Lacunary Statistical Limit and Cluster Points of Generalized Difference Sequences of Fuzzy Numbers

Abstract

The aim of present work is to introduce and study lacunary statistical limit and lacunary statistical cluster points for generalized difference sequences of fuzzy numbers. Some inclusion relations among the sets of ordinary limit points, statistical limit points, statistical cluster points, lacunary statistical limit points, and lacunary statistical cluster points for these type of sequences are obtained.

1. Introduction

The notion of statistical convergence of sequences of numbers was introduced by Fast [1] and Schoenberg [2] independently and latter discussed in [3–6], and so forth. In 1993, Fridy and Orhan [7] presented an interesting generalization of statistical convergence with the help of a lacunary sequence and called it lacunary statistical convergence or 𝑆_𝜃-convergence. Demirci [8] defined 𝑆_𝜃-limit and cluster points of number sequences and obtained some interesting results analogous to [4]. In past years, statistical convergence has also become an interesting area of research for sequences of fuzzy numbers. The credit goes to Nuray and Savaş [9] who first introduced statistical convergence of sequences of fuzzy numbers. After their pioneer work, many authors have made their contribution to study different generalizations of statistical convergence for sequences of fuzzy numbers (see [10–13], etc.).

Quite recently, statistical convergence of sequences of fuzzy numbers is studied with the help of the difference operator Δ. For instance, Bilgin [14] introduced strongly Δ-summable and Δ-statistical convergence of sequences of fuzzy numbers. Işik [15] studied some notions of generalized difference sequences of numbers. In 2006, Altin et al. [16] united lacunary sequences to introduce the concept of lacunary statistical convergence of generalized difference sequences of fuzzy numbers and obtained some interesting results. Some more work in this direction can be found in [17–19]. In present work, we continue with this study and introduce the concepts of lacunary statistical limit and cluster points of generalized difference sequences of fuzzy numbers. We obtain some relations among the sets of ordinary limit, points, lacunary statistical limit, and cluster points for these type of sequences.

2. Background and Preliminaries

We begin with the following terminology on fuzzy numbers. Given any interval 𝐴, we shall denote its end points by 𝐴,𝐴 and by 𝐷 the set of all closed bounded intervals on real line ℝ, that is, 𝐷={𝐴⊂ℝ∶𝐴=[𝐴,𝐴]}. For 𝐴,𝐵∈𝐷 we define 𝐴≤𝐵 if and only if 𝐴≤𝐵 and 𝐴≤𝐵. Moreover, the distance function 𝑑 defined by 𝑑(𝐴,𝐵)=max{|𝐴−𝐵|,|𝐴−𝐵|} is a Hausdorff metric on 𝐷 and (𝐷,𝑑) is a complete metric space. Also ≤ is a partial order on 𝐷.

A fuzzy number is a function 𝑋 from ℝ to [0,1] which is satisfying the following conditions: (i) 𝑋 is normal, that is, there exists 𝑥₀∈ℝ such that 𝑋(𝑥₀)=1; (ii) 𝑋 is fuzzy convex, that is, for any 𝑥,𝑦∈ℝ and 𝜆∈[0,1], 𝑋(𝜆𝑥+(1−𝜆)𝑦)≥min{𝑋(𝑥),𝑋(𝑦)}; (iii) 𝑋 is upper semicontinuous; and (iv) the closure of the set {𝑥∈ℝ∶𝑋(𝑥)>0} denoted by 𝑋⁰ is compact.

Properties (i)–(iv) imply that for each 𝛼∈(0,1], the 𝛼-level set, 𝑋^𝛼={𝑥∈ℝ∶𝑋(𝑥)≥𝛼}=[𝑋^𝛼,𝑋^𝛼], is a nonempty compact convex subset of ℝ. Let 𝐿(ℝ) denote the set of all fuzzy numbers. The linear structure of 𝐿(ℝ) induces an addition 𝑋+𝑌 and a scalar multiplication 𝜆𝑋 in terms of 𝛼-level sets by [𝑋+𝑌]^𝛼=[𝑋]^𝛼+[𝑌]^𝛼,[𝜆𝑋]^𝛼=𝜆[𝑋]^𝛼(𝑋,𝑌∈𝐿(ℝ),𝜆∈ℝ) (1) for each 𝛼∈[0,1]. Define a map 𝑑∶𝐿(ℝ)×𝐿(ℝ)→ℝ by 𝑑(𝑋,𝑌)=sup_𝛼∈[0,1]𝑑𝑋^𝛼,𝑌^𝛼. (2) Puri and Ralescu [20] proved that (𝐿(ℝ),𝑑) is a complete metric space. Also the ordered structure on 𝐿(ℝ) is defined as follows. For 𝑋,𝑌∈𝐿(ℝ), we define 𝑋≤𝑌 if and only if 𝑋^𝛼≤𝑌^𝛼 and 𝑋^𝛼≤𝑌^𝛼 for each 𝛼∈[0,1]. We say that 𝑋<𝑌 if 𝑋≤𝑌 and there exist 𝛼₀∈[0,1] such that 𝑋^𝛼0<𝑌^𝛼0 or 𝑋^𝛼0<𝑌^𝛼0. The fuzzy numbers 𝑋 and 𝑌 are said to be incomparable if neither 𝑋≤𝑌 nor 𝑌≤𝑋.

We next recall some definitions and results which form the base for present study. For any set 𝐾⊆ℕ, let 𝐾_𝑛 denote the set {𝑘∈𝐾∶𝑘≤𝑛} and |𝐾_𝑛| denote the number of elements in 𝐾_𝑛. The natural density 𝛿 of 𝐾 is defined by 𝛿(𝐾)=lim_𝑛𝑛⁻¹|𝐾_𝑛|. The natural density may not exist for each set 𝐾. But the upper density 𝛿 defined by 𝛿(𝐾)=limsup_𝑛𝑛⁻¹|𝐾_𝑛| always exists for each set 𝐾. Moreover, 𝛿(𝐾) different from zero means 𝛿(𝐾)>0. Besides that, 𝛿(𝐾^𝐶)=1−𝛿(𝐾) and if 𝐴⊆𝐵, then 𝛿(𝐴)≤𝛿(𝐵).

For any sequence 𝑋=(𝑋_𝑘) of fuzzy numbers, we write {𝑋_𝑘∶𝑘∈ℕ} to denote the range of 𝑋. If (𝑋_𝑘(𝑗)) is a subsequence of 𝑋 and 𝐾={𝑘(𝑗)∶𝑗∈ℕ}, then we abbreviate (𝑋_𝑘(𝑗)) by (𝑋)_𝐾. If 𝛿(𝐾)=0, (𝑋)_𝐾 is called a thin subsequence, otherwise if 𝛿(𝐾)≠0, (𝑋)_𝐾 is called nonthin subsequence of 𝑋.

For 𝑤, the set of all sequences of fuzzy numbers, the operator Δ^𝑚∶𝑤→𝑤 is defined by Δ⁰𝑋_𝑘=𝑋_𝑘,Δ¹𝑋_𝑘=𝑋_𝑘−𝑋_𝑘+1,Δ⋮=⋮_𝑚𝑋_𝑘=Δ¹Δ^𝑚−1𝑋_𝑘. (3)

Definition 1. A sequence 𝑋=(𝑋_𝑘) of fuzzy numbers is said to be Δ^𝑚-statistically convergent to a fuzzy number 𝑋₀, in symbol: 𝑆(Δ^𝑚)−lim_𝑘𝑋_𝑘=𝑋₀, if for each 𝜖>0, lim_𝑛→∞1𝑛𝑘∈ℕ∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀≥𝜖=0. (4) Let 𝑆(Δ^𝑚(𝑋)) denote the set of all Δ^𝑚-statistically convergent sequences of fuzzy numbers.

Definition 2. Let 𝑋=(𝑋_𝑘) be a sequence of fuzzy numbers. A fuzzy number 𝑋₀ is said to be a statistical limit point (s.l.p) of the generalized difference sequence (Δ^𝑚𝑋_𝑘) of fuzzy numbers provided that there is a nonthin subsequence of 𝑋 that is Δ^𝑚-convergent to 𝑋₀.
Let Λ_𝑆(Δ^𝑚(𝑋)) denote the set of all s.l.p. of the generalized difference sequence (Δ^𝑚𝑋_𝑘) of fuzzy numbers.

Definition 3. Let 𝑋=(𝑋_𝑘) be a sequence of fuzzy numbers. A fuzzy number 𝑌₀ is said to be a statistical cluster point (s.c.p) of the generalized difference sequence (Δ^𝑚𝑋_𝑘) of fuzzy numbers provided that, for each 𝜖>0, limsup_𝑛→∞1𝑛𝑘∈ℕ∶𝑑Δ^𝑚𝑋_𝑘,𝑌₀<𝜖>0. (5) Let Γ_𝑆(Δ^𝑚(𝑋)) denote the set of all s.c.p of the generalized difference sequence (Δ^𝑚𝑋_𝑘) of fuzzy numbers.

By a lacunary sequence we mean an increasing sequence 𝜃=(𝑘_𝑟) of positive integers such that 𝑘₀=0 and ℎ_𝑟=𝑘_𝑟−𝑘_𝑟−1→∞ as 𝑟→∞. The intervals determined by 𝜃=(𝑘_𝑟) will be denoted by 𝐼_𝑟=(𝑘_𝑟−1,𝑘_𝑟] whereas the ratio 𝑘_𝑟/𝑘_𝑟−1 is denoted by 𝑞_𝑟. Further, a lacunary sequence 𝜃^′=(𝑘^′_𝑟) is called a lacunary refinement of the lacunary sequence 𝜃=(𝑘_𝑟) if {𝑘_𝑟}⊆{𝑘^′_𝑟}.

Definition 4 (see [21]). Let 𝜃=(𝑘_𝑟) be a lacunary sequence. A sequence 𝑋=(𝑋_𝑘) of fuzzy numbers is said to be lacunary statistical convergent to a fuzzy number 𝑋₀ provided that for each 𝜖>0, lim_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑𝑋_𝑘,𝑌₀≥𝜖=0. (6) Let 𝑆_𝜃 denote the set of all lacunary statistically convergent sequences of fuzzy numbers.

Let 𝜃=(𝑘_𝑟) be a lacunary sequence and 𝑋=(𝑋_𝑘) a sequence of fuzzy numbers. If (𝑋)_𝐾 where 𝐾={𝑘(𝑗)∶𝑗∈ℕ} is a subsequence of 𝑋=(𝑋_𝑘) such that lim_𝑟→∞1ℎ_𝑟𝑘𝑗∈𝐼_𝑟∶𝑗∈ℕ=0, (7) we call (𝑋)_𝐾 a 𝜃-thin subsequence. On the other hand, (𝑋)_𝐾 is a 𝜃-nonthin subsequence of 𝑋 provided that limsup_𝑟→∞1ℎ_𝑟𝑘𝑗∈𝐼_𝑟∶𝑗∈ℕ>0. (8)

Definition 5. Let 𝜃=(𝑘_𝑟) be a lacunary sequence. A sequence 𝑋=(𝑋_𝑘) of fuzzy numbers is said to be lacunary Δ^𝑚-statistically convergent to a fuzzy number 𝑋₀, in symbol: 𝑆_𝜃(Δ^𝑚)−lim_𝑘𝑋_𝑘=𝑋₀, if for each 𝜖>0, lim_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀≥𝜖=0. (9) Let 𝑆_𝜃(Δ^𝑚(𝑋)) denote the set of all lacunary Δ^𝑚-statistically convergent sequences of fuzzy numbers.

We now consider the natural definitions of statistical limit and cluster points for generalized difference sequences of fuzzy numbers with respect to lacunary sequences.

3. Main Results

Definition 6. Let 𝜃=(𝑘_𝑟) be a lacunary sequence and 𝑋=(𝑋_𝑘) a sequence of fuzzy numbers. A fuzzy number 𝑋₀ is said to be a lacunary statistical limit point (l.s.l.p) of the generalized difference sequence (Δ^𝑚𝑋_𝑘) of fuzzy numbers provided that there is a 𝜃-nonthin subsequence of 𝑋 that is Δ^𝑚-convergent to 𝑋₀.

Let Λ_𝑆𝜃(Δ^𝑚(𝑋)) denote the set of all l.s.l.p. of the generalized difference sequence (Δ^𝑚𝑋_𝑘) of fuzzy numbers.

Definition 7. Let 𝜃=(𝑘_r) be a lacunary sequence and 𝑋=(𝑋_𝑘) a sequence of fuzzy numbers. A fuzzy number 𝑌₀ is said to be a lacunary statistical cluster point (l.s.c.p) of the generalized difference sequence (Δ^𝑚𝑋_𝑘) of fuzzy numbers provided that, for each 𝜖>0, limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑌₀<𝜖>0. (10) Let Γ_𝑆𝜃(Δ^𝑚(𝑋)) denote the set of all l.s.c.p of the generalized difference sequence (Δ^𝑚𝑋_𝑘) of fuzzy numbers.

Example 8. Let 𝜃=(𝑘_𝑟) be a lacunary sequence. We define a sequence of fuzzy numbers 𝑋=(𝑋_𝑘) as follows. For 𝑥∈ℝ, define 𝑋_𝑘(𝑥)=,𝑥−𝑘+1,if𝑘−1≤𝑥≤𝑘−𝑥+𝑘+1,if𝑘<𝑥≤𝑘+10,otherwiseif𝑘_𝑟−ℎ_𝑟+1≤𝑘≤𝑘_𝑟;𝑟∈ℕ𝑥−5,if5≤𝑥≤67−𝑥,if6<𝑥≤70,otherwise,otherwise. (11) Then, we obtain 𝑋_𝑘^𝛼=[𝑘−1+𝛼,𝑘+1−𝛼],if𝑘∈𝑘_𝑟−ℎ_𝑟+1,𝑘_𝑟[5+𝛼,7−𝛼]otherwise,Δ𝑋_𝑘^𝛼=[],𝑘−3+𝛼,1−2𝛼ifboth𝑘,𝑘+1∈_𝑟−ℎ_𝑟+1,𝑘_𝑟[],𝑘𝑘−8+2𝛼,𝑘−4−2𝛼if𝑘∈_𝑟−ℎ_𝑟+1,𝑘_𝑟[],𝑘butnot𝑘+1,−𝑘+3+2𝛼,−𝑘+7−2𝛼if𝑘+1∈_𝑟−ℎ_𝑟+1,𝑘_𝑟[]butnot𝑘,−2+2𝛼,2−2𝛼,otherwise. (12) Thus, for 𝑚=1, it is clear that the sequence Δ𝑋_𝑘 has two different subsequences which converge to 𝜇₁ and 𝜇₂, respectively, where [𝜇₁]^𝛼=[−3+𝛼,1−2𝛼] and [𝜇₂]^𝛼=[−2+2𝛼,2−2𝛼]. Hence, if 𝐿(Δ(𝑋)) denotes the set of ordinary limit points of (Δ𝑋_𝑘), then 𝐿(Δ(𝑋))={𝜇₁,𝜇₂}; however, Γ_𝑆𝜃(Δ(𝑋))={𝜇₂}.

Theorem 9. Let 𝜃=(𝑘_𝑟) be a lacunary sequence and 𝑋=(𝑋_𝑘) a sequence of fuzzy numbers. Then, one has Λ_𝑆𝜃(Δ^𝑚(𝑋))⊆Γ_𝑆𝜃(Δ^𝑚(𝑋)).

Proof. Suppose 𝑋₀∈Λ_𝑆𝜃(Δ^𝑚(𝑋)). By definition, there is a 𝜃-nonthin subsequence (𝑋_𝑘(𝑗)) of 𝑋=(𝑋_𝑘) which is Δ^𝑚-convergent to 𝑋₀, and therefore we have limsup_𝑟→∞1ℎ_𝑟𝑘𝑗∈𝐼_𝑟∶𝑗∈ℕ=𝑑>0. (13) Since, for every 𝜖>0, 𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀<𝜖⊇𝑘𝑗∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘(𝑗),𝑋₀<𝜖, (14) so we have the containment 𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀⊃𝑘𝑗<𝜖∈𝐼_𝑟−𝑘𝑗∶𝑗∈ℕ∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘(𝑗),𝑋₀≥𝜖. (15) Now, (𝑋_𝑘(𝑗)) is Δ^𝑚-convergent to 𝑋₀, which implies that, for every 𝜖>0, {𝑘(𝑗)∈𝐼_𝑟∶𝑑(Δ^𝑚𝑋_𝑘(𝑗),𝑋₀)≥𝜖} is finite for which we have limsup_𝑟→∞1ℎ_𝑟𝑘𝑗∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘(𝑗),𝑋₀≥𝜖=0. (16) Thus from (15), we obtain limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀<𝜖≥limsup_𝑟→∞1ℎ_𝑟𝑘𝑗∈𝐼_𝑟∶𝑗∈ℕ−limsup_𝑟→∞1ℎ_𝑟𝑘𝑗∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘(𝑗),𝑋₀≥𝜖≥limsup_𝑟→∞1ℎ_𝑟𝑘𝑗∈𝐼_𝑟∶𝑗∈ℕ=𝑑>0, (17) using (13) and (16). This shows that 𝑋₀∈Γ_𝑆𝜃(Δ^𝑚(𝑋)) and therefore the result is proved.

Theorem 10. Let 𝜃=(𝑘_𝑟) be a lacunary sequence. Then, for any sequence 𝑋=(𝑋_𝑘) of fuzzy numbers, one has Γ_𝑆𝜃(Δ^𝑚(𝑋))⊆𝐿(Δ^𝑚(𝑋)).

Proof. Assume 𝑌₀∈Γ_𝑆𝜃(Δ^𝑚(𝑋)). By definition, for each 𝜖>0 we have limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑌₀<𝜖>0. (18) We set (𝑋)_𝐾 a 𝜃-nonthin subsequence of 𝑋=(𝑋_𝑘) such that 𝐾={𝑘(𝑗)∈𝐼_𝑟∶𝑑(Δ^𝑚𝑋_𝑘,𝑌₀)<𝜖} for 𝜖>0. Since limsup_𝑟→∞(1/ℎ_𝑟)|{𝐾}|>0, it follows that 𝐾 is an infinite set. Thus we have a subsequence (𝑋)_𝐾 of 𝑋 that is Δ^𝑚-convergent to 𝑌₀. This shows that 𝑌₀∈𝐿(Δ^𝑚(𝑋)). Hence Γ_𝑆𝜃(Δ^𝑚(𝑋))⊆𝐿(Δ^𝑚(𝑋)).

Theorem 11. Let 𝜃=(𝑘_𝑟) be a lacunary sequence. If 𝑋=(𝑋_𝑘) and 𝑌=(𝑌_𝑘) are two sequences of fuzzy numbers such that lim_𝑟→∞(1/ℎ_𝑟)|{𝑘∈𝐼_𝑟∶𝑋_𝑘≠𝑌_𝑘}|=0, then Λ_𝑆𝜃(Δ^𝑚(𝑋))=Λ_𝑆𝜃(Δ^𝑚(𝑌)) and Γ_𝑆𝜃(Δ^𝑚(𝑋))=Γ_𝑆𝜃(Δ^𝑚(𝑌)).

Proof. We prove the theorem into two parts. In the first part we prove that Λ_𝑆𝜃(Δ^𝑚(𝑋))=Λ_𝑆𝜃(Δ^𝑚(𝑌)); however, in the second part we shall prove Γ_𝑆𝜃(Δ^𝑚(𝑋))=Γ_𝑆𝜃(Δ^𝑚(𝑌)).
Part (i). Let 𝑋₀∈Λ_𝑆𝜃(Δ^𝑚(𝑌)). By definition, there is a 𝜃-nonthin subsequence (𝑌)_𝐾 of 𝑌=(𝑌_𝑘) that is Δ^𝑚-convergent to 𝑋₀. Since lim_𝑟→∞(1/ℎ_𝑟)|{𝑘∈𝐼_𝑟∶𝑘∈𝐾and𝑋_𝑘≠𝑌_𝑘}|=0, it follows that limsup_𝑟→∞(1/ℎ_𝑟)|{𝑘∈𝐼_𝑟∶𝑘∈𝐾and𝑋_𝑘=𝑌_𝑘}|>0. Therefore, from the later set, we can yield a 𝜃-nonthin subsequence (𝑋)_𝐾′ of 𝑋=(𝑋_𝑘) that is Δ^𝑚-convergent to 𝑋₀. Hence, 𝑋₀∈Λ_𝑆𝜃(Δ^𝑚(𝑋)), and therefore we have Λ_𝑆𝜃(Δ^𝑚(𝑌))⊂Λ_𝑆𝜃(Δ^𝑚(𝑋)). Also by symmetry one get Λ_𝑆𝜃(Δ^𝑚(𝑋))⊂Λ_𝑆𝜃(Δ^𝑚(𝑌)). On combining we have Λ_𝑆𝜃(Δ^𝑚(𝑋))=Λ_𝑆𝜃(Δ^𝑚(𝑌)).
Part (ii). Let 𝑌₀∈Γ_𝑆𝜃(Δ^𝑚(𝑋)). By definition, for each 𝜖>0, limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑌₀<𝜖>0. (19) Since 𝑋_𝑘=𝑌_𝑘 for all most all 𝑘, it follows that, for each 𝜖>0, limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑌_𝑘,𝑌₀<𝜖>0. (20) This shows that 𝑌₀∈Γ_𝑆𝜃(Δ^𝑚(𝑌)) and therefore Γ_𝑆𝜃(Δ^𝑚(𝑋))⊂Γ_𝑆𝜃(Δ^𝑚(𝑌)). By symmetry, we see that Γ_𝑆𝜃(Δ^𝑚(𝑌))⊂Γ_𝑆𝜃(Δ^𝑚(𝑋)), whence Γ_𝑆𝜃(Δ^𝑚(𝑋))=Γ_𝑆𝜃(Δ^𝑚(𝑌)).

Theorem 12. Let 𝜃=(𝑘_𝑟) be a lacunary sequence. If 𝑋=(𝑋_𝑘) is a sequence of fuzzy numbers such that 𝑆_𝜃(Δ^𝑚)−lim_𝑘𝑋_𝑘=𝑋₀, then Λ_𝑆𝜃(Δ^𝑚(𝑋))=Γ_𝑆𝜃(Δ^𝑚(𝑋))={𝑋₀}.

Proof. We prove the theorem in two parts. In the first part, we prove that Λ_𝑆𝜃(Δ^𝑚(𝑋))={𝑋₀} whereas in the second part we obtain Γ_𝑆𝜃(Δ^𝑚(𝑋))={𝑋₀}.
Part (i). Suppose that Λ_𝑆𝜃(Δ^𝑚(𝑋))={𝑋₀,𝑌₀}, where 𝑋₀≠𝑌₀, that is, 𝑌₀ is a l.s.l.p. of the generalized difference sequence (Δ^𝑚𝑋_𝑘) different from 𝑋₀. Choose 𝜖>0 such that 0<𝜖<𝑑(𝑋₀,𝑌₀)/2. By definition there exist two 𝜃-nonthin subsequences (𝑋_𝑘(𝑗)) and (𝑋_𝑙(𝑖)) of the sequence 𝑋=(𝑋_𝑘) which are Δ^𝑚-convergent to 𝑋₀ and 𝑌₀, respectively. Since (𝑋_𝑙(𝑖)) is Δ^𝑚-convergent to 𝑌₀, so for each 𝜖>0, {𝑙(𝑖)∈𝐼_𝑛∶𝑑(Δ^𝑚𝑋_𝑙(𝑖),𝑌₀)≥𝜖} is a finite set for which limsup_𝑟→∞1ℎ_𝑟𝑙(𝑖)∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑙(𝑖),𝑌₀≥𝜖=0. (21) Further, we can write 𝑙(𝑖)∈𝐼_𝑟∶𝑖∈ℕ=𝑙(𝑖)∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑙(𝑖),𝑌₀∪𝑙(𝑖)<𝜖∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑙(𝑖),𝑌₀≥𝜖, (22) for which we have limsup_𝑟→∞1ℎ_𝑟𝑙(𝑖)∈𝐼_𝑟∶𝑖∈ℕ=limsup_𝑟→∞1ℎ_𝑟𝑙(𝑖)∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑙(𝑖),𝑌₀<𝜖+limsup_𝑟→∞1ℎ_𝑟𝑙(𝑖)∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑙(𝑖),𝑌₀≥𝜖. (23) Since (𝑋_𝑙(𝑖)) is nonthin, so, by use of (21), we have limsup_𝑟→∞1ℎ_𝑟𝑙(𝑖)∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑙(𝑖),𝑌₀<𝜖>0. (24) Since 𝑆_𝜃(Δ^𝑚)−lim_𝑘𝑋_𝑘=𝑋₀, so for each 𝜖>0lim_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀≥𝜖=0, (25) and therefore we can write limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀<𝜖>0. (26) Furthermore for 0<2𝜖<𝑑(𝑋₀,𝑌₀), 𝑙(𝑖)∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑙(𝑖),𝑌₀∩<𝜖𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀<𝜖=∅, (27) which immediately gives the containment 𝑙(𝑖)∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑙(𝑖),𝑌₀<𝜖⊂𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀≥𝜖 (28) for which we have limsup_𝑟→∞1ℎ_𝑟𝑙(𝑖)∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑙(𝑖),𝑌₀<𝜖≤limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀≥𝜖=0. (29) As left side of (29) cannot be negative, so we must have limsup_𝑟→∞1ℎ_𝑟𝑙(𝑖)∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑙(𝑖),𝑌₀<𝜖=0. (30) This contradicts (24). Hence, Λ_𝑆𝜃(Δ^𝑚(𝑋))={𝑋₀}.
Part (ii). Let 𝑍₀ be a l.s.c.p. of the generalized difference sequence (Δ^𝑚𝑋_𝑘) different from 𝑋₀, that is, Γ_𝑆𝜃(Δ^𝑚(𝑋))={𝑋₀,𝑍₀}, where 𝑋₀≠𝑍₀. Choose 𝜖 such that 0<𝜖<𝑑(𝑋₀,𝑍₀)/2. Since 𝑍₀ is a l.s.c.p of (Δ^𝑚𝑋_𝑘), so for each 𝜖>0 we have limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑍₀<𝜖>0. (31) Since {𝑘∈𝐼_𝑟∶𝑑(Δ^𝑚𝑋_𝑘,𝑋₀)<𝜖}∩{𝑘∈𝐼_𝑟∶𝑑(Δ^𝑚𝑋_𝑘,𝑍₀)<𝜖}=∅ for every 0<𝜖<𝑑(𝑋₀,𝑍₀)/2, it follows that {𝑘∈𝐼_𝑟∶𝑑(Δ^𝑚𝑋_𝑘,𝑋₀)≥𝜖}⊇{𝑘∈𝐼_𝑟∶𝑑(Δ^𝑚𝑋_𝑘,𝑍₀)<𝜖} for which we have limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑋₀≥𝜖≥limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑍₀<𝜖>0 (32) by (31), which is impossible as by (25) limsup_𝑟→∞(1/ℎ_𝑟)|{𝑘∈𝐼_𝑟∶𝑑(Δ^𝑚𝑋_𝑘,𝑋₀)≥𝜖}|=0. In this way we obtained a contradiction. Hence, Γ_𝑆𝜃(Δ^𝑚(𝑋))={𝑋₀}.

Theorem 13. Let 𝜃=(𝑘_𝑟) be a lacunary sequence and 𝑋=(𝑋_𝑘) a sequence of fuzzy numbers. Then one has the following:(i)if  liminf𝑞_𝑟>1, then Λ_𝑆𝜃(Δ^𝑚(𝑋))⊂Λ_𝑆(Δ^𝑚(𝑋)),(ii)if  limsup𝑞_𝑟<∞, then Λ_𝑆(Δ^𝑚(𝑋))⊂Λ_𝑆𝜃(Δ^𝑚(𝑋)),(iii)if  1<liminf𝑞_𝑟≤limsup𝑞_𝑟<∞, then Λ_𝑆(Δ^𝑚(𝑋))=Λ_𝑆𝜃(Δ^𝑚(𝑋)).

Proof. (i) Suppose liminf𝑞_𝑟>1; there exists a 𝛿>0 such that 𝑞_𝑟>1+𝛿 for sufficient large 𝑟, which implies that ℎ_𝑟/𝑘_𝑟≥𝛿/(1+𝛿). Assume that 𝑋₀∈Λ_𝑆𝜃(Δ^𝑚(𝑋)), then there is 𝜃-nonthin subsequence (𝑋_𝑘(𝑗)) of (𝑋_𝑘) that is Δ^𝑚-convergent to 𝑋₀ and limsup_𝑟→∞1ℎ_𝑟𝑘𝑗∈𝐼_𝑟∶𝑗∈ℕ=𝑑>0. (33) Since 1𝑘_𝑟𝑘𝑗≤𝑘_𝑟≥1∶𝑗∈ℕ𝑘_𝑟𝑘𝑗∈𝐼_𝑟=ℎ∶𝑗∈ℕ_𝑟𝑘_𝑟1ℎ_𝑟𝑘𝑗∈𝐼_𝑟≥𝛿∶𝑗∈ℕ11+𝛿ℎ_𝑟𝑘𝑗∈𝐼_𝑟∶𝑗∈ℕ, (34) it follows by (33) that limsup_𝑟→∞(1/𝑘_𝑟)|{𝑘(𝑗)≤𝑘_𝑟∶𝑗∈ℕ}|>0. Since (𝑋_𝑘(𝑗)) is already Δ^𝑚-convergent to 𝑋₀, so we have 𝑋₀∈Λ_𝑆(Δ^𝑚(𝑋)). Hence Λ_𝑆𝜃(Δ^𝑚(𝑋))⊂Λ_𝑆(Δ^𝑚(𝑋)).
(ii) If limsup𝑞_𝑟<∞, then there exists a real number 𝐻 such that 𝑞_𝑟<𝐻 for all 𝑟. Without loss of generality, we can assume 𝐻>1 (as otherwise 𝑘_𝑟<𝑘_𝑟−1). Now for all 𝑟,(ℎ_𝑟/𝑘_𝑟−1)=(𝑘_𝑟−𝑘_𝑟−1)/𝑘_𝑟−1=𝑞_𝑟−1≤𝐻−1. Let 𝑋₀∈Λ_𝑆(Δ^𝑚(𝑋)), then there is a set 𝐾={𝑘(𝑗)∶𝑗∈ℕ} with 𝛿(𝐾)≠0 and lim_𝑗Δ^𝑚𝑋_𝑘(𝑗)=𝑋₀. Let 𝑁_𝑟=|{𝑘∈𝐼_𝑟∶𝑘∈𝐾}|=|𝐾∩𝐼_𝑟| and 𝑡_𝑟=𝑁_𝑟/ℎ_𝑟. For any integer 𝑛 satisfying 𝑘_𝑟−1<𝑛≤𝑘_𝑟, we can write 1𝑛≤1|{𝑘≤𝑛∶𝑘∈𝐾}|𝑘_𝑟−1𝑘≤𝑘_𝑟=1∶𝑘∈𝐾𝑘_𝑟−1𝑁₁+𝑁₂+⋯+𝑁_𝑟=1𝑘_𝑟−1𝑡₁ℎ₁+𝑡₂ℎ₂+⋯𝑡_𝑟ℎ_𝑟=1∑^𝑟−1_𝑖=1ℎ_𝑖^𝑟−1_𝑖=1ℎ_𝑖𝑡_𝑖+ℎ_𝑟𝑘_𝑟−1𝑡_𝑟≤1∑^𝑟−1_𝑖=1ℎ_𝑖^𝑟−1_𝑖=1ℎ_𝑖𝑡_𝑖+(𝐻−1)𝑡_𝑟. (35) Suppose 𝑡_𝑟→0 as 𝑟→∞. Since 𝜃 is a lacunary sequence and the first part on the right side of above expression is a regular weighted mean transform of the sequence 𝑡=(𝑡_𝑟), therefore it too tends to zero as 𝑟→∞. Since 𝑛→∞ as 𝑟→∞, it follows that 𝛿(𝐾)=0 which is a contradiction as 𝛿(𝐾)≠0. Thus lim_𝑟→∞𝑡_𝑟≠0 and therefore 𝑋₀∈Λ_𝑆𝜃(Δ^𝑚(𝑋)). Hence Λ_𝑆(Δ^𝑚(𝑋))⊂Λ_𝑆𝜃(Δ^𝑚(𝑋)).
(iii) This is an immediate consequence of (i) and (ii).

Theorem 14. Let 𝜃=(𝑘_𝑟) be a lacunary sequence and 𝑋=(𝑋_𝑘) a sequence of fuzzy numbers. Then one has the following:(i)if  liminf𝑞_𝑟>1, then Γ_𝑆𝜃(Δ^𝑚(𝑋))⊂Γ_𝑆(Δ^𝑚(𝑋)),(ii)if  limsup𝑞_𝑟<∞, then Γ_𝑆(Δ^𝑚(𝑋))⊂Γ_𝑆𝜃(Δ^𝑚(𝑋)),(iii)if  1<liminf𝑞_𝑟≤limsup𝑞_𝑟<∞, then Γ_𝑆(Δ^𝑚(𝑋))=Γ_𝑆𝜃(Δ^𝑚(𝑋)).

Proof. The proof of the theorem can be obtain on the similar lines as that of the above theorem and therefore is omitted here.

Theorem 15. For any lacunary refinement 𝜃^′ of a lacunary sequence 𝜃=(𝑘_𝑟), Γ_𝑆𝜃(Δ^𝑚(𝑋))⊆Γ_𝑆𝜃′(Δ^𝑚(𝑋)) and Λ_𝑆𝜃(Δ^𝑚(𝑋))⊆Λ_𝑆𝜃′(Δ^𝑚(𝑋)).

Proof. Suppose each 𝐼_𝑟 of 𝜃 contains the points {𝑘^′_𝑟,𝑖}^𝑣(𝑟)_𝑖=1 of 𝜃^′ so that 𝑘_𝑟−1<𝑘^′_𝑟,1<𝑘^′_𝑟,2<⋯<𝑘^′_{𝑟,𝑣(𝑟)}=𝑘_𝑟, where 𝐼^′_𝑟=(𝑘^′_{𝑟,𝑖−1},𝑘^′_𝑟,𝑖]. Note that for all 𝑟, 𝑣(𝑟)≥1. Let (𝐼^∗_𝑗)^∞_𝑗=1 be the sequence of abutting intervals 𝐼^′_𝑟,𝑖 ordered by increasing right end points. Let 𝑌₀∈Γ_𝑆𝜃(Δ^𝑚(𝑋)), then for each 𝜖>0, limsup_𝑟→∞1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑌₀<𝜖>0. (36) As before, write ℎ^′_𝑟,𝑖=𝑘^′_𝑟,𝑖−𝑘^′_{𝑟,𝑖−1} and ℎ^′_𝑟,1=𝑘^′_𝑟,1−𝑘^′_𝑟−1. Now for each 𝜖>0, we can write 1ℎ_𝑟𝑘∈𝐼_𝑟∶𝑑Δ^𝑚𝑋_𝑘,𝑌₀=1<𝜖ℎ_{𝑟𝐼∗𝑗⊆𝐼𝑟}ℎ^∗_𝑗1ℎ^∗_𝑗𝑘∈𝐼^∗_𝑗∶𝑑Δ^𝑚𝑋_𝑘,𝑌₀=1<𝜖ℎ_{𝑟𝐼∗𝑗⊆𝐼𝑟}ℎ^∗_𝑗𝐶_𝜃′𝜒_𝐾𝑗, (37) where 𝜒_𝐾 is the characteristics function of the set 𝐾={𝑘∈𝐼^∗_𝑗∶𝑑(Δ^𝑚𝑋_𝑘,𝑌₀)<𝜖} and (𝐶_𝜃′𝜒_𝐾)_𝑗=|𝐾∩𝐼^∗_𝑗|/ℎ^∗_𝑗. Suppose lim_𝑗→∞(𝐶_𝜃′𝜒_𝐾)_𝑗=0. Then the right side of above expression is a regular weighted mean transform of (𝐶_𝜃′𝜒_𝐾)_𝑗 and therefore tends to zero as 𝑗→∞ which contradicts (36). Thus lim_𝑗→∞(𝐶_𝜃′𝜒_𝐾)_𝑗≠0, which shows that 𝑌₀∈Γ_𝑆𝜃′(Δ^𝑚(𝑋)). Hence Γ_𝑆𝜃(Δ^𝑚(𝑋))⊆Γ_𝑆𝜃′(Δ^𝑚(𝑋)).
Similarly, we can prove Λ_𝑆𝜃(Δ^𝑚(𝑋))⊆Λ_𝑆𝜃′(Δ^𝑚(𝑋)).

Acknowledgment

The authors are grateful to the referees for their valuable suggestions which improved the readability of the paper.