7/30/2019 Fuzzy Relational Equations

1/7

A F U Z Z Y R E L A T I O N A L E Q U A T I O N

I N

D Y N A M I C F U Z Z Y S Y S T E M S

M . K U R A N O ( C h i b a U n i v e r s i t y ) , M . Y A S U D A ( C h i b a U n i v e r s i t y ) ,

J . N A K A G A M I ( C h i b a U n i v e r s i t y ) & Y . Y O S H I D A ( K i t a k y u s h u U n i v e r s i t y )

A b s t r a c t : F o r a d y n a m i c f u z z y s y s t e m , t h e f u n d a m e n t a l m e t h o d i s t o a n a l y z e i t s r e c u r s i v e r e l a t i o n

o f t h e f u z z y s t a t e s . I t i s s i m i l a r t h a t t h e B e l l m a n e q u a t i o n i s t h e i m p o r t a n t t o o l i n t h e d y n a m i c

p r o g r a m m i n g . H e r e w e w i l l c o n s i d e r t h e e x i s t e n c e a n d t h e u n i q u e n e s s o f s o l u t i o n o f a f u z z y r e l a t i o n a l

e q u a t i o n . T w o e x a m p l e s , w h i c h s a t i s e s o u r c o n d i t i o n s , a r e g i v e n t o i l l u s t r a t e t h e r e s u l t s .

1 I n t r o d u c t i o n a n d n o t a t i o n s

W e u s e t h e n o t a t i o n s i n 4 ] . L e t X b e a c o m p a c t m e t r i c s p a c e . W e d e n o t e b y 2

X

t h e c o l l e c t i o n o f a l l

s u b s e t s o f X , a n d d e n o t e b y C ( X ) t h e c o l l e c t i o n o f a l l c l o s e d s u b s e t s o f X . L e t b e t h e H a u s d o r m e t r i c

o n 2

X

. T h e n i t i s w e l l - k n o w n ( 3 ] ) t h a t ( C ( X ) ) i s a c o m p a c t m e t r i c s p a c e . L e t F ( X ) b e t h e s e t o f a l l

f u z z y s e t s ~s X ! 0 1 ] w h i c h a r e u p p e r s e m i - c o n t i n u o u s a n d s a t i s f y s u p

x 2 X

~s ( x ) = 1 . L e t ~q X X !

0 1 ] b e a c o n t i n u o u s f u z z y r e l a t i o n o n X

I n t h i s p a p e r , w e c o n s i d e r t h e e x i s t e n c e a n d u n i q u e n e s s o f s o l u t i o n ~p 2 F ( X ) i n t h e f o l l o w i n g f u z z y

r e l a t i o n a l e q u a t i o n ( 1 . 1 ) f o r a g i v e n c o n t i n u o u s f u z z y r e l a t i o n ~q o n X ( s e e 4 ] ) :

~p ( y ) = s u p

x 2 X

f ~p ( x ) ~q ( x y ) g y 2 X ( 1 . 1 )

w h e r e a b : = m i n f a b g f o r r e a l n u m b e r s a a n d b . W e d e n e a m a p ~q

2

X

! 2

X

( 2 0 1 ] ) b y

~q

( D ) =

8

0 f o r s o m e x 2 D g f o r = 0 D 2 2

X

D 6= ;

X f o r 0 1 D = ;

( 1 . 2 )

w h e r e c l d e n o t e s t h e c l o s u r e o f a s e t . E s p e c i a l l y , w e p u t ~q

( x ) : = ~q

( f x g ) f o r x 2 X . W e n o t e t h a t ~q

C ( X ) ! C ( X )

L e m m a 1 . 1 ( 4 , L e m m a 2 ] ) . F o r e a c h 2 0 1 , t h e m a p ~q

C ( X ) ! C ( X ) i s c o n t i n u o u s w i t h r e s p e c t

t o

F o r ~s 2 F ( X ) , t h e - c u t ~s

2 0 1 ] i s d e n e d b y

~s

= f x 2 X ~s ( x ) g ( 6= 0 ) a n d ~s

0

: = c l f x 2 X ~s ( x ) > 0 g

L e m m a 1 . 2 .

( i ) F o r ~s 2 F ( X ) ~s s a t i s e s ( 1 . 1 ) i f a n d o n l y i f

~q

( ~s

) = ~s

2 0 1 ( 1 . 3 )

( i i ) W e s u p p o s e t h a t a f a m i l y o f s u b s e t s f D

2 0 1 g ( C ( X ) ) s a t i s e s t h e f o l l o w i n g c o n d i t i o n s ( a )

( b ) a n d ( c ) :

1

7/30/2019 Fuzzy Relational Equations

2/7

( a ) D

D

f o r 0 < 1 ;

( b ) l i m

"

D

= D

f o r 6= 0 ;

( c ) ~q

( D

) = D

f o r 2 0 1

T h e n ~s ( x ) : = s u p

2 0 1

f 1

D

( x ) g x 2 X , s a t i s e s ~s 2 F ( X ) a n d ( 1 . 1 ) , w h e r e 1

D

d e n o t e s t h e

c h a r a c t e r i s t i c f u n c t i o n o f a s e t D 2 2

X

P r o o f . ( i ) i s t r i v i a l . ( i i ) i s f r o m ( i ) a n d 4 , L e m m a 3 ] . 2

2 T h e e x i s t e n c e o f s o l u t i o n s

F o r 2 0 1 ] a n d x 2 X , a s e q u e n c e f ~q

k

( x ) g

k = 1 2

i s d e n e d i t e r a t i v e l y b y

~q

0

( x ) = f x g ~q

1

( x ) : = ~q

( x ) a n d ~q

k + 1

( x ) : = ~q

( ~q

k

( x ) ) f o r k = 1 2

T h e n , l e t G

( x ) =

S

1

k = 1

~q

k

( x ) a n d

F

( x ) =

1

k = 0

~q

k

( x ) = f x g G

( x ) ( 2 . 1 )

W e n o w c o n s i d e r a c l a s s o f i n v a r i a n t p o i n t s f o r t h i s i t e r a t i o n p r o c e d u r e , t h a t i s , x 2 G

( x ) . S o p u t

R

= f x 2 X x 2 G

( x ) g f o r 2 0 1 ( 2 . 2 )

E a c h s t a t e o f R

i s c a l l e d a s a n \ - r e c u r r e n t " s t a t e a n d i t i s s t u d i e d b y 7 ] . T h e f o l l o w i n g p r o p e r t i e s ( i )

a n d ( i i ) h o l d c l e a r l y :

( i ) ~q

( F

( x ) ) = G

( x ) f o r 2 0 1 ] a n d x 2 X

( i i ) R

R

f o r 0 < 1

L e m m a 2 . 1 . I f z 2 R

1

, t h e f o l l o w i n g ( i ) a n d ( i i ) h o l d :

( i ) ~q

( F

( z ) ) = F

( z ) f o r 2 0 1 ;

( i i ) F

( z ) F

( z ) f o r 0 < 1

P r o o f . S i n c e z 2 R

1

R

, w e h a v e

~q

( F

( z ) ) = G

( z ) = F

( z )

S o , w e o b t a i n ( i ) . ( i i ) i s t r i v i a l . 2

F o r z 2 R

1

, w e d e n e

^

F

( z ) =

\

<

c l f F

( z ) g ( 6= 0 ) a n d

^

F

0

( z ) : = c l f F

0

( z ) g ( 2 . 3 )

w h e r e c l f F

( z ) g d e n o t e s t h e c l o s u r e o f F

( z )

L e m m a 2 . 2 . I f z 2 R

1

, t h e f o l l o w i n g ( i ) ( i i ) a n d ( i i i ) h o l d :

( i ) ~q

(

^

F

( z ) ) =

^

F

( z ) f o r 2 0 1 ;

( i i )

^

F

( z )

^

F

( z ) f o r 0 < 1 ;

( i i i )

^

F

( z ) = l i m

"

^

F

( z ) f o r 6= 0

2

7/30/2019 Fuzzy Relational Equations

3/7

P r o o f . ( i i ) i s t r i v i a l f r o m L e m m a 2 . 1 a n d ( i i i ) i s a l s o t r i v i a l f r o m t h e d e n i t i o n . T o p r o v e ( i ) , l e t =

0 . F r o m L e m m a 2 . 1 ( i ) , w e h a v e ~q

0

( F

0

( z ) ) = F

0

( z ) . B y t h e c o n t i n u i t y o f ~q , w e c a n c h e c k ~q

0

( c l f F

0

( z ) g ) =

c l f F

0

( z ) g i n a s i m i l a r w a y t o t h e p r o o f o f 4 , L e m m a 1 ] . T h e r e f o r e , ~q

0

(

^

F

( z ) ) =

^

F

0

( z ) . L e t > 0 a n d y 2

~q

(

^

F

( z ) ) . B y L e m m a 1 . 1 , w e h a v e

y 2

\

<

~q

( c l f F

( z ) g ) =

1

\

n = 1

~q

( c l f F

( 1 = n ) _ 0

( z ) g )

F r o m t h e c o n t i n u i t y o f ~q , f o r n 1 , t h e r e e x i s t s x

n

2 F

( 1 = n ) _ 0

( z ) s u c h t h a t ~q ( x

n

y ) ? 1 = n B y

L e m m a 2 . 1 ( i ) ,

y 2 ~q

( 1 = n ) _ 0

( F

( 1 = n ) _ 0

( z ) ) = F

( 1 = n ) _ 0

( z ) c l f F

( 1 = n ) _ 0

( z ) g

f o r a l l n 1 . S o , y 2

^

F

( z ) . T h e r e f o r e , w e o b t a i n

~q

(

^

F

( z ) )

^

F

( z )

W h i l e , f r o m L e m m a 2 . 1 ( i ) , w e h a v e

c l f F

( z ) g ~q

( c l f F

( z ) g )

f o r < < . T h e n

^

F

( z ) =

\

<

c l f F

( z ) g

\

<

~q

( c l f F

( z ) g ) = ~q

( c l f F

( z ) g )

f o r < . S o , w e g e t

^

F

( z )

\

<

~q

( c l f F

( z ) g ) = ~q

\

<

c l f F

( z ) g

!

= ~q

(

^

F

( z ) )

T h e r e f o r e , w e c a n o b t a i n ( i ) . 2

L e t z 2 R

1

. S i n c e f

^

F

( z ) 2 0 1 g s a t i s e s t h e c o n d i t i o n s ( a ) { ( c ) o f L e m m a 1 . 2 ( i i ) , w e o b t a i n t h e

f o l l o w i n g t h e o r e m .

T h e o r e m 2 . 1 .

( i ) I f R

1

6= ; , t h e n t h e r e e x i s t s a s o l u t i o n o f ( 1 1 )

( i i ) L e t z 2 R

1

. D e n e a f u z z y s t a t e

~s

z

( x ) : = s u p

2 0 1

n

1

^

F

( z )

( x )

o

x 2 X ( 2 . 4 )

T h e n ~s

z

2 F ( X ) s a t i s e s ( 1 1 )

A s s u m e t h a t R

1

6= ; . W e i n t r o d u c e a n e q u i v a l e n t r e l a t i o n o n R

a s f o l l o w s : F o r z

1

z

2

2 R

z

1

z

2

m e a n s t h a t z

1

2 F

( z

2

) a n d z

2

2 F

( z

1

)

T h e n w e c o u l d i d e n t i f y t h e s t a t e s o f R

w h i c h i s e q u i v a l e n t w i t h r e s p e c t t o , a n d s o p u t

R

= R

=

L e m m a 2 . 3 . F o r z

1

z

2

2 R

1

z

1

z

2

i f a n d o n l y i f F

( z

1

) = F

( z

2

) f o r a l l 2 0 1

3

7/30/2019 Fuzzy Relational Equations

4/7

P r o o f . L e t z

1

z

2

. T h e n , w e h a v e z

1

2 F

1

( z

2

) F

( z

2

) f o r a n y 2 0 1 ] . F r o m t h e d e n i t i o n ( 2 . 1 ) o f

F

( z

1

) , w e o b t a i n F

( z

1

) F

( z

2

) S i n c e w e h a v e F

( z

2

) F

( z

1

) s i m i l a r l y , F

( z

1

) = F

( z

2

) h o l d s . T h e

r e v e r s e p r o o f i s t r i v i a l . 2

F r o m T h e o r e m 2 . 1 a n d L e m m a 2 . 3 , t h e n u m b e r o f s o l u t i o n s o f ( 1 . 1 ) i s g r e a t e r t h a n o r e q u a l s t o t h e n u m -

b e r o f \ 1 - r e c u r r e n t " s e t s . T o c o n s i d e r t h e c l a s s o f s o l u t i o n ( 1 . 1 ) , l e t P = f ~p 2 F ( X ) ~p i s a s o l u t i o n o f ( 1 . 1 ) g

T h e n P h a s t h e f o l l o w i n g p r o p e r t y :

T h e o r e m 2 . 2 . L e t ~p

k

2 P ( k = 1 2 l ) . T h e n :

( i ) P u t

~p ( x ) : = m a x

k = 1 2

~p

k

( x ) f o r x 2 X

T h e n ~p 2 P

( i i ) L e t f

k

2 0 1 k = 1 2 l g s a t i s f y m a x

k = 1 2

k

= 1 . P u t

~p ( x ) : = m a x

k = 1 2

f

k

~p

k

( x ) g f o r x 2 X

T h e n ~p 2 P

P r o o f . ( i i ) T a k i n g t h e - c u t o f ~p 2 F ( X ) , w e h a v e

~p

=

k

k

~p

k

T h e n ,

~q

( ~p

) = ~q

0

@

k

k

~p

k

1

A

=

k

k

~q

( ~p

k

) =

k

k

~p

k

= ~p

T h e r e f o r e , w e o b t a i n ( i i ) f r o m L e m m a 1 . 2 ( i ) . ( i ) i s p r o v e d s i m i l a r l y . 2

3 T h e u n i q u e n e s s o f s o l u t i o n s

I n t h i s s e c t i o n , w e d i s c u s s t h e u n i q u e n e s s o f s o l u t i o n s o f t h e e q u a t i o n ( 1 . 1 ) u n d e r c o n v e x i t y a n d c o m p a c t n e s s .

L e t B b e a c o n v e x s u b s e t o f a n n - d i m e n s i o n a l E u c l i d e a n s p a c e R

n

a n d C

c

( B ) t h e c l a s s o f a l l c l o s e d a n d

c o n v e x s u b s e t s o f B . T h r o u g h o u t t h i s s e c t i o n , w e a s s u m e t h a t t h e s t a t e s p a c e X i s a c o n v e x a n d c o m p a c t

s u b s e t o f R

n

. T h e f u z z y s e t ~s 2 F ( X ) i s c a l l e d c o n v e x i f i t s - c u t ~s

i s c o n v e x f o r e a c h 2 0 1 ] . L e t

F

c

( X ) = f ~s 2 F ~s i s c o n v e x g

B y a p p l y i n g K a k u t a n i ' s x e d p o i n t t h e o r e m ( 2 ] ) , w e h a v e t h e f o l l o w i n g .

L e m m a 3 . 1 . L e t 2 0 1 a n d ~q

( x ) i s c o n v e x f o r e a c h x 2 X . T h e n , f o r a n y A 2 C

c

( X ) w i t h A =

~q

( A ) , t h e r e e x i s t s a n x 2 X s u c h t h a t ~q ( x x )

P r o o f . T h e m a p ~q

A ! C

c

( A ) w i t h ~q

( x ) 2 C

c

( A ) f o r a l l x 2 A i s c o n t i n u o u s f r o m L e m m a 1 . 1 , s o

K a k u t a n i ' s x e d p o i n t t h e o r e m g u a r a n t e e s t h e e x i s t e n c e o f a n e l e m e n t x 2 A s u c h t h a t x 2 ~q

( x ) , w h i c h

i m p l i e s ~q ( x x ) . T h i s c o m p l e t e s t h e p r o o f . 2

W e a s s u m e t h a t ~q

( x ) i s c o n v e x f o r e a c h x 2 X . A s a c o n s e q u e n c e , w e h a v e a p r o p e r t y o f t h e s o l u t i o n s

o f ( 1 . 1 ) .

P r o p o s i t i o n 3 . 1 . L e t p 2 F

c

( X ) b e a s o l u t i o n o f ( 1 . 1 ) . T h e n , f o r e a c h 2 0 1 , t h e r e e x i s t s a n x 2 p

w i t h ~q ( x x )

P r o o f . B y L e m m a 1 . 2 , ~p

= ~q

( ~p

) f o r e a c h 2 0 1 ] . T h u s , L e m m a 3 . 1 c l e a r l y p r o v e s t h e d e s i r e d r e s u l t .

2

N o w , w e g i v e s u c i e n t c o n d i t i o n s f o r t h e u n i q u e n e s s o f s o l u t i o n s o f ( 1 . 1 ) . L e t U

= f x 2 X ~q ( x x )

g f o r 2 0 1

A s s u m p t i o n A . T h e f o l l o w i n g A 1 { A 3 h o l d .

4

7/30/2019 Fuzzy Relational Equations

5/7

A 1 . T h e s e t U

1

i s a o n e - p o i n t s e t , s a y u . T h a t i s , U

1

= f u g

A 2 U

F

( u ) f o r e a c h 2 0 1 ] , w h e r e u i s g i v e n b y A 1 a n d F

( u ) i s d e n e d b y ( 2 . 1 ) .

A 3 . L e t 2 0 1 ] a n d A 2 C

c

( X ) . I f A = ~q

( A ) , t h e n

A =

x 2 U

\ A

F

( x )

T h e o r e m 3 . 1 . U n d e r A s s u m p t i o n A , t h e e q u a t i o n ( 1 . 1 ) h a s a u n i q u e s o l u t i o n i n F

c

( X )

P r o o f . L e t ~p ~p 2 F

c

( X ) b e s o l u t i o n s o f ( 1 . 1 ) . B y L e m m a 3 . 1 , ~p

1

\ U

1

6= a n d ~p

1

\ U

1

6= . S i n c e U

1

i s a

o n e - p o i n t s e t , u 2 ~p

1

a n d u 2 ~p

1

. T h u s , b y A 3 , ~p

1

= F

1

( u ) a n d ~p

1

= F

1

( u ) , w h i c h i m p l i e s ~p

1

= ~p

1

W e

n o w s h o w t h a t ~p

=

^

F

( u ) f o r 0 < 1 . S i n c e u 2 ~p

= ~q

( ~p

) , i t h o l d s t h a t F

( u ) ~p

. T h e r e f o r e ,

s i n c e ~p

i s c l o s e d ,

^

F

( u ) =

<

c l f F

( u ) g

<

~p

( u ) = ~p

O n t h e o t h e r h a n d , w e h a v e

~p

S

x 2 U

\ ~p

c l f F

( x ) g ( f r o m A 3 )

S

x 2 F

( u ) \ ~p

c l f F

( x ) g ( f r o m A 2 )

S

x 2

^

F

( u )

c l f F

( x ) g

F r o m t h a t x 2

^

F

( u ) m e a n s

^

F

( x )

^

F

( u ) , i t h o l d s t h a t

~p

c l f F

( u ) g

^

F

( u )

T h e a b o v e s h o w s ~p

=

^

F ( u ) . S i m i l a r l y ~p

=

^

F

( u ) . T h u s , ~p

= ~p

. T h i s c o m p l e t e s t h e p r o o f . 2

4 N u m e r i c a l e x a m p l e s

H e r e t w o n u m e r i c a l e x a m p l e s o f S e c t i o n 2 a n d 3 a r e g i v e n t o c o m p r e h e n d c o m p u t a t i o n a l a s p e c t o f t h i s

p a p e r .

E x a m p l e 1 . L e t X = 0 1 ] . F o r a n y g 0 1 ! 0 1 ] , l e t

~q ( x y ) : = ( 1 ? y ? g ( x ) ) _ 0 ; x ; y 2 0 1

W e a s s u m e t h a t g ( ) i s s t r i c t l y i n c r e a s i n g a n d c o n t i n u o u s a n d t h a t t h e r e e x i s t s a u n i q u e x

0

2 0 1 ] w i t h

x

0

= g ( x

0

) . U n d e r t h e a b o v e c o n d i t i o n , R

1

= f x

0

g a n d f o r e a c h 2 0 1 )

U

= x

x

( 4 . 1 )

w h e n x

x

i s a u n i q u e s o l u t i o n o f x = g ( x ) ? ( 1 ? ) x = g ( x ) + ( 1 ? ) r e s p e c t i v e l y a n d x

= 0 x

= 1

i f t h e s o l u t i o n d o e s n o t e x i s t i n 0 , 1 ] .

C l e a r l y , U

i s a u n i q u e s o l u t i o n o f t h e e q u a t i o n A = ~p

( A ) i n C

c

( 0 1 ] ) , s o t h a t A s s u m p t i o n A i n S e c t i o n

3 h o l d s i n t h i s c a s e . T h u s , b y T h e o r e m 3 . 1 ,

~s ( x ) = s u p

2 0 1

f I

U

( x ) g ( 4 . 2 )

i s a u n i q u e c o n v e x s o l u t i o n o f ( 1 . 1 ) . F o r a c o n c r e t e e x a m p l e s u c h a s g ( x ) = ( 2 x

2

+ 1 ) = 4 , t h e n i t i s s e e n t h a t

R

1

= f ( 2 ?

p

2 ) = 2 g a n d

x

=

1 ?

p

5 = 2 ? 2

_ 0

x

=

1 3 = 4 <

1 ?

p

2 ? 3 = 2 3 = 4 1

5

7/30/2019 Fuzzy Relational Equations

6/7

B y ( 4 . 1 ) a n d ( 4 . 2 ) , t h e u n i q u e s o l u t i o n i s a s f o l l o w s ( F i g . 1 ) :

~s ( x ) =

? x

2

= 2 + x + 3 = 4 0 x 1 ?

p

2 = 2

x

2

= 2 ? x + 5 = 4 1 ?

p

2 = 2 < x 1

F i g . 1 T h e u n i q u e s o l u t i o n ~s

E x a m p l e 2 . T h i s e x a m p l e h a s t w o p e a k s f o r t h e f u z z y r e l a t i o n . L e t X = 0 1 ] a n d

~q ( x y ) = ( 1 ? y ? ( x

2

+ 1 ) = 4 ) _ ( 1 ? y ? ( x

2

+ 2 ) = 4 )

f o r x y 2 0 1 ] . T h e n , R

1

= f a b g , w h e r e a = 2 ?

p

3 b = 2 ?

p

2 . B y s i m p l e c a l c u l a t i o n , w e g e t

^

F

( a ) = x

a

x

a

] a n d

^

F

( b ) = x

b

x

b

f o r 2 0 1 ] , w h e r e

x

a

=

0 0 3 = 4

2 ?

p

7 ? 4 3 = 4 < 1

x

a

=

1 0 7 = 8

2 ?

p

4 ? 1 7 = 8 < 1

x

b

=

0 0 7 = 8

2 ?

p

6 ? 4 7 = 8 < 1

x

b

=

1 0 3 = 4

2 ?

p

4 ? 2 3 = 4 < 1

B y T h e o r e m 2 . 1 , t h e s o l u t i o n s o f ( 1 . 1 ) a r e g i v e n a s f o l l o w s ( F i g . 2 ) :

~s

a

( x ) =

8

7/30/2019 Fuzzy Relational Equations

7/7

F i g . 2 T h e u n i q u e s o l u t i o n s ~s

a

a n d ~s

b

R e f e r e n c e s

1 ] R . E . B e l l m a n a n d L . A . Z a d e h , D e c i s i o n - m a k i n g i n a f u z z y e n v i r o n m e n t , M a n a g e m e n t S c i . S e r B . 1 7

( 1 9 7 0 ) 1 4 1 - 1 6 4 .

2 ] N . D u n f o r d a n d J . T . S c h w a r t z , L i n e a r O p e r a t o r s , P a r t 1 : G e n e r a l T h e o r y ( I n t e r s c i e n c e P u b l i s h e r s ,

N e w Y o r k , 1 9 5 8 ) .

3 ] K . K u r a t o w s k i , T o p o l o g y I ( A c a d e m i c P r e s s , N e w Y o r k , 1 9 6 6 ) .

4 ] M . K u r a n o , M . Y a s u d a , J . N a k a g a m i a n d Y . Y o s h i d a , A l i m i t t h e o r e m i n s o m e d y n a m i c f u z z y s y s t e m s ,

F u z z y S e t s a n d S y s t e m s 5 1 ( 1 9 9 2 ) 8 3 - 8 8 .

5 ] V . N o v a k F u z z y S e t s a n d T h e i r A p p l i c a t i o n s ( A d a m H i l d e r , B r i s t o l - B o s t o n , 1 9 8 9 ) .

6 ] Y . Y o s h i d a , M . Y a s u d a , J . N a k a g a m i a n d M . K u r a n o , A l i m i t t h e o r e m i n s o m e d y n a m i c f u z z y s y s t e m s

w i t h a m o n o t o n e p r o p e r t y , t o a p p e a r i n F u z z y S e t s a n d S y s t e m s

7 ] Y . Y o s h i d a , T h e r e c u r r e n c e o f d y n a m i c f u z z y s y s t e m s , t o a p p e a r i n F u z z y S e t s a n d S y s t e m s

7