for Nonexpansive 8 Conlractive-Type Random Operators 577 (where A:2--(0, 1) is a measurable map), then there is a random coincidence point of f and T.
an(w)) (i-an(w)) d((n(w T(w, n(w))) < I I n(w) y I I < M(w)---O Theorem 3.2: Let X and T be as in Theorem 3.1.