Asymptotically Lacunary Statistical Equivalent Sequences of Fuzzy

Bol. Soc. Paran. Mat. c

SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm

(3s.) v. 30 2 (2012): 57–62. ISSN-00378712 in press doi:10.5269/bspm.v30i2.14007

Asymptotically Lacunary Statistical Equivalent Sequences of Fuzzy Numbers Ayhan Esi and Necdet Çatalbaş

abstract:

In this article we present the following definition which is natural combination of the definition for asymptotically equivalent and lacunary statistical convergence of fuzzy numbers. Let θ = (kr ) be a lacunary sequence. The two sequnces X = (Xk ) and Y = (Yk ) of fuzzy numbers are said to be asymptotically lacunary statistical equivalent to multiple L provided that for every ε > 0 1 Xk k ∈ Ir : d , L ≥ ε = 0. lim r hr Yk

Key Words: Asymptotically equivalent, lacunary sequence, fuzzy numbers. Contents

1 Introduction

57

2 Definitions and Notations

58

3 Main Results

59 1. Introduction

In 1993, Marouf [2] presented definitions for asymptotically equivalent sequences of real numbers. In 2003, Patterson [4] extended these definitions by presenting on asymptotically statistical equivalent analog of these definitions. For sequences of fuzzy numbers Savaş [5] introduced and studied asymptotically λ−statistical equivalent sequences. Later Esi and Esi [1] studied ∆−asymptotically equivalent sequences of fuzzy numbers. The goal of this paper is to extended the idea to apply to asymptotically equivalent and lacunary statistical convergence of fuzzy numbers. By a lacunary sequence θ = (kr ) ; r = 0, 1, 2, 3, ... where k0 = 0 , we shall mean an increasing sequence of nonnegative integers with hr = kr − kr−1 → ∞ as r → ∞. The intervals determined by θ will be denoted by Ir = (kr−1 , kr ] . the ratio kr kr−1 will be denoted by qr . Let D denote the set of all closed and bounded intervals on R , the real line. For X, Y ∈ D we define d (X, Y ) = max (|a1 − b1 | , |a2 − b2 |) where X = [a1 , a2 ] and Y = [b1 , b2 ] . It is known that (D, d) is a complete metric space. A fuzzy real number X is a fuzzy set on R , i.e. a mapping X :R → I (= [0, 1]) associating each real number t with its grade of membership X (t) . 2000 Mathematics Subject Classification: 40A05, 40C05, 46D25

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Typeset by BSM P style. c Soc. Paran. de Mat.

58

Ayhan Esi and Necdet Çatalbaş

The set of all upper - semi continous, normal,convex fuzzy real numbers is denoted by R (I) . Throughout the paper, by a fuzzy real number X ,we mean that X ∈ R (I) . α The α− cut or α− level set [X] of the fuzzy real number X , for 0 < α ≤ 1 α defined by [X] = {t ∈ R : X (t) ≥ α} ; for α = 0 , it is the closure of the strong 0−cut , i.e. closure of the set {t ∈ R : X (t) > 0} . The linear structure of R (I) induces addition X + Y and scalar multipliction µX , µ ∈ R, in terms of α− level α α α α α set , by [X + Y ] = [X] + [Y ] , [µX] = µ [X] for each α ∈ [0, 1]. Let d : R (I) × R (I) → R be defined by

α

α

d (X, Y ) = sup d ([X] , [Y ] ) . α∈(0,1]

Then d defines a metric on R (I) . It is well known that R (I) is complete with respect to d. A sequence (Xk ) of fuzzy real numbers is said to be convergent to the fuzzy real numbers X0 ,if for every ε > 0 , there exists n0 ∈ N such that d (Xk , X0 ) < ε for all k ≥ n0 .Nuray and Savaş [3] defined the notion of statistical convergence for sequences of fuzzy numbers as follows :A sequence of fuzzy numbers is said to be statistically convergent to the fuzzy real number X0 , if for evey ε > 0 lim

n→∞

1 k ≤ n : d (Xk , X0 ) ≥ ε = 0. n

2. Definitions and Notations

Definition 2.1 Let θ = (kr ) be lacunary sequence. The two sequences X = (Xk ) and Y = (Yk ) of fuzzy numbers are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every ε ≻ 0, 1 Xk lim k ∈ Ir : d , L ≥ ε = 0. r hr Yk S L (F ) denoted by X θ∼ Y and simply asymptotically lacunary statistical equivalent if L = 1 (where 1 is unity element for addition in R (I) ). Furthermore, let SθL (F ) denotes the set of X = (Xk ) and Y = (Yk ) of fuzzy numbers such that X

SθL (F )

∼

Y.

Definition 2.2 Let θ = (kr ) be lacunary sequence. The two sequences X = (Xk ) and Y = (Yk ) of fuzzy numbers are strong asymptotically lacunary statistical equivalent of multiple L provided that 1 X Xk lim , L = 0, d r hr Yk k∈Ir

59

Fuzzy Numbers

denoted by X

NθL (F )

∼

Y

and simply strong asymptotically lacunary statistical equiv-

alent if L = 1. In addition, let NθL (F ) denotes the set of X = (Xk ) and Y = (Yk ) of fuzzy numbers such that X

NθL (F )

∼

Y.

3. Main Results Theorem 3.1 Let θ = (kr ) be lacunary sequence. Then (a) If X

NθL (F )

∼

SθL (F )

Y then X

∼

Y, N L (F )

SθL (F )

(b) if X ∈ ℓ∞ (F ) and X ∼ Y then X θ∼ Y, (c) SθL (F ) ∩ ℓ∞ (F ) = NθL (F ) ∩ ℓ∞ (F ) where ℓ∞ (F ) the set of all bounded sequences of fuzzy numbers. Proof: (a) If ε > 0 and X X

d

k∈Ir

Therefore X

NθL (F )

∼

Y, then

Xk ,L ≥ Yk k∈Ir

SθL (F )

∼

Y.

&

X

X d Y k ,L ≥ε

d

Xk ,L Yk

k

Xk , L ≥ ε . ≥ ε. k ∈ Ir : d Yk

(b) Suppose that = (Xk ) and Y = (Yk ) ∈ ℓ∞ (F ) and X X Xk we can assume that d Yk , L ≤ T for all k.Given ε > 0 Xk 1 1 X d ,L = hr Yk hr k∈Ir

≤

X

Xk ,L ≥ε k∈Ir & d Yk

d

Xk 1 ,L + Yk hr

X

SθL (F )

∼

d

Y.Then

Xk ,L Yk

Xk ,L 1, then X SθL (F )

2 SL (F )

∼

Y implies X ∼ Y, where S L (F ) (see [5]) the set of X = (Xk ) and Y = (Yk ) of fuzzy numbers such that 1 Xk , L ≥ ε = 0. lim k ≤ n : d n n Yk

60


Proof: Suppose that lim inf r qr > 1, then there exists a δ > 0 such that qr ≥ 1 + δ for sufficiently large r, which implies

δ hr ≥ kr 1+δ S L (F )

if X

∼

Y , then for every ε > 0 and for sufficiently large r, we have

Xk Xk 1 1 d d ≥ k ≤ k : k ∈ I : , L ≥ ε , L ≥ ε r r kr Yk kr Yk 1 δ Xk . k ∈ Ir : d , L ≥ ε . ≥ 1 + δ hr Yk This complete the proof.

2

Theorem 3.3 Let θ = (kr ) be lacunary sequence with lim supr qr < ∞ , then X

SθL (F )

∼

Y implies X

SL (F )

∼

Y.

Proof: If lim supr qr < ∞,then there exists B > 0 such that qr < C for all r ≥ 1.Let X

SθL (F )

∼

Y and ε > 0.There exists B > 0 such that for every j ≥ B

AJ =

Xk 1 < ε. d , L ≥ ε k ∈ I : j hj Yk

we can also find K > 0 such that Aj < K for all j = 1, 2, 3, .... Now let n be any integer with kr−1 < n < kr ,where r ≥ B. Then

Xk 1 1 , L ≥ ε ≤ k≤n:d n Yk kr−1

k ≤ kr : d Xk , L ≥ ε Yk

Fuzzy Numbers

61

Xk Xk 1 ,L ≥ ε + , L ≥ ε = k ∈ I1 : d k ∈ I2 : d kr−1 Yk kr−1 Yk Xk 1 k ∈ Ir : d , L ≥ ε + ... + kr−1 Yk k2 − k1 k1 Xk Xk k ∈ I1 : d k ∈ I2 : d = ,L ≥ ε + , L ≥ ε kr−1 k1 Yk kr−1 (k2 − k1 ) Yk kB − kB−1 Xk + ... + k ∈ IB : d , L ≥ ε kr−1 (kB − kB−1 ) Yk kr − kr−1 Xk k ∈ Ir : d , L ≥ ε + ... + kr−1 (kr − kr−1 ) Yk k2 − k1 kB − kB−1 kr − kr−1 k1 A1 + A2 + ... + AB + ... + Ar = kr−1 kr−1 kr−1 kr−1 kB kr − kB ≤ sup Aj + sup Aj kr−1 kr−1 j≥1 jB kB ≤ K. + ε.C. kr−1 1

This complete the proof.

2

Theorem 3.4 Let θ = (kr ) be lacunary sequence 1 < lim inf r qr ≤ lim supr qr < ∞, then X

SL (F )

∼

Y ⇔X

SθL (F )

∼

Y.

Proof: The result follow from Theorem 3.2 and Theorem 3.3.

2

References 1. Esi, A. and Esi, A., On ∆−asymptotically equivalent sequences of fuzzy numbers, International Journal of Mathematics and Computation, 1(8)(2008), 29-35. 2. Marouf , M., Asymptotic equivalence and summability,Inf. J. Math.Sci.,16 (1993),755 -762. 3. Nuray, F. and Savaş, E., Statistical convergence of sequences of fuzzy real numbers, Math. Slovaca , 45(3) (1995), 269 -273. 4. Patterson , R.F., On asymptotically statistically equivalent sequences, Demonstratio Math., 36 (1) (2003), 149 -153. 5. Savaş , E ., On asymptotically λ−statistical equivalent sequence of fuzzy numbers, New Mathematics and Natural Computation, 3(3) (2007), 301 -306.

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Ayhan Esi Adiyaman University, Science and Art Faculty Department of Mathematics 02040,Adiyaman,Turkey. E-mail address: [email protected] and Necdet Çatalbaş Firat University, Science and Art Faculty Department of Mathematics 23199, Elazığ, Turkey. E-mail address: [email protected]