A Characterization for Compact Sets in the Space of Fuzzy Star-Shaped Numbers with 𝐿_𝑝 Metric

Abstract

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 star-shaped numbers in [13]. Especially, he studied the properties of 𝐿_𝑝 metric for 1≤𝑝<∞. The induced metric spaces (𝑆^𝑛₀,𝑑_𝑝) were shown to be separable, but not complete for 1≤𝑝<∞. Finally, compact sets in (𝑆^𝑛₀,𝑑_𝑝) 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 (𝑆^𝑛₀,𝑑_𝑝) 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 ℝ^𝑛, denote the Euclidean norm by ‖⋅‖, and denote the class of all nonempty compact sets in ℝ^𝑛 by 𝕂^𝑛. If 𝐴,𝐵 are nonempty compact sets in ℝ^𝑛, then the Hausdorff distance between 𝐴 and 𝐵 is given by 𝑑_𝐻(𝐴,𝐵)=max{sup_𝑎∈𝐴inf_𝑏∈𝐵‖𝑎−𝑏‖,sup_𝑏∈𝐵inf_𝑎∈𝐴‖𝑎−𝑏‖}.

Definition 1 (see [14]). An element 𝐾∈𝕂^𝑛 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 ker𝐾 of 𝐾 is the set of all points 𝑥∈𝐾 such that the line segment 𝑥𝑦⊂𝐾 for each 𝑦∈𝐾 and co𝐾 is the convex hull of 𝐾.

For a fuzzy set 𝑢∶ℝ^𝑛→[0,1], we suppose that (1) 𝑢 is normal; that is, there exists an 𝑥₀∈ℝ^𝑛 such that 𝑢(𝑥₀)=1, (2) 𝑢 is upper semicontinuous, (3) supp𝑢=cl{𝑥∈ℝ^𝑛∶𝑢(𝑥)>0} is compact, (4) 𝑢 is fuzzy star-shaped; that is, there exists 𝑥∈ℝ^𝑛 such that 𝑢 is fuzzy star-shaped with respect to 𝑥, namely, for any 𝑦∈ℝ^𝑛 and 0≤𝜆≤1, 𝑢𝜆𝑦+(1−𝜆)𝑥≥𝑢𝑦, (1) (4°) 𝑢 is fuzzy star-shaped with respect to the origin.

Definition 2 (see [16]). A fuzzy star-shaped number is a fuzzy set 𝑢∶ℝ^𝑛→[0,1] satisfying (1), (2), (3), and (4). Let 𝑆^𝑛 be the family of all fuzzy star-shaped numbers, and let 𝑆^𝑛₀ be the family of all fuzzy sets which satisfy (1), (2), (3), and (4°). Clearly, 𝑆^𝑛₀⊂𝑆^𝑛.

For a fuzzy star-shaped number 𝑢∈𝑆^𝑛, we define its 𝛼-level set as follows: [𝑢]^𝛼=𝑥∈ℝ^𝑛∶𝑢(𝑥)≥𝛼,if0<𝛼≤1,supp𝑢,if𝛼=0. (2)

Letting 𝑚∈ℝ^𝑛, we define 𝑚(𝑥)=1,if𝑥=𝑚,0,if𝑥≠𝑚. (3) It is obvious that 𝑚∈𝑆^𝑛 for each 𝑚∈ℝ^𝑛.

We will define addition and scalar multiplication of fuzzy shaped-numbers levelsetwise; that is, for 𝑢,𝑣∈𝑆^𝑛 and 𝑐≥0[𝑢+𝑣]^𝛼=[𝑢]^𝛼+[𝑣]^𝛼,[𝑐𝑢]^𝛼=𝑐[𝑢]^𝛼, (4) for each 𝛼∈[0,1]. It directly follows that 𝑢+𝑣,𝑐𝑢∈𝑆^𝑛.

Now, let us recall some properties of fuzzy star-shaped numbers which will be used in this paper.

Proposition 3. If 𝑢∈𝑆^𝑛, then(1)[𝑢]^𝛽⊆[𝑢]^𝛼⊆[𝑢]⁰, 0≤𝛼≤𝛽≤1,(2)[𝑢]^𝛼 is a compact set of ℝ^𝑛 for each 0≤𝛼≤1,(3)for any 𝛼∈(0,1], if {𝛼_𝑘} is an increasing sequence of real numbers in (0,1] converging to 𝛼, then [𝑢]^𝛼=⋂^∞_𝑘=1[𝑢]^𝛼𝑘.

The following property derives directly from Definitions 1 and 2.

Proposition 4 (see [14]). For any 𝑢∈𝑆^𝑛, 𝑢 is fuzzy star-shaped with respect to 𝑦 if and only if [𝑢]^𝛼 are star-shaped with respect to 𝑦 for all 𝛼∈[0,1].

Definition 5 (see [14]). Let ker(𝑢) be the totality of 𝑦∈ℝ^𝑛 such that 𝑢 is fuzzy star-shaped with respect to 𝑦. Define 𝑓ker(𝑢) by [𝑓ker(𝑢)]^𝛼=ker[𝑢]^𝛼 for a fuzzy star-shaped number 𝑢.

Proposition 6 (see [14]). For a fuzzy star-shaped number 𝑢, ker(𝑢) is a convex set in ℝ^𝑛, and 𝑓ker(𝑢) is a fuzzy convex set which is normal; that is, [𝑓ker(𝑢)]¹≠∅.

Definition 7 (see [14]). For each 1≤𝑝<∞, one defines 𝑑_𝑝(𝑢,𝑣)=¹₀𝑑_𝐻[𝑢]^𝛼,[𝑣]^𝛼𝑝d𝛼^1/𝑝, (5) for all 𝑢,𝑣∈𝑆^𝑛, then 𝑑_𝑝 is called the 𝐿_𝑝 metric on 𝑆^𝑛.

For the characterization of compact sets in (𝑆^𝑛,𝑑_𝑝), the following definitions will be used in the sequel.

Definition 8 (see [14]). One says that 𝑈⊂𝑆^𝑛 is uniformly support bounded if the support sets [𝑢]⁰ are bounded in ℝ^𝑛, uniformly for 𝑢∈𝑈; that is, there is a constant 𝒦>0 such that 𝑑_𝐻([𝑢]⁰,{𝜃})≤𝒦 holds for all 𝑢∈𝑈.

Definition 9 (see [14]). Let 𝑢∈𝑆^𝑛. If for any 𝜀>0, there exists 𝛿=𝛿(𝑢,𝜀)>0 such that for all 0≤ℎ<𝛿¹_ℎ𝑑_𝐻[𝑢]^𝛼,[𝑢]^{𝛼−ℎ𝑝}d𝛼<𝜀^𝑝,1≤𝑝<∞, (6) then one says that 𝑢 is 𝑝-mean left-continuous. If for nonempty 𝑈⊂𝑆^𝑛, the above inequality holds uniformly for 𝑢∈𝑈, one says that 𝑈 is 𝑝-mean equi-left-continuous.

Definition 10 (see [17]). One says that 𝑈⊂𝑆^𝑛 is uniformly 𝑝-mean bounded if there is a constant 𝒦>0 such that 𝑑_𝑝(𝑢,𝜃)≤𝒦 for all 𝑢∈𝑈, where 𝜃 denotes 𝑢∈𝑆^𝑛 with 𝑢(𝜃)=1, and 𝑢(𝑥)=0 for any 𝑥≠𝜃.

Remark 11. Note that uniform 𝑝-mean boundedness is weaker than uniform support boundedness, namely, if 𝑈⊂(𝑆^𝑛,𝑑_𝑝) 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.

Lemma 12. Let 𝑢∈𝑆^𝑛, then the following properties hold:(1)ker(𝑢)⊂[𝑢]¹,(2)for any 𝑚∈ker(𝑢), 𝑢−𝑚 exists, and for any 𝛼∈[0,1], one has [𝑢−𝑚]^𝛼=[𝑢]^𝛼−{𝑚},[𝑢+𝑚]^𝛼=[𝑢]^𝛼+{𝑚}, (7)(3)for any 𝑚∈ker(𝑢), 𝑢−𝑚∈𝑆^𝑛₀,(4)if {𝑚_𝑘}⊂ℝ^𝑛 converges to 𝑚₀∈ℝ^𝑛, then {𝑚_𝑘} converges to 𝑚₀ in (𝑆^𝑛,𝑑_𝑝).

Proof. (1) Let 𝑚∈ker(𝑢). By Definition 5, we know that for all 𝑥∈ℝ^𝑛𝑢(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 (𝑢−𝑚)(𝑧)=sup_{𝑧=𝑥−𝑦}min𝑢(𝑥),𝑚𝑦, (9) and so, it is clear that 𝑢−𝑚 exists for each 𝑚∈ker(𝑢). In the following, we infer that [𝑢−𝑚]^𝛼=[𝑢]^𝛼−{𝑚} for each 𝛼∈[0,1].
Let 𝛼∈[0,1]. For any 𝑧∈[𝑢]^𝛼−{𝑚}, there exists 𝑥₀∈[𝑢]^𝛼 such that 𝑧=𝑥₀−𝑚. Then (𝑢−𝑚)(𝑧)=sup_{𝑧=𝑥−𝑦}𝑢(𝑥),𝑚𝑦𝑢𝑥min≥min₀,𝑚(𝑚𝑥)=𝑢₀≥𝛼, (10) and so, 𝑧∈[𝑢−𝑚]^𝛼. On the other hand, for any 𝑧∈[𝑢−𝑚]^𝛼, by the Zadeh extension principle, we get (𝑢−𝑚)(𝑧)=sup_{𝑧=𝑥−𝑦}min𝑢(𝑥),𝑚𝑦=sup_{𝑧=𝑥−𝑚}𝑢(𝑥)≥𝛼. (11) Then, by the definition of the supremum, there exists 𝑥_𝑘∈[𝑢]⁰ for each 𝑘∈ℕ such that 𝑢𝑥_𝑘1≥𝛼−𝑘,𝑧=𝑥_𝑘−𝑚. (12)
From the compactness of [𝑢]⁰, {𝑥_𝑘} has a subsequence {𝑥_𝑘𝑗} converging to 𝑥₀∈[𝑢]⁰. Since 𝑢 is upper semicontinuous, then we have 𝑢𝑥₀≥lim_𝑗→∞𝑢𝑥_𝑘𝑗≥𝛼. (13) That is, 𝑥₀∈[𝑢]^𝛼; thus, 𝑧∈[𝑢]^𝛼−{𝑚}.
Applying the similar techniques, we obtain [𝑢+𝑚]^𝛼=[𝑢]^𝛼+{𝑚} for each 𝛼∈[0,1].
(3) By Proposition 4, to prove 𝑢−𝑚∈𝑆^𝑛₀, it is enough to verify that [𝑢−𝑚]^𝛼 is star-shaped with respect to the origin for each 𝛼∈[0,1], where 𝑚 is an arbitrary element of ker(𝑢).
For any 𝑥∈[𝑢−𝑚]^𝛼, 𝛼∈[0,1]; by statement (2), we have that 𝑥∈[𝑢]^𝛼−{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, 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}.
Consequently, [𝑢−𝑚]^𝛼 is star-shaped with respect to the origin for each 𝛼∈[0,1].
(4) Since {𝑚_𝑘}⊂ℝ^𝑛 converges to 𝑚₀∈ℝ^𝑛, then for every 𝜀>0, there exists an integer 𝑘₀ such that when 𝑘≥𝑘₀𝑚_𝑘−𝑚₀<𝜀. (15)
Thus, for the above 𝜀>0, we have 𝑑_𝑝𝑚_𝑘,𝑚₀=¹₀𝑑_𝐻𝑚_𝑘^𝛼,𝑚₀^𝛼𝑝d𝛼^1/𝑝=¹₀𝑑_𝐻𝑚_𝑘,𝑚₀^𝑝d𝛼^1/𝑝=¹₀𝑚_𝑘−𝑚₀^𝑝d𝛼^1/𝑝<𝜀, (16) whenever 𝑘≥𝑘₀. Therefore, {𝑚_𝑘} converges to 𝑚₀ in (𝑆^𝑛,𝑑_𝑝).

Lemma 13. For any 𝑢∈(𝑆^𝑛,𝑑_𝑝), 1≤𝑝<∞, 𝑢 is 𝑝-mean left-continuous.

Proof. The technique is similar to the proof of Lemma 3.1 in [16].

For every 𝑢∈𝑆^𝑛 and 𝑟∈[0,1], denote 𝑢^(𝑟)(𝑥)=𝑢(𝑥),if𝑢(𝑥)≥𝑟,0,if𝑢(𝑥)<𝑟. (17) Then, we have the following.

Lemma 14 (see [16]). A closed set 𝑈⊂(𝑆^𝑛₀,𝑑_𝑝), 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 {𝑟_𝑖} be a decreasing sequence in (0,1] converging to zero. For {𝑢_𝑘}⊂𝑈, if {𝑢^(𝑟𝑖)_𝑘∣𝑘=1,2,3,…} converges to 𝑢(𝑟_𝑖)∈𝑆^𝑛₀ in 𝑑_𝑝, then there exists a 𝑢₀∈𝑆^𝑛₀ such that 𝑢^(𝑟𝑖)₀^𝛼=𝑢𝑟_𝑖^𝛼,𝑟_𝑖<𝛼≤1. (18)

Lemma 15. For any 𝑢∈𝑆^𝑛,𝑚∈ℝ^𝑛, and 0<𝑟<1, one has 𝑚^(𝑟)=𝑚,(𝑢+𝑚)^(𝑟)=𝑢^(𝑟)+𝑚. (19)

Proof. Let 𝛼 be an arbitrary element in [0,1].
If 𝑟≤𝛼≤1, then [𝑚^(𝑟)]^𝛼=[𝑚]^𝛼={𝑚}. If 0<𝛼<𝑟, then [𝑚^(𝑟)]^𝛼=[𝑚]^𝑟={𝑚}=[𝑚]^𝛼. Hence, for each 𝛼∈(0,1],[𝑚^(𝑟)]^𝛼=[𝑚]^𝛼, and so, it follows that [𝑚^(𝑟)]⁰=[𝑚]⁰. This implies that 𝑚^(𝑟)=𝑚.
Similarly, if 𝑟≤𝛼≤1, then by Lemma 12(2), we have (𝑢+𝑚)^(𝑟)𝛼=[𝑢+𝑚]^𝛼=[𝑢]^𝛼+{𝑚}=𝑢^(𝑟)𝛼+{𝑚}=𝑢^(𝑟)+𝑚^𝛼. (20) If 0<𝛼<𝑟, then by Lemma 12(2), we have (𝑢+𝑚)^(𝑟)𝛼=[𝑢+𝑚]^𝑟=[𝑢]^𝑟+{𝑚}=𝑢^(𝑟)𝛼+{𝑚}=𝑢^(𝑟)+𝑚^𝛼. (21)
Hence, for each 𝛼∈(0,1],[(𝑢+𝑚)^(𝑟)]^𝛼=[𝑢^(𝑟)+𝑚]^𝛼, and so, it follows that [(𝑢+𝑚)^(𝑟)]⁰=[𝑢^(𝑟)+𝑚]⁰. This implies that (𝑢+𝑚)^(𝑟)=𝑢^(𝑟)+𝑚.

Now, we give the characterization of compact sets in (𝑆^𝑛,𝑑_𝑝) for 1≤𝑝<∞.

Theorem 16. A closed set 𝑈⊂(𝑆^𝑛,𝑑_𝑝), 1≤𝑝<∞ is compact if and only if(i)𝑈 is uniformly 𝑝-mean bounded,(ii)𝑈 is 𝑝-mean equi-left-continuous,(iii)let {𝑟_𝑖} be a decreasing sequence in (0,1] converging to zero. For {𝑢_𝑘}⊂𝑈, if {𝑢^(𝑟𝑖)_𝑘∣𝑘=1,2,3,…} converges to 𝑢(𝑟_𝑖)∈𝑆^𝑛 in 𝑑_𝑝, then there exists a 𝑢₀∈𝑆^𝑛 such that 𝑢^(𝑟𝑖)₀^𝛼=𝑢𝑟_𝑖^𝛼,𝑟_𝑖<𝛼≤1. (22)

Proof. Necessity: (1) Since 𝑈 is compact in (𝑆^𝑛,𝑑_𝑝), it follows that 𝑈 is a bounded set in (𝑆^𝑛,𝑑_𝑝). This implies that 𝑈 is uniformly 𝑝-mean bounded.
(2) Let 𝜀>0, and let 𝑢₁,𝑢₂,…,𝑢_𝑚∈𝑆^𝑛 be a (𝜀/3)-net of 𝑈, that is, for any 𝑢∈𝑈, there exists an element 𝑢_𝑖(1≤𝑖≤𝑚) satisfying 𝑑_𝑝(𝑢,𝑢_𝑖)<𝜀/3. By Lemma 12, 𝑢₁,𝑢₂,…,𝑢_𝑚 are 𝑝-mean left-continuous, and so, there exists 𝛿(𝜀)=min_{1≤𝑖≤𝑚}𝛿(𝑢_𝑖,𝜀)>0 such that ¹_ℎ𝑑_𝐻𝑢_𝑖^𝛼,𝑢_𝑖^{𝛼−ℎ𝑝}d𝜀𝛼<3^𝑝, (23) for 𝑖=1,2,…,𝑚 and 0≤ℎ<𝛿(𝜀).
Thus, for 𝑢∈𝑈, we can obtain by triangle inequality ¹_ℎ𝑑_𝐻[𝑢]^𝛼,[𝑢]^{𝛼−ℎ𝑝}d𝛼^1/𝑝≤¹_ℎ𝑑_𝐻[𝑢]^𝛼,𝑢_𝑖^𝛼𝑝d𝛼^1/𝑝+¹_ℎ𝑑_𝐻𝑢_𝑖^𝛼,𝑢_𝑖^{𝛼−ℎ𝑝}d𝛼^1/𝑝+¹_ℎ𝑑_𝐻𝑢_𝑖^𝛼−ℎ,[𝑢]^{𝛼−ℎ𝑝}d𝛼^1/𝑝≤𝑑_𝑝𝑢_𝑖,𝑢+¹_ℎ𝑑_𝐻𝑢_𝑖^𝛼,𝑢_𝑖^{𝛼−ℎ𝑝}d𝛼^1/𝑝+𝑑_𝑝𝑢_𝑖<𝜀,𝑢3+𝜀3+𝜀3=𝜀. (24) Therefore, 𝑈 is 𝑝-mean equi-left-continuous.
(3) According to the definition of 𝑢^(𝑟𝑖), [𝑢^(𝑟𝑖)]^𝛼=[𝑢]^𝛼 for each 𝑟_𝑖≤𝛼≤1, and [𝑢^(𝑟𝑖)]^𝛼=[𝑢]^𝑟𝑖 for each 0≤𝛼<𝑟_𝑖. Since 𝑢∈𝑆^𝑛, then from Proposition 4, there exists 𝑥∈ℝ^𝑛 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)∈𝑆^𝑛 for each 𝑖∈ℕ.
For {𝑢_𝑘}⊂𝑈, we assume that {𝑢^(𝑟𝑖)_𝑘}^∞_𝑘=1 converges to 𝑢(𝑟_𝑖)∈𝑆^𝑛 in 𝑑_𝑝. Since 𝑈 is compact in (𝑆^𝑛,𝑑_𝑝), then {𝑢_𝑘} has a subsequence {𝑢_𝑘𝑗} converging to 𝑢₀∈𝑆^𝑛 in 𝑑_𝑝. By Minkowski’s inequality, it follows that ¹_𝑟𝑖𝑑_𝐻𝑢₀^𝛼,𝑢𝑟_𝑖^𝛼𝑝d𝛼^1/𝑝≤¹_𝑟𝑖𝑑_𝐻𝑢₀^𝛼,𝑢_𝑘𝑗^𝛼𝑝d𝛼^1/𝑝+¹_𝑟𝑖𝑑_𝐻𝑢_𝑘𝑗^𝛼,𝑢𝑟_𝑖^𝛼𝑝d𝛼^1/𝑝=¹_𝑟𝑖𝑑_𝐻𝑢₀^𝛼,𝑢_𝑘𝑗^𝛼𝑝d𝛼^1/𝑝+¹_𝑟𝑖𝑑_𝐻𝑢^(𝑟𝑖)_𝑘𝑗^𝛼,𝑢𝑟_𝑖^𝛼𝑝d𝛼^1/𝑝. (25) But since lim_𝑗→∞¹_𝑟𝑖𝑑_𝐻𝑢₀^𝛼,𝑢_𝑘𝑗^𝛼𝑝d𝛼^1/𝑝=0,lim_𝑗→∞¹_𝑟𝑖𝑑_𝐻𝑢^(𝑟𝑖)_𝑘𝑗^𝛼,𝑢𝑟_𝑖^𝛼𝑝d𝛼^1/𝑝=0, (26) then we get ¹_𝑟𝑖𝑑_𝐻𝑢₀^𝛼,𝑢𝑟_𝑖^𝛼𝑝d𝛼^1/𝑝=0, (27) and this implies that 𝑑_𝐻𝑢₀^𝛼,𝑢𝑟_𝑖^𝛼=0, (28) a.e. in (𝑟_𝑖,1]. Thus, we obtain that 𝑢₀^𝛼=𝑢𝑟_𝑖^𝛼, (29) a.e. in (𝑟_𝑖,1]; that is, [𝑢₀]^𝛼=[𝑢(𝑟_𝑖)]^𝛼 for each (𝑟_𝑖,1]⧵Ω, where Ω⊂(𝑟_𝑖,1] has Lebesgue measure zero and clearly (𝑟_𝑖,1]⧵Ω is dense in (𝑟_𝑖,1]. Let 𝛼∈Ω, then there exists an increasing sequence 𝛼_𝑘↗𝛼 with 𝛼_𝑘∈(𝑟_𝑖,1]⧵Ω and [𝑢₀]^𝛼𝑘=[𝑢(𝑟_𝑖)]^𝛼𝑘. So 𝑑_𝐻𝑢₀^𝛼,𝑢𝑟_𝑖^𝛼≤𝑑_𝐻𝑢₀^𝛼,𝑢₀^𝛼𝑘+𝑑_𝐻𝑢₀^𝛼𝑘,𝑢𝑟_𝑖^𝛼𝑘+𝑑_𝐻𝑢𝑟_𝑖^𝛼𝑘,𝑢𝑟_𝑖^𝛼=𝑑_𝐻𝑢₀^𝛼,𝑢₀^𝛼𝑘+𝑑_𝐻𝑢𝑟_𝑖^𝛼𝑘,𝑢𝑟_𝑖^𝛼. (30) By Proposition 3, both terms on the right converge to zero as 𝑘→∞, so 𝑑_𝐻([𝑢₀]^𝛼,[𝑢(𝑟_𝑖)]^𝛼)=0, and hence [𝑢₀]^𝛼=[𝑢(𝑟_𝑖)]^𝛼 for each 𝛼∈Ω.
Consequently, we have [𝑢^(𝑟𝑖)₀]^𝛼=[𝑢₀]^𝛼=[𝑢(𝑟_𝑖)]^𝛼 for each 𝑟_𝑖<𝛼≤1.
Sufficiency: From Lemma 12(3), we have that {𝑢−𝑚∶𝑢∈𝑈,𝑚∈ker(𝑢)}⊂𝑆^𝑛₀. We divide the rest of the proof into two steps.
Step  1. We prove that {𝑢−𝑚∶𝑢∈𝑈,𝑚∈ker(𝑢)} satisfies (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 𝑢_𝑘∈𝑈 such that 𝑑_𝐻ker𝑢_𝑘,{𝜃}>𝑘. (31)
By Proposition 3(1) and Lemma 12(1), it follows that ¹₀𝑑_𝐻𝑢_𝑘^𝛼,{𝜃}^𝑝d𝛼^1/𝑝≥¹₀𝑑_𝐻𝑢_𝑘¹,{𝜃}^𝑝d𝛼^1/𝑝≥¹₀𝑑_𝐻ker𝑢_𝑘,{𝜃}^𝑝d𝛼^1/𝑝>¹₀𝑘^𝑝d𝛼^1/𝑝=𝑘,𝑘∈ℕ. (32) This contradicts condition (i).
Secondly, we conclude that {𝑢−𝑚∶𝑢∈𝑈,𝑚∈ker(𝑢)} is 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∈𝑈𝑑_𝐻(ker(𝑢),{𝜃})≤𝒦. (33) By the condition (i), there exists a 𝒦^′>0 such that for all 𝑢∈𝑈¹₀𝑑_𝐻[𝑢]^𝛼,{𝜃}^𝑝d𝛼^1/𝑝≤𝒦^′. (34)
Thus, by Lemma 12(2) for all 𝑢∈𝑈, we get ¹₀𝑑_𝐻[𝑢−𝑚]^𝛼,{𝜃}^𝑝d𝛼^1/𝑝=¹₀𝑑_𝐻[𝑢]^𝛼−{𝑚},{𝜃}^𝑝d𝛼^1/𝑝=¹₀𝑑_𝐻[𝑢]^𝛼,{𝑚}^𝑝d𝛼^1/𝑝≤¹₀𝑑_𝐻[𝑢]^𝛼,{𝜃}^𝑝d𝛼^1/𝑝+¹₀𝑑_𝐻({𝜃},{𝑚})^𝑝d𝛼^1/𝑝≤¹₀𝑑_𝐻[𝑢]^𝛼,{𝜃}^𝑝d𝛼^1/𝑝+¹₀𝑑_𝐻({𝜃},ker(𝑢))^𝑝d𝛼^1/𝑝≤𝒦^′+𝒦. (35) This implies that condition (1) of Lemma 14 is satisfied.
Thirdly, we verify that {𝑢−𝑚∶𝑢∈𝑈,𝑚∈ker(𝑢)} is 𝑝-mean equi-left-continuous. In fact, since 𝑈 is 𝑝-mean equi-left-continuous, then for every 𝜀>0, there exists a 𝛿>0 such that for all 0≤ℎ≤𝛿 and 𝑢∈𝑈¹_ℎ𝑑_𝐻[𝑢]^𝛼,[𝑢]^{𝛼−ℎ𝑝}d𝛼<𝜀^𝑝, (36) and so, we have ¹_ℎ𝑑_𝐻[𝑢−𝑚]^𝛼,[𝑢−𝑚]^{𝛼−ℎ𝑝}d𝛼=¹_ℎ𝑑_𝐻[𝑢]^𝛼−{𝑚},[𝑢]^𝛼−ℎ−{𝑚}^𝑝d𝛼=¹_ℎ𝑑_𝐻[𝑢]^𝛼,[𝑢]^{𝛼−ℎ𝑝}d𝛼<𝜀^𝑝. (37) Hence, condition (2) of Lemma 14 holds.
Finally, we prove that condition (3) of Lemma 14 is also satisfied. Let {𝑟_𝑖} be a decreasing sequence in (0,1] converging to zero. We suppose that {(𝑢_𝑘−𝑚_𝑘)^(𝑟𝑖)}(𝑢_𝑘∈𝑈,𝑚_𝑘∈ker(𝑢_𝑘)) converges to 𝑢^∗(𝑟_𝑖)∈𝑆^𝑛₀ in 𝑑_𝑝. Denote 𝑣_𝑘=𝑢_𝑘−𝑚_𝑘. Then, {𝑣^(𝑟𝑖)_𝑘} converges to 𝑢^∗(𝑟_𝑖)∈𝑆^𝑛₀ in 𝑑_𝑝. By Lemma 15, we have 𝑢^(𝑟𝑖)_𝑘=𝑣_𝑘+𝑚_𝑘^(𝑟𝑖)=𝑣^(𝑟𝑖)_𝑘+𝑚_𝑘. (38)
By the preceding proof, {ker(𝑢)∣𝑢∈𝑈} is uniformly bounded. Then, {𝑚_𝑘} has a subsequence {𝑚_𝑘𝑗} converging to 𝑚₀∈ℝ^𝑛, and so by Lemma 12(4), {𝑢^(𝑟𝑖)_𝑘𝑗}={𝑣^(𝑟𝑖)_𝑘𝑗+𝑚_𝑘𝑗} converges to 𝑢^∗(𝑟_𝑖)+𝑚₀ in (𝑆^𝑛,𝑑_𝑝). By condition (iii) and Lemma 12(2), there exists a 𝑢₀∈𝑆^𝑛 such that 𝑢^(𝑟𝑖)₀^𝛼=𝑢^∗𝑟_𝑖+𝑚₀^𝛼=𝑢^∗𝑟_𝑖^𝛼+𝑚₀, (39) wherever 𝑟_𝑖<𝛼≤1, and so by Lemma 12(2), we have 𝑢^∗𝑟_𝑖^𝛼=𝑢^(𝑟𝑖)₀^𝛼−𝑚₀=𝑢₀^𝛼−𝑚₀=𝑢₀−𝑚₀^𝛼=𝑢₀−𝑚₀^{(𝑟𝑖)𝛼}, (40) wherever 𝑟_𝑖<𝛼≤1.
Define 𝑢^∗₀=𝑢₀−𝑚₀. Then, by Lemma 12(3), it follows that 𝑢^∗₀∈𝑆^𝑛₀. Thus, condition (3) of Lemma 14 holds.
Step  2. We infer that 𝑈 is a compact set in (𝑆^𝑛,𝑑_𝑝).
Let {𝑢_𝑘}⊂𝑈. Since {𝑢−𝑚∶𝑢∈𝑈,𝑚∈ker(𝑢)} is a subset of (𝑆^𝑛₀,𝑑_𝑝) and satisfies condition (1)–(3) of Lemma 14, then {𝑢−𝑚∶𝑢∈𝑈,𝑚∈ker(𝑢)} is a compact set of (𝑆^𝑛₀,𝑑_𝑝), and so for fixed 𝑚_𝑘∈ker(𝑢_𝑘)  (𝑘∈ℕ), {𝑢_𝑘−𝑚_𝑘} has a subsequence {𝑢_𝑘𝑗−𝑚_𝑘𝑗} converging to 𝑢₀∈(𝑆^𝑛₀,𝑑_𝑝). By the proceeding proof, {ker(𝑢)∶𝑢∈𝑈} is uniformly bounded, so {𝑚_𝑘𝑗} has a subsequence {𝑚_{𝑘𝑗(𝑙)}} converging to 𝑚₀∈ℝ^𝑛. It is obvious that {𝑢_{𝑘𝑗(𝑙)}−𝑚_{𝑘𝑗(𝑙)}} converges to 𝑢₀∈(𝑆^𝑛₀,𝑑_𝑝).
Denote 𝑣_{𝑘𝑗(𝑙)}=𝑢_{𝑘𝑗(𝑙)}−𝑚_{𝑘𝑗(𝑙)}. Then, {𝑣_{𝑘𝑗(𝑙)}} converges to 𝑢₀∈(𝑆^𝑛₀,𝑑_𝑝). Therefore, by Lemma 12(4), we get that {𝑢_{𝑘𝑗(𝑙)}}={𝑣_{𝑘𝑗(𝑙)}+𝑚_{𝑘𝑗(𝑙)}} converges to 𝑢₀+𝑚₀∈(𝑆^𝑛,𝑑_𝑝), which completes 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 star-shaped 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).