VECTOR-VALUED FUZZY MULTIFUNCTIONS

(1)

VECTOR-VALUED

FUZZY

MULTIFUNCTIONS

ISMAT BEG

Lahore University

_of

Management

Sciences

Department

_of

Mathematics

5792

Lahore

Cantt.,

Pakistan

(Received

October, 1999;

Revised February,

2001)

Some

of the properties of vector-valued fuzzy multifunctions are studied. The notion of sum fuzzy

multifunction,

convex hull fuzzy

multifunction,

close convex hull fuzzy

multifunction,

and upper demicontinuous are

given, and some of the properties of these fuzzy multifunctions are

investi-gated.

Key

words:

Fuzzy

Multifunction,

Fuzzy

Topological

Vector Space,

Fuzzy

Topological

Space,

Fuzzy

Analysis.

AMS

subject classifications:

46S40,

47S40, 47H04, 04A72, 03E72,

46A99.

1. Introduction

In

the last three

decades,

the theory of multifunctions has advanced in a variety of ways. The theory of multifunctions was first codified by

Berge

[8].

Applications of this theory can be found in economic theory, noncooperative games, artificial intelli-gence,

medicine,

and existence of solutions for differential inclusions

(see

Aubin and Ekeland

_[2],

Klein and Thompson

_[15],

Aubin and Frankowska

_[3],

and the references

therein).

Recently, Heilpern

[11],

Butnariu

_[9],

Albrycht and Maltoka

[1],

Papageor-giou

[20],

Ozbakir and Aslim

_[19],

Tsiporkova-Hristoskova,

De Baets

and

Kerre

[22-24],

and

Beg

[4-7]

have started the study offuzzy multifunctions and hemicontinuous fuzzy multifunctions. The aim of this paper is to study properties of vector-valued fuzzy multifunctions. The notion ofsum fuzzy multifunction, and upper

demicontin-uous fuzzy multifunction are given, and some of the properties of these fuzzy multi-functions are investigated.

2. Preliminaries

Let X

be an arbitrary

(nonempty)

set.

A

fuzzy set

_(in

_X)is

a function with domain

X

and values in

[0,

1].

If

A

is a fuzzy set and x E

X,

the function

A(x)is

called the grade of membership of x in

A.

The fuzzy set

A

c,

defined by

AC(x)= 1-A(x),

is

(2)

called the complement of

A. A(x) <_ B(x)

for each x E

X.

and

Let

A

and

B

be fuzzy sets in

X. We

write

ACB

if

For

any family

{Ai}

I offuzzy sets in

X,

we define:

f’

_Ail

(x)=

inf

_Ai(x

ieI

_J

ieI

[

U

_IAiJ

(x)

sup

I

The family v of fuzzy sets in

X

is called a fuzzy

topology

for

X

(and

the pair

(X,

7")

afuzzy topological

space)"

(i)

v contains every constant fuzzy set

(function)

in

X;

(ii)

_U

_ie

_iAi

7- whenever each

A

7-(i

E

1);

and

(iii)

Afl

B

7" whenever

A,

B

E r.

The elements of r and their complements are called open and

closed,

respectively.

A

neighborhoods of a fuzzy set

A

ofa fuzzy topological space

X

is any fuzzy set

B

for which there is an open fuzzy set

V

satisfying

A

C

V

C

B. Any

open fuzzy set

V

that satisfies

A

C

V

is called an open neighborhood of

A. A

fuzzy set

A

in

(X,

r)

is called fuzzy

compact

if and only if open covering of

A

has a finite subcovering. Similarly,

we can define fuzzy Hausdorff spaces.

A

net

_(x;)

_e_A in a fuzzy topological space

(X,

7")

converges to a point x

_(denoted

by

_xx):

if given a neighborhood

V

of

x,

there exists a

_"0

E

A

such that

_x,x

E

V

whenever

,

>_

_"0"

A

point x

belongs

to the closure ofafuzzy subset

C

of

X

ifthereis

a net in

C

converging to x.

In general,

a net in a fuzzy

topological

space may

con-verge to several pointsbut in afuzzy Hausdorff space, the convergence is unique.

A

single-valued map

_f

from a fuzzy topological space

X

to a fuzzy topological space

Y

is called continuous at some xE

X

if

_f-

I(V)

is a neighborhood ofx or each neighborhood

V

of

_f(x).

_(Here

_f-l(v)is

the fuzzy set in

X

defined by

[(f-

I(V))(z)-

V(f(x))].

For

further details, werefer to

[8,

10, 16-18,

25-27].

3. Fuzzy

Multifunctions

A

fuzzy multifunction

_f

from a set

X

into a set

Y

assigns to each x in

X,

a fuzzy subset

f(x)

of

Y. We

denote this assignment by

_f:

X--,Y.

We

can identify

_f

with a

fuzzy subset

_GI

of

X

x

Y

and

[f(x)](y)

_G](x,y).

If

A

is a fuzzysubset of

X,

then thefuzzy set

_f(A)

in

Y

is defined by

[f(A)](y)

sup

_[Gl(x,y

A

A(x)].

xX

The graph

G

_f of

_f

is thefuzzy subset of

X

x

Y

associated with

f,

G]

{(x,y)

X

x

Y:[f(x)](y)

0}.

by

Definition 3.1" The upper inverse

_fu

ofa fuzzy multifunction

_f:

XY,

is defined

[fU(A)](x)

inf

_[(1-Gl(x,y))

V

A(y)].

(3)

by

Definition 3.2: The lower inverse

fg

of a fuzzy multifunction

_f"

X---+Y

is defined

[fg(A)](x)

_[Gi(x,y)

A

A(y)].

yEY

Definition 3.3: The fuzzy multifunction

_f:

XY

is fuzzy closed valued if

f(x)

is a

closed fuzzy set for each x. The terms fuzzy open valued and fuzzy compact valued

are defined similarly.

Definition 3.4:

A

fuzzy multifunction

_f:XY

between two fuzzy topological spaces

X

and

Y

is"

(a)

upper hemicontinuous at the point x, if for every open neighborhood

U

of

f(x),

f’(U)

is a neighborhood of x in

X.

The fuzzy multifunction

_f

is upper hemicontinuous on

X

if it is upper hemicontinuous at every point of

X;

(b)

lower hemicontinuous at

x,

if for every open fuzzy set

U

which intersects

f(x),

I(U)

is a neighborhood of x.

As above,

f

is lower hemicontinuous

on

X

if it is lower hemicontinuous at each point of

X;

(c)

continuousif it is both upper and lower hemicontinuous.

For

a more detailed account of the

concepts

outlined

above,

the reader is referred to

Beg

[5, 6]

and Tsiporkova-Hristoskova,

De

naets

and

Kerre

[23,

24].

4. Fuzzy

Topological

Vector Spaces

Let

E

be a vector space over

K,

where

K

denotes either the real or the complex

num-bers.

Let

_A1,A2,...,A

_n be fuzzy subsets of

E,

with

A

_{lxA2xA3x...xA}

_n denoting thefuzzy subset

A

in

E

n _defined _by

A(xl,x2,...,Xn)

min{A(x),A2(x2),...,An(xn)

}.

If

_f:En--+E

is defined by

_{f(xl,x2;...Xn)}

_x

₊

x₂

+...

+

_xn,

then the fuzzy set

_f(A)

in

E

is called the sum of the fuzzy sets

_{A1,A2,...,An,}

and it is denoted by

_A

₊

A2...

+

A

_n.

For

a fuzzy subset

A

of

E

and t a

scalar,

we denote

tA

as the image of

A

under the map g:

EE,

g(x)

ix. Ifa is a fuzzy set in

K

and

A

afuzzy set in

E,

then the image in

E

ofthe fuzzy set

axA,

a fuzzy subset of

KE[(a

A) (t,x)=

min{a(t),A(x)}]

under the map h:Kx

EE,

h(t,x)=

tx,

is denoted by

cA.

A

fuzzy set

A

in

E

is called convex if for each

_E[0,1],

_{[tA+(1-t)A] (x)< A(x).}

The

convex hull of a fuzzy set

B

is the smallest convex fuzzy set containing

B,

and is denoted by

_Co(B).

Given a topological space

(X,

7-),

the collection

w(7-)

of all fuzzy sets in

X,

which

are lower

semicontinuous,

as a function from

X

to

_[0,1]

equipped with the usual

topology,

is a fuzzy

topology

of

X.

The fuzzy topology

w(v)

is called the fuzzy

topology

generated by the usual

topology

7-. The fuzzy usual topology on

K

is the

fuzzy topology generated by the topology of

K.

Definition 4.1"

A

fuzzy linear topology on a vector space

E

over

K

is a fuzzy topology 7- on

E

such that the two mappings:

(4)

h:

K

E--E,

h(t,

x)

tx,

are continuous when

K

has the usual fuzzy topology, with

K

x

E,

E

x

E

being the corresponding product fuzzy topologies.

A

linear space with a fuzzy linear

topology

is called a fuzzy topological vector space.

Lemma

4.2:

In

a fuzzy topological vector space

X,

the algebraic sum

_of

a compact fuzzy set and a closedfuzzy set is closedfuzzy set.

Proof:

Let

A

be a compact fuzzy subset and

B

be a closed fuzzy subset of

X. Let

a net

_{x,x

₊

_Y,X}

in

A

₊

B

satisfy

_xx

+

yx--z.

Since

A

is compact fuzzy

set,

we can assume

(by

passing to a

_subnet)

that

_xA-x

E

A.

The continuity of the algebraic operations imply"

y)

(x

+

y)- x,x--z-

x y.

Since

B

is a closed fuzzy

subset, therefore, yEB. So

z=x+yA+B.

Hence

A

+

B

is a close fuzzy set.

Lemma

4.3:

In

X,

the algebraic sum

_of

two compact fuzzy sets is a compact fuzzy set.

Proof: Similar to

Lemma

4.2.

Theorem 4.4:

Let

K

be a compact fuzzy subset

_of

X. Suppose K

C

U,

where

U

is an openfuzzy subset. Then there is a neighborhood

W

_of

origin such that

K

₊

W

C

U.

Proof:

For

each

x

K,

there is a neighborhood

V

_x of origin such that

x

+

Vx

C

U.

Choose an open neighborhood

_Wx

oforigin so that

_Wx

₊

_Wx

C

Vx

for each x. Since

K

is a compact fuzzy

set,

there is a finite set

_{{Xl,X2,...,xn)}

of points with

K

C

_{[.Jni l(Xi}

₊

_Wx

."

Set W

n

_i=1

W

x

For

every x

K

there is some x

satisfying x x

+

_Wx.

F)r

this _xi,

+ w

+

w) c

+

x;

+ w

x

C x

₊

_Vxi

C

U. Hence

K+WCU.

Theorem 4.5:

Let

X

be afuzzy topological vector space.

_If

each

_Ai(i

co n act,

Co( [J

i--1

Proof: Since the continuous image of a compact fuzzy set is a compact fuzzy set and the

Hence

C0(

J

_

1Ai)

is a compact fuzzy set.

Definition 4.6:

A

fuzzy topological vector space

E

is called locally convexif it has

a base at the origin ofconvex fuzzy sets.

For basic concepts and details regarding fuzzy topological vector spaces, we refer to

[12-14,

17,

_181.

(5)

5. Vector-Valued

Fuzzy

Multifunctions

When the range space of a fuzzy multifunction is a vector space, then there are

additional natural operations onfuzzy multifunctions.

Definition 5.1" If f,g’X---Y are two fuzzy multifunctions, where

Y

is a vector space, then wedefine:

1. The sumfuzzy

_{multifunction}

_f

₊

g by

(f

/

g)(x)

f(x)

+

g(x)

{y

+

z:y E

f(x)

and z E

g(x)}.

The convexhullfuzzy

_{multifunction}

_co(f)

of

_f

by

(co(f))(x)

co(f(x)).

If

Y

is a fuzzy topological vector space, the closed convex hull fuzzy

multifunction

c(co(f)

of

_f

by

(cg(co(f)))(x)

c(co(f(x))

).

Lemmas

4.2 and 4.3 imply the following theorem.

Theorem 5.2:

Let

f,g:

X-Y

be twofuzzy

multifunctions

from

a fuzzy topological space

X

into afuzzy topological vector space

Y:

1.

_{If f}

is closed valued and g is compact-valued, then

_f

₊

g is closed valued. 2.

_If

_f

and g are compact

valued,

then

_f

₊

g is compact valued.

Theorem 5.3:

Let

f,g:

X---Y

be two fuzzy

_{multifunctions}

_from

X

into a fuzzy topological vector space

Y. _If

_f

and g are compact valued and upper hemicontinuous at apoint _Xo, then

_f

₊

g is upper hemicontinuous at x_o.

Proof:

Let

_f

and g be upper hemicontinuous fuzzy multifunctions at the point x_0.

Suppose

_f(xo)

₊

g(xo)

C

G,

where

G

is an open fuzzy subset ofY.

By

Theorem

4.4,

there is a neighborhood

V

oforigin such that

_f(xo)

₊

g(xo)

+

V

C

G.

Select an open neighborhood

W

of origin with

_W+W

C

Y.

Since

_f(xo)

C

_f(xo)

+W

and

f(xo)

+

W

is open, the upper hemicontinuity of

_f

at x₀

guarantees

the existence ofan

open neighborhood

N

₁ of x₀ such that

_f(N1)

C

_f(Xo)+

W.

Similarly, there existsan

open neighborhood

N

₂ of x₀ with

_g(N2)

Cg(x

_o+W.

Let

N=N

_1AN2,

then

N

is

an open neighborhood ofx₀and

(f

+

g)(N)

C

_f(N1)

+

g(N2)

C

f(xo)

+

W

₊

g(Xo)

+

W

C

G. It

further implies that

_f

/g is upper hemicontinuousfuzzy multifunctions at x_0.

Theorem 5.4:

Let

f,g:

X---Y

be two fuzzy

multifunctions

from

X

Y. _{If f}

and g are also lower

hemicon-tinuous fuzzy

_{multifunctions}

at a point Xo, then

f

/g is also lower hemicontinuous fuzzy

multifunctions

at x_O.

Proof:

Suppose

_[f(x0)

₊

g(x0)

N

U

=

,

where

U

is open fuzzy subset. Then there

are y in

f(xo)

and z

g(xo),

with y

₊

z

U. Thus,

there is an open neighborhood

V

oforigin such that y

₊

V

₊

z

+

V

C

U.

Since y

f(xo)

(y

+

V)

and

_f

is lower hemi-continuous fuzzy multifunction at _x0,

_f

(y

₊

V)

is a neighborhood ofx_0.

g

Similarly, g

(z/V)

is aneighborhood ofx_0.

Hence,

ifxf

(y/V)

N

(zWV),

then

_If(z)

₊

g(x)]

U

.

(6)

Theorem 5.5:

Let

_fi:

X-Y

(i

1,2,...,n)

be

(single-valued)

fuzzy

functions

from

X

Y,

and the fuzzy

multifunction

I:X--Y

be given by

_{f(x)-{fl(x),f2(x),...,fn(x)}.}

_If

each

_fi

is

continuous at apoint x_{o E}

X,

then thefuzzy

multifunction

f

is continuous at x_o.

Proof:

Suppose

_f(xo)=

{fl(xo,

f2(xo),...,fn(xo)}

C

U,

where

V

is an open fuzzy subset of

Y.

Then:

n

V

_f/-

_I(U)

i=1

is an open

neighborhood

of x₀ such that x E

V

implies

f(x)C

U. It

further implies that the fuzzy multifunction

_f

is upper hemicontinuous.

Next,

suppose

_f(xo)

r’l

W

-

for some open fuzzy subset

W

of

Y.

If

_fn(xo)

W,

then

P-fl(W)

is a neighborhood of _x0, and xG

P

implies

f(P)

MW

:-.

It

further implies that the fuzzy multifunction

_f

is lower semicontinuous.

Hence

the fuzzy multifunction

_f

iscontinuous.

Theorem 5.6:

Let

_fi:XY

_(i

_1,2,3,...,n)

be

(single-valued)

fuzzy

_functions

from

X

into a locally convex fuzzy topological vector space

Y,

and

_I:X--Y

be given by

f(x)-

{fl(x),f2(x),...,fn(x)}.

If

each

_fi

is continuous

at some point _Xo, then the convex hullfuzzy

_{multifunction}

_co(f)

is continuous at x_o.

Proof:

Suppose

that

_(co(f))(x)

C

U,

where

U

is an open fuzzysubset ofthe locally

convex fuzzy topological vector space

Y. By

Theorem

4.5,

(co(f))(x)

is compact. Theorem 4.4 further implies that there exists an open convex

neighborhood W

of origin satisfying

_(co(f))(Xo)

₊

W

C

U. From

_f(xo)

C

_f(xo)

+

W

and the upper hemi-continuity of

_f

at x₀

(Theorem 5.5),

there exists a neighborhood

V

of x₀ such that

f(x)

C

_f(xo)+

W

for each x

Y. So,

if x

V,

then

_(co(f))(x)

C

_(co(f))(Xo)+

W

C

U.

This also implies that

_co(f)

is an upper hemicontinuous fuzzy multifunction at x_0.

Next,

let

_{(co(f))(Xo)ClU}

_:/=

for some open fuzzy subset

V.

Pick

(i-

1,

2,

n),

with

n t

A

i-1

and

Aifi(xo)

GU,

i=1 i=1

The fuzzy function g:

XY

defined by

g(x)-

_E

=

lifi(x)

is continuous at x₀

(by

definition offuzzy topological vector

spaces).

This implies that there exists a neigh-borhood

V

ofx₀ such that x

V

implies

_i=

_llifi(x)

U. Therefore,

_((Co)(I))(x)

gl

_U

_:/:

_{for each x}

e

_V.

Thus

_Co(I)

is a lower

hemicontin-uous fuzzy multifunction at x_0.

Hence,

co(f)

is a continuous fuzzy multifunction at

X 0

Theorem 5.7:

Let

X

be a fuzzy topological space and

Y

be a locally convexfuzzy topological vector space. Let

_f:XY

be an upper hemicontinuous fuzzy

multifunc-tion at x.

_If

_{(cg(co(f)))(x)}

is compact, then

cg(co(f)

is an upper hemicontinuous fuzzy

multifunction

at x.

Proof:

Let

_{(c(co(I)))(x)}

C

P

for some open fuzzy set

P.

If

_{(c(co(I)))(x)}

is

com-pact, then there is a convex neighborhood

V

of origin with

v

₊

v

c

P

(by

definition of local convexity and Theorem

_4.4).

Lemma

4.2 further implies that

_{(c(co(I)))(x)}

₊

c(V)is

a closed convex fuzzy set. Since

_f

is an upper hemicontinuous fuzzy multifunction at x,

IU(f(x)+

V)is

a neighborhood of x. If

(7)

(cg(co(f)))(z)

C

(cg(co(f)))(x)

+

cg(V)

c

v

+

v

c

e.

Therefore,

(ce(co(f)))(P)

includes

_fu(f(x)+

_V).

Hence

_ce(co(f)),

is an upper hemi-continuousfuzzy multifunction at x.

Definition 5.8:

A

fuzzy multifunction

_f:X-+Y

from a fuzzy topological space

X

Y

is upper demicontinuous if

fU({y

E

Y:

h(y)

<

a where h is a continuous linear fuzzy singlevalued function from

Y

into

K})

is an open subset of

X.

Theorem 5.9:

A

compact valued fuzzy

_{multifunction}

_{f:X-Y from}

X

Y

is upper demicontinuous

_if

and only

_if

_ce(co(f)

is upper demicontinuous.

Proof:

Let

H

C

Y

be an open half fuzzy space of the form

H

{y

where g:Y-K is a continuous linear fuzzy single valued function. Since g is a linear andcontinuous fuzzy function,

max{g(y):

y

f(x)}

max{g(y):

y

_{(c.(co(f)))(x)}.}

It

implies

f(x)

C

H

if and only if

_{(ce(co(f)))()}

c

H.

demicontinuous if and only if

_ce(c0(f)

is also demicontinuous.

Hence

_f

is upper

References

[2]

[41

[5]

[1]

Albrycht,

a.

and

Maltoka,

M.,

On

fuzzy multivalued functions,

Fuzzy Sets

and

Systems

12

_(1984),

61-69.

Aubin, J.P.

and

Ekeland,

I.,

Applied Nonlinear Analysis, Wiley-Interscience 1984.

Aubin, J.P.

and

Frankowska,

H.,

Set-Valued Analysis, Birkhauser,

Boston

1990.

Beg,

I.,

Fixed points of fuzzy multivalued mappings with values in fuzzy ordered

sets, J. Fuzzy

Math. 6:1

(1998),

127-131.

Beg,

I.,

Continuity of fuzzy

multifunctions, J.

Appl. Math. and Stoch. Anal. 12:1

_(1999),

17-22.

[6]

Beg,

I., Fuzzy

closed

graph

fuzzy

multifunctions, Fuzzy Sets

Systems

115:3

(2000),

451-454.

[7]

Beg,

I.,

Operations on fuzzy multifunctions and fuzzy maximum principle, submitted.

[8]

Berge,

C.,

Topological

Spaces,

MacMillan,

New

York 1963. English trans, by

E.M. Patterson

of

Espaces

Topologiques Fonctions Multivoques,

Dunod,

Paris 1959.

[9]

Butanriu,

D.,

Fixed points for fuzzy mappings,

Fuzzy Sets

and

Systems

7

(1982),

191-207.

[10]

Chang,

C.L.,

Fuzzy

topological spaces,

J.

Math. Anal. Appl. 24

_(1968),

182-190.

[11]

Heilpern,

S.,

Fuzzy

mappings and fixed point

theorem, J.

(1981),

566-569.

[12]

Katsaras,

A.K.,

Fuzzy

topological vector spaces

I,

Fuzzy Sets

and

Systems

6

(1981),

85-95.

(8)

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

(1984),

143-154.

Katsaras,

A.K.

and Liu,

D.B.,

Fuzzy

vector spaces and fuzzy topological vector spaces,

J. _(1977),

135-146.

Klein, E.

and Thompson,

A.C.,

Theory

_of

Correspondences: Including Applica-tions to Mathematical Economics, John Wiley and

Sons

1984.

Li,

H.X.

and

Yen,

V.C.,

Fuzzy

Sets

and

Fuzzy

Decision Making,

CRC

Press,

London 1995.

Lowen, R., Fuzzy

topological spaces and fuzzy compactness,

J. (1976),

621-633.

Lowen, R., A

comparison of different compactness notions in fuzzy topological spaces,

J. _(1978),

445-454.

Ozbakir, O.B.

and Aslim,

G.,

On

some types of fuzzy continuous

multifunc-tions,

Bull. Cal. Math.

Soc.

88

(1996),

217-230.

Papageorgiou,

N.S., Fuzzy

topology

and fuzzy

multifunctions, J.

_(1985),

397-425.

Pu,

P.

and

Liu,

Y., Fuzzy

topology I:

Neighborhood structure of a fuzzy point andMoore-Smith convergence,

J. (1980),

571-599.

E.,

De

Baets,

B.

and

Kerre,

E., A

detailed study of direct and inverse images under fuzzy multivalued mappings,

J. Fuzzy

Math 3

(1995),

191-208.

E.,

De

Baets,

B.

and

Kerre, E., A

fuzzy inclusion based approach to upper inverse images under fuzzy multivaluedmappings,