Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 627314, 6 pages
http://dx.doi.org/10.1155/2013/627314
Research Article
A Characterization for Compact Sets in the Space of
Fuzzy Star-Shaped Numbers with 𝐿 𝑝 Metric
Zhitao Zhao1 and Congxin Wu2
1
2
School of Mathematical Sciences, Heilongjiang University, Harbin 150080, China
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Zhitao Zhao; zhitaozhao0808@126.com
Received 15 January 2013; Accepted 12 April 2013
Academic Editor: Marco Donatelli
Copyright © 2013 Z. Zhao and C. Wu. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By means of some auxiliary lemmas, we obtain a characterization of compact subsets in the space of all fuzzy star-shaped numbers
with 𝐿 𝑝 metric for 1 ≤ 𝑝 < ∞. The result further completes and develops the previous characterization of compact subsets given
by Wu and Zhao in 2008.
1. Introduction
Since the concept of fuzzy numbers was firstly introduced
in 1970s, it has been studied extensively from many different
aspects of the theory and applications such as fuzzy algebra,
fuzzy analysis, fuzzy topology, fuzzy logic, and fuzzy decision
making. Many applications restrict their description to fuzzy
numbers, often implicitly, because of powerful fuzzy convexity of fuzzy numbers. This fuzzy convexity is mainly reflected
on the convexity of the level sets of fuzzy numbers. However,
apart from possible applications, it is of independent interest
to see how far the supposition of convexity can be weakened
without losing too much structure. Star-shapedness is a
fairly natural extension to convexity. Surprisingly, many
topological properties of spaces of compact star-shaped sets
are similar to those of their compact convex counterparts. It is
well known that star-shapedness also plays an important role
in the theory and applications, such as nonsmooth analysis,
approximation problems and optimization problems, (see
[1–11]). Based on the importance of star-shapedness, as a
corresponding extension to fuzzy numbers, fuzzy star-shaped
numbers have been payed more and more attention, such as
Chanussot et al. [12], Diamond [13], Diamond and Kloeden
[14], Qiu et al. [15], and Wu and Zhao [16].
In this regard, Diamond has done a lot of work. In
1990, he firstly introduced the concept of fuzzy star-shaped
numbers and defined metrics on the space of fuzzy starshaped numbers in [13]. Especially, he studied the properties
of 𝐿 𝑝 metric for 1 ≤ 𝑝 < ∞. The induced metric spaces
(𝑆0𝑛 , 𝑑𝑝 ) were shown to be separable, but not complete for
1 ≤ 𝑝 < ∞. Finally, compact sets in (𝑆0𝑛 , 𝑑𝑝 ) were also characterized. Later on, Wu and Zhao [16] pointed out that the
characterization of Diamond was incorrect by a counterexample and gave a correct characterization for compact sets in
spaces (𝑆0𝑛 , 𝑑𝑝 ) for 1 ≤ 𝑝 < ∞.
The aim of this paper is to further improve the characterization of compact sets in [16] and obtain a characterization
of compact sets in the space of all fuzzy star-shaped numbers
with 𝐿 𝑝 metric for 1 ≤ 𝑝 < ∞. The result of this paper will
provide some help for future research on fuzzy star-shaped
numbers.
2. Preliminaries
In R𝑛 , denote the Euclidean norm by ‖ ⋅ ‖, and denote
the class of all nonempty compact sets in R𝑛 by K𝑛 . If
𝐴, 𝐵 are nonempty compact sets in R𝑛 , then the Hausdorff distance between 𝐴 and 𝐵 is given by 𝑑𝐻(𝐴, 𝐵) =
max{sup𝑎∈𝐴inf 𝑏∈𝐵 ‖𝑎 − 𝑏‖, sup𝑏∈𝐵 inf 𝑎∈𝐴‖𝑎 − 𝑏‖}.
2
Abstract and Applied Analysis
Definition 1 (see [14]). An element 𝐾 ∈ K𝑛 is star-shaped
relative to a point 𝑥 ∈ 𝐾 if for each 𝑦 ∈ 𝐾, the line segment
𝑥𝑦 joining 𝑥 to 𝑦 is contained in 𝐾. The ker 𝐾 of 𝐾 is the set
of all points 𝑥 ∈ 𝐾 such that the line segment 𝑥𝑦 ⊂ 𝐾 for each
𝑦 ∈ 𝐾 and co 𝐾 is the convex hull of 𝐾.
For a fuzzy set 𝑢 : R𝑛 → [0, 1], we suppose that
(1) 𝑢 is normal; that is, there exists an 𝑥0 ∈ R𝑛 such
that 𝑢(𝑥0 ) = 1,
(2) 𝑢 is upper semicontinuous,
𝑛
(3) supp 𝑢 = cl{𝑥 ∈ R : 𝑢(𝑥) > 0} is compact,
(4) 𝑢 is fuzzy star-shaped; that is, there exists 𝑥 ∈ R𝑛
such that 𝑢 is fuzzy star-shaped with respect to 𝑥,
namely, for any 𝑦 ∈ R𝑛 and 0 ≤ 𝜆 ≤ 1,
𝑢 (𝜆𝑦 + (1 − 𝜆) 𝑥) ≥ 𝑢 (𝑦) ,
Definition 2 (see [16]). A fuzzy star-shaped number is a fuzzy
set 𝑢 : R𝑛 → [0, 1] satisfying (1), (2), (3), and (4). Let 𝑆𝑛
be the family of all fuzzy star-shaped numbers, and let 𝑆0𝑛 be
the family of all fuzzy sets which satisfy (1), (2), (3), and (4∘ ).
Clearly, 𝑆0𝑛 ⊂ 𝑆𝑛 .
For a fuzzy star-shaped number 𝑢 ∈ 𝑆𝑛 , we define its 𝛼level set as follows:
(2)
if 𝑥 = 𝑚,
if 𝑥 ≠ 𝑚.
Proposition 6 (see [14]). For a fuzzy star-shaped number 𝑢,
ker(𝑢) is a convex set in R𝑛 , and 𝑓 ker(𝑢) is a fuzzy convex set
which is normal; that is, [𝑓 ker(𝑢)]1 ≠ 0.
Definition 7 (see [14]). For each 1 ≤ 𝑝 < ∞, one defines
1
𝑝
1/𝑝
𝑑𝑝 (𝑢, V) = (∫ 𝑑𝐻([𝑢]𝛼 , [V]𝛼 ) d𝛼)
0
,
(5)
for all 𝑢, V ∈ 𝑆𝑛 , then 𝑑𝑝 is called the 𝐿 𝑝 metric on 𝑆𝑛 .
For the characterization of compact sets in (𝑆𝑛 , 𝑑𝑝 ), the
following definitions will be used in the sequel.
Definition 8 (see [14]). One says that 𝑈 ⊂ 𝑆𝑛 is uniformly
support bounded if the support sets [𝑢]0 are bounded in R𝑛 ,
uniformly for 𝑢 ∈ 𝑈; that is, there is a constant K > 0 such
that 𝑑𝐻([𝑢]0 , {𝜃}) ≤ K holds for all 𝑢 ∈ 𝑈.
Definition 9 (see [14]). Let 𝑢 ∈ 𝑆𝑛 . If for any 𝜀 > 0, there exists
𝛿 = 𝛿(𝑢, 𝜀) > 0 such that for all 0 ≤ ℎ < 𝛿
Letting 𝑚 ∈ R𝑛 , we define
1,
̂ (𝑥) = {
𝑚
0,
Definition 5 (see [14]). Let ker(𝑢) be the totality of 𝑦 ∈ R𝑛
such that 𝑢 is fuzzy star-shaped with respect to 𝑦. Define
𝑓 ker(𝑢) by [𝑓 ker(𝑢)]𝛼 = ker[𝑢]𝛼 for a fuzzy star-shaped
number 𝑢.
(1)
(4∘ ) 𝑢 is fuzzy star-shaped with respect to the origin.
{𝑥 ∈ R𝑛 : 𝑢 (𝑥) ≥ 𝛼} , if 0 < 𝛼 ≤ 1,
[𝑢]𝛼 = {
supp 𝑢,
if 𝛼 = 0.
Proposition 4 (see [14]). For any 𝑢 ∈ 𝑆𝑛 , 𝑢 is fuzzy starshaped with respect to 𝑦 if and only if [𝑢]𝛼 are star-shaped with
respect to 𝑦 for all 𝛼 ∈ [0, 1].
1
(3)
𝑝
∫ 𝑑𝐻([𝑢]𝛼 , [𝑢]𝛼−ℎ ) d𝛼 < 𝜀𝑝 ,
ℎ
1 ≤ 𝑝 < ∞,
(6)
̂ ∈ 𝑆𝑛 for each 𝑚 ∈ R𝑛 .
It is obvious that 𝑚
We will define addition and scalar multiplication of fuzzy
shaped-numbers levelsetwise; that is, for 𝑢, V ∈ 𝑆𝑛 and 𝑐 ≥ 0
then one says that 𝑢 is 𝑝-mean left-continuous. If for
nonempty 𝑈 ⊂ 𝑆𝑛 , the above inequality holds uniformly for
𝑢 ∈ 𝑈, one says that 𝑈 is 𝑝-mean equi-left-continuous.
[𝑢 + V]𝛼 = [𝑢]𝛼 + [V]𝛼 ,
Definition 10 (see [17]). One says that 𝑈 ⊂ 𝑆𝑛 is uniformly
𝑝-mean bounded if there is a constant K > 0 such that
̂ ≤ K for all 𝑢 ∈ 𝑈, where 𝜃̂ denotes 𝑢 ∈ 𝑆𝑛 with
𝑑𝑝 (𝑢, 𝜃)
𝑢(𝜃) = 1, and 𝑢(𝑥) = 0 for any 𝑥 ≠ 𝜃.
[𝑐𝑢]𝛼 = 𝑐[𝑢]𝛼 ,
(4)
for each 𝛼 ∈ [0, 1]. It directly follows that 𝑢 + V, 𝑐𝑢 ∈ 𝑆𝑛 .
Now, let us recall some properties of fuzzy star-shaped
numbers which will be used in this paper.
𝑛
Proposition 3. If 𝑢 ∈ 𝑆 , then
(1) [𝑢]𝛽 ⊆ [𝑢]𝛼 ⊆ [𝑢]0 , 0 ≤ 𝛼 ≤ 𝛽 ≤ 1,
(2) [𝑢]𝛼 is a compact set of R𝑛 for each 0 ≤ 𝛼 ≤ 1,
(3) for any 𝛼 ∈ (0, 1], if {𝛼𝑘 } is an increasing sequence
of real numbers in (0, 1] converging to 𝛼, then [𝑢]𝛼 =
𝛼𝑘
⋂∞
𝑘=1 [𝑢] .
The following property derives directly from Definitions
1 and 2.
Remark 11. Note that uniform 𝑝-mean boundedness is
weaker than uniform support boundedness, namely, if 𝑈 ⊂
(𝑆𝑛 , 𝑑𝑝 ) is uniformly support bounded, then 𝑈 is uniformly
𝑝-mean bounded; however, the converse implication does
not hold.
3. Main Results
Before proving our main result, we demonstrate some auxiliary lemmas.
Abstract and Applied Analysis
3
Lemma 12. Let 𝑢 ∈ 𝑆𝑛 , then the following properties hold:
(1) ker(𝑢) ⊂ [𝑢]1 ,
̂ exists, and for any 𝛼 ∈ [0, 1],
(2) for any 𝑚 ∈ ker(𝑢), 𝑢 − 𝑚
one has
̂ 𝛼 = [𝑢]𝛼 − {𝑚} ,
[𝑢 − 𝑚]
(7)
̂ 𝛼 = [𝑢]𝛼 + {𝑚} ,
[𝑢 + 𝑚]
̂ ∈ 𝑆0𝑛 ,
(3) for any 𝑚 ∈ ker(𝑢), 𝑢 − 𝑚
̂𝑘 }
(4) if {𝑚𝑘 } ⊂ R𝑛 converges to 𝑚0 ∈ R𝑛 , then {𝑚
̂0 in (𝑆𝑛 , 𝑑𝑝 ).
converges to 𝑚
Proof. (1) Let 𝑚 ∈ ker(𝑢). By Definition 5, we know that for
all 𝑥 ∈ R𝑛
𝑢 (𝜆𝑥 + (1 − 𝜆) 𝑚) ≥ 𝑢 (𝑥) ,
0 ≤ 𝜆 ≤ 1.
(8)
Especially, the above inequality holds for each 𝑥 ∈ [𝑢]1 and
𝜆 = 0; that is, 𝑢(𝑚) ≥ 𝑢(𝑥) = 1. Thus, we get 𝑚 ∈ [𝑢]1 .
Since 𝑚 is an arbitrary element of ker(𝑢), then we have that
ker(𝑢) ⊂ [𝑢]1 .
(2) According to the Zadeh extension principle, we have
̂ (𝑦)} ,
̂ (𝑧) = sup min {𝑢 (𝑥) , 𝑚
(𝑢 − 𝑚)
𝑧=𝑥−𝑦
̂𝛼 =
Applying the similar techniques, we obtain [𝑢 + 𝑚]
[𝑢] + {𝑚} for each 𝛼 ∈ [0, 1].
̂ ∈ 𝑆0𝑛 , it is enough to
(3) By Proposition 4, to prove 𝑢 − 𝑚
𝛼
̂ is star-shaped with respect to the origin
verify that [𝑢 − 𝑚]
for each 𝛼 ∈ [0, 1], where 𝑚 is an arbitrary element of ker(𝑢).
̂ 𝛼 , 𝛼 ∈ [0, 1]; by statement (2), we have
For any 𝑥 ∈ [𝑢 − 𝑚]
𝛼
that 𝑥 ∈ [𝑢] − {𝑚}. So, we obtain that 𝑥 + 𝑚 ∈ [𝑢]𝛼 . Since
𝑚 is an arbitrary element of ker(𝑢), then by Definition 5 and
Proposition 4, the line segment 𝑚(𝑥 + 𝑚) joining 𝑚 to 𝑥 + 𝑚
is contained in [𝑢]𝛼 ; that is,
𝛼
(9)
̂ exists for each 𝑚 ∈ ker(𝑢). In
and so, it is clear that 𝑢 − 𝑚
̂ 𝛼 = [𝑢]𝛼 − {𝑚} for each
the following, we infer that [𝑢 − 𝑚]
𝛼 ∈ [0, 1].
Let 𝛼 ∈ [0, 1]. For any 𝑧 ∈ [𝑢]𝛼 −{𝑚}, there exists 𝑥0 ∈ [𝑢]𝛼
such that 𝑧 = 𝑥0 − 𝑚. Then
𝜆 (𝑥 + 𝑚) + (1 − 𝜆) 𝑚 = 𝜆𝑥 + 𝑚 ∈ [𝑢]𝛼 ,
0 ≤ 𝜆 ≤ 1. (14)
Thus, 𝜆𝑥 ∈ [𝑢]𝛼 − {𝑚} for all 0 ≤ 𝜆 ≤ 1. That is, the line
segment 𝜃𝑥 ⊂ [𝑢]𝛼 − {𝑚}.
̂ 𝛼 is star-shaped with respect to the
Consequently, [𝑢 − 𝑚]
origin for each 𝛼 ∈ [0, 1].
(4) Since {𝑚𝑘 } ⊂ R𝑛 converges to 𝑚0 ∈ R𝑛 , then for every
𝜀 > 0, there exists an integer 𝑘0 such that when 𝑘 ≥ 𝑘0
󵄩󵄩󵄩𝑚𝑘 − 𝑚0 󵄩󵄩󵄩 < 𝜀.
(15)
󵄩
󵄩
Thus, for the above 𝜀 > 0, we have
1
𝛼
1/𝑝
𝛼 𝑝
̂𝑘 , 𝑚
̂0 ) = (∫ 𝑑𝐻([𝑚
̂ 𝑘 ] , [𝑚
̂0 ] ) d𝛼)
𝑑𝑝 (𝑚
0
1
𝑝
1/𝑝
= (∫ 𝑑𝐻({𝑚𝑘 } , {𝑚0 }) d𝛼)
0
(16)
1/𝑝
1
󵄩
󵄩𝑝
= (∫ 󵄩󵄩󵄩𝑚𝑘 − 𝑚0 󵄩󵄩󵄩 d𝛼)
0
< 𝜀,
̂𝑘 } converges to 𝑚
̂0 in
whenever 𝑘 ≥ 𝑘0 . Therefore, {𝑚
(𝑆𝑛 , 𝑑𝑝 ).
̂ (𝑦)}
̂ (𝑧) = sup min {𝑢 (𝑥) , 𝑚
(𝑢 − 𝑚)
𝑧=𝑥−𝑦
(10)
Lemma 13. For any 𝑢 ∈ (𝑆𝑛 , 𝑑𝑝 ), 1 ≤ 𝑝 < ∞, 𝑢 is 𝑝-mean
left-continuous.
̂ 𝛼,
̂ 𝛼 . On the other hand, for any 𝑧 ∈ [𝑢 − 𝑚]
and so, 𝑧 ∈ [𝑢 − 𝑚]
by the Zadeh extension principle, we get
Proof. The technique is similar to the proof of Lemma 3.1
in [16].
̂ (𝑚)}
≥ min {𝑢 (𝑥0 ) , 𝑚
= 𝑢 (𝑥0 ) ≥ 𝛼,
̂ (𝑧) = sup min {𝑢 (𝑥) , 𝑚
̂ (𝑦)} = sup 𝑢 (𝑥) ≥ 𝛼.
(𝑢 − 𝑚)
𝑧=𝑥−𝑦
𝑧=𝑥−𝑚
(11)
Then, by the definition of the supremum, there exists 𝑥𝑘 ∈
[𝑢]0 for each 𝑘 ∈ N such that
1
𝑢 (𝑥𝑘 ) ≥ 𝛼 − ,
𝑘
𝑧 = 𝑥𝑘 − 𝑚.
(12)
From the compactness of [𝑢]0 , {𝑥𝑘 } has a subsequence
{𝑥𝑘𝑗 } converging to 𝑥0 ∈ [𝑢]0 . Since 𝑢 is upper semicontinuous, then we have
𝑢 (𝑥0 ) ≥ lim 𝑢 (𝑥𝑘𝑗 ) ≥ 𝛼.
𝑗→∞
That is, 𝑥0 ∈ [𝑢]𝛼 ; thus, 𝑧 ∈ [𝑢]𝛼 − {𝑚}.
(13)
For every 𝑢 ∈ 𝑆𝑛 and 𝑟 ∈ [0, 1], denote
𝑢(𝑟) (𝑥) = {
𝑢 (𝑥) , if 𝑢 (𝑥) ≥ 𝑟,
0,
if 𝑢 (𝑥) < 𝑟.
(17)
Then, we have the following.
Lemma 14 (see [16]). A closed set 𝑈 ⊂ (𝑆0𝑛 , 𝑑𝑝 ), 1 ≤ 𝑝 < ∞,
is compact if and only if
(1) 𝑈 is uniformly 𝑝-mean bounded,
(2) 𝑈 is 𝑝-mean equi-left-continuous,
(3) let {𝑟𝑖 } be a decreasing sequence in (0, 1] converging to
(𝑟 )
zero. For {𝑢𝑘 } ⊂ 𝑈, if {𝑢𝑘 𝑖 | 𝑘 = 1, 2, 3, . . .} converges
𝑛
to 𝑢(𝑟𝑖 ) ∈ 𝑆0 in 𝑑𝑝 , then there exists a 𝑢0 ∈ 𝑆0𝑛 such that
(𝑟 ) 𝛼
𝛼
[𝑢0 𝑖 ] = [𝑢 (𝑟𝑖 )] ,
𝑟𝑖 < 𝛼 ≤ 1.
(18)
4
Abstract and Applied Analysis
Lemma 15. For any 𝑢 ∈ 𝑆𝑛 , 𝑚 ∈ R𝑛 , and 0 < 𝑟 < 1, one has
̂
̂(𝑟) = 𝑚,
𝑚
̂ (𝑟) = 𝑢(𝑟) + 𝑚.
̂
(𝑢 + 𝑚)
(19)
Thus, for 𝑢 ∈ 𝑈, we can obtain by triangle inequality
1
ℎ
Proof. Let 𝛼 be an arbitrary element in [0, 1].
̂(𝑟) ]𝛼 = [𝑚]
̂ 𝛼 = {𝑚}. If 0 < 𝛼 < 𝑟,
If 𝑟 ≤ 𝛼 ≤ 1, then [𝑚
𝑟
(𝑟) 𝛼
̂ = {𝑚} = [𝑚]
̂ 𝛼 . Hence, for each 𝛼 ∈
̂ ] = [𝑚]
then [𝑚
𝛼
(𝑟) 𝛼
̂ , and so, it follows that [𝑚
̂(𝑟) ]0 = [𝑚]
̂ 0.
̂ ] = [𝑚]
(0, 1], [𝑚
(𝑟)
̂
̂ = 𝑚.
This implies that 𝑚
Similarly, if 𝑟 ≤ 𝛼 ≤ 1, then by Lemma 12(2), we have
1
𝛼
ℎ
1
𝛼
+ (∫ 𝑑𝐻([𝑢𝑖 ] , [𝑢𝑖 ]
̂ = [𝑢] + {𝑚}
̂ ] = [𝑢 + 𝑚]
[(𝑢 + 𝑚)
𝛼
𝛼
ℎ
(20)
) d𝛼)
1/𝑝
𝑝
, [𝑢]𝛼−ℎ ) d𝛼)
1
𝛼
≤ 𝑑𝑝 (𝑢𝑖 , 𝑢) + (∫ 𝑑𝐻([𝑢𝑖 ] , [𝑢𝑖 ]
(24)
1/𝑝
𝛼−ℎ 𝑝
) d𝛼)
ℎ
If 0 < 𝛼 < 𝑟, then by Lemma 12(2), we have
+ 𝑑𝑝 (𝑢𝑖 , 𝑢)
(𝑟) 𝛼
̂ 𝑟 = [𝑢]𝑟 + {𝑚}
̂ ] = [𝑢 + 𝑚]
[(𝑢 + 𝑚)
𝛼
𝛼−ℎ
+ (∫ 𝑑𝐻([𝑢𝑖 ]
1/𝑝
𝛼−ℎ 𝑝
ℎ
𝛼
̂ .
= [𝑢(𝑟) ] + {𝑚} = [𝑢(𝑟) + 𝑚]
1/𝑝
𝛼 𝑝
≤ (∫ 𝑑𝐻([𝑢]𝛼 , [𝑢𝑖 ] ) d𝛼)
1
(𝑟) 𝛼
1/𝑝
𝑝
(∫ 𝑑𝐻([𝑢]𝛼 , [𝑢]𝛼−ℎ ) d𝛼)
𝛼
̂ .
= [𝑢(𝑟) ] + {𝑚} = [𝑢(𝑟) + 𝑚]
(21)
<
𝜀 𝜀 𝜀
+ +
3 3 3
= 𝜀.
̂ 𝛼 , and
̂ (𝑟) ]𝛼 = [𝑢(𝑟) + 𝑚]
Hence, for each 𝛼 ∈ (0, 1], [(𝑢 + 𝑚)
(𝑟) 0
0
(𝑟)
̂ . This implies that
̂ ] = [𝑢 + 𝑚]
so, it follows that [(𝑢 + 𝑚)
̂ (𝑟) = 𝑢(𝑟) + 𝑚.
̂
(𝑢 + 𝑚)
Now, we give the characterization of compact sets in
(𝑆𝑛 , 𝑑𝑝 ) for 1 ≤ 𝑝 < ∞.
Theorem 16. A closed set 𝑈 ⊂ (𝑆𝑛 , 𝑑𝑝 ), 1 ≤ 𝑝 < ∞ is compact
if and only if
(i) 𝑈 is uniformly 𝑝-mean bounded,
(ii) 𝑈 is 𝑝-mean equi-left-continuous,
Therefore, 𝑈 is 𝑝-mean equi-left-continuous.
(3) According to the definition of 𝑢(𝑟𝑖 ) , [𝑢(𝑟𝑖 ) ]𝛼 = [𝑢]𝛼 for
each 𝑟𝑖 ≤ 𝛼 ≤ 1, and [𝑢(𝑟𝑖 ) ]𝛼 = [𝑢]𝑟𝑖 for each 0 ≤ 𝛼 < 𝑟𝑖 . Since
𝑢 ∈ 𝑆𝑛 , then from Proposition 4, there exists 𝑥 ∈ R𝑛 such that
[𝑢]𝛼 is star-shaped with respect to 𝑥 for each 0 ≤ 𝛼 ≤ 1. So,
[𝑢(𝑟𝑖 ) ]𝛼 is star-shaped with respect to 𝑥 for each 0 ≤ 𝛼 ≤ 1;
thus, it is obvious by Proposition 4 that 𝑢(𝑟𝑖 ) ∈ 𝑆𝑛 for each
𝑖 ∈ N.
(𝑟 )
For {𝑢𝑘 } ⊂ 𝑈, we assume that {𝑢𝑘 𝑖 }∞
𝑘=1 converges to
𝑢(𝑟𝑖 ) ∈ 𝑆𝑛 in 𝑑𝑝 . Since 𝑈 is compact in (𝑆𝑛 , 𝑑𝑝 ), then {𝑢𝑘 }
has a subsequence {𝑢𝑘𝑗 } converging to 𝑢0 ∈ 𝑆𝑛 in 𝑑𝑝 . By
Minkowski’s inequality, it follows that
1
(iii) let {𝑟𝑖 } be a decreasing sequence in (0, 1] converging to
(𝑟 )
zero. For {𝑢𝑘 } ⊂ 𝑈, if {𝑢𝑘 𝑖 | 𝑘 = 1, 2, 3, . . .} converges
𝑛
to 𝑢(𝑟𝑖 ) ∈ 𝑆 in 𝑑𝑝 , then there exists a 𝑢0 ∈ 𝑆𝑛 such that
𝛼 𝑝
𝛼
1/𝑝
(∫ 𝑑𝐻([𝑢0 ] , [𝑢 (𝑟𝑖 )] ) d𝛼)
𝑟𝑖
1
1/𝑝
𝛼 𝑝
𝛼
≤ (∫ 𝑑𝐻([𝑢0 ] , [𝑢𝑘𝑗 ] ) d𝛼)
𝑟𝑖
(𝑟 ) 𝛼
[𝑢0 𝑖 ]
𝛼
= [𝑢 (𝑟𝑖 )] ,
𝑟𝑖 < 𝛼 ≤ 1.
(22)
1
𝛼
1/𝑝
𝑝
𝛼
+ (∫ 𝑑𝐻([𝑢𝑘𝑗 ] , [𝑢 (𝑟𝑖 )] ) d𝛼)
(25)
𝑟𝑖
Proof. Necessity: (1) Since 𝑈 is compact in (𝑆𝑛 , 𝑑𝑝 ), it follows
that 𝑈 is a bounded set in (𝑆𝑛 , 𝑑𝑝 ). This implies that 𝑈 is
uniformly 𝑝-mean bounded.
(2) Let 𝜀 > 0, and let 𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ∈ 𝑆𝑛 be a (𝜀/3)net of 𝑈, that is, for any 𝑢 ∈ 𝑈, there exists an element
𝑢𝑖 (1 ≤ 𝑖 ≤ 𝑚) satisfying 𝑑𝑝 (𝑢, 𝑢𝑖 ) < 𝜀/3. By Lemma 12,
𝑢1 , 𝑢2 , . . . , 𝑢𝑚 are 𝑝-mean left-continuous, and so, there exists
𝛿(𝜀) = min1≤𝑖≤𝑚 𝛿(𝑢𝑖 , 𝜀) > 0 such that
1
𝛼
∫ 𝑑𝐻([𝑢𝑖 ] , [𝑢𝑖 ]
ℎ
𝜀 𝑝
) d𝛼 < ( ) ,
3
𝛼−ℎ 𝑝
for 𝑖 = 1, 2, . . . , 𝑚 and 0 ≤ ℎ < 𝛿(𝜀).
1
1/𝑝
𝛼 𝑝
𝛼
= (∫ 𝑑𝐻([𝑢0 ] , [𝑢𝑘𝑗 ] ) d𝛼)
𝑟𝑖
1
(𝑟 ) 𝛼
𝑗
𝑟𝑖
.
But since
1
𝛼
𝛼 𝑝
1/𝑝
lim (∫ 𝑑𝐻([𝑢0 ] , [𝑢𝑘𝑗 ] ) d𝛼)
(23)
1/𝑝
𝑝
𝛼
+ (∫ 𝑑𝐻([𝑢𝑘 𝑖 ] , [𝑢 (𝑟𝑖 )] ) d𝛼)
𝑗→∞
𝑟𝑖
lim (∫
𝑗→∞
1
𝑟𝑖
= 0,
1/𝑝
𝑝
𝛼
(𝑟 ) 𝛼
𝑑𝐻([𝑢𝑘 𝑖 ] , [𝑢 (𝑟𝑖 )] ) d𝛼)
𝑗
(26)
= 0,
Abstract and Applied Analysis
5
(27)
̂ : 𝑢 ∈ 𝑈, 𝑚 ∈ ker(𝑢)} is
Secondly, we conclude that {𝑢 − 𝑚
uniformly 𝑝-mean bounded. According to condition (i) and
the conclusion in last paragraph, we obtain that {ker(𝑢) : 𝑢 ∈
𝑈} is uniformly bounded, and then there exists a K > 0 such
that for all 𝑢 ∈ 𝑈
(28)
𝑑𝐻 (ker (𝑢) , {𝜃}) ≤ K.
(29)
By the condition (i), there exists a K󸀠 > 0 such that for all
𝑢∈𝑈
then we get
1
1/𝑝
𝛼 𝑝
𝛼
(∫ 𝑑𝐻([𝑢0 ] , [𝑢 (𝑟𝑖 )] ) d𝛼)
= 0,
𝑟𝑖
and this implies that
𝛼
𝛼
𝑑𝐻 ([𝑢0 ] , [𝑢 (𝑟𝑖 )] ) = 0,
a.e. in (𝑟𝑖 , 1]. Thus, we obtain that
𝛼
𝛼
[𝑢0 ] = [𝑢 (𝑟𝑖 )] ,
a.e. in (𝑟𝑖 , 1]; that is, [𝑢0 ]𝛼 = [𝑢(𝑟𝑖 )]𝛼 for each (𝑟𝑖 , 1] \ Ω, where
Ω ⊂ (𝑟𝑖 , 1] has Lebesgue measure zero and clearly (𝑟𝑖 , 1] \ Ω
is dense in (𝑟𝑖 , 1]. Let 𝛼 ∈ Ω, then there exists an increasing
sequence 𝛼𝑘 ↗ 𝛼 with 𝛼𝑘 ∈ (𝑟𝑖 , 1] \ Ω and [𝑢0 ]𝛼𝑘 = [𝑢(𝑟𝑖 )]𝛼𝑘 .
So
𝛼
1
0
≤ K󸀠 .
(34)
Thus, by Lemma 12(2) for all 𝑢 ∈ 𝑈, we get
1
𝛼
1/𝑝
𝑝
̂ 𝛼 , {𝜃}) d𝛼)
(∫ 𝑑𝐻([𝑢 − 𝑚]
𝑑𝐻 ([𝑢0 ] , [𝑢 (𝑟𝑖 )] )
0
𝛼
𝛼
𝛼
𝛼
≤ 𝑑𝐻 ([𝑢0 ] , [𝑢0 ] 𝑘 ) + 𝑑𝐻 ([𝑢0 ] 𝑘 , [𝑢 (𝑟𝑖 )] 𝑘 )
𝛼
1
𝛼
0
𝛼
𝛼𝑘
𝛼𝑘
1
0
1
(31)
1
0
1
0
󸀠
≤ K + K.
This implies that condition (1) of Lemma 14 is satisfied.
̂ : 𝑢 ∈ 𝑈, 𝑚 ∈ ker(𝑢)} is 𝑝Thirdly, we verify that {𝑢 − 𝑚
mean equi-left-continuous. In fact, since 𝑈 is 𝑝-mean equileft-continuous, then for every 𝜀 > 0, there exists a 𝛿 > 0 such
that for all 0 ≤ ℎ ≤ 𝛿 and 𝑢 ∈ 𝑈
1
𝑝
1
𝑝
1/𝑝
≥ (∫ 𝑑𝐻(ker (𝑢𝑘 ) , {𝜃}) d𝛼)
0
1/𝑝
0
This contradicts condition (i).
= 𝑘,
(36)
and so, we have
1/𝑝
0
1
𝑝
ℎ
≥ (∫ 𝑑𝐻([𝑢𝑘 ] , {𝜃}) d𝛼)
> (∫ 𝑘𝑝 d𝛼)
1/𝑝
1
+ (∫ 𝑑𝐻({𝜃} , ker (𝑢))𝑝 d𝛼)
∫ 𝑑𝐻([𝑢]𝛼 , [𝑢]𝛼−ℎ ) d𝛼 < 𝜀𝑝 ,
0
1
1/𝑝
0
1/𝑝
1
𝑝
≤ (∫ 𝑑𝐻([𝑢]𝛼 , {𝜃}) d𝛼)
(∫ 𝑑𝐻([𝑢𝑘 ] , {𝜃}) d𝛼)
1
(35)
1/𝑝
+ (∫ 𝑑𝐻({𝜃} , {𝑚})𝑝 d𝛼)
By Proposition 3(1) and Lemma 12(1), it follows that
𝛼
1/𝑝
0
̂:𝑢∈
Sufficiency: From Lemma 12(3), we have that {𝑢 − 𝑚
𝑈, 𝑚 ∈ ker(𝑢)} ⊂ 𝑆0𝑛 . We divide the rest of the proof into two
steps.
𝑑𝐻 (ker (𝑢𝑘 ) , {𝜃}) > 𝑘.
𝑝
≤ (∫ 𝑑𝐻([𝑢]𝛼 , {𝜃}) d𝛼)
By Proposition 3, both terms on the right converge to zero
as 𝑘 → ∞, so 𝑑𝐻([𝑢0 ]𝛼 , [𝑢(𝑟𝑖 )]𝛼 ) = 0, and hence [𝑢0 ]𝛼 =
[𝑢(𝑟𝑖 )]𝛼 for each 𝛼 ∈ Ω.
(𝑟 )
Consequently, we have [𝑢0 𝑖 ]𝛼 = [𝑢0 ]𝛼 = [𝑢(𝑟𝑖 )]𝛼 for each
𝑟𝑖 < 𝛼 ≤ 1.
̂ : 𝑢 ∈ 𝑈, 𝑚 ∈ ker(𝑢)} satisfies
Step 1. We prove that {𝑢 − 𝑚
(1)–(3) in Lemma 14.
Firstly, we show that if 𝑈 is uniformly 𝑝-mean bounded,
then {ker(𝑢) : 𝑢 ∈ 𝑈} is uniformly bounded. Otherwise, we
infer that for any 𝑘 > 0, 𝑘 ∈ N; we can find that 𝑢𝑘 ∈ 𝑈 such
that
1/𝑝
𝑝
= (∫ 𝑑𝐻([𝑢]𝛼 , {𝑚}) d𝛼)
𝛼
= 𝑑𝐻 ([𝑢0 ] , [𝑢0 ] ) + 𝑑𝐻 ([𝑢 (𝑟𝑖 )] , [𝑢 (𝑟𝑖 )] ) .
(30)
𝑝
1/𝑝
𝑝
= (∫ 𝑑𝐻([𝑢]𝛼 − {𝑚} , {𝜃}) d𝛼)
+ 𝑑𝐻 ([𝑢 (𝑟𝑖 )] 𝑘 , [𝑢 (𝑟𝑖 )] )
1
1/𝑝
𝑝
(∫ 𝑑𝐻([𝑢]𝛼 , {𝜃}) d𝛼)
(33)
𝑘 ∈ N.
(32)
𝑝
̂ 𝛼 , [𝑢 − 𝑚]
̂ 𝛼−ℎ ) d𝛼
∫ 𝑑𝐻([𝑢 − 𝑚]
ℎ
1
𝑝
= ∫ 𝑑𝐻([𝑢]𝛼 − {𝑚} , [𝑢]𝛼−ℎ − {𝑚}) d𝛼
ℎ
1
𝑝
= ∫ 𝑑𝐻([𝑢]𝛼 , [𝑢]𝛼−ℎ ) d𝛼 < 𝜀𝑝 .
ℎ
Hence, condition (2) of Lemma 14 holds.
(37)
6
Abstract and Applied Analysis
Finally, we prove that condition (3) of Lemma 14 is also
satisfied. Let {𝑟𝑖 } be a decreasing sequence in (0, 1] converging
̂𝑘 )(𝑟𝑖 ) } (𝑢𝑘 ∈ 𝑈, 𝑚𝑘 ∈ ker(𝑢𝑘 ))
to zero. We suppose that {(𝑢𝑘 −𝑚
∗
𝑛
̂𝑘 . Then,
converges to 𝑢 (𝑟𝑖 ) ∈ 𝑆0 in 𝑑𝑝 . Denote V𝑘 = 𝑢𝑘 − 𝑚
(𝑟 )
{V𝑘 𝑖 } converges to 𝑢∗ (𝑟𝑖 ) ∈ 𝑆0𝑛 in 𝑑𝑝 . By Lemma 15, we have
(𝑟 )
̂𝑘 )
𝑢𝑘 𝑖 = (V𝑘 + 𝑚
(𝑟𝑖 )
(𝑟 )
̂𝑘 .
= V𝑘 𝑖 + 𝑚
(38)
By the preceding proof, {ker(𝑢) | 𝑢 ∈ 𝑈} is uniformly
bounded. Then, {𝑚𝑘 } has a subsequence {𝑚𝑘𝑗 } converging to
(𝑟 )
(𝑟 )
̂𝑘𝑗 }
𝑚0 ∈ R𝑛 , and so by Lemma 12(4), {𝑢𝑘 𝑖 } = {V𝑘 𝑖 + 𝑚
𝑗
𝑗
∗
𝑛
̂
converges to 𝑢 (𝑟𝑖 ) + 𝑚0 in (𝑆 , 𝑑𝑝 ). By condition (iii) and
Lemma 12(2), there exists a 𝑢0 ∈ 𝑆𝑛 such that
𝛼
𝛼
𝛼
(𝑟 )
̂0 ] = [𝑢∗ (𝑟𝑖 )] + {𝑚0 } ,
[𝑢0 𝑖 ] = [𝑢∗ (𝑟𝑖 ) + 𝑚
(39)
wherever 𝑟𝑖 < 𝛼 ≤ 1, and so by Lemma 12(2), we have
(𝑟 ) 𝛼
𝛼
𝛼
[𝑢∗ (𝑟𝑖 )] = [𝑢0 𝑖 ] − {𝑚0 } = [𝑢0 ] − {𝑚0 }
𝛼
(𝑟𝑖 ) 𝛼
̂0 ] = [(𝑢0 − 𝑚
̂0 )
= [𝑢0 − 𝑚
(40)
] ,
wherever 𝑟𝑖 < 𝛼 ≤ 1.
̂0 . Then, by Lemma 12(3), it follows
Define 𝑢0∗ = 𝑢0 − 𝑚
that 𝑢0∗ ∈ 𝑆0𝑛 . Thus, condition (3) of Lemma 14 holds.
Step 2. We infer that 𝑈 is a compact set in (𝑆𝑛 , 𝑑𝑝 ).
̂ : 𝑢 ∈ 𝑈, 𝑚 ∈ ker(𝑢)} is a subset
Let {𝑢𝑘 } ⊂ 𝑈. Since {𝑢 − 𝑚
of (𝑆0𝑛 , 𝑑𝑝 ) and satisfies condition (1)–(3) of Lemma 14, then
̂ : 𝑢 ∈ 𝑈, 𝑚 ∈ ker(𝑢)} is a compact set of (𝑆0𝑛 , 𝑑𝑝 ), and so
{𝑢− 𝑚
̂𝑘 } has a subsequence
for fixed 𝑚𝑘 ∈ ker(𝑢𝑘 ) (𝑘 ∈ N), {𝑢𝑘 − 𝑚
̂𝑘𝑗 } converging to 𝑢0 ∈ (𝑆0𝑛 , 𝑑𝑝 ). By the proceeding
{𝑢𝑘𝑗 − 𝑚
proof, {ker(𝑢) : 𝑢 ∈ 𝑈} is uniformly bounded, so {𝑚𝑘𝑗 } has a
subsequence {𝑚𝑘𝑗(𝑙) } converging to 𝑚0 ∈ R𝑛 . It is obvious that
̂𝑘𝑗(𝑙) } converges to 𝑢0 ∈ (𝑆0𝑛 , 𝑑𝑝 ).
{𝑢𝑘𝑗(𝑙) − 𝑚
̂𝑘𝑗(𝑙) . Then, {V𝑘𝑗(𝑙) } converges to
Denote V𝑘𝑗(𝑙) = 𝑢𝑘𝑗(𝑙) − 𝑚
𝑢0 ∈ (𝑆0𝑛 , 𝑑𝑝 ). Therefore, by Lemma 12(4), we get that {𝑢𝑘𝑗(𝑙) } =
̂𝑘𝑗(𝑙) } converges to 𝑢0 + 𝑚
̂0 ∈ (𝑆𝑛 , 𝑑𝑝 ), which completes
{V𝑘𝑗(𝑙) + 𝑚
the proof.
4. Conclusion
Since star-shapedness has played an important role in the
theory and applications of classical analysis, such as nonsmooth analysis, approximation problems, and optimization
problems, then as an extension to fuzzy numbers, fuzzy
star-shaped numbers should be also payed more and more
attention. In this paper, we further complete and develop the
previous result in [16] and give a characterization of compact
subsets in the space of all fuzzy star-shaped numbers with 𝐿 𝑝
metric for 1 ≤ 𝑝 < ∞. The result of this paper will provide
some help for future research on the theory of fuzzy starshaped numbers.
Acknowledgments
The authors would like to thank the anonymous reviewers
for their helpful comments which improved the original
paper. The project is supported by the Youth Scientific Funds
of Heilongjiang University (no. QL201007) and NSFC (nos.
11201128 and 11226255).
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Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
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Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014