A note on modular hyperconvex space

Int. J.IndustrialMathematisVol. 2,No. 4(2010)271-278

A Note on Modular Hyperonvex Spae

H.R. Rahimi a

,S. Sadeghilemraski b

,M.S.Asgari

(a)Departmentof Mathematis,Faulty ofSiene,CenteralTehranBranh, IslamiAzadUniversity,

Tehran,Iran

(b)Department ofMathematis,NorthBranh, IslamiAzadUniversity,Tehran,Iran

()Department ofMathematis,Faultyof Siene,CenteralTehranBranh,IslamiAzad University,

Tehran,Iran

Reeived20Otober2010; revised6Deember2010; aepted 11Deember 2010.

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Reently, it has been obtaineda lot of resultson hyperonvex spae (see [1 , 2 , 3, 5℄). In

thispaperwe develop some of those resultsformodularhyperonvex spaes.

Keywords: Hyperonvexspae,Modularfuntion,Modularhyperonvexspae.

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1 Introdution

The theory of hyperonvex spae was initiated by Aronzajn and Panithpakdi [1 ℄. It

was proved that a hyperonvex spae is a nonexpansive absolute retrat ,i.e. it is a

nonexpansive retrat of any metri spae in whih it is isometrially embedded. Many

interestingresultsandappliationsofthetheoryofhyperonvexspaesinanotherbranh

of mathematisare thebase ofour motivation to study thissubjet. Forexample,it has

been used in probability and mathematial statistis, boundary-value problems [3 ℄, the

inverse funtion [10 ℄, and the existene of solutions of dierential equations [9 , 12 ℄. The

theoryof modularfuntionspaewasinitiated byNakanoin1950 inonnetion withthe

theoryof order spaesand redenedand generalizedbyLuxemburgand Orlizin1959.

The organization of this paper is the following: We start with introduingthe

de-nitions and notations whih will be used later. For onveniene of readers, we suggest

that one refers to [1, 2 , 4 , 5 , 13 ℄. Setion two startes with the proof of the existene

of the seletion of Lipshitz set valued mappingsT

from a modular hyperonvex spae

H

that hoose their values from the spae of the external modularhyperonvex subset

H

i.e "

(H

). Also we get interesting results by onsidering intersetion of the sets in

modularhyperonvexspaes suh asthe intersetion ofa modularadmissiblesubsetand

a externallymodularhyperonvexsubsetwithrespetto thehyperonvex modularspae

H

is externallymodularhyperonvex withrespetto H

:

Also we show thatwhen the ommunion of externally modularhyperonvex subsets of a

modularhyperonvexspae isnonempty.

2 Preliminaries

LetX beavetor spaeon R, afuntion:X ![0;+1℄isalledmodularifforevery

x;y inX , (i) (x) = 0 if and only if x = 0, (ii) (x) = (x) , for every 2 R where

jj=1,(iii)(x+y)(x)+(y)if+ =1and0;0,andisalledonvex

modular if, (x+y) (x)+(y) if + =1 and 0; 0. By a modular

spae we mean X

=fx 2 X : lim

!0

(x)= 0g, where is a modularfuntion on X:

Following Khamsi [3 ℄, for a modularspae X

, the sequene fx

n

g is alled -onvergent

to x if (x

n

;x) ! x, and it is alled -Cauhy if (x

n ;x

m

) ! 0 as n;m ! 0. We will

say that the modular funtion satises the Fatou property if (x) liminf

n (x

n ) as

x

n

! x ,where fx

n

g is a sequene in X

. A modular funtion is alled omplete if

every - Caushy sequene fx

n

g is - onvergent. A subset A of X

is alled - losed if

the-limitof a -onvergent sequene of A always belongsto A.By a -ballB

(x;r),we

mean fy2X

:(x y)rg.

Finally,a subsetA of X

isalled -boundedif

Æ

(A)=supf(x y):x;y2Ag<1:

We note that does not behave in general as a metri beause does not satisfy the

triangle inequality. For example - onvergent does not imply - Caushy. However,

-ballsare -losedina modularspaeX

ifandonlyifthey have Fatouproperty[5℄.

Denition 2.1. A modular spae X

is alled modular hyperonvex spae if, for any

olletionof pointsfx

g

2

ofX andforanyolletion fr

gof non-negativereal numbers

suh that (1=2(x

x

))r

+r

(; 2 ),it follows that T

2 B

(x

;r

)6=;:

If X

isa modularspaeweshowthefamilyof allthenonemptyandboundedsubset

X

byB

(X

):

Denition 2.2. Let X

be a modular spae suh that has Fatou property. A subset A of

X

isalled modular admissiblesubset if A isan intersetion of -losed balls in X

.

Denition 2.3. Suppose that X

isa modular spaeand C it's subset. We saythat C is

modular proximinal ,if for eah x2X

C\B

(x;dist

(x;C))6=

suh that

dist

(x;C)=inff(x y) : y2Cg

Denition 2.4. A subset E of modular spae(X

;) isalled externally modular

hyper-onvex (with respet toX

)if for eah family of elements fx

g

2 inX

andeah family

of positivereal number fr

g

2

suh that for every ; 2

dist

(x

;E)r

; ( 1

(x

x

))r

+r

the following holds

\

2 B

(x

;r

)\E 6=:

A

(X

)isthenotationofallthenonemptymodularadmissiblesubsetsX

and"

(X

)

is the notation of all the externally modularhyperonvex subsets X

and H

(X

) is the

notation ofall themodularhyperonvexsubsets X

:

Denition 2.5. Let A bea subset of a modular hyperonvex spaeX

, set

r

x

(A) = supf(x y) : y2Ag; x2X

;

r(A) = inffr

x

(A) : x2X

g;

R (A) = inffr

x

(A) : x2Ag;

diam(A) = supf(x y) : x;y2Ag;

C(A) = fx2X

: r

x

(A)=r(A)g;

C

A

(A) = fx2A: r

x

(A)=r(A)g;

ov

(A) = T

fB:B is a ball and BAg;

r(A)isalledthe redueof A(relativetoX

),diam(A) isalled thediameter of A,R (A)

isalled Chebyshev radius of A, C(A)is alled the Chebyshev enterof A, andov

(A) is

alled the over of A.

The readeran see theproof ofthefollowingLemma in[8℄.

Lemma 2.1. Let A be a -bounded subsetof modular hyperonvex spae X

,then:

1) ov

(A)=

T

fB

(x;r

x

(A)):x2X

g.

2) r

x (ov

(A))=r

x

(A), for any x2X

.

3) r(ov

(A))) =r(A).

4) r(A)=1=2(diam(A)).

5) diam(ov

(A))=diam(A).

6) If A=ov

(A), then r(A)=R (A). In partiular we haveR (A) =1=2(diam(A)).

If A is a subsetof modularspae X

; we denote the" losedneighborhood A with

N

"

(A)inwhih

N

"

(A)=fx2X

: dist

(x;A)"g:

Denition 2.6. Suppose that A is an arbitrary set. A map T from A to P(A) where

P(A) isthe power set of A, is alled set-valued mapping.

Denition 2.7. Let T

: X

! B

(X

) is a set-valued mapping. A seletion is a map

suh as T :X

!X

suh that T(x)2T

(x) for eah x2X

:

Denition 2.8. Suppose that (X

1 ;

1

) and(X

2 ;

2

) are modular spaes. we say the

map-ping T :X

1 !X

2

is lipshitzian whenthere exists 0 suh that for eah x;y 2X

1

the following satises

2

(T(x) T(y))

1

(x y):

Thesmallest that satises inthe above relationisalled lipshitzianonstant andwe

note it withLip(T): If =1 then we say the above map isnonexpansive.

Lemma2.2. (lemma2.2[8 ℄)IfAisaexternalmodularhyperonvexsubsetoraadmissible

modular subset of a modular hyperonvex spae H

, then A is the modular proximinal in

Lemma2.3. (lemma3.2[8℄)LetX

beamodularhyperonvexspaeandD= T

2 B

(x

;r

):

In this ase for eah ">0 we have

N

" (D)=

\

2 B

(x

;r

+"):

Denition2.9. If AandB arebounded andlosedsubsetsinmodularspaeX

wedene

Hausdor distane

H

as follows:

H

(A;B)=inff">0:AN

"

(B) ; B N

" (A)g:

Theorem 2.1. (Theorem 2.2 [8℄) If X

isa modular hyperonvex spae then we have

A

(X

)"

(X

)H

(X

):

3 Main results

Inthissetion,wedevelopsomeresultsgettingin[4,7 ℄,formodularhyperonvexspae.

Theorem3.1. LetH

beamodularhyperonvexspae,SisanarbitrarysetandT

:S!

"

(H

). Then there exists the map T : S ! H

suh that for eah x 2S; T(x) 2 T

(x)

and for eah x;y 2S we have

(T(x) T(y))

H (T

(x);T

(y)):

Proof: LetF denotethe olletionof allpairs(D,T), whereDS and foralld2D

and forall x;y inD we have

T :D !H

T(d)2T

(d)

(T(x) T(y))

H (T

(x);T

(y))

We note thatforeah x

0

2S;T(x

0 )2T

(x

0

); then(fx

0

g;T)2F: Thuswe have F 6=:

Now, we denetheorder relationon F asfollows

(D

1 ;T

1 )(D

2 ;T

2 ),D

1 D

2 ; T

2 j

D1 =T

1 :

Supposef(D

;T

)gis theinreasing hain in(F;):Soitfollowsthat( S

2 D

;T)2F

where Tj

D

=T

: It is learthat this memberis an upperboundfor above hain. with

deningorderbyZorn ;

slemma,themaximalelementsuhas(D;T)in(F;)exists. Now,

we show that D = S:Assume D 6=S, therefore there exists the element x

0

2 SnD: Let

~

D =D[fx

0 g:

Now, onsider thefollowingset:

J =( \

x2D B

(T(x);

H (T

(x);T

(x

0 ))))

\

T

(x

0 ):

Sine T

(x

0 )2"

(H

) we have J 6= ifand onlyifforeah x2D :

dist

(T(x);T

(x

0 ))

H (T

(x);T

(x

0 )):

by lemma 2.2 we have T

(x

0 ) 2 "

(H

) as a modular proximinal subset from H

: The

above is trueifand onlyifforeah x2D

B (T(x); (T

(x);T

(x )))\T

By thedenitionofHausdor distane

T

(x)N

H (T

(x);T

(x

0 ))+"

(T

(x

0 )):

However byassumptionT(x)2T

(x) soitmust be thease thatforeah ">0

B

(T(x);

H (T

(x);T

(x

0

))+")\T

(x

0 )6=:

Sine T

(x

0

) is amodularproximinialinH

, thisimpliesthat

B

(T(x);

H (T

(x);T

(x

0 )))\T

(x

0 )6=:

ThusJ 6=:Nowhoosey

0

2J anddene

~

T(x)=

y

0

ifx=x

0

T(x) ifx2D

On the other hand ( ~

T(x

0 )

~

T(x)) = (y

0

T(x))

H (T

(x);T

(x

0

))(8x 2 D):

Thus (D[fx

0 g;

~

T) 2 F and it has ontradition with maximality of (D;T): Therefore

D =S:

Corollary 3.1. Suppose that H

isa modular hyperonvexspaeand(M;

1

)isamodular

spaeandthe set-valuedmapping T

: M !"

(H

)isnonexpansive i.e,for eah x;y 2M

,

H (T

(x);T

(y))

1

(x y): Thenthe nonexpansive map T :M !H

existssuh that

for eah x2M; T(x)2T

(x):

Proof: By theorem 3.1and nonexpansive theT

; there existstheseletion T :M !

H

that foreah x2M; T(x)2T

(x)and

(T(x) T(y))

H (T

(x);T

(y))

1

(x y); (8x;y2M):

Therefore(T(x) T(y))

1

(x y),thusT isnonexpansive.

Theorem3.2. Let M

bea bounded modular spaeand(H

)

2

beadereasingfamilyof

nonempty modular hyperonvex subsets of M

, where istotally ordered. Then T

2 H

is nonempty and modular hyperonvex.

Proof: DeneF as thefollows:

F =fA=

2 A

; A

2A

(H

) and (A

) isdereasingandnonemptyg:

Sine M

isboundedsoH

isbounded. ThusH

2A

(H

):So H

isnotemptyand

dereasingthen

2 H

2F and F 6=:

SineH

ismodularhyperonvexthenA

(H

)isompatforevery 2 :ThusF satises

the assumptionsof zorn's lemma when orderedby set inlusion. Hene for every D 2F

there existsa minimalelement A2F suhthat, AD:

We laimthatifA=

2 A

isminimalthenthereexists

0

2 suhthatÆ(A

)=0for

every

0

, whereÆ(A)=diam(A). Let2 be xed. Forevery D M

dene

ov

(D)=

\

x2H

(x;r

ConsiderA 0 = 2 A 0 whereA 0 =ov (A )\A

if and A 0

=A

if.

Sine A 2 F then the family(A 0

) is dereasing . Let : Sine A

A andA =ov (A )\A

soA 0 A 0

:Henethefamily(A 0

)isdereasing. Ontheother

hand if then ov

(A

)\A

2 A

(H

): Sine H

H so A 0 2 A (H ): Thus A 0

2F:Sine A isminimalthisimpliesthat A=A 0 whih implies A =ov (A )\A

8:

Let x 2 H

and : Sine A

A

; then r

x (A

) r

x (A

): Now ov

(A ) = T x2H (x;r x (A

)), then we have ov

(A ) (x;r x (A

)), whihimplies

r x (ov (A ))r

x (A ): AdditionallyA ov (A )so r x (A )r

x (A

)r

x (ov

(A

))r

x (A

):

Thereforewehave r

x (A

)=r

x (A

) forevery x2H

. Usingthedenitionofr, weget

r

( A

)r(A

):

Leta2A

ands=r

a (A

), thena2ov

(A

). Sine A

ov (A )so a2 \ x2A

(x;s)\ov

(A

):

By hyperonvexityof H

, S =H \ \ x2A

(x;s)\ov

(A

)6=:

Letz2S

, thenz2 T

x2A

(x;s):Sine

A =H \ov (A ):

It followsthat r

z (A

)s;whihimplies

r(A

)s=r

a (A

)

forevery a2A

: Hene

r(A

)=r(A

) 8;2 :

Assume that Æ(A

) > 0 for every 2 : Set A 00

= C(A

) for every 2 : The family

(A 00

) is dereasing. Let and x 2 A 00

, then r

x (A

) =r(A

): Sine we proved that

r

z (A

)=r

z (A

) forevery z2H

thenr

x (A

)=r

x (A

)=r(A

)=r(A

),whih implies

thatx2A 00 : Therefore A 00 = 2 A 00 2F: Sine A 00

A and A is minimal, we get that A 00

= A: Therefore A

=C(A

) for every

2. Thisis inontradition to hyperonvexityofH

foreah 2 . Thusthere exists

0

2 suhthatÆ(A

)=0forevery

0 .SoA

=fagforevery

0

,whihimplies

thata2 T

2 H

6=:

In order to omplete the proof, we need to show that S = T

2 H

is modular

hyper-onvex. Let (

i )

i2I

be a family of balls entered in S suh that T

i2I

i

6= : Dene

D = T i2I i \H

for all 2 : Sine H

is modular hyperonvex and the family

(

i )

i2I

enteredinH

thenD

isnotemptyandD

2A

(H

):ThereforeD

ismodular

hyperonvex. the above proof shows that T

D

Lemma 3.1. Let H

bea modular hyperonvex spae and E H

be externally modular

hyperonvex related to H

: Suppose A is a modular admissible subset of H

: Then E\A

is externally modular hyperonvex related to H

:

Proof: Supposefx

gandfr

gsatisfy( 1

2 (x

x

))r

+r

anddist

(x

;E\A)

r

: Sine Aisadmissible,A= T

i2I

(x

i ;r

i

) and sine

(x

;r

)\A6=:Itfollows that

( 1

2 (x

x

i ))r

+r

i

foreahi2I:Sine A

(x

i ;r

i

) itfollowsthat

dist

(x

i

;E\A)r

i

; (

1

2 (x

i x

j ))r

i +r

j

8i;j2I:

Thereforebyexternalmodularhyperonvexityof E

( \

i

(x

i ;r

i ))\(

\

(x

;r

))\E = \

(x

;r

)\(A\E)6=:

ThusE\Ais externallymodularhyperonvexrelated to H

:

Theorem 3.3. Let fH

i

g be a dereasing hain of nonempty modular externally

hyper-onvex subsets of a bounded modular hyperonvex spae H

: Then \

i H

i

is nonempty and

externally modular hyperonvex in H

:

Proof: ByTheorem(3.2) and Theorem(2.1), we haveD = T

i H

i

6=:ToproveD is

externallymodularhyperonvex, let fx

gH andfr

gR satisfy

( 1

2 (x

x

))r

+r

; dist

(x

;D)r

:

SineH

ismodularhyperonvexweknowthatA= T

(x

;r

)6=:alsodist

(x

;D)

r

anddist

(x

;H

i )r

foreahi,sobyexternallymodularhyperonvexityofH

i

implies

thatforeahiwehaveA\H

i

6=:ByTheorem2.1andlemma3.1,fA\H

i

gisadereasing

hainof nonemptymodularhyperonvexsubsets of H

: Now by Theorem3.2, we have

\

i

(A\H

i

)=A\D6=:

Thus T

i H

i

is nonemptyand externallymodularhyperonvexinH

:

Aknowledgments

The authors would like to sinerely thank the referee for his/her valuable omments

and usefulsuggestions

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