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Fuzzy Sets and Systems 160 (2009) 1620 – 1631
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Some properties of the space of fuzzy-valued continuous functions
on a compact set夡
Jin-Xuan Fang∗ , Qiong-Yu Xue
Department of Mathematics, Nanjing Normal University, Nanjing, Jiangsu 210097, China
Received 1 August 2007; received in revised form 21 June 2008; accepted 31 July 2008
Available online 12 August 2008
Abstract
In this paper, a generalized form of the Bolzano theorem in classical analysis to fuzzy number space and a characterization of
compact subsets in fuzzy number space are given. Some properties of the fuzzy-valued continuous functions defined on a compact
set K are studied. Completeness of the space C(K , E1 ) of fuzzy-valued continuous functions on K endowed with the supremum
metric D is proved. A characterization of compact subsets in the space (C(K , E1 ), D) is presented, which is a generalization of the
Arzela–Ascoli theorem in classical analysis.
© 2008 Published by Elsevier B.V.
Keywords: Fuzzy number space; Fuzzy-valued continuous function; Completeness; Compactness
1. Introduction and preliminaries
The theory of fuzzy numbers is the foundation of fuzzy analysis as the status of theory of real numbers in classical
analysis. In 1986, Goetschel and Voxman [7] established the representation theorem of fuzzy numbers on R which
promoted the further development of theory of fuzzy numbers. Using this representation theorem, Wu and Ma [10–12]
studied the embedding problem of fuzzy number space. Zhang [16] studied the fuzzy number-valued fuzzy integral
of fuzzy number-valued function with respect to fuzzy number-valued fuzzy measure on the fuzzy set. Diamond and
Kloeden [3] systematically studied the metrics d∞ , d p and D∞ on fuzzy number space En and their properties. Wu
and Wu [13,14] prove the existence theorem of supremum and infimum in fuzzy number space E1 and studied some
properties of fuzzy-valued continuous function on [a, b]. Wu and Wang [15] discuss some properties of the convergence
of sequence of fuzzy numbers and obtain some fixed point theorems for increasing fuzzy mappings. Joo and Kim [8]
introduced the Skorokhod metric on fuzzy number space E1 and proved that E1 is separable and topologically complete
in the metric. Also, Ghil et al. [6] presented a characterization of compact subsets of E1 endowed with the Skorokhod
topology. Fang and Huang [4,5] improved and generalized some results in Wu and Wu [13,14] and investigated some
properties of fuzzy number space En endowed with the level-convergence topology. Recently, Aytar et al. [1] introduced
the concept of the core of a sequence of fuzzy numbers and prove that the core of a sequence of fuzzy numbers is the
interval [, ] where and are extreme limit points of the sequence.
夡 This work is supported by the National Natural Science Foundation of China (No.10671094).
∗ Corresponding author. Tel.: +86 25 83598394.
E-mail address: [email protected] (J.-X. Fang).
0165-0114/$ - see front matter © 2008 Published by Elsevier B.V.
doi:10.1016/j.fss.2008.07.014
J.-X. Fang, Q.-Y. Xue / Fuzzy Sets and Systems 160 (2009) 1620 – 1631
1621
In this paper, we continue to investigate the theory of fuzzy number space and the space of fuzzy-valued continuous
functions on a compact subset. The present paper is organized as follows. In Section 2, we generalize the Bolzano
theorem in classical analysis to fuzzy number space E1 and present a characterization of compact subsets in E1 endowed
with the metric d∞ . In Section 3, we investigate some properties of fuzzy-valued continuous functions defined on a
compact subset K and prove that the space C(K , E1 ) of fuzzy-valued continuous functions on K is complete in the
supremum metric D. In Section 4, we present a characterization of compact subsets in the space (C(K , E1 ), D), which
is a generalization of the Arzela–Ascoli theorem in classical analysis.
Throughout this paper, R is the real field, N denotes the set of all natural numbers and F(R) denotes the family of
all fuzzy subsets on R. For u ∈ F(R) and ∈ [0, 1], the -level set of u is defined by
[u] = {x ∈ R | u(x) }, ∈ (0, 1], [u]0 = {x ∈ R | u(x) > 0}.
We define E1 = {u ∈ F(R) | u satisfies(1).(4) below}
(1)
(2)
(3)
(4)
u is normal , i.e., there exists an x0 ∈ R with u(x0 ) = 1;
u is convex , i.e., u(x + (1 − )y) min{u(x), u(y)} for all x, y ∈ R1 , ∈ [0, 1];
u(x) is upper semi-continuous;
[u]0 is a compact set in R.
Every member u in E1 is called a fuzzy number and E1 is called the fuzzy number space. Obviously, if u ∈ E1 , then
the -level set [u] of u is a closed interval for each ∈ [0, 1]. We denote [u] = [u − (), u + ()]. A real number r can
be regarded as a fuzzy number r̃ defined by
1 if t = r,
r̃ (t) =
(1.1)
0 if t r.
Theorem 1.1 (Goetschel and Voxman [7]). Let u ∈ E1 and [u] = [u − (), u + ()], ∈ [0, 1]. Then the pair of
functions u − () and u + () has the following properties:
(1)
(2)
(3)
(4)
u − () is a bounded left continuous nondecreasing function on (0, 1];
u + () is a bounded left continuous nonincreasing function on (0, 1];
u − () and u + () are right continuous at = 0;
u − (1) u + (1).
Conversely, if the pair of functions () and () satisfies the above conditions (1)–(4), then there exists a unique u ∈ E1
such that [u] = [(), ()] for each ∈ [0, 1].
Theorem 1.2 (Goetschel and Voxman [7], Diamond and Kloeden [3]). For u, v ∈ E1 , define
d∞ (u, v) = sup max{|u − () − v − ()|, |u + () − v + ()|},
∈[0,1]
(1.2)
then d∞ is a metric on E1 , is called the supremum metric on E1 , and (E1 , d∞ ) is a complete metric space.
2. Generalization of the Bolzano theorem and characterization of compact subsets in (E1 , d∞ )
Definition 2.1. Let {u n } be a sequence in E1 and u ∈ E1 . We say that
(1) {u n } d∞ -converges to u, if limn→∞ d∞ (u n , u) = 0 (see [3]).
−
+
+
(2) {u n } level-wise converges to u, if limn→∞ u −
n () = u () and lim n→∞ u n () = u () for each ∈ [0, 1] (see
[15] or [5]).
+
−
+
Obviously, {u n } d∞ -converges to u ∈ E1 if and only if {u −
n ()} and {u n ()} converge uniformly to u () and u ()
on [0, 1], respectively. The d∞ -convergence implies the level-wise convergence, but the level-wise convergence does
not imply d∞ -convergence (see [15]).
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J.-X. Fang, Q.-Y. Xue / Fuzzy Sets and Systems 160 (2009) 1620 – 1631
Lemma 2.1. Let {u n } be a sequence in E1 which d∞ -converges to u ∈ E1 .
+
−
−
(1) If 0 ∈ (0, 1], {n } ⊂ (0, 0 ) and n → 0 , then limn→∞ u +
n (n ) = u (0 ), lim n→∞ u n (n ) = u (0 ).
+
+
−
−
(2) If {n } ⊂ (0, 1] and n → 0, then limn→∞ u n (n ) = u (0), limn→∞ u n (n ) = u (0).
−
+
−
Proof. Note that {u +
n ()} and {u n ()} are uniformly convergent u () and u () on [0, 1], respectively. By using the
+
−
left-continuity of u () and u () at 0 ∈ (0, 1] and the right-continuity of u + () and u − () at = 0, the conclusion
is proved easily. Definition 2.2. Let { f t }t∈T be a family of functions defined on [0, 1]. We say that
(1) { f t }t∈T is equi-left-continuous on (0, 1] if for each > 0 there exists = () > 0 such that | f t ( ) − f t ()| < for all t ∈ T whenever , ∈ (0, 1] with ∈ ( − , ].
(2) { f t }t∈T is equi-continuous on (0, 1] (or on [0, 1]) if for each > 0 there exists = () > 0 such that
| f t ( ) − f t ()| < for all t ∈ T whenever , ∈ (0, 1] (resp. , ∈ [0, 1]) with | − | < .
Remark 2.1. Actually, the equi-left continuity of { f t }t∈T on (0, 1] is equivalent to the equi-continuity of { f t }t∈T on
(0, 1].
In fact, if { f t }t∈T is equi-left-continuous on (0, 1], then for each > 0 there exists = () > 0 such that
| f t ( ) − f t ()| < /2 for all t ∈ T whenever , ∈ (0, 1] with ∈ ( − , ].
Taking 1 ∈ (0, ), then it is not difficult to prove that | f t ( ) − f t ()| < for all t ∈ T whenever , ∈ (0, 1] with
| − | < 1 .
When ∈ ( − 1 , ], it is evident that | f t ( ) − f t ()| < for all t ∈ T ; When ∈ (, + 1 ), let 0 = + 1 ,
then we have ∈ (0 − 1 , 0 ) ⊂ (0 − , 0 ) and ∈ (0 − , 0 ). It follows that
| f t ( ) − f t ()|| f t ( ) − f t (0 )| + | f t (0 ) − f t ()| < /2 + /2 = for all t ∈ T.
Therefore { f t }t∈T is equi-continuous on (0, 1].
−
Theorem 2.1. Let {u n } be a sequence of fuzzy numbers. If {u n } d∞ -converges to u in E1 , then {u +
n ()} and {u n ()} are
equi continuous on [0, 1].
Proof. We first prove that {u +
n ()} is equi-left-continuous on (0, 1]. If not, then there exist an 0 > 0, and {k } ⊂ (0, 1]
and a subsequence {u +
n k } such that
k
+
u+
n k (k − 1/2 ) u n k (k ) + 0 , k = 1, 2, . . . .
Put 0 = supn n , it is evident that there exists a subsequence {ki } such that ki → 0 . Note that {u +
n ()} converges
uniformly to u + () on [0, 1] and u + () is left-continuous on (0, 1] and
ki
+
u+
n k (ki − 1/2 )u n k (ki ) + 0 , i = 1, 2, . . . .
i
i
(2.1)
Letting i → ∞ in (2.1), by Lemma 2.1, we obtain u + (0 ) u + (0 ) + 0 , which is a contradiction. This shows that
+
{u +
n ()} is equi-left-continuous on (0, 1]. Thus, by Remark 2.1, we know that {u n ()} is equi-continuous on (0, 1].
+
Secondly, we prove that {u n ()} is equi-right-continuous at = 0, i.e., for each > 0 there exists = () ∈ (0, 1]
+
k
such that |u +
n () − u n (0)| < for all n ∈ N whenever ∈ (0, ). If not, then there exist an 0 > 0, and {k } ⊂ (0, 1/2 )
+
and a subsequence {u n k } such that
+
u+
n k (k ) u n k (0) − 0 , k = 1, 2, . . . .
(2.1)
Letting k → ∞ in (2.1) , by Lemma 2.1, we obtain u + (0)u + (0) − 0 , which is a contradiction. This shows that
{u +
n ()} is equi-right-continuous at = 0.
−
Therefore, we can conclude that {u +
n ()} is equi-continuous on [0, 1]. Similarly, we can prove {u n ()} is also
equi-continuous on [0, 1]. J.-X. Fang, Q.-Y. Xue / Fuzzy Sets and Systems 160 (2009) 1620 – 1631
1623
−
• Theorem 2.1 shows that if {u +
n ()} or {u n ()} is not equi-continuous on [0, 1], then {u n } does not d∞ -converge in
1
E . By using this fact, we can expediently determine the d∞ -convergence of some sequences in E1 . For example,
define
⎧
0,
x∈
/ [0, 1],
⎪
⎪
⎪
⎪
⎪
⎪
1
⎨
1,
x ∈ 0,
,
n = 1, 2, . . . .
u n (x) =
2
⎪
⎪
⎪
⎪
1
1
1
⎪
⎪
, x∈
,1 ,
⎩ +
2 4n
2
Obviously,
⎧
⎪
⎪
⎪
⎨ 1,
1
1
∈ 0, +
,
2 4n
+
u−
()
=
0,
u
()
=
n
n
⎪
1
1
⎪1
⎪
+
,1 .
⎩ , ∈
2
2 4n
It is easy to see that {u +
n ()} is not equi-continuous on [0, 1]. Hence, by Theorem 2.1, we conclude that {u n } does
not d∞ -converge in E1 . However, {u n } is point-wise convergent to u which is defined by
⎧
0, x ∈
/ [0, 1],
⎪
⎪
⎪
⎪
⎪
⎪
1
⎨
1, ∈ 0,
,
u(x) =
2
⎪
⎪
⎪
⎪
1
1
⎪
⎪
,1 .
⎩ , ∈
2
2
Obviously,
⎧
⎪
⎪
⎪
⎨ 1,
1
∈ 0,
,
2
−
+
u () = 0, u () =
⎪
1
1
⎪
⎪
,
∈
,
1
.
⎩
2
2
Definition 2.3 (Diamond and Kloeden [2]). A subset U of E1 is said to be uniformly support-bounded if there is a
constant L > 0 such that |u − (0)| L and |u + (0)| L for all u ∈ U .
Lemma 2.2. Let U be a subset of E1 . Then the following statements are equivalent:
(1) U is uniformly support-bounded.
(2) The families of functions {u + (·) | u ∈ U } and {u − (·) | u ∈ U } are uniformly bounded on [0, 1].
(3) U is d∞ -bounded, i.e., there is an L > 0 such that d∞ (u, 0̃) L for all u ∈ U .
Proof. Since −|u − (0)|u − (0)u − () u + () u + (0)|u + (0)| for all ∈ (0, 1], it is easy to prove that (1)
implies (2).
Since d∞ (u, 0̃) = sup∈[0,1] max{|u − ()|, |u + ()|}, it is easy to see that (2) ⇒ (3) and (3) ⇒ (1) are tenable. Lemma 2.3 (Rudin [9]). Let {gn } be a sequence of functions which is point-wise bounded on a countable set Q. Then
there exists a subsequence {gni } such that {gni (x)} is convergent for each x ∈ Q.
−
Theorem 2.2. Let {u n } be a sequence in E1 such that {u +
n ()} and {u n ()} are point-wise bounded and equi-leftcontinuous on (0, 1]. Then {u n } has a d∞ -convergent subsequence {u ni }.
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J.-X. Fang, Q.-Y. Xue / Fuzzy Sets and Systems 160 (2009) 1620 – 1631
−
Proof. Since {u +
n ()} and {u n ()} are equi-left-continuous on [0, 1], for each > 0, there exists a = () > 0
such that
+
− −
|u +
n ( ) − u n ()| < /3 |u n ( ) − u n ()| < /3 for all n ∈ N
(2.2)
whenever , ∈ (0, 1] with ∈ ( − , ].
−
Let Q denote the set of all rational numbers on [0, 1]. It is evident that {u +
n ()} and {u n ()} are point-wise bounded
+
−
on Q. So, by Lemma 2.3, there exists a subsequence {u ni } such that {u ni ()} and {u ni ()} are convergent for each ∈ Q.
−
In the following, we will prove that {u +
n i ()} and {u n i ()} are uniformly convergent on [0, 1].
For the above , take a enough big m ∈ N such that 1/m < . We consider a partition of [0, 1]:
i
, i = 1, 2, . . . , m.
m
It is evident that [i−1 , i ] ⊂ (i − , i ] (i = 1, 2, . . . , m). Note that
0 = 0 < 1 < · · · < m = 1 where i =
∀ ∈ (0, 1] ⇒ ∈ [i0 −1 , i0 ] ⊂ (i0 − , i0 ] for some i 0 (1i 0 m).
And then, by (2.2), we get
+
|u +
n i () − u n i (i 0 )| < /3 for all i ∈ N.
(2.3)
Since {u +
n i (i 0 )} is convergent, there exists N1 ∈ N such that
+
|u +
n i (i 0 ) − u n j (i 0 )| < /3 for all i, j N1 .
(2.4)
It follows from (2.3) and (2.4) that
+
+
+
+
+
|u +
n i () − u n j ()| |u n i () − u n i (i 0 )| + |u n i (i 0 ) − u n j (i 0 )|
+
+|u +
n j (i 0 ) − u n j ()| < /3 + /3 + /3 = for all i, j N1 and all ∈ (0, 1].
On the other hand, since {u +
n i (0)} is convergent, for the above > 0 there exists N2 ∈ N such that
−
|u +
n i (0) − u n j (0)| < for all i, j N2 .
Put N = max{N1 , N2 }, we have
+
|u +
n i () − u n j ()| < for all i, j N and all ∈ [0, 1],
−
which shows that {u +
n i ()} is uniformly convergent on [0, 1]. Similarly, we can prove that {u n i ()} is also uniformly
convergent on [0, 1]. Therefore we conclude that {u ni } is d∞ -convergent in (E1 , d∞ ). This completes the proof. It is not difficult to see that the conditions in Theorem 2.2 can be weakened reasonably. We have the following
theorem:
−
Theorem 2.2 . Let {u n } be a sequence in E1 . If it has a subsequence {u n k } such that {u +
n k ()} and {u n k ()} are point-wise
bounded and equi-left-continuous on (0, 1], then {u n } has a d∞ -convergent subsequence.
Corollary 2.1 (Bolzano theorem). Every bounded sequence of real numbers admits a convergent subsequence.
Proof. Suppose that {an } be a bounded sequence of real numbers. We consider a corresponding sequence {ãn } of
fuzzy numbers, where ãn is defined by (1.1). Since ãn+ () = ãn− () = an for all ∈ [0, 1], it is easy to see that (1)
{an } is bounded if and only if {ãn } is d∞ -bounded; (2) {ãn+ ()} and {ãn− ()} are equi-left-continuous on [0, 1]. So, by
d∞
Theorem 2.2, we can conclude that there exists a subsequence {ãni } of {ãn } such that ãni → u ∈ E1 , which implies that
limi→∞ ani = u + () = u − () for all ∈ [0, 1]. By the uniqueness of limit, we know that u = ã, where a ∈ R, and so
d∞
ãni −→ ã implies that ani → a. This completes the proof.
Remark 2.2. Corollary 2.1 shows that Theorem 2.2 (Theorem 2.2 ) is a generalization of the Bolzano theorem in
classical analysis to fuzzy number space.
J.-X. Fang, Q.-Y. Xue / Fuzzy Sets and Systems 160 (2009) 1620 – 1631
1625
Theorem 2.3. A subset U in (E1 , d∞ ) is compact if and only if the following three conditions are satisfied:
(1) U is uniformly support-bounded;
(2) U is a closed subset in (E1 , d∞ );
(3) {u + (·) | u ∈ U } and {u − (·) | u ∈ U } are equi-left-continuous on (0, 1].
Proof. Necessity: Suppose that U is a compact subset in (E1 , d∞ ). Note that every compact subset in a metric space
is bounded and closed. Hence U is d∞ -bounded and closed in (E1 , d∞ ). By Lemma 2.2, we know that U is uniformly
support-bounded. (1) and (2) are proved.
Suppose that {u + () | u ∈ U } is not equi-left-continuous on (0, 1]. Then there exist an 0 and sequences {n } in
(0, 1], {u n } in U such that
n
+
u+
n (n − 1/2 )u n (n ) + 0 for all n ∈ N.
(2.5)
Put 0 = supn n . It is evident that there exists a subsequence {n k } of {n } such that n k → 0 . Note that U is compact
in (E1 , d∞ ) and {u n } ⊂ U . Hence there exists a subsequence {u n k } of {u n } such that {u n k } d∞ -converges to u ∈ E1 .
By (2.5) we have
nk
+
u+
n k (n k − 1/2 ) u n k (n k ) + 0 for all k ∈ N.
(2.6)
Letting k → ∞ in (2.6), from Lemma 2.1 we get u + (0 )u + (0 ) + 0 , which is a contradiction. This shows that
{u + () | u ∈ U } is equi-left-continuous on (0, 1]. Similarly, we can prove that {u − () | u ∈ U } is equi-left-continuous
on (0, 1]. (3) is proved.
−
Sufficiency: Let {u n } be an arbitrary sequence in U. Then condition (1) implies that {u +
n ()} and {u n ()} are point+
−
wise bounded on [0, 1]. From the condition (3), we know that {u n ()} and {u n ()} are equi-left-continuous on (0, 1].
Hence, by Theorem 2.2, we infer that {u n } has a d∞ -convergent subsequence {u ni }. By (2), U is a closed subset of E1
and {u ni } ⊂ U . Hence there exists u ∈ U such that d∞ (u ni , u) → 0 as i → ∞. This shows that U is sequentially
compact in (E1 , d∞ ). The sequential compactness and the compactness in a metric space are equivalent. Therefore U
is a compact subset in (E1 , d∞ ). Remark 2.3. By using the embedding theory of fuzzy number space and the support functions of fuzzy numbers,
Diamond and Kloeden [3] proved a characterization of compact subsets in (En , d∞ ), i.e., Proposition 8.2.1 in [3]:
A closed set of U of (En , d∞ ) is compact if and only if U is uniformly support-bounded and U ∗ is equi-leftcontinuous in ∈ I uniformly in x ∈ S n−1 , where S n−1 is the unit sphere in Rn , U ∗ = {u ∗ | u ∈ U } is the
family of support functions, i.e., u ∗ (, p) = sup{
p, x | x ∈ [u] }, p ∈ S n−1
It is easy to see that the above Theorem 2.3 is an improvement of Proposition 8.2.1 in [3] when n = 1.
Obviously, condition (1) of Theorem 2.3 implies the condition
(1) {u + () | u ∈ U } and {u − () | u ∈ U } are point-wise bounded on [0, 1], i.e., for each ∈ [0, 1], supu∈U u + () <
+∞ and inf u∈U u − () > −∞.
Also, in the proof of Sufficiency, we only used condition (1) . Therefore, we have the following theorem.
Theorem 2.4. A subset U in (E1 , d∞ ) is compact if and only if the following three conditions are satisfied:
(1) {u + (·) | u ∈ U } and {u − (·) | u ∈ U } are point-wise bounded on [0, 1];
(2) U is a closed subset in (E1 , d∞ );
(3) {u + (·) | u ∈ U } and {u − (·) | u ∈ U } are equi-left-continuous on (0, 1].
Corollary 2.2. A subset B in R is compact if and only if it is bounded and closed.
Proof. Define B = {x̃ | x ∈ B}. It is evident that x̃ + () = x̃ − () = x for all ∈ [0, 1] and x ∈ B. Hence, it is easy
+
to see that {x̃ () | x ∈ B} and {x̃ − () | x ∈ B} are equi-left-continuous on [0, 1]. Moreover, it is not difficult to
prove that
B is bounded in R ⇔ B is d∞ -bounded in E1 ;
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J.-X. Fang, Q.-Y. Xue / Fuzzy Sets and Systems 160 (2009) 1620 – 1631
B is closed in R ⇔ B is closed in (E1 , d∞ );
B is compact in R ⇔ B is compact in (E1 , d∞ ).
Therefore the conclusion follows from Theorem 2.3 immediately. 3. Some properties of fuzzy-valued continuous functions on a compact set
In the following text, we always suppose that (X, d) is a metric space and K is a compact subset of X. A fuzzy-valued
function f on K, i.e., a mapping f from K into E1 , is said to be continuous at t0 ∈ K if for each > 0, there exists > 0
such that d∞ ( f (t), f (t0 )) < whenever t ∈ K with d(t, t0 ) < . We say that f is continuous on K if it is continuous
at each t ∈ K .
Definition 3.1. Let { f | ∈ [0, 1]} be a family of real-valued functions on K. We say that
(1) { f | ∈ [0, 1]} is equi-continuous at t0 ∈ K , if for each > 0 there exists > 0 such that | f (t) − f (t0 )| < for all ∈ [0, 1] whenever t ∈ K with d(t, t0 ) < .
(2) { f | ∈ [0, 1]} is equi-continuous on K, if for each > 0 there exists > 0 such that | f (t) − f (t )| < for all
∈ [0, 1] whenever t, t ∈ K with d(t, t ) < .
By (1.2) and Definition 3.1, it is easy to prove the following lemma:
Lemma 3.1. A fuzzy-valued function f : K → E1 is continuous at t0 ∈ K if and only if the families of functions
{[ f (·)]+ () | ∈ [0, 1]} and {[ f (·)]− () | ∈ [0, 1]} are equi-continuous at t0 ∈ K .
Definition 3.2. A fuzzy-valued function f : K → E1 is said to be uniformly continuous on K, if for each > 0, there
exists = () > 0 such that d∞ ( f (t), f (t )) < whenever t, t ∈ K with d(t, t ) < .
Theorem 3.1. Let f be a fuzzy valued function on the compact subset K, then the following statements are equivalent:
(1) f is continuous on K;
(2) The families of functions {[ f (·)]+ () | ∈ [0, 1]} and {[ f (·)]− () | ∈ [0, 1]} are equi-continuous on K;
(3) f is uniformly continuous on K.
Proof. (1) ⇒ (2). Since f : K → E1 is continuous, by Lemma 3.1, {[ f (·)]+ () | ∈ [0, 1]} is equi-continuous at
each t ∈ K , that is, for each t ∈ K and > 0, there exists t > 0 such that
|[ f (s)]+ () − [ f (t)]+ ()| < /2 for all ∈ [0, 1]
whenever s ∈ K with d(s, t) < t . Obviously, {O(t, t /2) | t ∈ K } is an open cover of K,where O(t, t /2) =
m
O(ti , ti /2). Let
{s ∈ X | d(s, t) < t /2}. Since K is compact, there exist ti ∈ K (1i m) such that K ⊂ i=1
= min{t1 /2, . . . , tm /2}. For simplification, we write ti = i . Note that ∀t, t ∈ K ,
d(t, t ) < ⇒ ∃i 0 (1i 0 m) such that d(t, ti0 ) < i0 /2 and d(t , ti0 ) < i0 .
So, we have
|[ f (t )]+ () − [ f (t)]+ ()| |[ f (t )]+ () − [ f (ti0 )]+ ()| + |[ f (ti0 )]+ () − [ f (t)]+ ()|
< /2 + /2 = for all ∈ [0, 1]
whenever t, t ∈ K with d(t, t ) < , which shows that {[ f (·)]+ () | ∈ [0, 1]} is equi-continuous on K. Similarly, we
can prove that {[ f (·)]− () | ∈ [0, 1]} is equi-continuous on K. (2) is proved.
(2) ⇒ (3). By (2), we know that for each > 0 there exists = () > 0 such that whenever s, t ∈ K with
d(s, t) < ,
|[ f (s)]+ () − [ f (t)]+ ()| < , |[ f (s)]− () − [ f (t)]− ()| < for all ∈ [0, 1].
This implies that d∞ ( f (s), f (t)) whenever s, t ∈ K with d(s, t) < . Therefore f is uniformly continuous on K.
J.-X. Fang, Q.-Y. Xue / Fuzzy Sets and Systems 160 (2009) 1620 – 1631
1627
(3) ⇒ (1). It is obvious. Definition 3.3. A fuzzy-valued function f : K → E1 is said to be level-continuous at t0 ∈ K if
lim [ f (t)]− () = [ f (t0 )]− (),
t→t0
lim [ f (t)]+ () = [ f (t0 )]+ () for all ∈ [0, 1].
t→t0
If f is level-continuous at each t ∈ K , then we say that f is level-continuous on K.
Remark 3.1. It is evident that the continuity of a fuzzy-valued function imply the level-continuity, but the converse
does not hold in general, see Example 5.1 in [4].
Theorem 3.2. Let f : K → E1 be a fuzzy-valued function. If f is level-continuous on K, then u = supt∈K f (t) must
exist in E1 and
u − () = sup[ f (t)]− (), u + () = sup[ f (t)]+ () for each ∈ [0, 1].
t∈K
t∈K
A similar conclusion for the infimum is also true.
Proof. Similar to the proof of Theorem 5.1 in [4]. Let C(K , E1 ) denote the family of all fuzzy-valued continuous functions on K. We define the mapping D : C(K , E1 )×
C(K , E1 ) → R by
D( f, g) = sup d∞ ( f (t), g(t)),
f, g ∈ C(K , E1 )
(3.1)
t∈K
where d∞ (·, ·) is the supremum metric on E1 , and then
d∞ ( f (t), g(t)) = sup max{|[ f (t)]+ () − [g(t)]+ ()|, |[ f (t)]− () − [g(t)]− ()|}
∈[0,1]
(3.2)
Remark 3.2. Put (t) = d∞ ( f (t), g(t)), t ∈ K . Since
|d∞ ( f (t), g(t)) − d∞ ( f (t0 ), g(t0 ))|d∞ ( f (t), f (t0 )) + d∞ (g(t), g(t0 )),
by the continuity of f and g, it is easy to see that the function (t) ∈ C(K , R), and so supt∈K (t) exists in R by
Theorem 3.2. Hence the definition of D( f, g) is reasonable.
By (3.1) and (3.2), we easily obtain the following result.
Theorem 3.3. (C(K , E1 ), D) is a metric space, and { f n } D-converges to f in C(K , E1 ) if and only if {[ f n (t)]+ ()} and
{[ f n (t)]− ()} uniformly converge to [ f (t)]+ () and [ f (t)]− () on K × [0, 1], respectively.
Theorem 3.4. Let { f n } ⊂ C(K , E1 ) and f : K → E be a fuzzy-valued function. If { f n } converges uniformly to f on K
in the metric d∞ , then f ∈ C(K , E1 ).
Proof. Since { f n } converges uniformly to f on K in the metric d∞ , for each > 0 there exists N ∈ N such that
d∞ ( f n (t), f (t)) < /3 for all t ∈ K and n N . Note that f N ∈ C(K , E1 ). Hence for the above > 0 and each t ∈ K ,
exists > 0 such that d∞ ( f N (s), f N (t)) < /3 for all s ∈ K with d(s, t) < . So we have
d∞ ( f (s), f (t)) d∞ ( f (s), f N (s)) + d∞ ( f N (s), f N (t)) + d∞ ( f N (t), f (t))
< /3 + /3 + /3 = for all s ∈ K with d(s, t) < .
This shows that f is continuous at each t ∈ K . Therefore f ∈ C(K , E1 ). Theorem 3.5. (C(K , E1 ), D) is a complete metric space.
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Proof. By Theorem 3.3, we need only to prove the completeness of (C(K , E1 ), D). Suppose that { f n } is a D-Cauchy
sequence in C(K , E1 ). Then for each > 0, there exists N ∈ N such that D( f n , f m ) < whenever n, m N , which
implies that
d∞ ( f n (t), f m (t)) < for all t ∈ K ,
(3.3)
whenever n, m N . This shows that for each t ∈ K , { f n (t)} is a Cauchy sequence in (E1 , d∞ ). By Theorem i.1, (E1 , d∞ )
is complete, hence, we can define mapping f : K → E as follows:
f (t) = lim f n (t), t ∈ K .
n→∞
Letting m → ∞ in (3.3), it follows that d∞ ( f n (t), f (t)) for all t ∈ K and n N . This shows that { f n } converges to
f (t) uniformly on K in the metric d∞ . So, from Theorem 3.4, we conclude that f ∈ C(K , E1 ) and { f n } D-converges
to f. Therefore (C(K , E1 ), D) is complete. 4. A characterization of compact subsets in (C(K , E1 ), D)
Lemma 4.1. Let Q be a countable subset of K and { f n } be a sequence of fuzzy-valued functions on K satisfying the
following two conditions:
(1) { f n } is point-wise d∞ -bounded on Q;
(2) For each x ∈ Q, {[ f n (x)]+ (·)} and {[ f n (x)]− (·)} are equi-left-continuous on (0, 1].
Then { f n } has a subsequence which is point-wise d∞ -convergent on Q.
Proof. Let Q = {x1 , x2 , . . .}. By condition (1), we know that { f n (x1 )}∞
n=1 is d∞ -bounded, and so, by Lemma 2.2,
{[ f n (x1 )]+ (·)} and {[ f n (x1 )]− (·)} are uniformly bounded on [0, 1], and so also point-wise bounded on [0, 1]. By condition (2), we know that {[ f n (x1 )]+ (·)} and {[ f n (x1 )]− (·)} are equi-left-continuous on (0, 1]. Hence, by
∞
Theorem 2.2, there exists a subsequence S1 = { f 1n }∞
n=1 of { f n } such that { f 1n (x 1 )}n=1 d∞ -converges. Similarly,
∞
∞
+
−
we know that {[ f 1n (x2 )] (·)}n=1 and {[ f 1n (x2 )] (·)}n=1 are point-wise bounded on [0, 1] and equi-left-continuous
∞
∞
on (0, 1], hence by Theorem 2.2 there exists a subsequence S2 = { f 2n }∞
n=1 of { f 1n }n=1 such that { f 2n (x 2 )}n=1 d∞ converges. Continuing this process, we get an array of sequences of the form
S1 = { f 11 , f 12 , f 13 , f 14 , . . .},
S2 = { f 21 , f 22 , f 23 , f 24 , . . .},
S3 = { f 31 , f 32 , f 33 , f 34 , . . .},
.. .. .. .. .. .. ..
.......
From this construction, it is easy to see that (a) Sk is a subsequence of Sk−1 (k = 2, 3, . . . ; ) (b) for each k = 1, 2, . . .,
{ f kn (xk )} is d∞ -convergent as n → ∞. We consider the diagonal sequence S : f 11 , f 22 , f 33 , . . .. It is clear that S is a
subsequence of { f n } and S (except the anterior k − 1 terms of it) is also the subsequence of Sk (k = 1, 2, . . .). Therefore,
By (b) we can affirm that for each xk ∈ Q, { f nn (xk )} is d∞ -convergent as n → ∞. Remark 4.1. Condition (2) in Lemma 4.1 can be weakened as
(2) There exists a subsequence { f n k } such that for each x ∈ Q, {[ f n k (x)]+ (·)} and {[ f n k (x)]− (·)} are equi-leftcontinuous on (0, 1].
The conclusion still holds.
Theorem 4.1. A closed subset F in (C(K , E1 ), D) is compact if and only if the following conditions are satisfied:
(1) As a family of fuzzy-valued functions on K, F is uniformly d∞ -bounded on K;
(2) F is equi-continuous on K, i.e., for each > 0, there exists > 0 such that D( f (t), f (t )) < for all f ∈ F
whenever t, t ∈ K with d(t, t ) < ;
(3) For each t ∈ K , {[ f (t)]+ (·) | f ∈ F} and {[ f (t)]− (·) | f ∈ F} are equi-left-continuous on (0, 1].
J.-X. Fang, Q.-Y. Xue / Fuzzy Sets and Systems 160 (2009) 1620 – 1631
1629
Proof. Necessity: Assume that F is compact in (C(K , E1 ), D). Note that (C(K , E1 ), D) is a metric space. Hence F is
D-bounded, and so it is not difficult to see that F, as a family of fuzzy-valued functions on K, is uniformly d∞ -bounded
on K, i.e., the conclusion (1) holds.
Now we prove that F is equi-continuous on K. Since F is a compact subset in the metric space (C(K , E1 ), D), it
is totally bounded. So, for each > 0 we can find a finite /3-net { f 1 , . . . , f n } of F, where { f 1 , . . . , f n } ⊂ F. By
Theorem 3.1, we know that every f k (k = 1, 2, . . . , n) is uniformly continuous on K, and so there exists > 0 such
that whenever t, t ∈ K with d(t, t ) < d∞ ( f k (t), f k (t )) < /3, k = 1, 2, . . . , n.
Since { f 1 , . . . , f n } is an /3-net of F, for each f ∈ F, we can choose some f k (1 k n) such that D( f k , f ) < /3,
i.e., d∞ ( f k (t), f (t)) < /3 for all t ∈ K . Thus, we have
d∞ ( f (t), f (t ))d∞ ( f (t), f k (t)) + d∞ ( f k (t), f k (t )) + d∞ ( f k (t ), f (t )) < whenever t, t ∈ K with d(t, t ) < . This shows that F, as a family of fuzzy-valued functions on K, is equi-continuous
on K. (2) is proved.
Since F is a compact subset of C(K , E1 ), D), it is not difficult to prove that for each t ∈ K , { f (t) | f ∈ F} is a
compact subset of (E1 , d∞ ). So, by Theorem 2.3, we know that {[ f (t)]+ (·) | f ∈ F} and {[ f (t)]− (·) | f ∈ F} are
equi-left-continuous on (0, 1]. The conclusion (3) is proved.
Sufficiency: Assume that F ⊂ (C(K , E1 ), D) satisfies conditions (1)–(3) and { f n } is a sequence in F. By condition
(2), for every > 0 there exists = () > 0 such that
d∞ ( f n (t), f n (t )) < /3 for all n ∈ N
t, t (4.1)
d(t, t )
whenever
∈ K with
< . Since K is compact, it is separable, and so has a countably dense subset Q.
It is evident
that
{O(t,
)
|
t
∈
Q}
is an open cover of K. Hence there exist tk ∈ Q (k = 1, 2, . . . , m) such that
O(t
,
).
K ⊂ m
k
k=1
By conditions (1) and (3), we know that { f n } is point-wise d∞ -bounded on Q, and for each t ∈ Q, {[ f n (t)]+ (·)} and
{[ f n (t)]− (·)} are equi-left-continuous on (0, 1]. Thus, from Lemma 4.1 we know that there exists a subsequence { f ni }
which is point-wise convergent on Q. In the following, we prove that { f ni } is uniformly d∞ -convergent on K.
Since { f ni } is d∞ -convergent at each tk (k = 1, 2, . . . , m), for the above there exists N ∈ N such that
d∞ ( f ni (tk ), f n j (tk )) < /3 for all i, j N , k = 1, 2, . . . , m.
(4.2)
Note that ∀ t ∈ K ⇒ ∃k0 (1 k0 m) such that t ∈ O(tk0 , ). So, by (4.1) and (4.2), we have
d∞ ( f ni (t), f n j (t)) d∞ ( f ni (t), f ni (tk0 )) + d∞ ( f ni (tk0 ), f n j (tk0 ))
+d∞ ( f n j (tk0 ), f n j (t)) < /3 + /3 + /3 = for all i, j N and t ∈ K . This shows that { f ni } is uniformly d∞ -convergent on K, i.e., { f ni } D-converge in
(C(K , E1 ), D). Note that F is a closet subset of (C(K , E1 ), D). Hence F is sequentially compact in (C(K , E1 ), D).
The sequential compactness and the compactness in a metric space are equivalent. Therefore F is a compact subset in
(C(K , E1 ), D). From the proof of Theorem 4.1, it is not difficult to see that the condition (1) in the theorem can be replaced with a
more weak condition, that is, we have the following theorem:
Theorem 4.2. A closed subset F in (C(K , E1 ), D) is compact if and only if the following conditions are satisfied:
(1) For each t ∈ K , the set { f (t) | f ∈ F} is d∞ -bounded.
(2) F is equi-continuous on K.
(3) For each t ∈ K , {[ f (t)]+ (·) | f ∈ F} and {[ f (t)]− (·) | f ∈ F} are equi-left-continuous on (0, 1].
We denote the set of all continuous real-valued functions on K by C(K , R). The metric on C(K , R) is defined by
( f, g) = max | f (t) − g(t)|,
t∈K
f, g ∈ C(K , R).
Put R = {r̃ | r ∈ R}. Then C(K , R) is a subspace of C(K , E).
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J.-X. Fang, Q.-Y. Xue / Fuzzy Sets and Systems 160 (2009) 1620 – 1631
Lemma 4.2. Let K be a compact subset in the metric space (X, d), and let F be a family of real functions defined on
= { f˜ | f ∈ F}, where f˜(t) = K and F
f (t), t ∈ K . Then
(1)
(2)
(3)
(4)
(5)
(6)
(7)
{[ f˜(t)]+ (·) | f ∈ F} and {[ f˜(t)]− (·) | f ∈ F} are equi-continuous on [0, 1].
for each t ∈ K , { f (t) | f ∈ F} is bounded in R iff { f˜ | f ∈ F} is d∞ -bounded in E1 .
is uniformly d∞ -bonded on K.
F is uniformly bounded on K iff F
˜
f ∈ C(K , R) iff f ∈ C(K , R).
is a closet subset in C(K , F is a closed subset in C(K , R) iff F
R).
is a compact subset in C(K , F is a compact subset in C(K , R) iff F
R).
as a family of fuzzy-valued functions on K, is equi-continuous on K.
F is equi-continuous on K iff F,
Proof. Note that [ f˜(t)]+ () = [ f˜(t)]− () ≡ f (t) for ∈ [0, 1]. Hence it is evident that the conclusion (1) holds.
˜
= | f (t)| for all f ∈ F and t ∈ K , where (t) ≡ 0. It is easy to see that the conclusions
Note that d∞ ( f˜(t), (t))
(2) and (3) are tenable. Note that
f (x), f (x0 )) = | f (x) − f (x0 )| for all x, x0 ∈ K .
d∞ ( f˜(x), f˜(x0 )) = d∞ (
It is easy to prove the conclusions (4)–(7). We only prove (5) here.
Then there exists { f n } ⊂ F such that d∞ ( f˜n (t), g(t)) → 0
Suppose F is a closed subset in C(K , R) and g ∈ cl( F).
on K uniformly, which implies that
u.c.
f n (t) −→[g(t)]+ () = [g(t)]− () for all t ∈ K and ∈ [0, 1].
D
This implies that there exists an f ∈ C(K , R) such that g = f˜, and so f˜n → f˜. Equivalently, we have f n → f .
This shows that F
is closed in
Note that { f n } ⊂ F and F is closed in C(K , R). Hence f ∈ F, and so g = f˜ ∈ F.
(C(K , E1 ), D).
is a closed subset in (C(K , E1 ), D) and f ∈ cl(F). Then there exists a sequence { f n } in F
Contrarily, suppose that F
D
and f˜n →
is closed in (C(K , E1 ), D),
such that f n (t) → f (t) on K uniformly. This implies that { f˜n } ⊂ F
f˜. Since F
and so f ∈ F. This shows that F is closed in (C(K , R), ). (5) is proved. we have f˜ ∈ F,
By using Lemma 4.2, Theorems 4.1 and 4.2, it is easy to prove the Arzela–Ascoli theorem and its improvement.
Corollary 4.1 (Arzela–Ascoli). A closed subset F in C(K , R) is compact if and only if the following conditions are
satisfied:
(1) F is uniformly bounded on K;
(2) F is equi-continuous on K.
Corollary 4.2. A closed subset F in C(K , R) is compact if and only if the following conditions are satisfied:
(1) For each t ∈ K , the set { f (t) | f ∈ F} is bounded in R;
(2) F is equi-continuous on K.
Remark 4.2. From the above discussions, we know that Theorem 4.1 is a generalization of the Arzela–Ascoli theorem
in classical analysis, and Theorem 4.2 is a improvement of Theorem 4.1.
5. Conclusion
It is well known that the Bolzano theorem and the Arzela–Ascoli theorem are two famous theorems having many
important applications in classical analysis. The main purpose of this paper is to establish the generalized versions of
these two theorems in fuzzy setting. Theorems 2.2 (Theorem 2.2 ) and 4.1 (Theorem 4.2) are our main results. Moreover,
the equi-continuity of d∞ -convergent sequence of fuzzy numbers is proved. We also present a characterization of
compact subsets in the fuzzy number space (E1 , d∞ ), investigate some properties of fuzzy-valued continuous functions
defined on a compact subset K of metric space and prove that the space C(K , E1 ) of fuzzy-valued continuous functions
on K is complete in the supremum metric D. These studies are expected to be useful for development of fuzzy analysis.
J.-X. Fang, Q.-Y. Xue / Fuzzy Sets and Systems 160 (2009) 1620 – 1631
1631
Acknowledgments
The authors thank the referees for their valuable comments and helpful suggestions.
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