Cyclic classes and an ergodic theorem in dynamic fuzzy systems

W~,;,,~!t 'w ELSEVIER

FUZZY

sets and systems Fuzzy Sets and Systems 95 (1998) 189-199

Cyclic classes and an ergodic theorem in dynamic fuzzy systems Yuji Yoshida* Faculty of Economics and Business Administration, Kitakyushu University, 4-2-1 Kitagata, Kokuraminami, Kitakyushu 802, Japan Received January 1995; revised September 1996

Abstract This paper analyses cyclic behavior of dynamic fuzzy systems defined by fuzzy relations on a metric space. For the system, we give an ergodic theorem with a strata structure. © 1998 Elsevier Science B.V.

Keywords. Cyclic class; Ergodic theorem; Dynamic fuzzy system; Recurrence; Fuzzy relation

I. Introduction Limit theorems of a sequence of fuzzy sets defined successively by fuzzy relations are first studied by Bellman and Zadeh [1]. Kurano et al. [3] and Yoshida et al. [10], under a contraction condition and a monotone condition, respectively, studied the limiting behavior of fuzzy states defined by the dynamic fuzzy system with a compact space. A notion of recurrent sets for the dynamic fuzzy system is given by Yoshida [7]. This paper analyses the cyclic behavior of the dynamic fuzzy system and gives an ergodic theorem which is different from the ergodic theorems of the classical Markov chains. In Section 2, by an approach of Chung [2], we show the existence of cyclic classes for recurrent sets in the dynamic fuzzy system in preparation for an ergodic theorem in Section 3. Next, as a property in dynamic fuzzy systems, we prove that the period of a cyclic class at a grade is a devisor of the period of the cyclic class at the greater grades. In Section 3, we show a correspondence between a limiting possibility of the fuzzy transition and a period of the cyclic classes. By use of these results, we give an ergodic theorem which has different properties from the ergodic theorems of the classical Markov chains. In the dynamic fuzzy system, there exist a finite/countable number of grades to which the fuzzy transition periodically converges, and the states are classified into cyclic classes which has a strata structure with respect to the grades. In Section 4, a numerical example is given to illustrate our idea.

* Fax: +81-93-964-4000; e-mail: [email protected]. 0165-0114/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved PII S0165-01 14(96)00303-X

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2. Cyclic classes for dynamic fuzzy system Let a state space E be a complete metric space. Let a time space by ~ := {0, 1,2 .... }. Let A be an upper semi-continuous fuzzy relation on E × E satisfying the following normality condition: supA(x,y)=l

(yEE)

and

supA(x,y)=l

xcE

(xEE).

yEE

We denote a fuzzy set on E by its membership function g : E ~ [0, 1]. Kurano et al. [3], under a contractive condition, studied the limiting behavior of a sequence of fuzzy states {s'~}n°°__0 which is given as follows: An initial fuzzy state go is a given fuzzy set on E, and we define

gn+t(y):=sup{~n(x)AA(x,y)},

yEE, n = 0 , 1 , 2 ....

(2.1)

xEE

This is called a dynamic fuzzy system (see [3, 6, 8, 9]). We define a sequence of fuzzy relations, {A"},~0, on E × E as follows: For x, y E E, 1 ifx=y

0 if x # y '

q°(x'Y):=

Al(x,y):=A(x,y),

A"+l(x, y ) := sup{An(x,z) A A(z, y)},

n = 1,2, 3 . . . . .

(2.2)

zEE

where the operation A means a A b = min{a, b} for real numbers a and b. Then, An(x, y) means the possibility to transit from a point x to a point y at the nth step. The sequence of fuzzy state {s-~}n~0 in (2.1) can be written as

g~(y)=sup{go(x)AA~(x,y)},

yEE, n = 0 , 1 , 2 , . . .

(2.3)

xEE

In this paper, we discuss the cyclic property of the fuzzy relations {An(x, Y)}~0. Let x,y E E. For a E (0, 1], we write x ~,~ y if there exists a positive integer n such that A~(x,y)>~a, and we also write x ~'0 Y if there exists a positive integer n such that A~(x, y ) > 0.

Definition 2.1. For a E [0, 1], we call a non-empty set A(C E) a-connected if x~y

and

y~-*~x

for allx, y E A .

We define A~(x):= {y E E lx ~ is not empty. Next we define

y and y ~-~,~x} for x E E and a E [0, 1]. A~(x) is clearly a-connected if it

~{positive integers n Iqn(x,x) >ja} for xEE, a > 0, l(x, a) := [{positive integers n I qn(x, x) > 0} for x E E, a = 0.

(2.4)

Then we put dx,~ := g c d / ( x , a ) (x E E, a E [0, 1]), where gcd means the greatest common divisor of a set of positive integers. We call dx,~ a period of a state x with a grade a. The following lemmas hold for a E [0, 1] and x E E such that l(x,a) # O.

Lemma 2.1. (i) I f m, n E l(x, a), then m -4- n E l(x, a). (ii) I f m E l(x, a), then mn E l(x, a) for all positive integers n. Proof. (i) Let a > 0 and m, nEl(x,a). Then

Am+n(x,x) = sup{Am(x, y) A An(y,x)} >~Am(X,X) /~ qn(x,x)>~a. yEE

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Therefore, m + n E l(x,~) and so (i) holds. We can easily check the case of ~ = 0 similarly. (ii) is trivial from (i). [] The following lemma is simple, but it is important in Section 3 to show that the ergodicity of dynamic fuzzy systems has a strata structure classified by a finite number of grades. Lemma 2.2. A family of periods, {dx,~ [c~ E [0, 1]}, has the following properties: (i) dx,~, divides dx,~ if O<.Ga' < e ~ l , (ii) the map ~ ( E ( 0 , I]) ~ dx,~(E{1,2,3 .... }) is a non-decreasing left-continuous piecewise constant

function. Proof. (i) Let a, ctt satisfy 0~< ~' < ~ < 1. Since l(x,~ t) D l(x, et), we have

{ d i d is a common divisor of l ( x , a ' ) } C { d i d is a common divisor of l(x,~)}. Therefore, dx,~, <~dx,~, and so dx,~, divides dx,~. (ii) From the definition of l(x, ~), we have

liml(x,a')= ('~ l(x, od)=l(x,~). Therefore lim~,T~ dx,~, =dx,~. We obtain the assertion (ii) since the map ~ H dx,~ takes discrete values.

[]

We can check the following lemmas and corollary in similar ways to [2, Theorems 1.3.2 and 1.3.3]. Lemma 2.3. I f x - ~ y and y ~*~ x, then dx,~ =dy,~. Lemma 2.4. Let ~ > 0 and y EA~,(x). Then there exists a unique integer nx,y,~ ( 0 4 nx,y,~
the following: (i) if 4m(x,y)>~, then rn =_ nx, y,~(mod dx,~). (ii) Further, there exists a positive integer Nx, y,~ such that ~"d~+"x,,~(x,y)>~ for all n>>-Nx,y,~. Corollary 2.1. Let y EAo(x). Then there exists a unique integer nx, y,O (O~nx, y,O < dx, o) satisfying the fol-

lowing: (i) if 4m(x, y) > O, then m = nx, y,O (mod dx, o). (ii) Further there exists a positive integer Nx, y,0 such that 4ndx,°+"x,v.°(x,y) > 0 for all n >~Nx,y,0. Let a E [0, 1] and x E E satisfy l(x, a) # O. For a positive integer d, we put subsets of A~(x) by

D~,m(X):=

S{YEA~(x)[~n(x,y)>~a for some n satisfying n - m ( m o d d)} [ { y E Ao(x) ] ~n(x, y ) > 0 for some n satisfying n =-- m ( m o d d ) }

i f a > O, if ~ = 0

(2.5)

for m = 0 , 1,2 . . . . . d - 1. If {D~,m(X)}m=O,1,2,...,d-1 are disjoint and their union equals to A~(x), then we call them subclasses of A~(x) with the period d. For convenience, we put

D~,n(x):=D~,m(X)

i f n = m(mod d)

for n E ~.

To investigate a cyclic property for the dynamic fuzzy system, we use the following notations and definitions (see [2-4]):

~(x):=-{yEE[~(x,y)>~}

forxEE

and ~E(O, 1].

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Y. Yoshida/FuzzySets and Systems 95 (1998) 189-199

~n 0(3 For aE(O, 1] and x E E , we define a sequence o f {q~(x)}n= 1 by

and ¢]n+l(x):~-- U

~ls(x):=~(x)

qs(Y)

for n = O , l , 2 ....

y E 4~(x)

Definition 2.2. For a E (0, 1], we call a non-empty set A(C E ) a-closed if ~ ( x ) c A for all x E A. Definition 2.3. Subclasses D~,m (m = 0, 1,2 . . . . . d - 1) with a period d are called cyclic classes if ~ ( y ) CD~,m+n for all m = 0 , 1 , 2 . . . . . d - 1; n = 1,2,3 .... ; yED~,m. From Lemma 2.4(i) and Corollary 2.1(i), we can define subclasses {Cs,,(x)},=O, h2,...,d~,~-i of As(x) with the period d~,~ by {y EA~(x) l ~ ( x , y ) ~ a

C~,m(X) : =

for some n satisfying n -- m (mod dx,~)}

if a > 0 ,

{y E Ao(X) I ~"(x, y ) > 0 for some n satisfying n -z m (rood dx, o)}

if a = O

(2.6)

for m = 0 , 1 , 2 . . . . . dx,~ - 1.

Theorem 2.1. (i) If y EA~(x), then As(y)--A~(x) and dy, s =dx,~. (ii) Let m, n E ~ . If yEC~,,~(x), then C~,n(y)----C~,,+m(X). (iii) If A~(x) is a-closed, then C~,m(X) ( m = 0 , 1 , 2 . . . . . dx,~- 1) are cyclic classes. Proof. We can check this theorem in a similar way to [2, Theorem 1.3.4]. From Theorem 2.1, {C~,n(X)}n=O, 1,2,...,dx,_ 1 does not depend on the choice of x. Therefore, we denote it by {C~,n}n=O,hz,...,a~-1. Further, we also have the following properties. L e m m a 2.5. Let x EE. If {De~,m}m=O,1,2,..,,d-1 be subclasses of A~(x) with a period d, then d divides ds. Proof. Let e E (0, 1] and let m E l(x, ~). Then qm(x,x) ~ 0~. Since {De, m}rn=O,1,2,...,d_l are subclasses o f A~(x), D~,o={yEA~(x)lOn(x,y)>~a for some n satisfying n ~ 0 ( m o d d)}. By renumbering {O~,m}m=O,~,2,...,d-~ if necessary, we may assume that C~,0 nD~,0 ¢ 0. For y E C~,0 ND~,0, there exists n such that

gt~(x,y)>~a

and

n = 0(rood d).

(2.7)

Therefore, om+n(x,y))Om(x,x) /~ ~n(x,y)>lC & We have m + n ~ 0 (mod d) since {Ds, l}l~O,l,2,...,d-I a r e disjoint. Together with (2.7), we obtain m ~- 0 ( m o d d ) , namely, d divides all m E l(x, ~). Thus d divides d~. We can check the case o f a -- 0 similarly. []

Theorem 2.2. Let A~(x) be a-closed. Let {Ds, m}m=O,l,2,._,d-1 be subclasses of A~(x) with a period d. if"

{O~,m}m=O,l,2,.,.,d-i are cyclic classes, then each D~,m is the union of d~/d sets from {C~,n},=0,1,2,...,a~-i. Proof. Without loss o f generality, we may assume that x EC~,oND~,o. Then {C~,nnDa, m}n=o,l,2,...,d_l; ,,=0,1,2,...,d-1 are subclasses of A~(x) with a period dad. Suppose that C~,m, ADs, m, ~L C~,m' for some m/. Let y E C~,m,\(C~,m, nOa, m, ). Since y ~ (C~,m, ND~,,~,), we have

~ta'a+m'(x,y) < ~ for all l = 1,2 ....

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This contradicts y E C~,,n,. Therefore, we obtain Cot, m C D~,m for all m = 0, 1..... d - 1. Using the cyclic assumption for {D~,,n}m=o,k2,...,a-1, we have C~,mcD~,m=D~,, if 0~< n
U

U

m=O, 1,2,...,d--I l=O,l,2,...,n'

U

1

m=O,l,2,...,d-I

we obtain D~,m = Ul=o,l,2,...,,,-1C~,ta+m for rn=O, 1,2 ..... d -

1. []

3. An ergodic theorem In this section, we consider the periodic limit of the sequence of the fuzzy transitions, {q~n (X, Y)}n=0, and discuss the ergodic property. Define ~(x):= sup~n(x,x)

for xCE.

(3.1)

n>~l

We put

A*:={x~E]~(x)>O}=

{xEE

s u p 0 " ( x , x ) > 0 ) ={xCEIxEAo(x)}.

(3.2)

n>~l

Then, A* equals to the union of all positive recurrent sets (see [7, Theorem 3.1]). By the results of Section 2, we have the following limit theorem on A* for the dynamic fuzzy system. Theorem 3.1. Let x E A*. Put d ( x ) : = lim gcd{n~>l I~n(x,x)>~}. ~T~(x)

I f d(x) < oo, then we have nlirn~ qnd(x }(X, X)

= ~(X).

Proof. Let ~ satisfy 0 < ~ < ~(x). Then x CA~(x). Therefore, there exists a period d~ of Adx). Let _2~ := inf l i m n ~ d ~ ( x , x ) and 2~ := sup limn~4nd'(x,x). From Lemma 2.4(ii), we have c~~<2~ ~<,~ ~< sup qn(x,x) = cffx). n>~l

Therefore, ~(x)= lim ).~ = lim 2~. ~T~(x) -

(3.3)

~q'~(x)

From Lemma 2.2(i), there exists a limit d ( x ) : = lira d~= lim gcd{n~>l It~"(x,x)~>~}. ~T~(x) ~t~(x) Suppose that d(x)
~Y~x)

n---,~

n~w

~T~(x)

n~

Similarly, we also have ~(x)= sup l i m , _ ~ ~"d(~}(x,x). Therefore the proof is completed.

[]

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Y.

YoshidalFuzzy Sets and Systems 95 (1998) 189-199

Further, we need some notations in Yoshida [6, p. 42] to describe the ergodic property. Put a path space ~ E and write a path by co=(~o(0),co(1),~o(2) .... )El2. J¢' denotes the a-field generated by all by O : = I~,=0 Borel subsets of f2. We define a map Xn(co) := og(n) for n E ~ and 09 = (co(0), co(1 ), ~o(2).... ) E (2. Then Xn(og) means a point at time n when the system transits along the path 09. For y E E, the first hitting time of the point y is defined by a{y}(O~):= inf{n>~l IX,(~o)=y},

O~E~2,

where the infimum of the empty set is understood to be +oo. For an initial state x E E and an J#-measurable fuzzy set h E ~-(O), we define a fuzzy expectation by Ex( h )

:= f(o~c~:~(o>=x}h(~°) dP(og),

where/5 is the following possibility measure: /5(A) := sup A q(Xn(fD)'Xn+I((D))'

A E J/[

coE A nG[~

and f dP denotes Sugeno integral [5]. We also put Px(A):= Ex(1 A) for A E J / a n d x E E. The fuzzy expectation is an extension of the fuzzy transition {~n}~ 0 and admits stopping times depending on paths. We note that a possibility of transition from x(EE) to y ( E E ) at the nth step becomes Px(Xn = y ) = ~n(x, y). Define the following possibilities: Fro(x, y)

:=

Px(a{y} = m),

m = 1,2 ..... x, y E E,

(3.4)

and 7(x,y):= sup~m(x,y)=Px(a{y} < oo),

x, y E E .

(3.5)

m~>l

Then, by [7, Proposition 2.1(ii)], we obtain 7(x, y) = sup ~m(x, y). m~>l

Theorem 3.2. Let y E A* and y E Ao(x) such that d(y) < oo. Then there exists a positive integer d(x, y), which divides d(y), and an integer n(x,y) ( 0 ~ n(x,y) < d(x,y) ) such that nlirn qnd(x'y)+n(x'Y)(x, y ) ~ ~(X, y ) /~ o~(y ). Proof. From the definition of 7(x, y) and a(y), it is trivial that sup lim ~n(x, y) ~
(3.6)

n----~oo

Let a satisfy 0 < ~t < ?(x,y) A ~(y). Then y EA~(x). From Lemma 2.4, there exists a positive integer Nx, y,~ such that qndy'~+nx'Y'~(x,y)~o~ for all n ~ N x , y,~. Therefore, inf lim

qndy'a+rlx'Y'~(X,y) >~cc

(3.7)

Y. Yoshida/Fuzzy Sets and Systems 95 (1998) 189-199

195

We define

d(x,y)'--

lim

7TT(x,Y) A ~(y)

dy,~ and

n(x,y):=

lim

sTy(x, y) A ~(y)

nx, y,~.

Then d(x, y) divides d(y) from Lemma 2.2(i) since ct < ~(y). Letting c~T y(x, y ) A ~(y) in (3.7), in the same reason as the proof of Theorem 3.1, we obtain inf lim n--, oc~

qnd(x'y)+n(x'Y)(x, y) >~7(x, y) A ~(y).

Together with (3.6), we get this theorem.

[]

Finally, we give an ergodic theorem regarding {qm(x,y)}m~=l for the dynamic fuzzy system, which has a strata structure with respect to a finite number of grades and which is different from the ergodic theorems of Markov chains [2, Theorems 1.6.1 and 1.6.4]. By Lemma 2.2, for yEA* and yEAo(x) = U~>0A~(x) such that d(y) < oo, there exist a finite number of grades {~ili = 0, 1,2 . . . . . k} satisfying the following: 0 = ~0 < ~1 < ~2 < ' ' "

ay,

(dy,~,~,

d(y)

< ~k =

:~(Y),

(3.8)

if ~i < ~ < ~ i + 1 for i ( = 0 , 1 , 2 . . . . . k - 2 ) ,

if ~k-1 < ~ < ~k.

(3.9)

Further, there exists a positive integer k r (k'<~ k) satisfying O~k'-I < 7(x, y) A ~(y)<~ CCk',

(3.10)

dy,~=d(x,y)

(3.11)

ifc~k,-1 < ~ < ~ k ' .

Then, by replacing ~k, by 7(x,y) A ~(y) A ~k,, from Lemma 2.4(i) we have

nx,y,~ =

nx,y.... ~ if c~i < ct~
(3.12)

For simplicity, we write di := dv,~, and ni := nx,y,~, for i = 1,2 . . . . . U - 1 and we also write dk, := d(x,y) and nk, := n(x, y). Theorem 3.3 (Ergodic theorem). Let x, y E E satisfy yCAo(x) and d(y) < ~ . (i) I f y f[ A*, then there exists a positive integer N such that

~m(x, y) = 0 for all m >-N. (ii) I f y e A * , then we have the followin9 three cases: (a) Case of m =_ n(x, y) (mod d(x, y)): nlirn qnd(x'y)+n(x'Y)( x, Y) = 7(x, y) A ~(y). (b) Let i = 1,2 . . . . . k ' - 1. Case of m ~ ni+l (mod di+l ) and m = ni (mod di): There exists a positive

integer Ni such that (tm(x,Y) = ~i for all m E J ( i ) satisfyin9 m~Ni, where J(i) := {m I m is a positive inteoer satisfyin9 m ~ ni+l (mod di+l ) and m = ni (rood di)}.

Y Yoshida/Fuzzy Sets and Systems 95 (1998) 189-199

196

(c) Case of m ~ nl (mod dr): There exists a positive integer No such that

(tm(x, y) = 0 for all m satisfying m>~No. Proof. (i) If y ~ A*, then c~(y) : 0. Then, there exists a positive integer N such that ~m(x, y) = 0 for all

m >~N. (ii.a) is from Theorem 3.2. (ii.b) Let i satisfy 1 ~< i < U. Let ~ satisfy ~/<~<~i+l. From Lemmas 2.2(ii) and 2.4, we have dy,~=dy, ai+,=di+l

and

(3.11)

nx, y,:~=nx, y.... , = n i + l .

From (3.11) and Lemma 2.4(i) and (ii), there exists a positive integer Ni such that

c~i~m(x,y)
for all m E g ( i ) satisfying rn~>N/.

(3.12)

Since Ni depends only on x,y and eel+l, letting c~,[c~i in (3.12),

~m(x,y) = o~i for all mEJ(i) satisfying m>~N/.

(3.13)

Therefore we get (ii.b). (ii.c) Similar to the proof of (ii.b), for i = 0, we get

~tm(x,y) = c~0 = 0

for all m satisfying m ~ nl (rood dl).

Thus we obtain this theorem.

[]

4. Numerical examples We consider a one-dimensional numerical example with a one-dimensional state space E -- ~, where denotes the set of all real numbers. Then, ~ ( x ) are bounded closed intervals of ~ (~ E (0, 1], x E ~) and is written by ~ ( x ) = [min ~ ( x ) , max ~(x)], where m in F (max F ) denotes the minimum (maximum re sp.) point of a interval F C ~. We give a fuzzy relation, which is not monotone in the sense of [7], by ~(x,y)=(1-lx+y3l)v0,

x, yEIR.

(4.1)

Then ~(x, y) satisfies the conditions in Section 2 (see Fig. 1). In this paper, we call a set {xEl~l~/(x,x):-o~ } e-slice for each ctE[0,1]. Figs. 2 and 3 show its 0.7-slice and 0.5-slice, respectively. First, we consider the cyclic property in Section 2. In this example, we have F~:= { y E N

n~>Isupqn(Y'Y)~>e}

= {YENIq(Y'Y) Vq2(Y'Y))e}

= {xE R Imin~(x)~< - x ~ < m a x ~ ( x ) }

for c~E(0, 1].

(4.2)

Then Fo.5 is a closed interval. However, Fo.7 becomes a union of three closed intervals (see Figs. 2 and 3). Therefore, we write it by Fo.7 = Fo.7,o U Fo.7,1 U Fo.7,2. Calculating them, we have Fo.7,o ~ [-1.1254,-0.7865], and Fo.5 ~ [-1.1915, 1.1915].

Fo.7,1 ~ [-0.3389,0.3389],

Fo.7,2~ [0.7865, 1.1254],

E YoshidalFuzzy Sets and Systems 95 (1998) 189-199

1 0.8 (~0.6 (1.4 0.; I

Y Fig. 1. The fuzzy relation ~(x, y).

y~x

Y

Y

.

-'2 ' ' ~

~

o 1

2 x

~(~)

y----x

Fig. 2. The 0.7-slice { ( x , y ) l,~(x, y ) = 0.7).

Y

y=x

y= X

~

?o.5x() y=--x

Fig. 3. The 0.5-slice {(x, ),) I c~(x, y ) = 0.5 }.

197

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Y. Yoshida/Fuzzy Sets and Systems 95 (1998) 189-199

~ I

I

,

t

',

0.4

/i i o-

.... I!

-2

-a-b-1

.=(1-h-y+y31)vo a ~ 1.3247 b ~ 1.1547 c ~ 0.5774

2 i

i

i

v ~ 0.6151 .

-c

Zba

,

ff

2

Fig. 4. The graph of a = a(y).

For s = 0.7, F0.7,1 and Fo.7,o L-JF0.7,2 a r e 0.7-connected, which is 0.7-recurrent sets in the sense of Yoshida [7]. Fo.7,1 is not 0.7-closed. Fo.7,o tO F0.7, 2 is 0.7-closed, and Fo.7,0 and Fo.7,2 are cyclic classes of Fo.7,0 tO Fo.7,2. The periods are 1 dY'°7 =

2

yCFo.7,1, ifycFo.7,oUFo.7,2. if

(4.3)

For s = 0.5, Fo.s is 0.5-connected and 0.5-closed. We also have the period dy,o.5 = 1 for Next, we consider the ergodic property in Section 3. In this example, we have

s(y) = q(y,y) Vgt2(y,y)

= (1 - [ -

y+y3

I)V0

for y E ~ .

yEFo.5. (4.4)

Fig. 4 shows the graph. Then we put the union of all positive recurrent sets by

(-a,a)

:= A* = U F~ = { y E N I s ( y ) > 0} ~ (-1.3247, 1.3247).

(4.5)

~>0

We also put 2 b:= ~

1.1547,

1 c:= ~0.5774,

v:= 1

2 ~ - ~ ~0.6151.

From Fig. 4, we obtain the following three cases: (C.1) If y ~ [ - a , a], then

qm(y,y)=O

for a l l m =

(C.2) If y C ( - b , - c ) U S 0 = O, dy

'~

O~1 = V,

(c,b),

1,2 .... we may take

S2 = (1 -- I -- Y + y3 I) V O,

Sdl = 1

ifSo<:~Sh

],d2

if sl < a < e 2 ,

2

where v is a constant given in Fig. 4. Then we have the period following cases: (a) Case of m - 0 (mod 2):

nlirn qnX2(y,Y)

= ~2 = (1 -- I - Y + Y3I) V 0.

(4.6) (4.7)

d(y)

= 2. By Theorem 3.3, we get the

(4.8)

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199

(b) Case of m -= 1 (mod2): There exists a positive integer N1 such that

~m(y, y ) = Ctl = v

(C.3) If y E [ - a , - b ] ct0--0,

for all odd m satisfying m>~N1.

(4.9)

t3 [ - c , c ] U [b,a], we may take

cq = ( 1 - I - y + y 3 l ) V 0 ,

(4.10)

and dy,~ = dl = 1 if s0 < c~ < cq. Then we have the period d ( y ) -- 1. By Theorem 3.3, we get lim ~ m ( y , y ) = c q

=(1-

I-y+ya])VO.

(4.11)

m ----+ O o

Acknowledgements The author is grateful to a n o n y m o u s referees for their useful comments.

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