AFS Advances in Fuzzy Systems 1687-711X 1687-7101 Hindawi Publishing Corporation 459370 10.1155/2012/459370 459370 Research Article Lacunary Statistical Limit and Cluster Points of Generalized Difference Sequences of Fuzzy Numbers Kumar Pankaj 1 Kumar Vijay 1 Bhatia S. S. 2 Honda Katsuhiro 1 Department of Mathematics Haryana College of Technology and Management Haryana Kaithal 136027 India hctmkaithal-edu.org 2 School of Mathematics and Computer Application Thapar University Punjab, Patiala 147004 India thapar.edu 2012 18 7 2012 2012 20 04 2012 14 06 2012 2012 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 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  and Schoenberg  independently and latter discussed in , and so forth. In 1993, Fridy and Orhan  presented an interesting generalization of statistical convergence with the help of a lacunary sequence and called it lacunary statistical convergence or Sθ-convergence. Demirci  defined Sθ-limit and cluster points of number sequences and obtained some interesting results analogous to . 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  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 , etc.).

Quite recently, statistical convergence of sequences of fuzzy numbers is studied with the help of the difference operator Δ. For instance, Bilgin  introduced strongly Δ-summable and Δ-statistical convergence of sequences of fuzzy numbers. Işik  studied some notions of generalized difference sequences of numbers. In 2006, Altin et al.  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 . 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 , that is, D={A:A=[A_,A¯]}. For A,BD we define AB 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 to [0,1] which is satisfying the following conditions: (i) X is normal, that is, there exists x0 such that X(x0)=1; (ii) X is fuzzy convex, that is, for any x,y 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:X(x)>0} denoted by X0 is compact.

Properties (i)–(iv) imply that for each α(0,1], the α-level set, Xα={x:X(x)α}=[X_α,X¯α], is a nonempty compact convex subset of . Let L() denote the set of all fuzzy numbers. The linear structure of L() induces an addition X+Y and a scalar multiplication λX in terms of α-level sets by (1)[X+Y]α=[X]α+[Y]α,[λX]α=λ[X]α(X,YL(R),λR) for each α[0,1]. Define a map d¯:L()×L() by (2)d¯(X,Y)=supα[0,1]d(Xα,Yα). Puri and Ralescu  proved that (L(),d¯) is a complete metric space. Also the ordered structure on L() is defined as follows. For X,YL(), we define XY if and only if X_αY_α and X¯αY¯α for each α[0,1]. We say that X<Y if XY 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 XY nor YX.

We next recall some definitions and results which form the base for present study. For any set K, let Kn denote the set {kK:kn} and |Kn| denote the number of elements in Kn. The natural density δ of K is defined by δ(K)=limnn-1|Kn|. The natural density may not exist for each set K. But the upper density δ¯ defined by δ¯(K)=limsupnn-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 AB, then δ¯(A)δ¯(B).

For any sequence X=(Xk) of fuzzy numbers, we write {Xk:k} to denote the range of X. If (Xk(j)) is a subsequence of X and K={k(j):j}, then we abbreviate (Xk(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 Δm:ww is defined by (3)Δ0Xk=Xk,Δ1Xk=Xk-Xk+1,=ΔmXk=Δ1(Δm-1Xk).

Definition 1.

A sequence X=(Xk) of fuzzy numbers is said to be Δm-statistically convergent to a fuzzy number X0, in symbol: S(Δm)-limkXk=X0, if for each ϵ>0, (4)limn1n|{kN:d¯(ΔmXk,X0)ϵ}|=0. Let S(Δm(X)) denote the set of all Δm-statistically convergent sequences of fuzzy numbers.

Definition 2.

Let X=(Xk) 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 (ΔmXk) of fuzzy numbers provided 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 generalized 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 fuzzy numbers provided that, for each ϵ>0, (5)limsupn1n|{kN:d¯(ΔmXk,Y0)<ϵ}|>0. Let ΓS(Δm(X)) denote the set of all s.c.p of the generalized difference sequence (ΔmXk) of fuzzy numbers.

By a lacunary sequence we mean an increasing sequence θ=(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=(kr) is called a lacunary refinement of the lacunary sequence θ=(kr) if {kr}{kr}.

Definition 4 (see [<xref ref-type="bibr" rid="B21">21</xref>]).

Let θ=(kr) be a lacunary sequence. A sequence X=(Xk) of fuzzy numbers is said to be lacunary statistical convergent to a fuzzy number X0 provided that for each ϵ>0, (6)limr1hr|{kIr:d¯(Xk,Y0)ϵ}|=0. Let Sθ denote the set of all lacunary statistically convergent sequences of fuzzy numbers.

Let θ=(kr) be a lacunary sequence and X=(Xk) a sequence of fuzzy numbers. If (X)K where K={k(j):j} is a subsequence of X=(Xk) such that (7)limr1hr|{k(j)Ir:jN}|=0, we call (X)K a θ-thin subsequence. On the other hand, (X)K is a θ-nonthin subsequence of X provided that (8)limsupr1hr|{k(j)Ir:jN}|>0.

Definition 5.

Let θ=(kr) be a lacunary sequence. A sequence X=(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, (9)limr1hr|{kIr:d¯(ΔmXk,X0)ϵ}|=0. Let Sθ(Δm(X)) denote the set of all lacunary Δm-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=(kr) be a lacunary sequence and X=(Xk) 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 (ΔmXk) of fuzzy numbers provided that there is a θ-nonthin subsequence of X that is Δm-convergent to X0.

Let ΛSθ(Δm(X)) denote the set of all l.s.l.p. of the generalized 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 is said to be a lacunary statistical cluster point (l.s.c.p) of the generalized difference sequence (ΔmXk) of fuzzy numbers provided that, for each ϵ>0, (10)limsupr1hr|{kIr:d¯(ΔmXk,Y0)<ϵ}|>0. Let ΓSθ(Δm(X)) denote the set of all l.s.c.p of the generalized difference sequence (ΔmXk) of fuzzy numbers.

Example 8.

Let θ=(kr) be a lacunary sequence. We define a sequence of fuzzy numbers X=(Xk) as follows. For x, define (11)Xk(x)={x-k+1,ifk-1xk-x+k+1,ifk<xk+10,otherwise},ifkr-[hr]+1kkr;  rNx-5,if5x67-x,if6<x70,otherwise},otherwise. Then, we obtain (12)[Xk]α={[k-1+α,k+1-α],ifk[kr-[hr]+1,kr][5+α,7-α]otherwise,[ΔXk]α={[-3+α,1-2α],ifbothk,k+1[kr-[hr]+1,kr][k-8+2α,k-4-2α],ifk[kr-[hr]+1,kr]butnotk+1,[-k+3+2α,-k+7-2α],ifk+1[kr-[hr]+1,kr]butnotk,[-2+2α,2-2α],otherwise. Thus, for m=1, it is clear that the sequence ΔXk 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 (ΔXk), then L(Δ(X))={μ1,μ2}; however, ΓSθ(Δ(X))={μ2}.

Theorem 9.

Let θ=(kr) be a lacunary sequence and X=(Xk) a sequence of fuzzy numbers. Then, one has ΛSθ(Δm(X))ΓSθ(Δm(X)).

Proof.

Suppose X0ΛSθ(Δm(X)). By definition, there is a θ-nonthin subsequence (Xk(j)) of X=(Xk) which is Δm-convergent to X0, and therefore we have (13)limsupr1hr|{k(j)Ir:jN}|=d>0. Since, for every ϵ>0, (14){kIr:d¯(ΔmXk,X0)<ϵ}{k(j)Ir:d¯(ΔmXk(j),X0)<ϵ}, so we have the containment (15){kIr:d¯(ΔmXk,X0)<ϵ}{k(j)Ir:jN}-{k(j)Ir:d¯(ΔmXk(j),X0)ϵ}. Now, (Xk(j)) is Δm-convergent to X0, which implies that, for every ϵ>0, {k(j)Ir:d¯(ΔmXk(j),X0)ϵ} is finite for which we have (16)limsupr1hr|{k(j)Ir:d¯(ΔmXk(j),X0)ϵ}|=0. Thus from (15), we obtain (17)limsupr1hr|{kIr:d¯(ΔmXk,X0)<ϵ}|limsupr1hr|{k(j)Ir:jN}|-limsupr1hr|{k(j)Ir:d¯(ΔmXk(j),X0)ϵ}|limsupr1hr|{k(j)Ir:jN}|=d>0, using (13) and (16). This shows that X0ΓSθ(Δm(X)) and therefore the result is proved.

Theorem 10.

Let θ=(kr) be a lacunary sequence. Then, for any sequence X=(Xk) of fuzzy numbers, one has ΓSθ(Δm(X))L(Δm(X)).

Proof.

Assume Y0ΓSθ(Δm(X)). By definition, for each ϵ>0 we have (18)limsupr1hr|{kIr:d¯(ΔmXk,Y0)<ϵ}|>0. We set (X)K a θ-nonthin subsequence of X=(Xk) such that K={k(j)Ir:d¯(ΔmXk,Y0)<ϵ} for ϵ>0. Since limsupr(1/hr)|{K}|>0, it follows that K is an infinite set. Thus we have a subsequence (X)K of X that is Δm-convergent to Y0. This shows that Y0L(Δm(X)). Hence ΓSθ(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 that limr(1/hr)|{kIr:XkYk}|=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 part we prove that ΛSθ(Δm(X))=ΛSθ(Δm(Y)); however, in the second part we shall prove ΓSθ(Δm(X))=ΓSθ(Δm(Y)).

Part (i). Let X0ΛSθ(Δm(Y)). By definition, there is a θ-nonthin subsequence (Y)K of Y=(Yk) that is Δm-convergent to X0. Since limr(1/hr)|{kIr:kKandXkYk}|=0, it follows that limsupr(1/hr)|{kIr:kKandXk=Yk}|>0. Therefore, from the later set, we can yield a θ-nonthin subsequence (X)K of X=(Xk) that is Δm-convergent to X0. Hence, X0ΛSθ(Δm(X)), and therefore we have ΛSθ(Δm(Y))ΛSθ(Δm(X)). Also by symmetry one get ΛSθ(Δm(X))ΛSθ(Δm(Y)). On combining we have ΛSθ(Δm(X))=ΛSθ(Δm(Y)).

Part (ii). Let Y0ΓSθ(Δm(X)). By definition, for each ϵ>0, (19)limsupr1hr|{kIr:d¯(ΔmXk,Y0)<ϵ}|>0. Since Xk=Yk for all most all k, it follows that, for each ϵ>0, (20)limsupr1hr|{kIr:d¯(ΔmYk,Y0)<ϵ}|>0. This shows that Y0ΓSθ(Δm(Y)) and therefore ΓSθ(Δm(X))ΓSθ(Δm(Y)). By symmetry, we see that ΓSθ(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, we prove that ΛSθ(Δm(X))={X0} whereas in the second part we obtain ΓSθ(Δm(X))={X0}.

Part (i). Suppose that ΛSθ(Δm(X))={X0,Y0}, where X0Y0, that is, Y0 is a l.s.l.p. of the generalized difference sequence (ΔmXk) different from X0. Choose ϵ>0 such that 0<ϵ<d¯(X0,Y0)/2. By definition there exist two θ-nonthin subsequences (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 (21)limsupr1hr|{l(i)Ir:d¯(ΔmXl(i),Y0)ϵ}|=0. Further, we can write (22){l(i)Ir:iN}={l(i)Ir:d¯(ΔmXl(i),Y0)<ϵ}{l(i)Ir:d¯(ΔmXl(i),Y0)ϵ}, for which we have (23)limsupr1hr|{l(i)Ir:iN}|=limsupr1hr|{l(i)Ir:d¯(ΔmXl(i),Y0)<ϵ}|+limsupr1hr|{l(i)Ir:d¯(ΔmXl(i),Y0)ϵ}|. Since (Xl(i)) is nonthin, so, by use of (21), we have (24)limsupr1hr|{l(i)Ir:d¯(ΔmXl(i),Y0)<ϵ}|>0. Since Sθ(Δm)-limkXk=X0, so for each ϵ>0(25)limr1hr|{kIr:d¯(ΔmXk,X0)ϵ}|=0, and therefore we can write (26)limsupr1hr|{kIr:d¯(ΔmXk,X0)<ϵ}|>0. Furthermore for 0<2ϵ<d¯(X0,Y0), (27){l(i)Ir:d¯(ΔmXl(i),Y0)<ϵ}{kIr:d¯(ΔmXk,X0)<ϵ}=, which immediately gives the containment (28){l(i)Ir:d¯(ΔmXl(i),Y0)<ϵ}{kIr:d¯(ΔmXk,X0)ϵ} for which we have (29)limsupr1hr|{l(i)Ir:d¯(ΔmXl(i),Y0)<ϵ}|limsupr1hr|{kIr:d¯(ΔmXk,X0)ϵ}|=0. As left side of (29) cannot be negative, so we must have (30)limsupr1hr|{l(i)Ir:d¯(ΔmXl(i),Y0)<ϵ}|=0. 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 X0Z0. Choose ϵ such that 0<ϵ<d¯(X0,Z0)/2. Since Z0 is a l.s.c.p of (ΔmXk), so for each ϵ>0 we have (31)limsupr1hr|{kIr:d¯(ΔmXk,Z0)<ϵ}|>0. Since {kIr:d¯(ΔmXk,X0)<ϵ}{kIr:d¯(ΔmXk,Z0)<ϵ}= for every 0<ϵ<d(X0,Z0)/2, it follows that {kIr:d¯(ΔmXk,X0)ϵ}{kIr:d¯(ΔmXk,Z0)<ϵ} for which we have (32)limsupr1hr|{kIr:d¯(ΔmXk,X0)ϵ}|limsupr1hr|{kIr:d¯(ΔmXk,Z0)<ϵ}|>0 by (31), which is impossible as by (25) limsupr(1/hr)|{kIr: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:

if  liminfqr>1, then ΛSθ(Δm(X))ΛS(Δm(X)),

if  limsupqr<, then ΛS(Δm(X))ΛSθ(Δm(X)),

if  1<liminfqrlimsupqr<, then ΛS(Δm(X))=ΛSθ(Δm(X)).

Proof.

(i) Suppose liminfqr>1; there exists a δ>0 such that qr>1+δ for sufficient large r, which implies that hr/krδ/(1+δ). Assume that X0ΛSθ(Δm(X)), then there is θ-nonthin subsequence (Xk(j)) of (Xk) that is Δm-convergent to X0 and (33)limsupr1hr|{k(j)Ir:jN}|=d>0. Since (34)1kr|{k(j)kr:jN}|1kr|{k(j)Ir:jN}|=hrkr1hr|{k(j)Ir:jN}|(δ1+δ)1hr|{k(j)Ir:jN}|, it follows by (33) that limsupr(1/kr)|{k(j)kr:j}|>0. Since (Xk(j)) is already Δm-convergent to X0, so we have X0ΛS(Δm(X)). Hence ΛSθ(Δm(X))ΛS(Δm(X)).

(ii) If limsupqr<, then there exists a real number H such that qr<H for all r. Without loss of generality, we can assume H>1 (as otherwise kr<kr-1). Now for all r,(hr/kr-1)=(kr-kr-1)/kr-1=qr-1H-1. Let X0ΛS(Δm(X)), then there is a set K={k(j):j} with δ(K)0 and limjΔmXk(j)=X0. Let Nr=|{kIr:kK}|=|KIr| and tr=Nr/hr. For any integer n satisfying kr-1<nkr, we can write (35)1n|{kn:kK}|1kr-1|{kkr:kK}|=1kr-1{N1+N2++Nr}=1kr-1{t1h1+t2h2+trhr}=1i=1r-1hii=1r-1hiti+hrkr-1tr1i=1r-1hii=1r-1hiti+(H-1)tr. Suppose tr0 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=(tr), 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 limrtr0 and therefore X0ΛSθ(Δm(X)). Hence ΛS(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:

if  liminfqr>1, then ΓSθ(Δm(X))ΓS(Δm(X)),

if  limsupqr<, then ΓS(Δm(X))ΓSθ(Δm(X)),

if  1<liminfqrlimsupqr<, then ΓS(Δm(X))=ΓSθ(Δm(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=(kr), ΓSθ(Δm(X))ΓSθ(Δm(X)) and ΛSθ(Δm(X))ΛSθ(Δm(X)).

Proof.

Suppose each Ir of θ contains the points {kr,i}i=1v(r) of θ so that kr-1<kr,1<kr,2<<kr,v(r)=kr, where Ir=(kr,i-1,kr,i]. Note that for all r, v(r)1. Let (Ij*)j=1 be the sequence of abutting intervals Ir,i ordered by increasing right end points. Let Y0ΓSθ(Δm(X)), then for each ϵ>0, (36)limsupr1hr|{kIr:d¯(ΔmXk,Y0)<ϵ}|>0. As before, write hr,i=kr,i-kr,i-1 and hr,1=kr,1-kr-1. Now for each ϵ>0, we can write (37)1hr|{kIr:d¯(ΔmXk,Y0)<ϵ}|=1hrIj*Irhj*1hj*|{kIj*:d¯(ΔmXk,Y0)<ϵ}|=1hrIj*Irhj*(CθχK)j, where χK is the characteristics function of the set K={kIj*:d¯(ΔmXk,Y0)<ϵ} and (CθχK)j=|KIj*|/hj*. Suppose limj(CθχK)j=0. Then the right side of above expression is a regular weighted mean transform of (CθχK)j and therefore tends to zero as j which contradicts (36). Thus limj(CθχK)j0, which shows that Y0ΓSθ(Δm(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 valuable suggestions which improved the readability of the paper.

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