* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

# Download Asymptotically Lacunary Statistical Equivalent Sequences of Fuzzy

Survey

Document related concepts

Mathematical proof wikipedia, lookup

Mathematics of radio engineering wikipedia, lookup

List of important publications in mathematics wikipedia, lookup

Infinitesimal wikipedia, lookup

Ethnomathematics wikipedia, lookup

Law of large numbers wikipedia, lookup

Foundations of mathematics wikipedia, lookup

Non-standard calculus wikipedia, lookup

Collatz conjecture wikipedia, lookup

Large numbers wikipedia, lookup

Series (mathematics) wikipedia, lookup

Non-standard analysis wikipedia, lookup

Fundamental theorem of algebra wikipedia, lookup

Georg Cantor's first set theory article wikipedia, lookup

Hyperreal number wikipedia, lookup

Transcript

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 57 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 <ε k∈Ir & d Yk T Xk +ε k ∈ I : d , L ≥ ε r hr Yk N L (F ) Therefore X θ∼ Y. (c) It follows from (a) and (b). Theorem 3.2 Let θ = (kr ) be lacunary sequence with lim inf r qr > 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 Ayhan Esi and Necdet Çatalbaş 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. 62 Ayhan Esi and Necdet Çatalbaş 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]