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Random fixed point theorems for composites ofacyclic multifunctionsLiaqat Ali Khan aa Department of Mathematics King Abdul Aziz University PO Box 80203 Jeddah21589 Saudi ArabiaPublished online 15 Feb 2007

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RANDOM FIXED POINT THEOREMS FORCOMPOSITES OF ACYCLIC

MULTIFUNCTIONS

Liaqat Ali Khan

Department of Mathematics King Abdul Aziz University

PO Box 80203 Jeddah-21589 Saudi Arabia

E-mail akliaqathotmailcom

ABSTRACT

In this paper we obtain random versions of KakutanindashFan type fixed

point theorems for a class V1c of multifunctions which contains

Kakutani factorizable maps and composites of acyclic maps As

applications we derive some random approximation theorems

1 INTRODUCTION

The study of random fixed point theory was initiated by the Prague school

of probabilistics in the fifties see eg Spacek (20) and Hans (5) Recently the

interest on the subject was revived especially after the publication of article of

Barucha-Reid (2) and later of Itoh (7) On the other hand random approximation

received further attention after the appearance of papers by Sehgal and Waters

(17) Sehgal and Singh (16) Papageorgiou (12) Lin (10) etc For more recent

contribution see (111181921) among others

The aim of this paper is to obtain random versions of KakutanindashFan type

fixed point theorem and Fan type approximation theorem for a class V1c of

multifunctions on Frechet and normed spaces This class contains compact acyclic

maps admissable maps Kakutani factorizable multifunctions and composites of

acyclic maps Our main results are the randomizations of most of the results of a

925

Copyright q 2001 by Marcel Dekker Inc wwwdekkercom

STOCHASTIC ANALYSIS AND APPLICATIONS 19(6) 925ndash931 (2001)

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recent paper by Park Singh and Watson (14) In particular these results extend

some results of Itoh (7) and of Sehgal and Singh (16) in some respect

2 PRELIMINARIES

Throughout this paper (V S ) denotes a measurable space with S a

s-algebra of subsets of V and for any nonempty set Y 2Y denotes the family of

all nonempty subsets of Y If X is a Frechet space (ie a complete metrizable

locally convex space) we may assume that the topology of X is generated by a

countable family pn of continuous seminorms with pn pn11 for all n $ 1 and

a metric d on X is given by

dethx yTHORN frac14X1nfrac141

cnpnethx 2 yTHORN

1 1 pnethx 2 yTHORN

for all x y [ X where cn 0 andP1

nfrac141cn 1Let S be a nonempty subset of a Frechet space X A mapping G V 2S is

called measurable (resp weakly measurable ) if for each closed (resp open)

subset A of X G21ethATHORN frac14 v [ V GethvTHORN ndash f [ S A mapping f V S is

called a measurable selector of G V 2S if f is measurable and for each

v [ V fethvTHORN [ GethvTHORN A map T V S 2X is called a random operator if for

each x [ S the map Teth xTHORN V 2X is measurable A measurable map f V S

is called a random fixed point of T V S 2X if fethvTHORN [ TethvfethvTHORNTHORN for each

v [ VFor any nonempty bounded subset B of a Frechet space X Kuratowski

measure a of noncompactness is defined by frac12aethBTHORNethpnTHORN frac14 infc 0 B can be

covered by a finite number of sets whose diameters with respect to pn are c

(cf Sadovskii (15)) A mapping F S 2X is called (a) condensing if for any

nonempty bounded subset B of X with frac12aethBTHORNethpnTHORN 0 frac12aethFethBTHORNethpnTHORN

frac12aethBTHORNethpnTHORN for all n $ 1 where FethBTHORN frac14 ltFethyTHORN y [ B (b) upper

semicontinuous (usc) if for all open subsets V of X y [ S FethyTHORN V is

open in S (c) lower semicontinuous (lsc) if for all open subsets V of X

F21ethVTHORN frac14 y [ S FethyTHORN V ndash f is open in X (d) continuous if it is both

usc and lsc (e) compact if F(S ) is contained in a compact subset of X A

random operator T V S 2X is called continuous compact or condensing

if for each v [ V Tethv THORN S 2X is so We mention that every compact map

is condensing

Let S be a nonempty subset of a normed space X and let x [ X Define

dethx STHORN frac14 infkx 2 yk y [ S and let

PSethxTHORN frac14 y [ S kx 2 yk frac14 dethx S

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the set of best S-approximants of x The set-valued map PS X 2S is called the

mectric projection If PS is a single-valued map it is called a proxmity map S is

called approximatively compact if for each x [ X every sequence xn S

such that kx 2 xnk dethx STHORN has a subnet that converges to a point of S A Banach

space X is said to have the strong Oshman property if the metric projection on

every closed convex subset is continuous X is said to have property (H) ((4) p

20) if X is strictly convex and whenever xn X is such that kxnk kxk and

xn weakly to x then xn x For any x [ X the inward set IS (x ) of S at x is

defined by ISethxTHORN frac14 x 1 rethy 2 xTHORN y [ S r $ 0 A nonempty topological space

Y is called acyclic if all its reduced ech homological groups over rationals vanish

Every convex or starshaped subset of X is acyclic

If X and Y are topogical spaces we define in steps a class V1c ethX YTHORN of all

composites of certain multifunctions as follows F [ VethX YTHORN if F X 2Y is

usc and for each x [ X F(x ) is compact and an acyclic set F [ VcethX YTHORN if

F [ VethX YTHORN and F frac14 Fn+Fn21+ +F0 where each Fi [ VethXiXi11THORN and X frac14

X0 Y frac14 Xn21 i frac14 0 1hellip n 2 1 F [ V1c ethXYTHORN if for any s-compact subset K

of X there is a G [ VcethX YTHORN such that GethxTHORN FethxTHORN for all x [ K Note that V1c

contains acyclic maps admissible maps Kakutani multifunctions and Kakutani

factorizable multifunctions For detail see ((9) (14) etc) We mention that the

composites of acyclic maps need not be an acyclic map ((3) p 205)

3 MAIN RESULTS

We will prove some new random fixed point and approximation theorems

concerning composites of acyclic multifunctions In fact we give stochastic

analogue of most of the results contained in a recent paper of Park Singh and

Watson (14) We begin by establishing a general random fixed point theorem in

the setting of Frechet spaces

Theorem 31 Let S be a closed convex subset of a separable Frechet space x

and let T V S 2S be a continuous compact random operator such that for

each v [ V the map T(v ) belongs to V1c ethS STHORN Then T has a random fixed

point

Proof Define G V 2S by

GethvTHORN frac14 x [ S x [ Tethv xTHORN

Then by ((14) Corollary 1) G(v ) is nonempty for each v [ V Now let

xj D with dethxj Tethv xjTHORNTHORN 0 for any v [ V where D is a countable dense

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 927

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subset of S Set B frac14 xj j $ 1 and gn frac14 frac12aethTethBTHORNethpnTHORN for n $ 1 Then for any

1 0 there exist sets B1hellipBk X with

dethBiTHORN gi 11

2and TethvBTHORN lt

k

ifrac141Bi

Let for each i Ai be an 1=2-neighborhood of Bi and choose an integer N such that

dethxj Tethv xjTHORNTHORN 1=2 for all j $ N Then xj j $ N ltkifrac141Ai and dethAiTHORN

di 1 1 Since 1 is arbitrary

frac12aethBTHORNethpnTHORN frac14 frac12aethxj j $ NTHORNethpnTHORN gn frac14 frac12aethTethvBTHORNTHORNethpnTHORN

for each n $ 1 This implies that frac12aethBTHORNethpnTHORN frac14 0 for each n $ 1 Since S is

complete the set B is precompact and so xj has a convergent subsequence

Using this observation we can next proceed exactly as in ((1) Theorem 31) and

show that for any closed subset C of X

G21ethCTHORN frac14 1

nfrac141lt1

jfrac141v [ V dethxj Tethv xjTHORNTHORN

1

n

Since for each j $ 1 the map v Tethv xjTHORN is measurable it is weakly

measurable by ((6) Proposition) so that by ((6) Theorem 33) the map

v dethxj Tethv xjTHORNTHORN is measurable Thus G21ethCTHORN [ S So G is measurable and

hence again by ((6) Theorem 21) G is weakly measurable Now by ((6)

Theorem 56) or (8) there exists a measurable selector f V S of G Clearly

f is a random fixed point of T This completes the proof A

Remark 32 The above theorem extends some random fixed point theorems in

the literature to Frechet spaces (see (7) Corollaries 22 and 23)

As an application of the above theorem we now obtain a random version of

a very interesting Fan type approximation theorem given in ((14) Theorem 3)

Theorem 33 Let S be an approximatively compact convex subset of a

separable Banach space X having property (H) and let T V S 2X be a

continuous compact random operator such that for each v [ V the map T(v )

belongs to V1c ethSXTHORN Then there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Proof The proximity map P frac14 PS X S defined by kx 2 Pxk frac14 dethx STHORN x [X is well-defined and continuous ((14) Lemma 1) and so P [ V1

c ethSXTHORN Since it

is clear that V1c is closed under composition (9) we have for each v [ V

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PTethv THORN [ V1c ethS STHORN and PT(v ) is compact Define F V S 2S by

Fethv xTHORN frac14 PTethv xTHORN v [ V and x [ S

By Theorem 31 there exists a measurable map f V S such that fethvTHORN [FethvfethvTHORNTHORN frac14 PTethvfethvTHORNTHORN for each v [ V that is

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORN STHORN

for all v [ V where h V S is a measurable selector of T( f()) Now we

can replace S by cl 2 ISethfethvTHORNTHORN in the above eqaution by the argument used in the

proof of ((13) Theorems 1 and 3)

Since every uniformly convex space is reflexive and has property (H) we

immediately have A

Corollary 34 Let S be an approximatively compact convex subset of a

separable uniformly convex space X and let T V S 2X be a continuous

compact random operator such that for each v [ V the map T(v ) belongs to

V1c ethSXTHORN Then there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Remark 35 It is interesting to compare the above results to Theorem 2 of

Sehgal and Singh (16) In fact Sehgal and Singh (16) were the first who obtained

random best approximation results using convex-valued continuous random

operators In this paper we have proved similar results for operators which are

not necessarily convex-valued This leads to the discovery of some new random

approximation results

We now deduce a random fixed point theorem for nonself maps

Corollary 36 Let X S and T be as in Theorem 33 Suppose in addition that

the following boundary condition is satisfied

(A) for each v [ V and each x [ S with x Tethv xTHORN there exists z depending

on v and x in cl 2 ISethxTHORN such that kx 2 yk kz 2 yk for all x ndash y [ Tethv xTHORNThen T has a random fixed point

Proof By Theorem 33 there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f()) If fethvTHORN TethvfethvTHORNTHORN for some v [ V the condition (A) implies that there exists a z [

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 929

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cl 2 ISethfethvTHORN such that kfethvTHORN2 yk kz 2 yk for all y ndash fethvTHORN a contradiction

This f is a random fixed point of T

The next result is a random version of another approximation theorem of

((14) Theorem 4) A

Theorem 37 Let S be a closed convex subset of a separable reflexive Banach

space X with the strong Oshman property and satisfying condition (H) and let

T V S 2X be a continuous compact random operator such that for each

v [ V the map T(v ) belongs to V1c ethSXTHORN Then there exists a measurable map

f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Proof Let P X S be the proximity map as in Theorem 33 Then in view of

the strong Oshman property P [ V1c ethX STHORN and hence for each v [ V the map

PTethv THORN [ V1c ethS STHORN and is compact Hence by Theorem 31 then there exists a

measurable map f V S such that for each v [ VfethvTHORN [ PTethvfethvTHORNTHORN It

follows that the conclusion of Theorem 37 holds A

Finally the author wishes to thank Dr Naseer Shahzad for pointing out an

error in an earlier proof of Theorem 33 and also for useful discussions

REFERENCES

1 Beg I Shahzad N A General Fixed Point Theorem for a Class of

Continuous Random Operators New Zealand J Math 1997 26 21ndash24

2 Bharucha-Reid AT Fixed Point Theorems in Probabilistic Analysis Bull

Am Math Soc 1976 82 641ndash657

3 Carbone A Conti G Multivalued Maps and the Existence of Best

Approximation J Approx Theory 1991 64 203ndash208

4 Fitzpatrick PM Petryshyn WV Fixed Point Theorems for Multivalued

Noncompact Acyclic Mappings Pacific J Math 1974 54 17ndash23

5 Hans O Reduzierende Zufallige Transformationen Czech Math J 1957

7 154ndash158

6 Himmelberg CJ Measurable Relations Fund Math 1975 87 53ndash72

7 Itoh S Random Fixed Point Theorems with an Application to Random

Differential Equations in Banach Spaces J Math Anal Appl 1979 67

261ndash273

8 Kuratowski K Ryll-Nardzewski C A General Theorem on Selector Bull

Acad Polon Sci Ser Sci Math Astronom Phys 1965 13 397ndash403

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9 Lassonde M Fixed Points for Kakutani Factorizable Multifunctions

J Math Anal Appl 1990 152 46ndash60

10 Lin TC Random Approximations and Random Fixed Point Theorems for

Continuous 1-Set-Contractive Random Maps Proc Am Math Soc 1995

123 1167ndash1176

11 Li-Shan L Some Random Approximations and Random Fixed Point

Theorems for 1-Set-Contractive Random Operators Proc Am Math Soc

1997 125 515ndash521

12 Papageorgiou NS Random Fixed Point Theorems for Measurable

Multifunctions in Banach Space Proc Am Math Soc 1986 97 507ndash514

13 Park S Fixed Point Theorems on Compact Convex Sets in Topological

Vector Spaces Contemp Math Amer Math Soc Providence RI 1988

72 183ndash191

14 Park S Singh SP Watson B Some Fixed Point Theorems for

Composites of Acyclic Maps Proc Am Math Soc 1994 121 1151ndash1158

15 Sadovskii BN Limit-Compact and Condensing Operators Russ Math

Sur 1972 27 85ndash155

16 Sehgal VM Singh SP On Random Approximations and a Random

Fixed Point Theorem for Set Valued Mappings Proc Am Math Soc 1985

95 91ndash94

17 Sehgal VM Waters C Some Random Fixed Point Theorems for

Condensing Operators Proc Am Math Soc 1984 90 425ndash429

18 Shahzad N Random Fixed Point Theorems for Various Classes of 1-Set-

Contractive Maps in Banach Spaces J Math Anal Appl 1996 203

712ndash718

19 Shahzad N Khan LA Random Fixed Points of 1-Set-Contractive

Random Maps in Frechet Spaces J Math Anal Appl 1999 231 68ndash75

20 Spacek A Zufallige Gleichungen Czech Math J 1955 5 462ndash466

21 Xu HK Some Random Fixed Point Theorems for Condensing and

Nonexpansive Operators Proc Am Math Soc 1990 110 395ndash400

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recent paper by Park Singh and Watson (14) In particular these results extend

some results of Itoh (7) and of Sehgal and Singh (16) in some respect

2 PRELIMINARIES

Throughout this paper (V S ) denotes a measurable space with S a

s-algebra of subsets of V and for any nonempty set Y 2Y denotes the family of

all nonempty subsets of Y If X is a Frechet space (ie a complete metrizable

locally convex space) we may assume that the topology of X is generated by a

countable family pn of continuous seminorms with pn pn11 for all n $ 1 and

a metric d on X is given by

dethx yTHORN frac14X1nfrac141

cnpnethx 2 yTHORN

1 1 pnethx 2 yTHORN

for all x y [ X where cn 0 andP1

nfrac141cn 1Let S be a nonempty subset of a Frechet space X A mapping G V 2S is

called measurable (resp weakly measurable ) if for each closed (resp open)

subset A of X G21ethATHORN frac14 v [ V GethvTHORN ndash f [ S A mapping f V S is

called a measurable selector of G V 2S if f is measurable and for each

v [ V fethvTHORN [ GethvTHORN A map T V S 2X is called a random operator if for

each x [ S the map Teth xTHORN V 2X is measurable A measurable map f V S

is called a random fixed point of T V S 2X if fethvTHORN [ TethvfethvTHORNTHORN for each

v [ VFor any nonempty bounded subset B of a Frechet space X Kuratowski

measure a of noncompactness is defined by frac12aethBTHORNethpnTHORN frac14 infc 0 B can be

covered by a finite number of sets whose diameters with respect to pn are c

(cf Sadovskii (15)) A mapping F S 2X is called (a) condensing if for any

nonempty bounded subset B of X with frac12aethBTHORNethpnTHORN 0 frac12aethFethBTHORNethpnTHORN

frac12aethBTHORNethpnTHORN for all n $ 1 where FethBTHORN frac14 ltFethyTHORN y [ B (b) upper

semicontinuous (usc) if for all open subsets V of X y [ S FethyTHORN V is

open in S (c) lower semicontinuous (lsc) if for all open subsets V of X

F21ethVTHORN frac14 y [ S FethyTHORN V ndash f is open in X (d) continuous if it is both

usc and lsc (e) compact if F(S ) is contained in a compact subset of X A

random operator T V S 2X is called continuous compact or condensing

if for each v [ V Tethv THORN S 2X is so We mention that every compact map

is condensing

Let S be a nonempty subset of a normed space X and let x [ X Define

dethx STHORN frac14 infkx 2 yk y [ S and let

PSethxTHORN frac14 y [ S kx 2 yk frac14 dethx S

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the set of best S-approximants of x The set-valued map PS X 2S is called the

mectric projection If PS is a single-valued map it is called a proxmity map S is

called approximatively compact if for each x [ X every sequence xn S

such that kx 2 xnk dethx STHORN has a subnet that converges to a point of S A Banach

space X is said to have the strong Oshman property if the metric projection on

every closed convex subset is continuous X is said to have property (H) ((4) p

20) if X is strictly convex and whenever xn X is such that kxnk kxk and

xn weakly to x then xn x For any x [ X the inward set IS (x ) of S at x is

defined by ISethxTHORN frac14 x 1 rethy 2 xTHORN y [ S r $ 0 A nonempty topological space

Y is called acyclic if all its reduced ech homological groups over rationals vanish

Every convex or starshaped subset of X is acyclic

If X and Y are topogical spaces we define in steps a class V1c ethX YTHORN of all

composites of certain multifunctions as follows F [ VethX YTHORN if F X 2Y is

usc and for each x [ X F(x ) is compact and an acyclic set F [ VcethX YTHORN if

F [ VethX YTHORN and F frac14 Fn+Fn21+ +F0 where each Fi [ VethXiXi11THORN and X frac14

X0 Y frac14 Xn21 i frac14 0 1hellip n 2 1 F [ V1c ethXYTHORN if for any s-compact subset K

of X there is a G [ VcethX YTHORN such that GethxTHORN FethxTHORN for all x [ K Note that V1c

contains acyclic maps admissible maps Kakutani multifunctions and Kakutani

factorizable multifunctions For detail see ((9) (14) etc) We mention that the

composites of acyclic maps need not be an acyclic map ((3) p 205)

3 MAIN RESULTS

We will prove some new random fixed point and approximation theorems

concerning composites of acyclic multifunctions In fact we give stochastic

analogue of most of the results contained in a recent paper of Park Singh and

Watson (14) We begin by establishing a general random fixed point theorem in

the setting of Frechet spaces

Theorem 31 Let S be a closed convex subset of a separable Frechet space x

and let T V S 2S be a continuous compact random operator such that for

each v [ V the map T(v ) belongs to V1c ethS STHORN Then T has a random fixed

point

Proof Define G V 2S by

GethvTHORN frac14 x [ S x [ Tethv xTHORN

Then by ((14) Corollary 1) G(v ) is nonempty for each v [ V Now let

xj D with dethxj Tethv xjTHORNTHORN 0 for any v [ V where D is a countable dense

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 927

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subset of S Set B frac14 xj j $ 1 and gn frac14 frac12aethTethBTHORNethpnTHORN for n $ 1 Then for any

1 0 there exist sets B1hellipBk X with

dethBiTHORN gi 11

2and TethvBTHORN lt

k

ifrac141Bi

Let for each i Ai be an 1=2-neighborhood of Bi and choose an integer N such that

dethxj Tethv xjTHORNTHORN 1=2 for all j $ N Then xj j $ N ltkifrac141Ai and dethAiTHORN

di 1 1 Since 1 is arbitrary

frac12aethBTHORNethpnTHORN frac14 frac12aethxj j $ NTHORNethpnTHORN gn frac14 frac12aethTethvBTHORNTHORNethpnTHORN

for each n $ 1 This implies that frac12aethBTHORNethpnTHORN frac14 0 for each n $ 1 Since S is

complete the set B is precompact and so xj has a convergent subsequence

Using this observation we can next proceed exactly as in ((1) Theorem 31) and

show that for any closed subset C of X

G21ethCTHORN frac14 1

nfrac141lt1

jfrac141v [ V dethxj Tethv xjTHORNTHORN

1

n

Since for each j $ 1 the map v Tethv xjTHORN is measurable it is weakly

measurable by ((6) Proposition) so that by ((6) Theorem 33) the map

v dethxj Tethv xjTHORNTHORN is measurable Thus G21ethCTHORN [ S So G is measurable and

hence again by ((6) Theorem 21) G is weakly measurable Now by ((6)

Theorem 56) or (8) there exists a measurable selector f V S of G Clearly

f is a random fixed point of T This completes the proof A

Remark 32 The above theorem extends some random fixed point theorems in

the literature to Frechet spaces (see (7) Corollaries 22 and 23)

As an application of the above theorem we now obtain a random version of

a very interesting Fan type approximation theorem given in ((14) Theorem 3)

Theorem 33 Let S be an approximatively compact convex subset of a

separable Banach space X having property (H) and let T V S 2X be a

continuous compact random operator such that for each v [ V the map T(v )

belongs to V1c ethSXTHORN Then there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Proof The proximity map P frac14 PS X S defined by kx 2 Pxk frac14 dethx STHORN x [X is well-defined and continuous ((14) Lemma 1) and so P [ V1

c ethSXTHORN Since it

is clear that V1c is closed under composition (9) we have for each v [ V

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PTethv THORN [ V1c ethS STHORN and PT(v ) is compact Define F V S 2S by

Fethv xTHORN frac14 PTethv xTHORN v [ V and x [ S

By Theorem 31 there exists a measurable map f V S such that fethvTHORN [FethvfethvTHORNTHORN frac14 PTethvfethvTHORNTHORN for each v [ V that is

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORN STHORN

for all v [ V where h V S is a measurable selector of T( f()) Now we

can replace S by cl 2 ISethfethvTHORNTHORN in the above eqaution by the argument used in the

proof of ((13) Theorems 1 and 3)

Since every uniformly convex space is reflexive and has property (H) we

immediately have A

Corollary 34 Let S be an approximatively compact convex subset of a

separable uniformly convex space X and let T V S 2X be a continuous

compact random operator such that for each v [ V the map T(v ) belongs to

V1c ethSXTHORN Then there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Remark 35 It is interesting to compare the above results to Theorem 2 of

Sehgal and Singh (16) In fact Sehgal and Singh (16) were the first who obtained

random best approximation results using convex-valued continuous random

operators In this paper we have proved similar results for operators which are

not necessarily convex-valued This leads to the discovery of some new random

approximation results

We now deduce a random fixed point theorem for nonself maps

Corollary 36 Let X S and T be as in Theorem 33 Suppose in addition that

the following boundary condition is satisfied

(A) for each v [ V and each x [ S with x Tethv xTHORN there exists z depending

on v and x in cl 2 ISethxTHORN such that kx 2 yk kz 2 yk for all x ndash y [ Tethv xTHORNThen T has a random fixed point

Proof By Theorem 33 there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f()) If fethvTHORN TethvfethvTHORNTHORN for some v [ V the condition (A) implies that there exists a z [

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 929

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cl 2 ISethfethvTHORN such that kfethvTHORN2 yk kz 2 yk for all y ndash fethvTHORN a contradiction

This f is a random fixed point of T

The next result is a random version of another approximation theorem of

((14) Theorem 4) A

Theorem 37 Let S be a closed convex subset of a separable reflexive Banach

space X with the strong Oshman property and satisfying condition (H) and let

T V S 2X be a continuous compact random operator such that for each

v [ V the map T(v ) belongs to V1c ethSXTHORN Then there exists a measurable map

f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Proof Let P X S be the proximity map as in Theorem 33 Then in view of

the strong Oshman property P [ V1c ethX STHORN and hence for each v [ V the map

PTethv THORN [ V1c ethS STHORN and is compact Hence by Theorem 31 then there exists a

measurable map f V S such that for each v [ VfethvTHORN [ PTethvfethvTHORNTHORN It

follows that the conclusion of Theorem 37 holds A

Finally the author wishes to thank Dr Naseer Shahzad for pointing out an

error in an earlier proof of Theorem 33 and also for useful discussions

REFERENCES

1 Beg I Shahzad N A General Fixed Point Theorem for a Class of

Continuous Random Operators New Zealand J Math 1997 26 21ndash24

2 Bharucha-Reid AT Fixed Point Theorems in Probabilistic Analysis Bull

Am Math Soc 1976 82 641ndash657

3 Carbone A Conti G Multivalued Maps and the Existence of Best

Approximation J Approx Theory 1991 64 203ndash208

4 Fitzpatrick PM Petryshyn WV Fixed Point Theorems for Multivalued

Noncompact Acyclic Mappings Pacific J Math 1974 54 17ndash23

5 Hans O Reduzierende Zufallige Transformationen Czech Math J 1957

7 154ndash158

6 Himmelberg CJ Measurable Relations Fund Math 1975 87 53ndash72

7 Itoh S Random Fixed Point Theorems with an Application to Random

Differential Equations in Banach Spaces J Math Anal Appl 1979 67

261ndash273

8 Kuratowski K Ryll-Nardzewski C A General Theorem on Selector Bull

Acad Polon Sci Ser Sci Math Astronom Phys 1965 13 397ndash403

KHAN930

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rary

] at

11

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ber

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ORDER REPRINTS

9 Lassonde M Fixed Points for Kakutani Factorizable Multifunctions

J Math Anal Appl 1990 152 46ndash60

10 Lin TC Random Approximations and Random Fixed Point Theorems for

Continuous 1-Set-Contractive Random Maps Proc Am Math Soc 1995

123 1167ndash1176

11 Li-Shan L Some Random Approximations and Random Fixed Point

Theorems for 1-Set-Contractive Random Operators Proc Am Math Soc

1997 125 515ndash521

12 Papageorgiou NS Random Fixed Point Theorems for Measurable

Multifunctions in Banach Space Proc Am Math Soc 1986 97 507ndash514

13 Park S Fixed Point Theorems on Compact Convex Sets in Topological

Vector Spaces Contemp Math Amer Math Soc Providence RI 1988

72 183ndash191

14 Park S Singh SP Watson B Some Fixed Point Theorems for

Composites of Acyclic Maps Proc Am Math Soc 1994 121 1151ndash1158

15 Sadovskii BN Limit-Compact and Condensing Operators Russ Math

Sur 1972 27 85ndash155

16 Sehgal VM Singh SP On Random Approximations and a Random

Fixed Point Theorem for Set Valued Mappings Proc Am Math Soc 1985

95 91ndash94

17 Sehgal VM Waters C Some Random Fixed Point Theorems for

Condensing Operators Proc Am Math Soc 1984 90 425ndash429

18 Shahzad N Random Fixed Point Theorems for Various Classes of 1-Set-

Contractive Maps in Banach Spaces J Math Anal Appl 1996 203

712ndash718

19 Shahzad N Khan LA Random Fixed Points of 1-Set-Contractive

Random Maps in Frechet Spaces J Math Anal Appl 1999 231 68ndash75

20 Spacek A Zufallige Gleichungen Czech Math J 1955 5 462ndash466

21 Xu HK Some Random Fixed Point Theorems for Condensing and

Nonexpansive Operators Proc Am Math Soc 1990 110 395ndash400

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 931

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Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081SAP120000754

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

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ORDER REPRINTS

the set of best S-approximants of x The set-valued map PS X 2S is called the

mectric projection If PS is a single-valued map it is called a proxmity map S is

called approximatively compact if for each x [ X every sequence xn S

such that kx 2 xnk dethx STHORN has a subnet that converges to a point of S A Banach

space X is said to have the strong Oshman property if the metric projection on

every closed convex subset is continuous X is said to have property (H) ((4) p

20) if X is strictly convex and whenever xn X is such that kxnk kxk and

xn weakly to x then xn x For any x [ X the inward set IS (x ) of S at x is

defined by ISethxTHORN frac14 x 1 rethy 2 xTHORN y [ S r $ 0 A nonempty topological space

Y is called acyclic if all its reduced ech homological groups over rationals vanish

Every convex or starshaped subset of X is acyclic

If X and Y are topogical spaces we define in steps a class V1c ethX YTHORN of all

composites of certain multifunctions as follows F [ VethX YTHORN if F X 2Y is

usc and for each x [ X F(x ) is compact and an acyclic set F [ VcethX YTHORN if

F [ VethX YTHORN and F frac14 Fn+Fn21+ +F0 where each Fi [ VethXiXi11THORN and X frac14

X0 Y frac14 Xn21 i frac14 0 1hellip n 2 1 F [ V1c ethXYTHORN if for any s-compact subset K

of X there is a G [ VcethX YTHORN such that GethxTHORN FethxTHORN for all x [ K Note that V1c

contains acyclic maps admissible maps Kakutani multifunctions and Kakutani

factorizable multifunctions For detail see ((9) (14) etc) We mention that the

composites of acyclic maps need not be an acyclic map ((3) p 205)

3 MAIN RESULTS

We will prove some new random fixed point and approximation theorems

concerning composites of acyclic multifunctions In fact we give stochastic

analogue of most of the results contained in a recent paper of Park Singh and

Watson (14) We begin by establishing a general random fixed point theorem in

the setting of Frechet spaces

Theorem 31 Let S be a closed convex subset of a separable Frechet space x

and let T V S 2S be a continuous compact random operator such that for

each v [ V the map T(v ) belongs to V1c ethS STHORN Then T has a random fixed

point

Proof Define G V 2S by

GethvTHORN frac14 x [ S x [ Tethv xTHORN

Then by ((14) Corollary 1) G(v ) is nonempty for each v [ V Now let

xj D with dethxj Tethv xjTHORNTHORN 0 for any v [ V where D is a countable dense

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 927

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subset of S Set B frac14 xj j $ 1 and gn frac14 frac12aethTethBTHORNethpnTHORN for n $ 1 Then for any

1 0 there exist sets B1hellipBk X with

dethBiTHORN gi 11

2and TethvBTHORN lt

k

ifrac141Bi

Let for each i Ai be an 1=2-neighborhood of Bi and choose an integer N such that

dethxj Tethv xjTHORNTHORN 1=2 for all j $ N Then xj j $ N ltkifrac141Ai and dethAiTHORN

di 1 1 Since 1 is arbitrary

frac12aethBTHORNethpnTHORN frac14 frac12aethxj j $ NTHORNethpnTHORN gn frac14 frac12aethTethvBTHORNTHORNethpnTHORN

for each n $ 1 This implies that frac12aethBTHORNethpnTHORN frac14 0 for each n $ 1 Since S is

complete the set B is precompact and so xj has a convergent subsequence

Using this observation we can next proceed exactly as in ((1) Theorem 31) and

show that for any closed subset C of X

G21ethCTHORN frac14 1

nfrac141lt1

jfrac141v [ V dethxj Tethv xjTHORNTHORN

1

n

Since for each j $ 1 the map v Tethv xjTHORN is measurable it is weakly

measurable by ((6) Proposition) so that by ((6) Theorem 33) the map

v dethxj Tethv xjTHORNTHORN is measurable Thus G21ethCTHORN [ S So G is measurable and

hence again by ((6) Theorem 21) G is weakly measurable Now by ((6)

Theorem 56) or (8) there exists a measurable selector f V S of G Clearly

f is a random fixed point of T This completes the proof A

Remark 32 The above theorem extends some random fixed point theorems in

the literature to Frechet spaces (see (7) Corollaries 22 and 23)

As an application of the above theorem we now obtain a random version of

a very interesting Fan type approximation theorem given in ((14) Theorem 3)

Theorem 33 Let S be an approximatively compact convex subset of a

separable Banach space X having property (H) and let T V S 2X be a

continuous compact random operator such that for each v [ V the map T(v )

belongs to V1c ethSXTHORN Then there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Proof The proximity map P frac14 PS X S defined by kx 2 Pxk frac14 dethx STHORN x [X is well-defined and continuous ((14) Lemma 1) and so P [ V1

c ethSXTHORN Since it

is clear that V1c is closed under composition (9) we have for each v [ V

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PTethv THORN [ V1c ethS STHORN and PT(v ) is compact Define F V S 2S by

Fethv xTHORN frac14 PTethv xTHORN v [ V and x [ S

By Theorem 31 there exists a measurable map f V S such that fethvTHORN [FethvfethvTHORNTHORN frac14 PTethvfethvTHORNTHORN for each v [ V that is

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORN STHORN

for all v [ V where h V S is a measurable selector of T( f()) Now we

can replace S by cl 2 ISethfethvTHORNTHORN in the above eqaution by the argument used in the

proof of ((13) Theorems 1 and 3)

Since every uniformly convex space is reflexive and has property (H) we

immediately have A

Corollary 34 Let S be an approximatively compact convex subset of a

separable uniformly convex space X and let T V S 2X be a continuous

compact random operator such that for each v [ V the map T(v ) belongs to

V1c ethSXTHORN Then there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Remark 35 It is interesting to compare the above results to Theorem 2 of

Sehgal and Singh (16) In fact Sehgal and Singh (16) were the first who obtained

random best approximation results using convex-valued continuous random

operators In this paper we have proved similar results for operators which are

not necessarily convex-valued This leads to the discovery of some new random

approximation results

We now deduce a random fixed point theorem for nonself maps

Corollary 36 Let X S and T be as in Theorem 33 Suppose in addition that

the following boundary condition is satisfied

(A) for each v [ V and each x [ S with x Tethv xTHORN there exists z depending

on v and x in cl 2 ISethxTHORN such that kx 2 yk kz 2 yk for all x ndash y [ Tethv xTHORNThen T has a random fixed point

Proof By Theorem 33 there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f()) If fethvTHORN TethvfethvTHORNTHORN for some v [ V the condition (A) implies that there exists a z [

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 929

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cl 2 ISethfethvTHORN such that kfethvTHORN2 yk kz 2 yk for all y ndash fethvTHORN a contradiction

This f is a random fixed point of T

The next result is a random version of another approximation theorem of

((14) Theorem 4) A

Theorem 37 Let S be a closed convex subset of a separable reflexive Banach

space X with the strong Oshman property and satisfying condition (H) and let

T V S 2X be a continuous compact random operator such that for each

v [ V the map T(v ) belongs to V1c ethSXTHORN Then there exists a measurable map

f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Proof Let P X S be the proximity map as in Theorem 33 Then in view of

the strong Oshman property P [ V1c ethX STHORN and hence for each v [ V the map

PTethv THORN [ V1c ethS STHORN and is compact Hence by Theorem 31 then there exists a

measurable map f V S such that for each v [ VfethvTHORN [ PTethvfethvTHORNTHORN It

follows that the conclusion of Theorem 37 holds A

Finally the author wishes to thank Dr Naseer Shahzad for pointing out an

error in an earlier proof of Theorem 33 and also for useful discussions

REFERENCES

1 Beg I Shahzad N A General Fixed Point Theorem for a Class of

Continuous Random Operators New Zealand J Math 1997 26 21ndash24

2 Bharucha-Reid AT Fixed Point Theorems in Probabilistic Analysis Bull

Am Math Soc 1976 82 641ndash657

3 Carbone A Conti G Multivalued Maps and the Existence of Best

Approximation J Approx Theory 1991 64 203ndash208

4 Fitzpatrick PM Petryshyn WV Fixed Point Theorems for Multivalued

Noncompact Acyclic Mappings Pacific J Math 1974 54 17ndash23

5 Hans O Reduzierende Zufallige Transformationen Czech Math J 1957

7 154ndash158

6 Himmelberg CJ Measurable Relations Fund Math 1975 87 53ndash72

7 Itoh S Random Fixed Point Theorems with an Application to Random

Differential Equations in Banach Spaces J Math Anal Appl 1979 67

261ndash273

8 Kuratowski K Ryll-Nardzewski C A General Theorem on Selector Bull

Acad Polon Sci Ser Sci Math Astronom Phys 1965 13 397ndash403

KHAN930

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ded

by [

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Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

ORDER REPRINTS

9 Lassonde M Fixed Points for Kakutani Factorizable Multifunctions

J Math Anal Appl 1990 152 46ndash60

10 Lin TC Random Approximations and Random Fixed Point Theorems for

Continuous 1-Set-Contractive Random Maps Proc Am Math Soc 1995

123 1167ndash1176

11 Li-Shan L Some Random Approximations and Random Fixed Point

Theorems for 1-Set-Contractive Random Operators Proc Am Math Soc

1997 125 515ndash521

12 Papageorgiou NS Random Fixed Point Theorems for Measurable

Multifunctions in Banach Space Proc Am Math Soc 1986 97 507ndash514

13 Park S Fixed Point Theorems on Compact Convex Sets in Topological

Vector Spaces Contemp Math Amer Math Soc Providence RI 1988

72 183ndash191

14 Park S Singh SP Watson B Some Fixed Point Theorems for

Composites of Acyclic Maps Proc Am Math Soc 1994 121 1151ndash1158

15 Sadovskii BN Limit-Compact and Condensing Operators Russ Math

Sur 1972 27 85ndash155

16 Sehgal VM Singh SP On Random Approximations and a Random

Fixed Point Theorem for Set Valued Mappings Proc Am Math Soc 1985

95 91ndash94

17 Sehgal VM Waters C Some Random Fixed Point Theorems for

Condensing Operators Proc Am Math Soc 1984 90 425ndash429

18 Shahzad N Random Fixed Point Theorems for Various Classes of 1-Set-

Contractive Maps in Banach Spaces J Math Anal Appl 1996 203

712ndash718

19 Shahzad N Khan LA Random Fixed Points of 1-Set-Contractive

Random Maps in Frechet Spaces J Math Anal Appl 1999 231 68ndash75

20 Spacek A Zufallige Gleichungen Czech Math J 1955 5 462ndash466

21 Xu HK Some Random Fixed Point Theorems for Condensing and

Nonexpansive Operators Proc Am Math Soc 1990 110 395ndash400

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 931

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ded

by [

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] at

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ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081SAP120000754

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

ORDER REPRINTS

subset of S Set B frac14 xj j $ 1 and gn frac14 frac12aethTethBTHORNethpnTHORN for n $ 1 Then for any

1 0 there exist sets B1hellipBk X with

dethBiTHORN gi 11

2and TethvBTHORN lt

k

ifrac141Bi

Let for each i Ai be an 1=2-neighborhood of Bi and choose an integer N such that

dethxj Tethv xjTHORNTHORN 1=2 for all j $ N Then xj j $ N ltkifrac141Ai and dethAiTHORN

di 1 1 Since 1 is arbitrary

frac12aethBTHORNethpnTHORN frac14 frac12aethxj j $ NTHORNethpnTHORN gn frac14 frac12aethTethvBTHORNTHORNethpnTHORN

for each n $ 1 This implies that frac12aethBTHORNethpnTHORN frac14 0 for each n $ 1 Since S is

complete the set B is precompact and so xj has a convergent subsequence

Using this observation we can next proceed exactly as in ((1) Theorem 31) and

show that for any closed subset C of X

G21ethCTHORN frac14 1

nfrac141lt1

jfrac141v [ V dethxj Tethv xjTHORNTHORN

1

n

Since for each j $ 1 the map v Tethv xjTHORN is measurable it is weakly

measurable by ((6) Proposition) so that by ((6) Theorem 33) the map

v dethxj Tethv xjTHORNTHORN is measurable Thus G21ethCTHORN [ S So G is measurable and

hence again by ((6) Theorem 21) G is weakly measurable Now by ((6)

Theorem 56) or (8) there exists a measurable selector f V S of G Clearly

f is a random fixed point of T This completes the proof A

Remark 32 The above theorem extends some random fixed point theorems in

the literature to Frechet spaces (see (7) Corollaries 22 and 23)

As an application of the above theorem we now obtain a random version of

a very interesting Fan type approximation theorem given in ((14) Theorem 3)

Theorem 33 Let S be an approximatively compact convex subset of a

separable Banach space X having property (H) and let T V S 2X be a

continuous compact random operator such that for each v [ V the map T(v )

belongs to V1c ethSXTHORN Then there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Proof The proximity map P frac14 PS X S defined by kx 2 Pxk frac14 dethx STHORN x [X is well-defined and continuous ((14) Lemma 1) and so P [ V1

c ethSXTHORN Since it

is clear that V1c is closed under composition (9) we have for each v [ V

KHAN928

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PTethv THORN [ V1c ethS STHORN and PT(v ) is compact Define F V S 2S by

Fethv xTHORN frac14 PTethv xTHORN v [ V and x [ S

By Theorem 31 there exists a measurable map f V S such that fethvTHORN [FethvfethvTHORNTHORN frac14 PTethvfethvTHORNTHORN for each v [ V that is

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORN STHORN

for all v [ V where h V S is a measurable selector of T( f()) Now we

can replace S by cl 2 ISethfethvTHORNTHORN in the above eqaution by the argument used in the

proof of ((13) Theorems 1 and 3)

Since every uniformly convex space is reflexive and has property (H) we

immediately have A

Corollary 34 Let S be an approximatively compact convex subset of a

separable uniformly convex space X and let T V S 2X be a continuous

compact random operator such that for each v [ V the map T(v ) belongs to

V1c ethSXTHORN Then there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Remark 35 It is interesting to compare the above results to Theorem 2 of

Sehgal and Singh (16) In fact Sehgal and Singh (16) were the first who obtained

random best approximation results using convex-valued continuous random

operators In this paper we have proved similar results for operators which are

not necessarily convex-valued This leads to the discovery of some new random

approximation results

We now deduce a random fixed point theorem for nonself maps

Corollary 36 Let X S and T be as in Theorem 33 Suppose in addition that

the following boundary condition is satisfied

(A) for each v [ V and each x [ S with x Tethv xTHORN there exists z depending

on v and x in cl 2 ISethxTHORN such that kx 2 yk kz 2 yk for all x ndash y [ Tethv xTHORNThen T has a random fixed point

Proof By Theorem 33 there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f()) If fethvTHORN TethvfethvTHORNTHORN for some v [ V the condition (A) implies that there exists a z [

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 929

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cl 2 ISethfethvTHORN such that kfethvTHORN2 yk kz 2 yk for all y ndash fethvTHORN a contradiction

This f is a random fixed point of T

The next result is a random version of another approximation theorem of

((14) Theorem 4) A

Theorem 37 Let S be a closed convex subset of a separable reflexive Banach

space X with the strong Oshman property and satisfying condition (H) and let

T V S 2X be a continuous compact random operator such that for each

v [ V the map T(v ) belongs to V1c ethSXTHORN Then there exists a measurable map

f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Proof Let P X S be the proximity map as in Theorem 33 Then in view of

the strong Oshman property P [ V1c ethX STHORN and hence for each v [ V the map

PTethv THORN [ V1c ethS STHORN and is compact Hence by Theorem 31 then there exists a

measurable map f V S such that for each v [ VfethvTHORN [ PTethvfethvTHORNTHORN It

follows that the conclusion of Theorem 37 holds A

Finally the author wishes to thank Dr Naseer Shahzad for pointing out an

error in an earlier proof of Theorem 33 and also for useful discussions

REFERENCES

1 Beg I Shahzad N A General Fixed Point Theorem for a Class of

Continuous Random Operators New Zealand J Math 1997 26 21ndash24

2 Bharucha-Reid AT Fixed Point Theorems in Probabilistic Analysis Bull

Am Math Soc 1976 82 641ndash657

3 Carbone A Conti G Multivalued Maps and the Existence of Best

Approximation J Approx Theory 1991 64 203ndash208

4 Fitzpatrick PM Petryshyn WV Fixed Point Theorems for Multivalued

Noncompact Acyclic Mappings Pacific J Math 1974 54 17ndash23

5 Hans O Reduzierende Zufallige Transformationen Czech Math J 1957

7 154ndash158

6 Himmelberg CJ Measurable Relations Fund Math 1975 87 53ndash72

7 Itoh S Random Fixed Point Theorems with an Application to Random

Differential Equations in Banach Spaces J Math Anal Appl 1979 67

261ndash273

8 Kuratowski K Ryll-Nardzewski C A General Theorem on Selector Bull

Acad Polon Sci Ser Sci Math Astronom Phys 1965 13 397ndash403

KHAN930

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

ORDER REPRINTS

9 Lassonde M Fixed Points for Kakutani Factorizable Multifunctions

J Math Anal Appl 1990 152 46ndash60

10 Lin TC Random Approximations and Random Fixed Point Theorems for

Continuous 1-Set-Contractive Random Maps Proc Am Math Soc 1995

123 1167ndash1176

11 Li-Shan L Some Random Approximations and Random Fixed Point

Theorems for 1-Set-Contractive Random Operators Proc Am Math Soc

1997 125 515ndash521

12 Papageorgiou NS Random Fixed Point Theorems for Measurable

Multifunctions in Banach Space Proc Am Math Soc 1986 97 507ndash514

13 Park S Fixed Point Theorems on Compact Convex Sets in Topological

Vector Spaces Contemp Math Amer Math Soc Providence RI 1988

72 183ndash191

14 Park S Singh SP Watson B Some Fixed Point Theorems for

Composites of Acyclic Maps Proc Am Math Soc 1994 121 1151ndash1158

15 Sadovskii BN Limit-Compact and Condensing Operators Russ Math

Sur 1972 27 85ndash155

16 Sehgal VM Singh SP On Random Approximations and a Random

Fixed Point Theorem for Set Valued Mappings Proc Am Math Soc 1985

95 91ndash94

17 Sehgal VM Waters C Some Random Fixed Point Theorems for

Condensing Operators Proc Am Math Soc 1984 90 425ndash429

18 Shahzad N Random Fixed Point Theorems for Various Classes of 1-Set-

Contractive Maps in Banach Spaces J Math Anal Appl 1996 203

712ndash718

19 Shahzad N Khan LA Random Fixed Points of 1-Set-Contractive

Random Maps in Frechet Spaces J Math Anal Appl 1999 231 68ndash75

20 Spacek A Zufallige Gleichungen Czech Math J 1955 5 462ndash466

21 Xu HK Some Random Fixed Point Theorems for Condensing and

Nonexpansive Operators Proc Am Math Soc 1990 110 395ndash400

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 931

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081SAP120000754

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

ORDER REPRINTS

PTethv THORN [ V1c ethS STHORN and PT(v ) is compact Define F V S 2S by

Fethv xTHORN frac14 PTethv xTHORN v [ V and x [ S

By Theorem 31 there exists a measurable map f V S such that fethvTHORN [FethvfethvTHORNTHORN frac14 PTethvfethvTHORNTHORN for each v [ V that is

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORN STHORN

for all v [ V where h V S is a measurable selector of T( f()) Now we

can replace S by cl 2 ISethfethvTHORNTHORN in the above eqaution by the argument used in the

proof of ((13) Theorems 1 and 3)

Since every uniformly convex space is reflexive and has property (H) we

immediately have A

Corollary 34 Let S be an approximatively compact convex subset of a

separable uniformly convex space X and let T V S 2X be a continuous

compact random operator such that for each v [ V the map T(v ) belongs to

V1c ethSXTHORN Then there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Remark 35 It is interesting to compare the above results to Theorem 2 of

Sehgal and Singh (16) In fact Sehgal and Singh (16) were the first who obtained

random best approximation results using convex-valued continuous random

operators In this paper we have proved similar results for operators which are

not necessarily convex-valued This leads to the discovery of some new random

approximation results

We now deduce a random fixed point theorem for nonself maps

Corollary 36 Let X S and T be as in Theorem 33 Suppose in addition that

the following boundary condition is satisfied

(A) for each v [ V and each x [ S with x Tethv xTHORN there exists z depending

on v and x in cl 2 ISethxTHORN such that kx 2 yk kz 2 yk for all x ndash y [ Tethv xTHORNThen T has a random fixed point

Proof By Theorem 33 there exists a measurable map f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f()) If fethvTHORN TethvfethvTHORNTHORN for some v [ V the condition (A) implies that there exists a z [

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 929

Dow

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ded

by [

UQ

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rary

] at

11

03 1

5 N

ovem

ber

2014

ORDER REPRINTS

cl 2 ISethfethvTHORN such that kfethvTHORN2 yk kz 2 yk for all y ndash fethvTHORN a contradiction

This f is a random fixed point of T

The next result is a random version of another approximation theorem of

((14) Theorem 4) A

Theorem 37 Let S be a closed convex subset of a separable reflexive Banach

space X with the strong Oshman property and satisfying condition (H) and let

T V S 2X be a continuous compact random operator such that for each

v [ V the map T(v ) belongs to V1c ethSXTHORN Then there exists a measurable map

f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Proof Let P X S be the proximity map as in Theorem 33 Then in view of

the strong Oshman property P [ V1c ethX STHORN and hence for each v [ V the map

PTethv THORN [ V1c ethS STHORN and is compact Hence by Theorem 31 then there exists a

measurable map f V S such that for each v [ VfethvTHORN [ PTethvfethvTHORNTHORN It

follows that the conclusion of Theorem 37 holds A

Finally the author wishes to thank Dr Naseer Shahzad for pointing out an

error in an earlier proof of Theorem 33 and also for useful discussions

REFERENCES

1 Beg I Shahzad N A General Fixed Point Theorem for a Class of

Continuous Random Operators New Zealand J Math 1997 26 21ndash24

2 Bharucha-Reid AT Fixed Point Theorems in Probabilistic Analysis Bull

Am Math Soc 1976 82 641ndash657

3 Carbone A Conti G Multivalued Maps and the Existence of Best

Approximation J Approx Theory 1991 64 203ndash208

4 Fitzpatrick PM Petryshyn WV Fixed Point Theorems for Multivalued

Noncompact Acyclic Mappings Pacific J Math 1974 54 17ndash23

5 Hans O Reduzierende Zufallige Transformationen Czech Math J 1957

7 154ndash158

6 Himmelberg CJ Measurable Relations Fund Math 1975 87 53ndash72

7 Itoh S Random Fixed Point Theorems with an Application to Random

Differential Equations in Banach Spaces J Math Anal Appl 1979 67

261ndash273

8 Kuratowski K Ryll-Nardzewski C A General Theorem on Selector Bull

Acad Polon Sci Ser Sci Math Astronom Phys 1965 13 397ndash403

KHAN930

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

ORDER REPRINTS

9 Lassonde M Fixed Points for Kakutani Factorizable Multifunctions

J Math Anal Appl 1990 152 46ndash60

10 Lin TC Random Approximations and Random Fixed Point Theorems for

Continuous 1-Set-Contractive Random Maps Proc Am Math Soc 1995

123 1167ndash1176

11 Li-Shan L Some Random Approximations and Random Fixed Point

Theorems for 1-Set-Contractive Random Operators Proc Am Math Soc

1997 125 515ndash521

12 Papageorgiou NS Random Fixed Point Theorems for Measurable

Multifunctions in Banach Space Proc Am Math Soc 1986 97 507ndash514

13 Park S Fixed Point Theorems on Compact Convex Sets in Topological

Vector Spaces Contemp Math Amer Math Soc Providence RI 1988

72 183ndash191

14 Park S Singh SP Watson B Some Fixed Point Theorems for

Composites of Acyclic Maps Proc Am Math Soc 1994 121 1151ndash1158

15 Sadovskii BN Limit-Compact and Condensing Operators Russ Math

Sur 1972 27 85ndash155

16 Sehgal VM Singh SP On Random Approximations and a Random

Fixed Point Theorem for Set Valued Mappings Proc Am Math Soc 1985

95 91ndash94

17 Sehgal VM Waters C Some Random Fixed Point Theorems for

Condensing Operators Proc Am Math Soc 1984 90 425ndash429

18 Shahzad N Random Fixed Point Theorems for Various Classes of 1-Set-

Contractive Maps in Banach Spaces J Math Anal Appl 1996 203

712ndash718

19 Shahzad N Khan LA Random Fixed Points of 1-Set-Contractive

Random Maps in Frechet Spaces J Math Anal Appl 1999 231 68ndash75

20 Spacek A Zufallige Gleichungen Czech Math J 1955 5 462ndash466

21 Xu HK Some Random Fixed Point Theorems for Condensing and

Nonexpansive Operators Proc Am Math Soc 1990 110 395ndash400

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 931

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081SAP120000754

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

ORDER REPRINTS

cl 2 ISethfethvTHORN such that kfethvTHORN2 yk kz 2 yk for all y ndash fethvTHORN a contradiction

This f is a random fixed point of T

The next result is a random version of another approximation theorem of

((14) Theorem 4) A

Theorem 37 Let S be a closed convex subset of a separable reflexive Banach

space X with the strong Oshman property and satisfying condition (H) and let

T V S 2X be a continuous compact random operator such that for each

v [ V the map T(v ) belongs to V1c ethSXTHORN Then there exists a measurable map

f V S such that

kfethvTHORN2 hethvTHORNk frac14 dethhethvTHORNTHORN cl 2 ISethfethvTHORNTHORN

for all v [ V where h V S is a measurable selector of T( f())

Proof Let P X S be the proximity map as in Theorem 33 Then in view of

the strong Oshman property P [ V1c ethX STHORN and hence for each v [ V the map

PTethv THORN [ V1c ethS STHORN and is compact Hence by Theorem 31 then there exists a

measurable map f V S such that for each v [ VfethvTHORN [ PTethvfethvTHORNTHORN It

follows that the conclusion of Theorem 37 holds A

Finally the author wishes to thank Dr Naseer Shahzad for pointing out an

error in an earlier proof of Theorem 33 and also for useful discussions

REFERENCES

1 Beg I Shahzad N A General Fixed Point Theorem for a Class of

Continuous Random Operators New Zealand J Math 1997 26 21ndash24

2 Bharucha-Reid AT Fixed Point Theorems in Probabilistic Analysis Bull

Am Math Soc 1976 82 641ndash657

3 Carbone A Conti G Multivalued Maps and the Existence of Best

Approximation J Approx Theory 1991 64 203ndash208

4 Fitzpatrick PM Petryshyn WV Fixed Point Theorems for Multivalued

Noncompact Acyclic Mappings Pacific J Math 1974 54 17ndash23

5 Hans O Reduzierende Zufallige Transformationen Czech Math J 1957

7 154ndash158

6 Himmelberg CJ Measurable Relations Fund Math 1975 87 53ndash72

7 Itoh S Random Fixed Point Theorems with an Application to Random

Differential Equations in Banach Spaces J Math Anal Appl 1979 67

261ndash273

8 Kuratowski K Ryll-Nardzewski C A General Theorem on Selector Bull

Acad Polon Sci Ser Sci Math Astronom Phys 1965 13 397ndash403

KHAN930

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

ORDER REPRINTS

9 Lassonde M Fixed Points for Kakutani Factorizable Multifunctions

J Math Anal Appl 1990 152 46ndash60

10 Lin TC Random Approximations and Random Fixed Point Theorems for

Continuous 1-Set-Contractive Random Maps Proc Am Math Soc 1995

123 1167ndash1176

11 Li-Shan L Some Random Approximations and Random Fixed Point

Theorems for 1-Set-Contractive Random Operators Proc Am Math Soc

1997 125 515ndash521

12 Papageorgiou NS Random Fixed Point Theorems for Measurable

Multifunctions in Banach Space Proc Am Math Soc 1986 97 507ndash514

13 Park S Fixed Point Theorems on Compact Convex Sets in Topological

Vector Spaces Contemp Math Amer Math Soc Providence RI 1988

72 183ndash191

14 Park S Singh SP Watson B Some Fixed Point Theorems for

Composites of Acyclic Maps Proc Am Math Soc 1994 121 1151ndash1158

15 Sadovskii BN Limit-Compact and Condensing Operators Russ Math

Sur 1972 27 85ndash155

16 Sehgal VM Singh SP On Random Approximations and a Random

Fixed Point Theorem for Set Valued Mappings Proc Am Math Soc 1985

95 91ndash94

17 Sehgal VM Waters C Some Random Fixed Point Theorems for

Condensing Operators Proc Am Math Soc 1984 90 425ndash429

18 Shahzad N Random Fixed Point Theorems for Various Classes of 1-Set-

Contractive Maps in Banach Spaces J Math Anal Appl 1996 203

712ndash718

19 Shahzad N Khan LA Random Fixed Points of 1-Set-Contractive

Random Maps in Frechet Spaces J Math Anal Appl 1999 231 68ndash75

20 Spacek A Zufallige Gleichungen Czech Math J 1955 5 462ndash466

21 Xu HK Some Random Fixed Point Theorems for Condensing and

Nonexpansive Operators Proc Am Math Soc 1990 110 395ndash400

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 931

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081SAP120000754

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

ORDER REPRINTS

9 Lassonde M Fixed Points for Kakutani Factorizable Multifunctions

J Math Anal Appl 1990 152 46ndash60

10 Lin TC Random Approximations and Random Fixed Point Theorems for

Continuous 1-Set-Contractive Random Maps Proc Am Math Soc 1995

123 1167ndash1176

11 Li-Shan L Some Random Approximations and Random Fixed Point

Theorems for 1-Set-Contractive Random Operators Proc Am Math Soc

1997 125 515ndash521

12 Papageorgiou NS Random Fixed Point Theorems for Measurable

Multifunctions in Banach Space Proc Am Math Soc 1986 97 507ndash514

13 Park S Fixed Point Theorems on Compact Convex Sets in Topological

Vector Spaces Contemp Math Amer Math Soc Providence RI 1988

72 183ndash191

14 Park S Singh SP Watson B Some Fixed Point Theorems for

Composites of Acyclic Maps Proc Am Math Soc 1994 121 1151ndash1158

15 Sadovskii BN Limit-Compact and Condensing Operators Russ Math

Sur 1972 27 85ndash155

16 Sehgal VM Singh SP On Random Approximations and a Random

Fixed Point Theorem for Set Valued Mappings Proc Am Math Soc 1985

95 91ndash94

17 Sehgal VM Waters C Some Random Fixed Point Theorems for

Condensing Operators Proc Am Math Soc 1984 90 425ndash429

18 Shahzad N Random Fixed Point Theorems for Various Classes of 1-Set-

Contractive Maps in Banach Spaces J Math Anal Appl 1996 203

712ndash718

19 Shahzad N Khan LA Random Fixed Points of 1-Set-Contractive

Random Maps in Frechet Spaces J Math Anal Appl 1999 231 68ndash75

20 Spacek A Zufallige Gleichungen Czech Math J 1955 5 462ndash466

21 Xu HK Some Random Fixed Point Theorems for Condensing and

Nonexpansive Operators Proc Am Math Soc 1990 110 395ndash400

COMPOSITES OF ACYCLIC MULTIFUNCTIONS 931

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081SAP120000754

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014

Order now

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081SAP120000754

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request PermissionReprints Here link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Dow

nloa

ded

by [

UQ

Lib

rary

] at

11

03 1

5 N

ovem

ber

2014