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Mathematica Slovaca Fatih Nuray; Ekrem Savaş Statistical convergence of sequences of fuzzy numbers Mathematica Slovaca, Vol. 45 (1995), No. 3, 269--273 Persistent URL: http://dml.cz/dmlcz/129143 Terms of use: © Mathematical Institute of the Slovak Academy of Sciences, 1995 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz rVblhematica Slovaca ©1995 . , . ol -- z-^^-x K, ~ ^^^ ~-~ Math. SlOVaca, 4 5 ( 1 9 9 5 ) , NO. 3, 2 6 9 - 2 7 3 Mathematical Institute Slovák Academy of Sciences STATISTICAL CONVERGENCE OF SEQUENCES OF FUZZY NUMBERS FATIH NURAY* — (Communicated EKREM by Anatolij SAVA§** Dvurečenskij) A B S T R A C T . In this paper, t h e concepts of statistically convergent and statis tically Cauchy sequences of fuzzy numbers have been introduced and discussed. Also /(p)-spaces of sequences of fuzzy numbers have been introduced. 1. I n t r o d u c t i o n a n d b a c k g r o u n d Let D denote the set of all closed bounded intervals A = [A, A] on the real line R. For A,B eD define A < B <=> A<B<mdA<B, d(A, B) = max(|A - J3|, \A - B\) . It is easy to see that d defines a metric on D and (D,d) is a complete metric space. Also < is a partial order in D. A fuzzy number is a fuzzy subset of the real line R which is bounded, convex and normal. Let L(R) denote the set of all fuzzy numbers which are upper semicontinuous and have compact support. In other words, if X E £ ( R ) , then, for any a G [0,1], Xa is compact, where x a f t:X(t)>a •{ t: X(t) if a e (0,1], >0 if a = 0. sup d(Xa,Ya). Define d: L(R) x L(R) -> R by d(X,Y)= 0<a<l For X, Y e L(R) define X < Y if and only if Xa < Ya for any a G [0,1]. We now recall the following definitions which were given in [3]. A M S S u b j e c t C l a s s i f i c a t i o n (1991): P r i m a r y 40A05, 40D25. K e y w o r d s : fuzzy number, statistical convergence, Cauchy sequence, sequence space. 269 FATIH NURAY — EKREM SAVA$ DEFINITION 1.1. A sequence X = {Xk} of fuzzy numbers is a function X from the set of all positive integers into L(R). The fuzzy number Xk denotes the value of the function at k G N and is called the kth term of the sequence. DEFINITION 1.2. A sequence X = {Xk} of fuzzy numbers is said to be convergent to the fuzzy number XQ if for every e > 0 there exists a positive integer ko such that d(Xk, X0) < S for k > k0 . Let c denote the set of all convergent sequences of fuzzy numbers. X = {Xk} is said to be a Cauchy sequence if for every s > 0 there exists ko £ N such that d(Xk, Xm) < e for k, m > ko • Let C denote the set of all Cauchy sequences of fuzzy numbers. It is easy to see that c C C. It is known ([3]) that L(R) is a complete metric space with the metric d. 2. S t a t i s t i c a l c o n v e r g e n c e Recall ([5]) that the "natural density" of a set K of positive integers is defined by S(K) = liml/n|{fc < n : k e K}\, where |{fc < n : k e K}\ denotes the n number of elements of K not exceeding n . We shall be particularly concerned with integer sets having natural density zero. To facilitate this, we introduce the following notation: If X = {Xk} is a sequence that satisfies some property P for all k except a set of natural density zero, then we say that Xk satisfies P for "almost all fc" and we abbreviate this by "a.a. k". The concept of statistical convergence of real or complex sequences was introduced by F a s t [1] and studied by several authors including [7], [6], [2]. In this section, we shall introduce and discuss the concepts of statistically convergent and statistically Cauchy sequences of fuzzy numbers. DEFINITION 2 . 1 . A sequence X = {Xk} of fuzzy numbers is said to be statistically convergent to the fuzzy number XQ, written as st-limK^- = XQ , if for every e > 0, l i m l / n | { k < n : d(Xk, X0) > e}\ = 0 , n J ' i.e., d(XkjXo) <e a.a. A;. It is clear that limX& = XQ implies st-limXfc = XQ. k 270 ' STATISTICAL CONVERGENCE OF SEQUENCES OF FUZZY NUMBERS DEFINITION 2.2. A sequence X = {Xk} of fuzzy numbers is a statistically Cauchy sequence if for every e > 0 there exists a number N ( = N(e)) such that l i m l / n | { k < n : d(Xk,XN) > e}\ = 0, J n ' i.e., d(Xk,XN) <e a.a. fc. THEOREM 2 . 1 . A sequence X = {Kjb} of fuzzy numbers is statistically vergent if and only if X = {Xk} is a statistically Cauchy sequence. con- P r o o f . Suppose that st-limX^ = X0 and e > 0. Then d(Xk,X0) < e/2 and if N is chosen so that d(XN,X0) d(Xk,XN) <d(Xk,X0) a.a k, < e/2, then we have +d(XN,X0) < e/2 + e/2 = e a.a fc . Hence, X is statistically Cauchy. Conversely, suppose that X is a statistically Cauchy sequence. Then d(Xk,XN) < e/2 a.a. k. Chose jV such that d(XN,X0) < e/2, then for every e > 0, we have d(Xk, XQ) < d(Xk, XN) + d(XN, XQ) < e/2 + e/2 = e a.a. k . Hence st-limX/- = X0. D THEOREM 2.2. If X = {Xk} is a sequence of fuzzy numbers for which there is a convergent sequence Y = {1^} such that Xk = Yk a.a. k, then X is statistically convergent. P r o o f . Let X = {Xk} be a sequence of fuzzy numbers such that lim Yk = k X0. Suppose e > 0. Then for each n {k<n: d(Xk,X0)>e}Q{k<n: Xk + Yk} U {k < n : d(Yk,X0)>e} since lim Yk = X0, the latter set contains a fixed number of integers, say 5 = s(e). Then l i m l / n | { / c < n : d(Xk, X0) > e}\ < lim 1/n \{k < n : Xk ^ Yk}\ + l i m 5 / n = 0 n J ' n because Xk = Yk a.a. k. Hence d(Xk,X0) convergent to X0 . n < e a.a. k, so X is statistically • 271 FATIH NURAY — EKREM SAVA§ 3. /(p)-spaces of sequences of fuzzy n u m b e r s In [4], N a n d a numbers as follows: lp = {X introduced and discussed Zp-spaces of sequences of fuzzy J2[d(Xk,0)]P = {Xk}: <oo, Z<p<oo}. k In this section, we shall introduce Z(P)-spaces of sequences of fuzzy numbers. We have l(p) = {x = {Xk}: ^[d(X fc ,0)] Pfc <oo}, k where (pk) is a bounded sequence of strictly positive real numbers. If pk = p for all k, then l(p) = lp, which was introduced by N a n d a . We have the following result. THEOREM 3 . 1 . l(p) is a complete metric space with the metric g defined by l/M Pк ß(X,Y)=íj2[d(Xk,Yk)] where X = {Xk} and Y = {Yk} are sequences of fuzzy numbers which are in l(p) and M = maxf 1, supp^ J . P r o o f . It is straightforward to see that g is a metric on l(p). To show that l(p) is complete in this metric, let {X 7 } be a Cauchy sequence in l(p). Then for each fixed fc, \1/M / Pk d(Xlxi)<[YJ[d(XlYi)] ) ^ =Q(X\Xn, k ' it follows that {Xlk} is a Cauchy sequence in L(R). But ( L ( R ) , d ) is complete. Hence HmX£ = Xk for each k. Put X = {Xk}. We shall show that limX* = X i i and X E l(p). We know that ^(X^O) is bounded, say, g(X\0) / t pk < L. Now, for any t, \VM (J2id(^0)] \ <g(X\0)<L. Letting i —> oo, and then t —* oo, we obtain / t \T[d(xk,o)r) 272 \VM <L, STATISTICAL CONVERGENCE OF SEQUENCES OF FUZZY NUMBERS and this shows that X G l(p). It remains to show g(Xl,X) —• 0. Let e > 0 be given. Then there is a integer N such that g(Xl,Xj) < e for ij > N. Therefore for any £, l^[d(XlXi)]Pkj <e(X\Xl)<e for iJ>N. Letting j —• oo we obtain f J2[d(XlXk)]Pk\ <e for i>N. Since t is arbitrary, we let t —• oo and obtain g(Xl,X) completes the proof. < e for i > N. This D Acknowledgement We are grateful to the referee for his careful reading of this paper and several valuable remarks. REFERENCES [1] [2] [3] [4] [5] F A S T , H . : Sur la convergence siatistique, Colloq. M a t h . 2 (1951), 241-244. F R I D Y , J. A . : On statistical convergence, Analysis 5 (1985), 301-313. M A T L O K A , M . : Sequences of fuzzy numbers, BUSEFAL 2 8 (1986), 28-37. NANDA, S.: On sequences of fuzzy numbers, Fuzzy Sets and Systems 3 3 (1989), 123-126. NIVEN, I . — Z U C K E R M A N , H. S.: An Introduction to the Theory of Numbers (4th edition), J o h n Wiley and Sons, New York, 1980. [6] SALAT, T . : On statistically convergent sequences of real numbers, M a t h . Slovaca 3 0 (1980), 139-150. [7] S C H O E N B E R G , I. J . : The integrability of certain functions and related summability methods, Amer. M a t h . Monthly 6 6 (1959), 361-375. Received April 8, 1993 Revised J u n e 17, 1993 * Cumhuriyet Universitesi Iktisadi ve Idari Bil. Fak. Sivas TURKEY Fax: (346) 2262256 ' Department of Mathematics Firat University TR-23169 Elazig TURKEY 273