LacunaryStatisticalLimitandClusterPointsofGeneralized ...downloads.hindawi.com/journals/afs/2012/459370.pdf · Advances in Fuzzy Systems 3 3.MainResults Deﬁnition 6. Let θ =(k

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2012, Article ID 459370, 6 pagesdoi:10.1155/2012/459370

Research Article

Lacunary Statistical Limit and Cluster Points of GeneralizedDifference Sequences of Fuzzy Numbers

Pankaj Kumar,1 Vijay Kumar,1 and S. S. Bhatia2

1 Department of Mathematics, Haryana College of Technology and Management, Haryana, Kaithal 136027, India2 School of Mathematics and Computer Application, Thapar University, Punjab, Patiala 147004, India

Correspondence should be addressed to Pankaj Kumar, [email protected]

Received 20 April 2012; Accepted 14 June 2012

Academic Editor: Katsuhiro Honda

Copyright © 2012 Pankaj Kumar et al. 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.

The aim of present work is to introduce and study lacunary statistical limit and lacunary statistical cluster points for generalizeddifference 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 areobtained.

1. Introduction

The notion of statistical convergence of sequences ofnumbers was introduced by Fast [1] and Schoenberg [2]independently and latter discussed in [3–6], and so forth. In1993, Fridy and Orhan [7] presented an interesting general-ization of statistical convergence with the help of a lacunarysequence and called it lacunary statistical convergence or Sθ-convergence. Demirci [8] defined Sθ-limit and cluster pointsof number sequences and obtained some interesting resultsanalogous to [4]. In past years, statistical convergence hasalso become an interesting area of research for sequencesof fuzzy numbers. The credit goes to Nuray and Savas [9]who first introduced statistical convergence of sequences offuzzy numbers. After their pioneer work, many authors havemade their contribution to study different generalizations ofstatistical convergence for sequences of fuzzy numbers (see[10–13], etc.).

Quite recently, statistical convergence of sequences offuzzy numbers is studied with the help of the differenceoperator Δ. For instance, Bilgin [14] introduced stronglyΔ-summable and Δ-statistical convergence of sequences offuzzy numbers. Isik [15] studied some notions of generalizeddifference sequences of numbers. In 2006, Altin et al.[16] united lacunary sequences to introduce the conceptof lacunary statistical convergence of generalized differencesequences 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 andintroduce the concepts of lacunary statistical limit and clusterpoints 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 thesetype 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,Aand by D the set of all closed bounded intervals on real lineR, that is, D = {A ⊂ R : A = [A,A]}. For A,B ∈ D we defineA ≤ B if and only if A ≤ B and A ≤ B. Moreover, the distancefunction d defined by d(A,B) = max{|A − B|, |A − B|} is aHausdorff 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] whichis satisfying the following conditions: (i) X is normal, thatis, there exists x0 ∈ R such that X(x0) = 1; (ii) X is fuzzyconvex, 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 byX0 is compact.

2 Advances in Fuzzy Systems

Properties (i)–(iv) imply that for each α ∈ (0, 1], theα-level set, Xα = {x ∈ R : X(x) ≥ α} = [Xα,X

α], is a

nonempty compact convex subset of R. Let L(R) denote theset of all fuzzy numbers. The linear structure of L(R) inducesan addition X + Y and a scalar multiplication λX in terms ofα-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) = supα∈[0,1]

d(Xα,Yα). (2)

Puri and Ralescu [20] proved that (L(R),d) is a completemetric space. Also the ordered structure on L(R) is definedas follows. For X ,Y ∈ L(R), we define X ≤ Y if and only ifXα ≤ Yα and X

α ≤ Yα

for each α ∈ [0, 1]. We say that X < Yif X ≤ Y and there exist α0 ∈ [0, 1] such that Xα0 < Yα0

or Xα0 < Y

α0 . The fuzzy numbers X and Y are said to beincomparable if neither X ≤ Y nor Y ≤ X .

We next recall some definitions and results which formthe base for present study. For any set K ⊆ N, let Kn denotethe set {k ∈ K : k ≤ n} and |Kn| denote the numberof elements in Kn. The natural density δ of K is definedby δ(K) = limnn−1|Kn|. The natural density may not existfor each set K . But the upper density δ defined by δ(K) =lim supnn

−1|Kn| always exists for each set K . Moreover, δ(K)different from zero means δ(K) > 0. Besides that, δ(KC) =1− δ(K) and if A ⊆ B, then δ(A) ≤ δ(B).

For any sequence X = (Xk) of fuzzy numbers, we write{Xk : k ∈ N} to denote the range of X . If (Xk( j)) is asubsequence of X and K = {k( j) : j ∈ N}, then weabbreviate (Xk( j)) by (X)K . If δ(K) = 0, (X)K is called a thinsubsequence, otherwise if δ(K) /= 0, (X)K is called nonthinsubsequence of X .

For w, the set of all sequences of fuzzy numbers, theoperator Δm : w → w is defined by

Δ0Xk = Xk,

Δ1Xk = Xk − Xk+1,

... = ...

ΔmXk = Δ1(Δm−1Xk).

(3)

Definition 1. A sequence X = (Xk) of fuzzy numbers is saidto be Δm-statistically convergent to a fuzzy number X0, insymbol: S(Δm)− limkXk = X0, if for each ε > 0,

limn→∞

1n

∣∣∣{k ∈ N : d(ΔmXk,X0) ≥ ε

}∣∣∣ = 0. (4)

Let S(Δm(X)) denote the set of all Δm-statistically convergentsequences of fuzzy numbers.

Definition 2. Let X = (Xk) be a sequence of fuzzy numbers. Afuzzy number X0 is said to be a statistical limit point (s.l.p) of

the generalized difference sequence (ΔmXk) of fuzzy numbersprovided that there is a nonthin subsequence of X that is Δm-convergent to X0.

Let ΛS(Δm(X)) denote the set of all s.l.p. of the general-ized difference sequence (ΔmXk) of fuzzy numbers.

Definition 3. Let X = (Xk) 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 (ΔmXk) of fuzzynumbers provided that, for each ε > 0,

lim supn→∞

1n

∣∣∣{k ∈ N : d(ΔmXk,Y0) < ε

}∣∣∣ > 0. (5)

Let ΓS(Δm(X)) denote the set of all s.c.p of the generalizeddifference sequence (ΔmXk) of fuzzy numbers.

By a lacunary sequence we mean an increasing sequenceθ = (kr) of positive integers such that k0 = 0 and hr = kr −kr−1 → ∞ as r → ∞. The intervals determined by θ = (kr)will be denoted by Ir = (kr−1, kr] whereas the ratio kr/kr−1

is denoted by qr . Further, a lacunary sequence θ′ = (k′r) iscalled a lacunary refinement of the lacunary sequence θ =(kr) if {kr} ⊆ {k′r}.

Definition 4 (see [21]). Let θ = (kr) be a lacunary sequence.A sequence X = (Xk) of fuzzy numbers is said to be lacunarystatistical convergent to a fuzzy number X0 provided that foreach ε > 0,

limr→∞

1hr

∣∣∣{k ∈ Ir : d(Xk,Y0) ≥ ε

}∣∣∣ = 0. (6)

Let Sθ denote the set of all lacunary statistically convergentsequences of fuzzy numbers.

Let θ = (kr) be a lacunary sequence and X = (Xk) asequence of fuzzy numbers. If (X)K where K = {k( j) : j ∈N} is a subsequence of X = (Xk) such that

limr→∞

1hr

∣∣{k

(j) ∈ Ir : j ∈ N}∣∣ = 0, (7)

we call (X)K a θ-thin subsequence. On the other hand, (X)Kis a θ-nonthin subsequence of X provided that

lim supr→∞

1hr

∣∣{k

(j) ∈ Ir : j ∈ N}∣∣ > 0. (8)

Definition 5. Let θ = (kr) be a lacunary sequence. A sequenceX = (Xk) of fuzzy numbers is said to be lacunary Δm-statistically convergent to a fuzzy number X0, in symbol:Sθ(Δm)− limkXk = X0, if for each ε > 0,

limr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,X0) ≥ ε

}∣∣∣ = 0. (9)

Let Sθ(Δm(X)) denote the set of all lacunary Δm-statisticallyconvergent sequences of fuzzy numbers.

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

Advances in Fuzzy Systems 3

3. Main Results

Definition 6. Let θ = (kr) be a lacunary sequence and X =(Xk) a sequence of fuzzy numbers. A fuzzy number X0 issaid to be a lacunary statistical limit point (l.s.l.p) of thegeneralized difference sequence (ΔmXk) of fuzzy numbersprovided that there is a θ-nonthin subsequence of X that isΔm-convergent to X0.

Let ΛSθ (Δm(X)) denote the set of all l.s.l.p. of thegeneralized difference sequence (ΔmXk) of fuzzy numbers.

Definition 7. Let θ = (kr) be a lacunary sequence and X =(Xk) a sequence of fuzzy numbers. A fuzzy number Y0 issaid to be a lacunary statistical cluster point (l.s.c.p) of thegeneralized difference sequence (ΔmXk) of fuzzy numbersprovided that, for each ε > 0,

lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,Y0) < ε

}∣∣∣ > 0. (10)

Let ΓSθ (Δm(X)) denote the set of all l.s.c.p of the generalizeddifference sequence (ΔmXk) of fuzzy numbers.

Example 8. Let θ = (kr) be a lacunary sequence. We define asequence of fuzzy numbers X = (Xk) as follows. For x ∈ R,define

Xk(x) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

x − k + 1, if k − 1 ≤ x ≤ k

−x + k + 1, if k < x ≤ k + 1

0, otherwise

⎫⎪⎪⎬

⎪⎪⎭,

if kr −[√

hr]

+ 1 ≤ k ≤ kr ; r ∈ Nx − 5, if 5 ≤ x ≤ 6

7− x, if 6 < x ≤ 7

0, otherwise

⎫⎪⎪⎬

⎪⎪⎭, otherwise.

(11)

Then, we obtain

[Xk]α =⎧⎨

⎩[k−1+α, k + 1−α], if k ∈

[kr−

[√hr]

+1, kr]

[5 + α, 7− α] otherwise,

[ΔXk]α =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[−3 + α, 1− 2α],

if both k, k + 1 ∈[kr −

[√hr]

+ 1, kr]

[k − 8 + 2α, k − 4− 2α],

if k ∈[kr −

[√hr]

+ 1, kr]

but not k + 1,

[−k + 3 + 2α,−k + 7− 2α],

if k + 1 ∈[kr −

[√hr]

+ 1, kr]

but not k,

[−2 + 2α, 2− 2α], otherwise.

(12)

Thus, for m = 1, it is clear that the sequence ΔXk hastwo 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 ordinarylimit points of (ΔXk), then L(Δ(X)) = {μ1,μ2}; however,ΓSθ (Δ(X)) = {μ2}.

Theorem 9. Let θ = (kr) be a lacunary sequence andX = (Xk) a sequence of fuzzy numbers. Then, one hasΛSθ (Δm(X)) ⊆ ΓSθ (Δm(X)).

Proof. Suppose X0 ∈ ΛSθ (Δm(X)). By definition, there is aθ-nonthin subsequence (Xk( j)) of X = (Xk) which is Δm-convergent to X0, and therefore we have

lim supr→∞

1hr

∣∣{k(j) ∈ Ir : j ∈ N}∣∣ = d > 0. (13)

Since, for every ε > 0,{k∈Ir : d(ΔmXk,X0)<ε

}⊇{k(j)∈Ir : d

(ΔmXk( j),X0

)<ε}

,

(14)

so we have the containment{k ∈ Ir : d(ΔmXk,X0) < ε

}

⊃ {k( j) ∈ Ir : j ∈ N}

−{k(j) ∈ Ir : d

(ΔmXk( j),X0

)≥ ε

}.

(15)

Now, (Xk( j)) is Δm-convergent to X0, which implies that, forevery ε > 0, {k( j) ∈ Ir : d(ΔmXk( j),X0) ≥ ε} is finite forwhich we have

lim supr→∞

1hr

∣∣∣{k(j) ∈ Ir : d

(ΔmXk( j),X0

)≥ ε

}∣∣∣ = 0.

(16)

Thus from (15), we obtain

lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,X0) < ε

}∣∣∣

≥ lim supr→∞

1hr

∣∣{k

(j) ∈ Ir : j ∈ N}∣∣

− lim supr→∞

1hr

∣∣∣{k(j) ∈ Ir : d

(ΔmXk( j),X0

)≥ ε

}∣∣∣

≥ lim supr→∞

1hr

∣∣{k

(j) ∈ Ir : j ∈ N}∣∣ = d > 0,

(17)

using (13) and (16). This shows that X0 ∈ ΓSθ (Δm(X)) andtherefore the result is proved.

Theorem 10. Let θ = (kr) be a lacunary sequence. Then,for any sequence X = (Xk) of fuzzy numbers, one hasΓSθ (Δm(X)) ⊆ L(Δm(X)).

Proof. Assume Y0 ∈ ΓSθ (Δm(X)). By definition, for each ε >0 we have

lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,Y0) < ε

}∣∣∣ > 0. (18)

We set (X)K a θ-nonthin subsequence of X = (Xk) suchthat K = {k( j) ∈ Ir : d(ΔmXk,Y0) < ε} for ε > 0.Since lim supr→∞(1/hr)|{K}| > 0, it follows that K is aninfinite set. Thus we have a subsequence (X)K of X that isΔm-convergent to Y0. This shows that Y0 ∈ L(Δm(X)). HenceΓSθ (Δm(X)) ⊆ L(Δm(X)).


Theorem 11. Let θ = (kr) be a lacunary sequence. If X = (Xk)and Y = (Yk) are two sequences of fuzzy numbers such thatlimr→∞(1/hr)|{k ∈ Ir : Xk /=Yk}| = 0, then ΛSθ (Δm(X)) =ΛSθ (Δm(Y)) and ΓSθ (Δm(X)) = ΓSθ (Δm(Y)).

Proof. We prove the theorem into two parts. In the first partwe prove that ΛSθ (Δm(X)) = ΛSθ (Δm(Y)); however, in thesecond part we shall prove ΓSθ (Δm(X)) = ΓSθ (Δm(Y)).

Part (i). Let X0 ∈ ΛSθ (Δm(Y)). By definition, there isa θ-nonthin subsequence (Y)K of Y = (Yk) that is Δm-convergent to X0. Since limr→∞(1/hr)|{k ∈ Ir : k ∈K and Xk /=Yk}| = 0, it follows that lim supr→∞(1/hr)|{k ∈Ir : k ∈ K and Xk = Yk}| > 0. Therefore, from the laterset, we can yield a θ-nonthin subsequence (X)K ′ of X =(Xk) that is Δm-convergent to X0. Hence, X0 ∈ ΛSθ (Δm(X)),and therefore we have ΛSθ (Δm(Y)) ⊂ ΛSθ (Δm(X)). Alsoby symmetry one get ΛSθ (Δm(X)) ⊂ ΛSθ (Δm(Y)). Oncombining we have ΛSθ (Δm(X)) = ΛSθ (Δm(Y)).

Part (ii). Let Y0 ∈ ΓSθ (Δm(X)). By definition, for eachε > 0,

lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,Y0) < ε

}∣∣∣ > 0. (19)

Since Xk = Yk for all most all k, it follows that, for each ε > 0,

lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmYk,Y0) < ε

}∣∣∣ > 0. (20)

This shows that Y0 ∈ ΓSθ (Δm(Y)) and thereforeΓSθ (Δm(X)) ⊂ ΓSθ (Δm(Y)). By symmetry, we see thatΓSθ (Δm(Y)) ⊂ ΓSθ (Δm(X)), whence ΓSθ (Δm(X)) =ΓSθ (Δm(Y)).

Theorem 12. Let θ = (kr) be a lacunary sequence. If X = (Xk)is a sequence of fuzzy numbers such that Sθ(Δm) − limkXk =X0, then ΛSθ (Δm(X)) = ΓSθ (Δm(X)) = {X0}.

Proof. We prove the theorem in two parts. In the first part, weprove that ΛSθ (Δm(X)) = {X0}whereas in the second part weobtain ΓSθ (Δm(X)) = {X0}.

Part (i). Suppose that ΛSθ (Δm(X)) = {X0,Y0}, whereX0 /=Y0, that is, Y0 is a l.s.l.p. of the generalized differencesequence (ΔmXk) different from X0. Choose ε > 0 such that0 < ε < d(X0,Y0)/2. By definition there exist two θ-nonthinsubsequences (Xk( j)) and (Xl(i)) of the sequence X = (Xk)which are Δm-convergent to X0 and Y0, respectively. Since(Xl(i)) is Δm-convergent to Y0, so for each ε > 0, {l(i) ∈ In :d(ΔmXl(i),Y0) ≥ ε} is a finite set for which

lim supr→∞

1hr

∣∣∣{l(i) ∈ Ir : d

(ΔmXl(i),Y0

) ≥ ε}∣∣∣ = 0. (21)

Further, we can write

{l(i) ∈ Ir : i ∈ N} ={l(i) ∈ Ir : d

(ΔmXl(i),Y0

)< ε}

∪{l(i) ∈ Ir : d

(ΔmXl(i),Y0

) ≥ ε}

,

(22)

for which we have

lim supr→∞

1hr|{l(i) ∈ Ir : i ∈ N}|

= lim supr→∞

1hr

∣∣∣{l(i) ∈ Ir : d

(ΔmXl(i),Y0

)< ε}∣∣∣

+ lim supr→∞

1hr

∣∣∣{l(i) ∈ Ir : d

(ΔmXl(i),Y0

) ≥ ε}∣∣∣.

(23)

Since (Xl(i)) is nonthin, so, by use of (21), we have

lim supr→∞

1hr

∣∣∣{l(i) ∈ Ir : d

(ΔmXl(i),Y0

)< ε}∣∣∣ > 0. (24)

Since Sθ(Δm)− limkXk = X0, so for each ε > 0

limr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,X0) ≥ ε

}∣∣∣ = 0, (25)

and therefore we can write

lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,X0) < ε

}∣∣∣ > 0. (26)

Furthermore for 0 < 2ε < d(X0,Y0),{l(i) ∈ Ir : d

(ΔmXl(i),Y0

)< ε}

∩{k ∈ Ir : d(ΔmXk,X0) < ε

}= ∅,

(27)

which immediately gives the containment{l(i)∈Ir : d

(ΔmXl(i),Y0

)<ε}⊂{k∈Ir : d(ΔmXk,X0)≥ε

}

(28)

for which we have

lim supr→∞

1hr

∣∣∣{l(i) ∈ Ir : d

(ΔmXl(i),Y0

)< ε}∣∣∣

≤ lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,X0) ≥ ε

}∣∣∣ = 0.

(29)

As left side of (29) cannot be negative, so we must have

lim supr→∞

1hr

∣∣∣{l(i) ∈ Ir : d

(ΔmXl(i),Y0

)< ε}∣∣∣ = 0. (30)

This contradicts (24). Hence, ΛSθ (Δm(X)) = {X0}.Part (ii). Let Z0 be a l.s.c.p. of the generalized difference

sequence (ΔmXk) different from X0, that is, ΓSθ (Δm(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 (ΔmXk), so for each ε > 0we have

lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,Z0) < ε

}∣∣∣ > 0. (31)

Since {k ∈ Ir : d(ΔmXk,X0) < ε} ∩ {k ∈ Ir : d(ΔmXk,Z0) <ε} = ∅ for every 0 < ε < d(X0,Z0)/2, it follows that

Advances in Fuzzy Systems 5

{k ∈ Ir : d(ΔmXk,X0) ≥ ε} ⊇ {k ∈ Ir : d(ΔmXk,Z0) < ε} forwhich we have

lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,X0) ≥ ε

}∣∣∣

≥ lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,Z0) < ε

}∣∣∣ > 0

(32)

by (31), which is impossible as by (25)lim supr→∞(1/hr)|{k ∈ Ir : d(ΔmXk,X0) ≥ ε}| = 0.In this way we obtained a contradiction. Hence,ΓSθ (Δm(X)) = {X0}.

Theorem 13. Let θ = (kr) be a lacunary sequence and X =(Xk) a sequence of fuzzy numbers. Then one has the following:

(i) if lim inf qr > 1, then ΛSθ (Δm(X)) ⊂ ΛS(Δm(X)),

(ii) if lim sup qr <∞, then ΛS(Δm(X)) ⊂ ΛSθ (Δm(X)),

(iii) if 1 < lim inf qr ≤ lim sup qr < ∞, thenΛS(Δm(X)) = ΛSθ (Δm(X)).

Proof. (i) Suppose lim inf qr > 1; there exists a δ > 0 suchthat qr > 1+δ for sufficient large r, which implies that hr/kr ≥δ/(1 + δ). Assume that X0 ∈ ΛSθ (Δm(X)), then there is θ-nonthin subsequence (Xk( j)) of (Xk) that is Δm-convergentto X0 and

lim supr→∞

1hr

∣∣{k

(j) ∈ Ir : j ∈ N}∣∣ = d > 0. (33)

Since

1kr

∣∣{k(j) ≤ kr : j ∈ N}∣∣

≥ 1kr

∣∣{k(j) ∈ Ir : j ∈ N}∣∣

= hrkr

1hr

∣∣{k

(j) ∈ Ir : j ∈ N}∣∣

≥(

δ

1 + δ

)1hr

∣∣{k(j) ∈ Ir : j ∈ N}∣∣,

(34)

it follows by (33) that lim supr→∞(1/kr)|{k( j) ≤ kr : j ∈N}| > 0. Since (Xk( j)) is already Δm-convergent to X0, so wehave X0 ∈ ΛS(Δm(X)). Hence ΛSθ (Δm(X)) ⊂ ΛS(Δm(X)).

(ii) If lim sup qr < ∞, then there exists a real numberH such that qr < H for all r. Without loss of generality,we can assume H > 1 (as otherwise kr < kr−1). Nowfor all r, (hr/kr−1) = (kr − kr−1)/kr−1 = qr − 1 ≤H − 1. Let X0 ∈ ΛS(Δm(X)), then there is a set K ={k( j) : j ∈ N} with δ(K) /= 0 and lim jΔmXk( j) = X0.

Let Nr = |{k ∈ Ir : k ∈ K}| = |K ∩ Ir| and tr = Nr/hr .For any integer n satisfying kr−1 < n ≤ kr , we can write

1n|{k ≤ n : k ∈ K}| ≤ 1

kr−1|{k ≤ kr : k ∈ K}|

= 1kr−1

{N1 + N2 + · · · + Nr}

= 1kr−1

{t1h1 + t2h2 + · · · trhr}

= 1∑r−1

i=1 hi

r−1∑

i=1

hiti +hrkr−1

tr

≤ 1∑r−1

i=1 hi

r−1∑

i=1

hiti + (H − 1)tr .

(35)

Suppose tr → 0 as r → ∞. Since θ is a lacunary sequence andthe first part on the right side of above expression is a regularweighted mean transform of the sequence t = (tr), thereforeit too tends to zero as r → ∞. Since n → ∞ as r → ∞, itfollows that δ(K) = 0 which is a contradiction as δ(K) /= 0.Thus limr→∞tr /= 0 and therefore X0 ∈ ΛSθ (Δm(X)). HenceΛS(Δm(X)) ⊂ ΛSθ (Δm(X)).

(iii) This is an immediate consequence of (i) and (ii).

Theorem 14. Let θ = (kr) be a lacunary sequence and X =(Xk) a sequence of fuzzy numbers. Then one has the following:

(i) if lim inf qr > 1, then ΓSθ (Δm(X)) ⊂ ΓS(Δm(X)),

(ii) if lim sup qr <∞, then ΓS(Δm(X)) ⊂ ΓSθ (Δm(X)),

(iii) if 1 < lim inf qr ≤ lim sup qr <∞, then ΓS(Δm(X)) =ΓSθ (Δm(X)).

Proof. The proof of the theorem can be obtain on the similarlines as that of the above theorem and therefore is omittedhere.

Theorem 15. For any lacunary refinement θ′ of a lacunarysequence θ = (kr), ΓSθ (Δm(X)) ⊆ ΓSθ′ (Δ

m(X)) andΛSθ (Δm(X)) ⊆ ΛSθ′ (Δ

m(X)).

Proof. Suppose each Ir of θ contains the points {k′r,i}v(r)i=1 of

θ′ so that kr−1 < k′r,1 < k′r,2 < · · · < k′r,v(r) = kr , whereI′r = (k′r,i−1, k′r,i]. Note that for all r, v(r) ≥ 1. Let (I∗j )∞j=1 bethe sequence of abutting intervals I′r,i ordered by increasingright end points. Let Y0 ∈ ΓSθ (Δm(X)), then for each ε > 0,

lim supr→∞

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,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. Nowfor each ε > 0, we can write

1hr

∣∣∣{k ∈ Ir : d(ΔmXk,Y0) < ε

}∣∣∣

= 1hr

∑

I∗j ⊆Irh∗j

1h∗j

∣∣∣{k ∈ I∗j : d(ΔmXk,Y0) < ε

}∣∣∣

= 1hr

∑

I∗j ⊆Irh∗j(Cθ′χK

)

j,

(37)

where χK is the characteristics function of the set K ={k ∈ I∗j : d(ΔmXk,Y0) < ε} and (Cθ′χK ) j = |K ∩ I∗j |/h∗j .Suppose lim j→∞(Cθ′χK ) j = 0. Then the right side of aboveexpression is a regular weighted mean transform of (Cθ′χK ) jand therefore tends to zero as j → ∞ which contradicts(36). Thus lim j→∞(Cθ′χK ) j /= 0, which shows that Y0 ∈ΓSθ′ (Δ

m(X)). Hence ΓSθ (Δm(X)) ⊆ ΓSθ′ (Δm(X)).

Similarly, we can prove ΛSθ (Δm(X)) ⊆ ΛSθ′ (Δm(X)).

Acknowledgment

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

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