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Internat. J. Math. & Math. Sci. VOL. ii NO. 3 (1988) 551-560 551 RANDOM FIXED POINTS AND RANDOM DIFFERENTIAL INCLUSIONS NIKOLAOS S. PAPAGEORGIOU Unlverslty of Callfornla 1015 Department of Mathematlcs Davls, California 95616 (Received July 28, 1987 ABSTRACT. using and in revised form October 19, 1987) In this paper, first, we study random best approximations to random sets, point techniques, obtaining this way stochastic analogues of earlier fixed deterministic results by Broder-Petryshyn, KyFan and Reich. Then we prove two fixed point theorems for random mmltifunctlons with stochastic domain that satisfy certain tangential conditions. Finally we consider a random differential inclusion wth upper semlcontinuous orlentor field and establish the existence of random solutions. KEY WORDS AND PHRASES. Random flxed point, Random solutions, Random Differential Incluslon and Abstract Differential Equations. 1980 AMS CLASSIFICATION CODE. I. 60H25, 47HI0. INTRODUCTION. Random fixed point theorems are stochastic generalizations of point theorems and are needed in the study of random equations. classical fixed Their study was initiated by the Prague school of probabillsts, with the works of Hans [I] and Spacek [2]. Recently the interest in these problems was revived by the survey article of Bharucha-Reld [3]. Since then, there has been a lot of activity in this area and several interesting results have appeared. In this paper, approximations and we will study will derive random fixed points in connection with random stochastic analogues of some results by Browder- Petryshyn [4], KyFan [5] and Reich [6]. proved by Engl We also extend a random fixed point theorem [7] and finally we prove the existence of a solution for a random differential inclusion with an upper semlcontlnuous orlentor field, extending this way an earlier result of the author [8] (theorem 5.1). For the corresponding deterministic theory, we refer to the recent books of Goebel-Relch [9] for fixed points (in connection with the study of the geometry of the underlying space) and of Aubln-Cellna [I0] for differential ncluslons. Another nice wDrk, bringing together the two main mathematical branches considered in this note, namely fixed point theory and differential equations, where an interesting approach to fixed point theory existence theory of abstract differential equations. is the paper of Reich is presented, through [II], the N.S. PAPAGEORGIOU 552 PRELIMINARIES 2. (R,E) be a measurable space and X a separable Banach space. Let Throughout this work, we will be using the following notations: {Ac_X: Pf(c)(X) (convex)} nonempty, closed and nonempty, compact, convex} Pkc(X) --{At_X: Pf(X) K:R Let be UcX open, we have that _a We say that K(.) is measur.able, if for all multlfunction. [mE:K(m) K (U) 0 U}E. It can be shown see Himmelberg that the above definition of measurability of K(.) is equivalent to saying that [I0] zEX, any for map the d(z,K(m)) 0 Furthermore, if there exists a complete inf{Iz-xH :xK(m)} measurable. is E, then the {(re,x) ExX:xEK(m)}ExB(X), above two o-finite measure on properties are equivalent to saying that GrK where Schal [12] and Engl [7], we will say Following B(X) is the Borel o-field of X. and there exists a countable set is it measurable if is that K: separable, Pf(X) It is not difficult to show that if K(.) is measurable DcX s.t. cl(DO K(m))--K(m). for all mER, then K(.) is a cl(intK(m)) with nonempty, closed values and K() separable multifunctlon. This is the case for example, when K(.) has closed, convex, solid values. Y,Z Let be topological Hausdorff spaces + G (U) Pf(X). Recall that (Pf(X),h) We Pf(X) and let F:GrK Pf(X). Hausdorff metric on Let and K: multlfunction with stochastic domain K(.), if K(.) say that F(.,.) compact e.t.c.) on fixed point (.) Finally, if will denote by of is a S 3. + is a measurable o-finite measure on G the {gLI(X):g()G()-a.e.}. and only if xeX and is said to be u.s.c. (continuous, u.s.c. Maps with stochastic domain were introduced by Engl [7]. K(m). F(.,.) mE, x(m)K(m) and x(0)EF(,x(m)). we a random a is is measurable and for all Such an F(.,.) {m:xK(m), F(,x)0U}E. F(m,.) is (continuous, compact e.t.c.), if for all ER, random be is a complete metric space. UcX open, we have A 2Z{} G:Y let We say that G(.) is upper semlcontlnuous (u.s.c.), if for all Uc._Z Also by h( ,.) we will denote the {yY:G(y)c_U} is open in Y. multlfunction. open, two inf{llx :xEG() set of l x: map and integrable G:R X s.t. Pf(X) selectors for all is measurable, of G(.) i.e. It is easy to check that this set is nonempty if belongs in L+ RANDOM APPROXIMATIONS AND RANDOM FIXED POINTS. We will start with a random version of proposition 2.3 of Reich [6], which in KyFan [5] (theorem 2). turn was an extension of an earlier very interesting result of In this section (R,E,) is a complete o-finite measure space. Also recall It is llf(x)-f(y)l llx-yH for all x, yEX. on a Goebel-Reich the [9]) that metric (see projection for known closed, example well convex set in a Hilbert space, is nonexpanslve. That’s why in theorems 3.1, 3.2, 3.3 that a map f:X X is nonexpansive, if 553 RANDOM FIXED POINTS AND RANDOM DIFFERENTIAL INCLUSIONS and 3.4, that follow and involve the metric projection (either in their statement or proof), we assume that the ambient space is a Hilbert space. If X is a separable Hilbert space, K: Pfc(X) in their is a separable THEOREM 3.1. X is a random, nonexpanslve map, with stochastic domain f:GrK multlfunction and , s, K(.) s.t. for all f(,K()) Then there exists is bounded. x: X x()K() and llx()-f()ll- d(f(0,x()),K()). From theorem 3.4 of [13], we know that there exists s.t. for all PROOF. Caratheodory extension of IGrK continuous and f(.,.) f)" Let We have already mentioned that (,x) (i.e. p():X p()(.) K(t) is measurable, measurable :xX X (,x) x also easy is K(). de the metric projection on is nonexpanslve and it is a to show Let p(t0)(z) is measurable. zsX, C(t0) =conv D (pof)(,y), where D is the yeD C() is a Hence countable set postulated from the separability of K(.). is and C(t0) (pof)(,.):C() measurable multlfunction. For every [14]), (see every for that conv(pof)(,K(t)). C() Note that , So from Broler [15], we know that it has a fixed point. nonexpanslve. multlfunctlon L: Pf(X) [xC(m):(pof) (,x)--x} L(0) {xc() (po) (,x)--x} {(m,x)exX: (po) (,x) GtL (pof) (,x) (,x) But x} N GrC is measurable in Applying theorem 3 of Salnt-Beuve [16], we get e. 0 Hence it is and continuous in x. Also since C(.) is measurable, jointly measurable. for all Consider the defined by: x: GrCEExB(X). X GrLEExB(X). Thus measurable s.t. x()eL(m) x()eK(m) and llx()-f(,x(m))l] --d(f(,x()),K(m)) If K(.) is bounded values, the assumption of the range of f(,.) can Therefore we have: REMARK I. be dropped. REMARK 2. Another result in the direction of theorem 3.1 above with a different set of hypotheses, can be found in [17] (theorem 4). We have a similar result for condensing maps. Recall that f:X X if it is continuous and for all BcX nonempty, Y-condenslng, (B)>O, ((f(B))<Y(B), where "((.) THEOREM 3.2. , f:GrK is said to be bounded s.t. is the Kuratowskl measure of noncompactness. If X is separable Hilbert space, Pfc(X) K:l is separable and X is a random condensing map with stochastic domain K(.) s.t. for all x: X measurable s.t. for all Then there exists f(,K()) is bounded. + d(f(.x()),K()). Ix()-f(,x()) Is the same as in theorem 3.1, using this time the fixed point of Furl- el x()sK() and PROOF. Vignoli [18]. Using theorem 3.1, we can have the following random version of a fixed point due to Broder-Petryshyn THEOREM 3.3. If X is a separable Hilbert space, bounded values and K(.) s.t. [4]. for every f:GrK X xebdK() K:R / Pfc(X) is separable with is a random, nonexpansive map with stochastic domain for which p(,f(,x)) x, we have f(,x) x. 554 N.S. PAPAGEORGIOU Then f(..) admits a random fixed point. x:R X d(f(,x()), K()))= Uf(,x())-p(,f(,x))ll. x() p(,f(,x())). If x(w)ebdK(), Since the best approximation is unique, Otherwise we must have that then by hypothesis x() f(,x()). p(,f(,x())) x(), wen. f(,x()) f(,x())eK() REMARK. In the previous theorem, we can instead assume that for all eR f(,.) Then in the proof we have to use theorem 3.2. is condensing on K(). (see remark I), we know that there exists From theorem 3.1 PROOF. lx()-f(,x())U measurable s.t. Now we pass to multlfunctlons and prove the following random fixed point theorem. THEOREM 3.4. F:GrK Pfc(X) K:R If X is a separable Hilbert space, is h-contlnuous, an stochastic domain K(.) s.t. for all Pfc(X) random Y-condenslng, multlfunctlon xebdK(), F(,x) ei% and for is separable and 0 p -I F(,K()) is bounded. Then F(.,.) admits a random fixed point. PROOF. Let G:RxX Pfc(X) be the multlfunctlon defined by From our hypotheses on F(.,.), we see that F(,p(,x)). G(,x) G(,x) is h-contlnuous. Also we claim that G(,.) measurable, while x So let BcX be nonempty, bounded, with y(B)>O. We have .Y-condenslng. c__{x} with (,x) and Y(GC,B)) YCFC,p(,B))) < yCPC,B)) C(00) {x p(,.) Then F(.,), G(,.):C(0) clearly we have C() that cOnvnU L(m) F(,xn) {xeC() Then consider the multifunction defined by measurable. of theorem men, is nonexpanslve. Note that if C(). is the countable set postulated from the separability of K(.) and exploiting convF(,K()). n n.>l the h-continuity of From is YCB) the last inequality being a consequence of the fact that Let G(,x) is Himmelberg-Port e r-Van 19 ], Vle ck we know C() is xeG(,x)}. that for al 1 {(m,x)exX:d(x,G(,x)) 0}0 GrCeZxB(X). Again theorem 3 of Saint-Beuve [16], produces a measurable map x: + X s.t. x(m)eL(), for all R. Let ()=p(,x()). Clearly (.) is measurable and ()bdK(). -I Then x() F(m,()) -x(m)ep p (,(m)) and x()eG(,x()) N (m) x(m) x(m)eK(m) and x()eF(m,x(m)) i.e. x(.) is the desired F(,()) L(m)0. Also note that GrL -l(,x(m)) random fixed point. REMARK. If there m dependence no is of the data in the previous theorem (deterministic case), then we can relax the hypotheses on F(.) and simply assume that F(.) is closed, Y-condensing and with bounded range. Also in the deterministic case, the theorem can be proved for general Banach spaces, if we assume approximatively w-compact and F(.) is w-u.s.c., with w-compact range. analogous to the random mult ifunction u.s.c, theorem. case and in the general [6], which tells us that the proposition 2.1 of Pelch and eventually apply the Banach space, we that K is The proof is have to use metric projection on K is a w- Kakutani-KyFa n Both those deterministic versions of theorem 3.4, fixed point extend theorem 3.3 of [6]. Note that the second deterministic result that was stated in general Banach spaces, can not be extended to the random case, since as it was illustrated Reich with a counter example in u.s.c, in [20], a multifunction G(m,x) x, is not in general jointly measurable. which is measurable in , w- RANDOM FIXED POINTS AND RANDOM DIFFERENTIAL INCLUSIONS 2Xx {} is F:K F(K) compact, if l(x,K)={zX:z=x+(y-x) for some translation cone" tangent the to of the Recall that a ,altifunction Also if KcX is convex is compact. yX and >0}. xK, we define So l(x,K) is nothing else, but the "Bouligand the well known from nonsmooth analysis, (see Aubin-Ekeland [21]). to K at x nonexpansiveness x of point [7]. theorem 8 of Engl result extends The next 555 metric the projection, Since we do not need any more the result can be stated for general separable Banach spaces. K: THEOREM 3.5. If X is separable Banach space, Pfc(X) is separable and F:GrK is a compact, u.s.c., random multifunction with stochastic domain Pfc(X) PROOF. H:GrK . F(.,.) a random fixed point. From proposition 5 of Engl [7], we know that there exists F(,x) K(.) s.t. Pfc(X) l(x,K()) c a multifunctlon s.t. (i) (m,x)gGrK, H(,x)cF(,x) for each (il) for every (iii) H(.,.) Then let Then L: 2 X m, H(m.,) is u.s.c, on K(m) is jointly measurable be defined by {xeK(m):xeH(,x)} L(m) From theorem 3.1 of Reich [6], we know that for all me, L()0. Observe that: {(m,x)eRxX:xeK(m), xeH(,x)} CrL ((RxD) N GrH) proJ xX where D=-{(x,y)eXxX:x=y}. But note that GrHer.xB(X)xB(X). So Then using the theorem in section 39.1V of Kuratowski (xD) N GrHelxB(D). get that prOJRxx(RXD) GrFeExB(X). Hence GrLeZxB(X). x:R X measurable x(o)sH(to,x(o))cF(o,x(o)). 3 Saint-Beuve [16], we get x(to)eL(o), osn 4. [22], we Once again, through theorem s.t. RANDOM DIFFERENTIAL INCLUSIONS. Let interval (,-,) in R+, be a complete probability space, T=[0,b] X a finite dimensional Banach space and a nonempty, closed, bounded x o : X measurable. We consider the following random differential inclusion: x(m,t)eF(,t,x(,t)) a.e. x(,0) for all wen x () o By random solution of (*), we understand a stochastic process absolutely continuous realizations, satisfying (*) a.e. in t, for all x:xT X, with e. In this section we present a theorem on the existence of random solutions of (*), N.S. PAPAGEORGIOU 556 that generalizes theorem 5.1 of [8]. If F :xTxX Pf (X) is a multifunction s.t. measurable is F(m,t,x) (m,t,x) F(,t,x) is u.s.c. for all (,t)eRxT, x a(m,t)+b(m,t) xUa.e. for all JER, with a( ,.) ,b( ,.) measurable Then (*) admits a random solution. a(,.) ,b( THEOREM 4. I. (2) (3) and IF(,t,x) l .)L We will by determining an a priori bound for the random solutions of PROOF. (*) So let x(.,.) be a random solution. meR, Fixing we have: Applying Gronwall’s inequality, we get that: (11 x ()11 +ila(,.)II o IIx(,tll l) exp lib(w,.)II F(m,t,x) Set (m,t,x) (.,.,.) It is easy to check that IIxII if (.,.) random orientor field (,.)L+I. measurable and ( > M() has the same measurability and semicontinuity properties as F(.,.,.) and furthermore we have that Let M() if Ilxll M()x) F(m,t,- --(,t), with M() l(,t,x) la(m,t)+b(m,t)M(m) We will consider (*) with the ). W(m)cC(T,X) be defined by: t W(m) Define g(s)ds, tT, llg(t)ll (m,t) a.e.} x () + {xC(T,X):x(t) o :xC(T,X)xLI(x) 0 C(T,X) by: t (,x,g)(t) Clearly measurable. g(s)ds-x(t) x ()+ o 0 (.,.,.) is measurable Also if (m)=B(O, (,.)III) is the closed ball ll(m,.)lll, then (.) is measurable and the origin with radius GrW ((,x)exC(T,X): GRWgr-xB(C(T,X)), in and continuous in (x,g). ll(m,x,g)ll 0, in d(g,())=0} (since g=x, see Kuratowski [22], 39.1V). So is jointly LI(X), centered at 557 RANDOM FIXED POINTS AND RANDOM DIFFERENTIAL INCLUSIONS Furthermore, a simple application of the Arzela-Ascoli theorem, tells us that for me, every W(m) is a compact subset of C(T,X). Next let (m,x) Define :GrW 2 C(T’X)xLI(X) [.} be the multlfunctlon defined by: {(y,f)eC(T,X)xLl(X):y(t)--x o (m)+ :xC(r,X)xLl(X) //R+ by: (,x, f) d(f,S t I^F(, f(s)ds, teT, feS 0 ,x( (m,. ,x(.) Note that: d(f,s^ F(,. ,x(.)) inf{ [If-h heS F(,.,x(.)) b inf{ llf(t)-h(t)ll dt:heS (,. ,x(.)) :ze(0,t,x(t))}dt 0 b inf{ llf(t)-z 0 b d(f(t),(m,t,x(t))dt 0 d(z,(m,t,y)) a (m,t,y) measurable and z d(z,(,t,y)) is continuous. Hence (w,t,y,z) d(z,(m,t,y)) is measurable. A/so the evaluation map e t(x(.))=x(t), is continuous from TxC(T,X) into (t,x(.)) X. Hence we deduce that (m,t,x(.)) d(f(t),F(m,t,x(t))) is measurable from xTxC(T,X) into Rewrite (.,.) as follows: But by hypothesis (I), IR+ (,x) {(y,f)eC(T,X)xLI(x) [(,y,f) ll P_cC(T,X)xLI(x) Let be defined O, by P={(y,f):=f}. first variable is one-to-one, continuous. So if 6(,x,f) for fixed R(,x) (m,x), proj , R(m,.):W() C(T,X) W(m). W(m)c__C( T, X) is compact, C(T,X)xC(T,X). So let (x n it 0} Then the projection to the P) N EZxB(C(T,X))xB(C(T,X)). GrRer-xB(C(T,X))xB(C(T,X)). then Note that We claim that it is u.s.c. suffices ,yn)GrR(m O} Thus by Kuratowskl [22] (39.1V): prOJxC(T,X)xC(T,rR=proJxC(T,X)xC(T,X)( (GrW {(m,x,y,f):y(0)--x (), o (,x,f) .) to show n)l s.t. that (Xn,Yn) To show this, since GrR(m, .) is closed in (x,y) By definition: t Yn(t)--Xo()+ of f (s)ds tT f eS F(w,. ,Xn(.) From the Dunford-Pettis compactness criterion, we deduce that sequentially w-compact in LI(x). {f w n n)l is So by passing to a subsequence if necessary, we N.S. PAPAGEORGIOU 558 may assne that f f in L (X). n f(t)econv llmffn(t)} n) I=c The inclusion last cony Using theorem 3.1 of [23], we have that: llm being a (.,.,.). convexity of the values of F(m,t,Xn(m)) consequence of So feS a.e. _c F(m,t,x(t)) the upper semlcontinuity and the a.e. Also F(m,. ,x(.)) t y(t) x (m)+ f f(s)ds, teT 0 (x,y)eGrR(,.) R(m .) is Let L:R 2 L(m) Since for C(T’x) from W(m) into W(m). u.s.c, be defined by: {xeC(T,X):xeR(m,x), xeW(m)} fixed wen, R(m,.):W(m)/ W(m) L(m)@, fixed point theorem, we have that is u.s.c., for every wen. from the Kakutani-KyFan Then as in the proof of theorem 3.5, we have: prOJnxC(T,X)((nxD) GrL 0GrR)eExB(C(T,X)) Apply theorem 3 of Salnt-Beuve [16], to get D {(x,y)eC(T,X)xC(T,X):xffiy}. wen, r(m)eL(m). Set x(m,t)fr(m) (t). for all s.t. C(T,X) measurable, Clearly this is a random solution of (*) with orlentor field F. 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