Compactness and convexity on the space of fuzzy sets II

Nonlinear Analysis 57 (2004) 639 – 653

www.elsevier.com/locate/na

Compactness and convexity on the space of fuzzy sets II Yun Kyong Kim∗ Department of Information and Communication Engineering, Dongshin University, 252 Daehodong Naju, Chonnam 520-714, South Korea Received 19 September 2003; accepted 10 March 2004

Abstract In this paper, we establish some characterizations of convex and relatively compact subsets of the space F(Rp ) of normal, upper-semicontinuous and compactly supported fuzzy sets in Rp equipped with the Skorokhod metric. ? 2004 Elsevier Ltd. All rights reserved. MSC: 47H04; 52A22; 53C65; 54C60; 58C60 Keywords: Fuzzy sets; Relatively compact subsets; Convex subsets; The Skorokhod metric; The Hausdor: metric

1. Introduction This work is a continuation of [9]. There we studied a criteria for which the convex hull of K is also relatively compact when K is a relatively compact subset of the space F(Rp ) of normal, upper-semicontinuous and compactly supported fuzzy sets in Rp endowed with the Skorokhod metric. Also, some examples were given to illustrate the criteria. In this paper, we establish another characterizations of convex, relatively compact subsets of F(Rp ) as a continuation of the above work [9], and investigate the relationship between convergences in the Skorokhod metric and the supremum metric. It is expected that the results have considerable potential usefulness on studying analytical properties of fuzzy valued functions. In fact, the present paper was motivated by the ∗

Tel.: +82-613303312; fax: +82-613302909. E-mail addresses: [email protected], [email protected] (Y.K. Kim).

0362-546X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.03.005

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Y.K. Kim / Nonlinear Analysis 57 (2004) 639 – 653

work of Da:er [2] which considered the similar problems on D[0; 1]. It turns out that the techniques used there is also useful in the case of our concerns. 2. Preliminaries In this section, we review some preliminary results and notations introduced in Kim [9]. Let P(Rp ) denote the family of non-empty compact subsets of the Euclidean space p R . Then the space P(Rp ) is metrizable by the Hausdor: metric deDned by
 dH (A; B) = max sup inf |a − b|; sup inf |a − b| ; a∈A b∈B

b∈B a∈A

where | · | denotes the Euclidean norm. It is well-known that P(Rp ) is complete and separable with respect to the Hausdor: metric dH (see Debreu [3]). The addition and scalar multiplication in P(Rp ) are deDned as usual: A ⊕ B = {a + b: a ∈ A; b ∈ B};

A = { a: a ∈ A} for A; B ∈ P(Rp ) and ∈ R. Let F(Rp ) denote the family of all fuzzy sets u˜ : Rp → [0; 1] with the following properties: (1) u˜ is normal, i.e., there exists x ∈ Rp such that u(x) ˜ = 1; (2) u˜ is upper semicontinuous; (3) supp u˜ = {x ∈ Rp : u(x) ˜ ¿ 0} is compact, where AG denotes the closure of a set p A⊂R . For a fuzzy set u˜ in Rp , if we deDne  {x: u(x) ˜ ¿ ; } if 0 ¡  6 1; L u˜ = supp u˜ if  = 0; then, it follows immediately that u˜ ∈ F(Rp ) if and only if L u˜ ∈ P(Rp ) for each  ∈ [0; 1]. This fact implies that u˜ ∈ F(Rp ) is completely determined by the family {L u: ˜  ∈ [0; 1]} of sets in P(Rp ). Lemma 2.1. Suppose that P(Rp ) is endowed with the Hausdor3 metric dH . For u˜ ∈ F(Rp ), we de5ne ˜ Fu˜ : [0; 1] → P(Rp ) by Fu˜() = L u: Then the followings hold: (1) Fu˜ is non-increasing, i.e.,  6  implies Fu˜() ⊃ Fu˜(), (2) Fu˜ is left continuous on (0; 1], (3) Fu˜ has right-limits on [0; 1) and Fu˜ is right-continuous at 0.

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Conversely, if G : [0; 1] → P(Rp ) is a function satisfying the above conditions (1)–(3), then there exists a unique v˜ ∈ F(Rp ) such that G() = L v˜ for all  ∈ [0; 1]. Proof. See Lemma 2.1 of Kim [9]. Note that the above Lemma 2.1 is stronger than the Negoita–Ralescu representation theorem for a general fuzzy set. (For details, see Lemma 6.5.1 of HHohle and Sostak [6]). The function Fu˜ will be called the level multifunction of u. ˜ The right-limit of Fu˜ at  ∈ [0; 1) is {x ∈ Rp : u(x) ˜ ¿ }, which will be denoted by L+ u. ˜ Thus, if we deDne ˜ L+ u) ˜ ju˜() = dH (L u; then the level multifunction Fu˜ deDned in Lemma 2.1 is continuous at  if and only if ju˜() = 0. The addition and scalar multiplication in F(Rp ) are deDned as usual: ˜ v(y)); ˜ (u˜ ⊕ v)(z) ˜ = sup min(u(x);  ( u)(z) ˜ =

x+y=z

u(z= ) ˜

if = 0;

˜ 0(z)

if = 0;

for u; ˜ v˜ ∈ F(Rp ) and ∈ R, where 0˜ = I{0} is the indicator function of {0}. Then it is well-known that for each  ∈ [0; 1], ˜ = L u˜ ⊕ L v˜ L (u˜ ⊕ v)

and

L ( u) ˜ = L u: ˜

Lemma 2.2. For each u˜ ∈ F(Rp ) and  ¿ 0, there exist a partition 0=0 ¡ 1 ¡ · · · ¡ r = 1 of [0; 1] such that + u; ˜ Li u) ˜ ¡ dH (Li−1

for all i = 1; 2; : : : ; r:

Proof. See Lemma 2.3 of Joo and Kim [8]. For u˜ ∈ F(Rp ) and 0 ¡ ! ¡ 1, we deDne + u; ˜ Li u); ˜ wu˜(!) = inf max dH (Li−1

16i6r

where the inDmum is taken over all partitions 0=0 ¡ 1 ¡ · · · ¡ r of [0; 1] satisfying i − i−1 ¿ ! for all i. Then Lemma 2.2 is equivalent to the assertion that lim wu˜(!) = 0

!→0

for each u˜ ∈ F(Rp ). Also, if we let J (u) ˜ = {: ju˜() ¿ }; ˜ is Dnite. then Lemma 2.2 implies that J (u)

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Now, the supremum metric d∞ on F(Rp ) is deDned by d∞ (u; ˜ v) ˜ = sup dH (L u; ˜ L v): ˜ 0661

Also, the norm of u˜ is deDned as ˜ = sup |x|: ˜ 0) u ˜ = d∞ (u; x∈L0 u˜

Then it is well-known that F(Rp ) is complete but is not separable with respect to the metric d∞ (see Diamond and Kloeden [4]). Joo and Kim [7,8] introduced the Skorokhod metric ds on F(Rp ) which makes it a separable metric space, and characterized the relatively compact subsets of F(Rp ) in terms of wu˜(!). This characterization is needed for the proofs of the main results. Denition 2.3. Let T denote the class of strictly increasing, continuous mapping of [0; 1] onto itself. For u; ˜ v˜ ∈ F(Rp ), we deDne ds (u; ˜ v) ˜ = inf { ¿ 0: there exists a t ∈ T such that sup |t() − | 6 

0661

and

˜ t(v)) ˜ 6 }; d∞ (u;

where t(v) ˜ denotes the composition of v˜ and t. Theorem 2.4. Let K be a subset of F(Rp ). Then K is relatively compact in the ds -metric topology if and only if sup { u : ˜ u˜ ∈ K} ¡ ∞

(2.1)

lim sup {wu˜(!): u˜ ∈ K} = 0:

(2.2)

and !→0

Proof. See Theorems 4.1 and 4.3 of Joo and Kim [8]. 3. Main results First we recall that the space of fuzzy numbers in Rp (i.e., the space of u˜ ∈ F(Rp ) such that u˜ is fuzzy convex) can be embedded into a Banach space (see [1,5,10]). But F(Rp ) cannot be embedded into a Banach space since the distributed law does not ˜ hold, i.e., ( 1 + 2 )u˜ may not be equal to 1 u˜ ⊕ 2 u. Nevertheless, we can deDne the concept of convexity on F(Rp ) as in the case of a vector space. That is, A ⊂ F(Rp ) is said to be convex if u˜ ⊕ (1 − )v˜ ∈ A whenever u; ˜ v˜ ∈ A and 0 6 6 1. Also, the convex hull co(A) of A ⊂ F(Rp ) is deDned to be the intersection of all convex sets that contain A. Then as in the case of a vector space, we can easily show that co(A) is equal to the family of consisting of all fuzzy sets in the form 1 u˜ 1 ⊕ · · · ⊕ n u˜ n , whereu˜ 1 ; : : : ; u˜ n are any elements of A; 1 ; : : : ; n n are nonnegative real numbers satisfying i=1 i = 1 and n = 2; 3; : : : .

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643

Throughout the remainder of this section, we assume that the space F(Rp ) is endowed the ds -metric topology. Let us denote by C(F(Rp )) the collection of all relatively compact subsets K of F(Rp ) for which the convex hull co(K) is also relatively compact. And we deDne, for A ⊂ F(Rp ) and  ¿ 0, J (A) = { ∈ (0; 1): sup ju˜() ¿ }: u∈A ˜

The following theorem was established by Kim [9] with the notation S (A) instead of J (A). Theorem 3.1. Let K be a relatively compact subset of F(Rp ). Then K ∈ C(F(Rp )) if and only if J (K) is 5nite for every  ¿ 0. First, we start with some results which can be obtained as an application of Theorem 3.1. Lemma 3.2. Let K; K1 ; K2 ⊂ F(Rp ) and ∈ R be not zero. Then (1) (2) (3) (4)

J (K1 ⊕ K2 ) ⊂ J=2 (K1 ) ∪ J=2 (K2 ). J ( K) = J=| | (K). J (K1 ∪ K2 ) = J (K1 ) ∪ J (K2 ). J (K1 ∩ K2 ) ⊂ J (K1 ) ∩ J (K2 ).

Proof. (1) follows from the fact that for each u˜ 1 ∈ K1 ; u˜ 2 ∈ K2 , ju˜ 1 ⊕u˜ 2 () = dH (L u˜ 1 ⊕ L u˜ 2 ; L+ u˜ 1 ⊕ L+ u˜ 2 ) 6 dH (L u˜ 1 ; L+ u˜ 1 ) + dH (L u˜ 2 ; L+ u˜ 2 ) = ju˜ 1 () + ju˜ 2 (): Also, (2) follows from the fact that j u˜() = | |ju˜(). (3) and (4) are trivial. Theorem 3.3. Let K; K1 ; K2 ∈ C(F(Rp )) and ∈ R. Then (1) (2) (3) (4)

K1 ⊕ K2 ∈ C(F(Rp )).

K ∈ C(F(Rp )). K1 ∪ K2 ∈ C(F(Rp )). K1 ∩ K2 ∈ C(F(Rp )).

Proof. (1) First we must show that K1 ⊕ K2 is relatively compact. Let {w˜ n } be a sequence in K1 ⊕ K2 . Then there exists a sequence {u˜ n } in K1 and a sequence {v˜n } in K2 such that w˜ n = u˜ n ⊕ v˜n . Since K1 and K2 are relatively compact, {u˜ n } and {v˜n } have a convergent subsequence. Without loss of generality, we may assume that lim ds (u˜ nk ; u˜ 0 ) = 0

k→∞

and

lim ds (v˜nk ; v˜0 ) = 0:

k→∞

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By Theorem 3.6 below, lim d∞ (u˜ nk ⊕ v˜nk ; u˜ 0 ⊕ v˜0 )

k→∞

6 lim d∞ (u˜ nk ; u˜ 0 ) + lim d∞ (v˜nk ; v˜0 ) = 0; k→∞

k→∞

which implies limk→∞ ds (u˜ nk ⊕ v˜nk ; u˜ 0 ⊕ v˜0 ) = 0. Thus, {w˜ n } has a convergent subsequence and so K1 ⊕ K2 is relatively compact. Hence K1 ⊕ K2 ∈ C(F(Rp )) by Theorem 3.1 and Lemma 3.2 (1). The remainder parts follow immediately from Lemma 3.2. Remark 1. In spite of Theorem 3.3 (1), if K1 ∈ C(F(Rp )) or K2 ∈ C(F(Rp )), then K1 ⊕ K2 may not be relatively compact even though K1 and K2 are relatively compact. This follows from the fact that the addition F(Rp ) is not continuous. For example, let  1 if x = 0;    u˜ n (x) = 12 + 1=n if 0 ¡ |x| 6 1;    0 elsewhere and v˜n (x) =

 1      

if x = 0;

1 2

if 0 ¡ |x| 6 1 − 1=n;

0

elsewhere:

Then {u˜ n } and {v˜n } converge to the same limit u, ˜ where  1 if x = 0;    u(x) ˜ = 12 if 0 ¡ |x| 6 1;    0 elsewhere: To see this, let tn be a function in T such that tn ( 12 + 1=n) = 12 and linear in [0; 12 + 1=n) and ( 12 + 1=n; 1]. Then sup |tn () − | = 1=n → 0

0661

as n → ∞

˜ = 0, which implies ds (u˜ n ; u) ˜ 6 1=n → 0 as n → ∞. and d∞ (tn (u˜ n ); u) ˜ = 1=n → 0 as n → ∞, we have ds (v˜n ; u) ˜ → 0 as Now for {v˜n }, since d∞ (v˜n ; u) n → ∞. Thus, K1 = {u˜ n : n = 2; 3; : : :} and K2 = {v˜n : n = 2; 3; : : :} are relatively compact. Note that K1 ∈ C(F(Rp )) and K2 ∈ C(F(Rp )) since J (K1 ) = { 12 + 1=n: n = 2; 3; : : :} for 0 ¡  ¡ 1 and J (K2 ) ⊂ { 12 }. Now we prove that {u˜ n ⊕ v˜n } does not have a convergent subsequence. To prove this, we Drst note that if {u˜ n ⊕ v˜n } has a convergent subsequence, then the

Y.K. Kim / Nonlinear Analysis 57 (2004) 639 – 653

limit is u˜ ⊕ u˜ = 2u. ˜ Since  1       1 + 1=n 2 (u˜ n ⊕ v˜n )(x) = 1    2    0

645

if x = 0; if 0 ¡ |x| 6 1; if 1 ¡ |x| 6 2 − 1=n; elsewhere;

we have

 {0}    L (u˜ n ⊕ v˜n ) = {x: |x| 6 1}    {x: |x| 6 2 − 1=n}

Also, for each t ∈ T ,  {0} L t(2u) ˜ = {x: |x| 6 2} Thus for each t ∈ T ,



d∞ (u˜ n ⊕ v˜n ; t(2u)) ˜ =

if if

1 2 1 2

+ 1=n ¡  6 1; ¡  6 12 + 1=n;

if 0 6  6 12 :

if t

1

1

if t

1

2

otherwise;

¡  6 1;

if 0 6  6 t 12 : 2

2

6 12 + 1=n;

which implies that ds (u˜ n ⊕ v˜n ; 2u) ˜ ¿ 1. Hence {u˜ n ⊕ v˜n } does not have a convergent subsequence and so K1 ⊕ K2 is not relatively compact. Now for A ⊂ F(Rp ) and  ¿ 0, let   A =

u˜ ∈ A : sup ju˜() ¿  : ∈(0;1)

Theorem 3.4. If A is 5nite, then J (A) is 5nite. Proof. By assumption, there exists at most Dnite many u˜ 1 ; : : : ; u˜ n ∈ A such that sup∈(0; 1) ju˜ i () ¿ . Then for each u˜ i ; J (u˜ i ) is Dnite by Lemma 2.2. Thus, n J (A) = J (u˜ i ) i=1

is Dnite. This completes the proof. The following example shows that the converse is not true. Example 1. For ∈ (0; 1),  1 − |x|=

   v˜ (x) = 12    0

let if |x| 6 =2; if =2 ¡ |x| 6 1; otherwise:

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Y.K. Kim / Nonlinear Analysis 57 (2004) 639 – 653

and A = {v˜ : ∈ (0; 1)}. Then  {x: |x| 6 1} L v˜ = {x: |x| 6 (1 − )}

if 0 6  6 12 ; if 12 ¡  6 1:



 Thus, J (A) ⊂ 12 for any  ¿ 0. But for each ∈ (0; 1); jv˜ 12 = 1 − =2 ¿ 12 and so A is inDnite set for any 0 ¡  6 12 . Corollary 3.5. If K is a relatively compact subset of F(Rp ) and K is 5nite for each  ¿ 0, then K ∈ C(F(Rp )). Theorem 3.6. Let K be a relatively compact and convex subset of F(Rp ). If {u˜ n } is a sequence of K, then for some u˜ 0 ∈ F(Rp ), lim ds (u˜ n ; u˜ 0 ) = 0 if and only if lim d∞ (u˜ n ; u˜ 0 ) = 0: n→∞

n→∞

Proof. The suLciency is trivial. To prove the necessity, we assume that lim ds (u˜ n ; u˜ 0 ) = 0;

n→∞

but lim sup d∞ (u˜ n ; u˜ 0 ) ¿  ¿ 0: n→∞

Then there is a subsequence {u˜ nk } of {u˜ n } such that d∞ (u˜ nk ; u˜ 0 ) ¿  for all k: This means that there is a corresponding sequence {nk } in [0; 1] such that dH (Lnk u˜ nk ; Lnk u˜ 0 ) ¿  for all k: By compactness of [0; 1]; {nk } has a further subsequence converging to 0 ∈ [0; 1]. For the sake of convenience, we assume that dH (Ln u˜ n ; Ln u˜ 0 ) ¿  for all n

(3.1)

and limn→∞ n = 0 . For given + ¿ 0, we choose ! ¿ 0 such that dH (L0+ u˜ 0 ; L0 +2! u˜ 0 ) ¡ +

and

dH (L0 u˜ 0 ; L0 −2! u˜ 0 ) ¡ +:

(3.2)

Since limn→∞ ds (u˜ n ; u˜ 0 ) = 0, there exists a sequence of functions {tn } in T such that lim

sup |tn () − | = 0

n→∞ 0661

Y.K. Kim / Nonlinear Analysis 57 (2004) 639 – 653

647

and lim d∞ (tn (u˜ n ); u˜ 0 ) = 0:

n→∞

Thus we can Dnd n0 such that n ¿ n0 , sup |tn () − | ¡ !;

0661

(3.3)

and d∞ (tn (u˜ n ); u˜ 0 ) = sup dH (L u˜ n ; Ltn () u˜ 0 ) ¡ +: 0661

(3.4)

Let n = tn−1 (0 ). Then (3.3) implies that for n ¿ n0 , 0 − ! ¡ n ¡ 0 + !: Now we divide into three cases and continue to prove the necessity. Case 1: Suppose n = 0 for almost all n, say n = 0 for n ¿ n1 . If 0 ¡  ¡ 0 + !, then we take + = =4 in (3.2) and (3.4), and choose n0 and ! ¿ 0 accordingly, and put n2 = max(n0 ; n1 ). Then (3.3) implies that 0 ¡ tn () ¡ 0 + 2! for n ¿ n2 . By (3.2) and (3.4), we have, for n ¿ n2 , dH (L u˜ n ; L u˜ 0 ) 6 dH (L u˜ n ; Ltn () u˜ 0 ) + dH (Ltn () u˜ 0 ; L0+ u˜ 0 ) + dH (L0+ u˜ 0 ; L u˜ 0 ) ¡ =4 + =4 + =4 = 3=4: Thus, sup

0 ¡¡0 +!

dH (L u˜ n ; L u˜ 0 ) ¡ :

(3.5)

Similarly, it can be proved that sup

0 −!¡60

dH (L u˜ n ; L u˜ 0 ) ¡ ;

(3.6)

which together with (3.5), implies that sup

0 −!¡¡0 +!

dH (L u˜ n ; L u˜ 0 ) ¡ :

(3.7)

But this contradicts to assumption (3.1) since limn→∞ n = 0 . Case 2: Suppose that ju˜ 0 (0 ) = 0. We take + = =4 in (3.2) and (3.4) and choose n0 and ! ¿ 0 accordingly. Then (3.2) implies that dH (L0 −2! u˜ 0 ; L0 +2! u˜ 0 ) ¡ =2:

(3.8)

Let n1 be such that 0 − ! ¡ n ¡ 0 + ! for n ¿ n1

(3.9)

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Y.K. Kim / Nonlinear Analysis 57 (2004) 639 – 653

and put n2 = max(n0 ; n1 ). Then (3.1), (3.8) and (3.9) yield that for any t ∈ T satisfying sup0661 |t() − | ¡ !, d∞ (t(u˜ n ); u˜ 0 ) ¿ dH (Lt(n ) t(u˜ n ); Lt(n ) u˜ 0 ) ¿ dH (Ln u˜ n ; Ln u˜ 0 ) − dH (Ln u˜ 0 ; Lt(n ) u˜ 0 ) ¿  − dH (Ln u˜ 0 ; Lt(n ) u˜ 0 ) ¿  − =2 = =2 for all n ¿ n2 . This implies that for n ¿ n2 , ds (u˜ n ; u˜ 0 ) ¿ =2; which contradicts to the hypothesis limn→∞ ds (u˜ n ; u˜ 0 ) = 0. Case 3: Suppose that n = 0 inDnitely often and ju˜ 0 (0 ) = c ¿ 0. Then by choosing a further subsequence, we may assume n = 0 for all n. In (3.2) and (3.4), we take + = c=4 and choose n0 and ! accordingly. Then we have, for n ¿ n0 , c = dH (L0 u˜ 0 ; L0+ u˜ 0 ) 6 dH (L0 u˜ 0 ; Ln u˜ n ) + dH (Ln u˜ n ; Ln+ u˜ n ) + dH (Ln+ u˜ n ; L0+ u˜ 0 ) ¡ c=2 + dH (Ln u˜ n ; Ln+ u˜ n ): Thus ju˜ n (n ) ¿ c=2 and so, n ∈ Jc=2 (K) for n ¿ n0 . But since limn→∞ n = 0 and n = 0 for all n, we have that {n : n ¿ n0 } is inDnite. This contradicts to Theorem 3.1. Therefore, we complete the proof. Corollary 3.7. If K ∈ C(F(Rp )), then K is relatively compact in the d∞ -topology. Proof. If {u˜ n } is a sequence in K, then there exists a subsequence {u˜ nk } such that lim ds (u˜ nk ; u˜ 0 ) = 0

n→∞

for some u˜ 0 ∈ F(Rp ). Since {u˜ n } is a sequence in co(K), we have by Theorem 3.6, lim d∞ (u˜ nk ; u˜ 0 ) = 0;

n→∞

which gives the desired result. Now we will prove that a characterization of relatively compact sets in F(Rp ) obtained by Joo and Kim [9] can be sharpened considerably if our attention is restricted to C(F(Rp )). The following two lemmas are needed in the proof. Lemma 3.8. Let {u˜ n } be a sequence in F(Rp ) such that limn→∞ ds (u˜ n ; u˜ 0 ) = 0 for some u˜ 0 ∈ F(Rp ). If {n } and {n } are sequences in [0; 1] such that lim n = lim n = 0 ;

n→∞

n→∞

Y.K. Kim / Nonlinear Analysis 57 (2004) 639 – 653

649

and dH (Ln u˜ n ; Ln u˜ n ) ¿  ¿ 0 for su
and

ds (u˜ n ; u˜ 0 ) ¡ !:

Then for each n ¿ n0 , there is tn ∈ T such that sup |tn () − | ¡ !

0661

and

d∞ (tn (u˜ n ); u˜ 0 ) ¡ !:

Thus for each n ¿ n0 , we have 0 − 2! ¡ tn (n ); tn (n ) ¡ 0 + 2!. If either 0 − 2! ¡ tn (n ); tn (n ) 6 0 , or 0 ¡ tn (n ); tn (n ) ¡ 0 + 2!, then dH (Ltn (n ) u˜ 0 ; Ltn (n ) u˜ 0 ) ¡ ( − 0 )=4: If 0 − 2! ¡ tn (n ) 6 0 ¡ tn (n ) ¡ 0 + 2!, then dH (Ltn (n ) u˜ 0 ; Ltn (n ) u˜ 0 ) ¡ 0 + ( − 0 )=2: In any case, dH (Ltn (n ) u˜ 0 ; Ltn (n ) u˜ 0 ) ¡ 0 + ( − 0 )=2: Thus for each n ¿ n0 , we have dH (Ln u˜ n ; Ln u˜ n ) 6 dH (Ln u˜ n ; Ltn (n ) u˜ 0 ) + dH (Ltn (n ) u˜ 0 ; Ltn (n ) u˜ 0 ) + dH (Ltn (n ) u˜ 0 ; Ln u˜ n ) ¡ ! + 0 + ( − 0 )=2 + ! ¡ : This yields a contradiction and completes the proof. Lemma 3.9. Let K ∈ C(F(Rp )) and {u˜ n } be a sequence in K such that limn→∞ ds (u˜ n ; u˜ 0 ) = 0 for some u˜ 0 ∈ F(Rp ). Then ju˜ 0 () ¿  implies  ∈ J (K). Proof. Suppose that there exists 0 ∈ J (K) such that ju˜ 0 (0 ) = 0 ¿ . Then since J (K) is Dnite, we have min{| − 0 |:  ∈ J (K)} = + ¿ 0:

650

Y.K. Kim / Nonlinear Analysis 57 (2004) 639 – 653

We choose n0 such that for n ¿ n0 ,   0 −  : ds (u˜ n ; u˜ 0 ) ¡ min +; 2 Then there exists tn ∈ T such that sup |tn () − | ¡ +;

0661

(3.10)

and d∞ (tn (u˜ n ); u˜ 0 ) ¡

0 −  : 2

(3.11)

Put n = tn−1 (0 ). Then by (3.11), 0 = dH (L0 u˜ 0 ; L0+ u˜ 0 ) 6 dH (L0 u˜ 0 ; Ln u˜ n ) + dH (Ln u˜ n ; Ln+ u˜ n ) + dH (Ln+ u˜ n ; L0+ u˜ 0 ) ¡ (0 − ) + ju˜ n (n ): Thus, ju˜ n (n ) ¿ . But this is a contradiction since n ∈ J (K) by (3.10). G ∈ C(F(Rp ), Corollary 3.10. Let K ⊂ F(Rp ). Then K ∈ C(F(Rp ) if and only if K G denotes the closure of K in the ds -topology. where K G Now if  ∈ J (K), G then there exists u˜ ∈ K G Proof. It is obvious that J (K) ⊂ J (K). G Thus the corolsuch that ju˜() ¿ . By Lemma 3.9,  ∈ J (K) and so J (K) = J (K). lary follows immediately from Theorem 3.1. Theorem 3.11. Let K be a subset of F(Rp ). Then K ∈ C(F(Rp )) if and only if the following two conditions hold: (1) sup { u ˜ : u˜ ∈ K} ¡ ∞. (2) For each  ¿ 0, there exists a 5nite partition 0 = 0 ¡ 1 ¡ · · · ¡ r = 1 of [0; 1], such that + u; sup max dH (Li−1 ˜ Li u) ˜ 6 :

16i6r u∈K ˜

Proof. SuLciency follows immediately from Theorems 2.4 and 3.1 since condition (2) of the theorem implies (2:2) and J (K) ⊂ {1 ; : : : ; r }: To prove the necessity, we assume that K ∈ C(F(Rp )). Then condition (1) is trivial. If condition (2) is false, then we can Dnd 0 ¿ 0 such that for any partition P: 0 = 0 ¡ 1 ¡ · · · ¡ r = 1 of [0; 1], there is u˜ ∈ K satisfying + u; dH (Li−1 ˜ Li u) ˜ ¿ 0

Y.K. Kim / Nonlinear Analysis 57 (2004) 639 – 653

651

for some i. We take 1 ¡ 0 and consider a sequence P1 ; P2 ; : : : of partitions of [0; 1] such that (1) P1 contains all points of J1 (K); (2) for each n, Pn+1 is a reDnement of Pn ; (3) Pn → 0, where Pn denotes the length of longest subinterval of Pn . For each n, let in be a point in Pn such that for some u˜ n ∈ K, dH (Li+ −1 u˜ n ; Lin u˜ n ) ¿ 0 : n

Let -n = in and n be a point in (in −1 ; in ) such that dH (Ln u˜ n ; L-n u˜ n ) ¿ 0 :

(3.12)

Since K is relatively compact, the sequence {u˜ n } contains a subsequence which converges to u˜ 0 ∈ F(Rp ). By compactness of [0; 1], the sequence {n } contains a subsequence converging to 0 in [0; 1]. Thus, we may assume that ds (u˜ n ; u˜ 0 ) → 0

and

lim n = 0 :

n→∞

Since Pn → 0, we have limn→∞ -n = 0 . By Lemma 3.8, ju˜ 0 (0 ) ¿ 0 ¿ 1 and so, by Lemma 3.9, 0 ∈ J1 (K). By construction of Pn ; 0 ∈ Pn for every n. Since d∞ (u˜ n ; u˜ 0 ) → 0 by Theorem 3.6, we can Dnd n0 such that d∞ (u˜ n ; u˜ 0 ) ¡ 0 =3 for n ¿ n0 : Then we can Dnd ! ¿ 0 such that dH (L0+ u˜ 0 ; L0 +! u˜ 0 ) ¡ 0 =3 and dH (L0 u˜ 0 ; L0 −! u˜ 0 ) ¡ 0 =3: Since 0 ∈ Pn for every n, there are only two possibility that either 0 ¡ n inDnitely often, or -n 6 0 inDnitely often. If 0 ¡ n inDnitely often, then we choose k ¿ n0 so that 0 ¡ k ¡ -k ¡ 0 + !. Then dH (Lk u˜ k ; L-k u˜ k ) 6 dH (Lk u˜ k ; Lk u˜ 0 ) + dH (Lk u˜ 0 ; L-k u˜ 0 ) + dH (L-k u˜ 0 ; L-k u˜ k ) ¡ 0 =3 + 0 =3 + 0 =3 = 0 : But this is a contradiction to (3.12). If -n 6 0 inDnitely often, then we Dnd k ¿ n0 so that 0 − ! ¡ k ¡ -k 6 0 . Then we reach to the contradiction by the same inequality. This completes the proof. We consider two examples to illustrate the above Theorem 3.11.

652

Y.K. Kim / Nonlinear Analysis 57 (2004) 639 – 653

Example 2. Let us deDne  1    u˜ n (x) = 12 + 1=n    0

u˜ n : Rp → [0; 1] by if x = 0; if 0 ¡ |x| 6 1; elsewhere:

We recall that K1 = {u˜ n : n = 2; 3; : : :} is relatively compact but K1 ∈ C(F(Rp ) by Remark 1. Since



L u˜ n =

{x: |x| 6 1}

if 0 6  6 12 + 1=n;

{0}

if

1 2

+ 1=n ¡  6 1

it can be proved that for any Dnite partition 0 = 0 ¡ 1 ¡ · · · ¡ r = 1 of [0; 1], + u ˜ n ; Li u˜ n ) = 1: sup max dH (Li−1

n 16i6r

Example 3. Let us deDne v˜n : Rp → [0; 1] by  1 − |x| if |x| 6 1 − 1=n;    if 1 − 1=n ¡ |x| 6 1; v˜n (x) = 1=n    0 elsewhere: Then

 L v˜n =

{x: |x| 6 1}

if 0 6  6 1=n;

{x: |x| 6 1 − }

if 1=n ¡  6 1:

It is obvious that ds (v˜n ; v) ˜ 6 d∞ (v˜n ; v) ˜ = 1=n → 0 where

 v(x) ˜ =

1 − |x|

if |x| 6 1;

0

elsewhere:

as n → ∞;

Thus, K = {v˜n : n = 1; 2; : : :} is relatively compact. Since J (K) = {1=n: n ¡ 1=} is Dnite for each  ¿ 0, it follows from Theorem 3.1 that K ∈ C(F(Rp ). Now we show that K satisDes two conditions of Theorem 3.11. It is trivial that supn v˜n = 1. For each  ¿ 0, we choose n0 such that 1=n0 ¡ . And then, we take a partition 0 = 0 ¡ 1 = 1=n0 ¡ · · · ¡ r = 1 of [0; 1] so that 
1 1 1 ; : : : ; 3 ; 2 ⊂ {2 ; : : : ; r−1 } n0 − 1 and i − i−1 6 1=n0 for all i. Then we can obtain that for each n, + v ˜n ; Li v˜n ) ¡  max dH (Li−1

16i6r

Y.K. Kim / Nonlinear Analysis 57 (2004) 639 – 653

from the fact that dH (L+ v˜n ; L v˜n ) =



0

if [; ] ⊂ [0; 1=n];

−

if [; ] ⊂ [1=n; 1]:

653

This yields the desired result. References [1] W. Congxin, M. Ming, Embedding problem of fuzzy number space: part 5, Fuzzy Sets and Systems 55 (1993) 313–318. [2] P.Z. Da:er, On compact convex subsets of D[0,1], Rocky Mountain J. Math. 11 (1981) 501–510. [3] G. Debreu, Integration of correspondences, Proceedings of the Fifth Berkeley Symposium, Mathematical Statistics and Probability, Vol. 2, 1966, pp. 351–372. [4] P. Diamond, P.E. Kloeden, Metric Spaces of Fuzzy Sets, World ScientiDc Publishing, Singapore, London, 1994. [5] J.-X. Fang, H. Huang, Some properties of the level convergence topology on fuzzy number space E n , Fuzzy Sets and Systems, 140 (2003) 509–517. [6] U. HHohle, A.P. Sostak, Axiomatic foundations of Dxed-basis fuzzy topology, in: U. HHohle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets—Logic, Topology, and Measure Theory, Kluwer Academic, Dordrecht/Norwell, MA, 1999, pp. 123–272. [7] S.Y. Joo, Y.K. Kim, The Skorokhod topology on space of fuzzy numbers, Fuzzy Sets and Systems 111 (2000) 497–501. [8] S.Y. Joo, Y.K. Kim, Topological properties on the space of fuzzy sets, J. Math. Anal. Appl. 246 (2000) 576–590. [9] Y.K. Kim, Compactness and convexity on the space of fuzzy sets, J. Math. Anal. Appl. 264 (2001) 122–132. [10] J. Wu, An embedding theorem for fuzzy numbers on Banach spaces and its applications, Fuzzy Sets and Systems 129 (2002) 57–63.