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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
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( 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
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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
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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
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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
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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
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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