Nörlund and Riesz mean of sequences of fuzzy real numbers

Applied Mathematics Letters 23 (2010) 651–655

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Nörlund and Riesz mean of sequences of fuzzy real numbers Binod Chandra Tripathy a , Achyutananda Baruah b,∗ a

Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Paschim Boragaon, Garchuk, Guwahati-781035, India

b

Department of Mathematics, North Gauhati College, College Nagar, Guwahati–781031, India

article

info

Article history: Received 11 August 2009 Received in revised form 10 February 2010 Accepted 11 February 2010 Keywords: Nörlund mean Riesz mean Fuzzy real number Regular method

abstract In this article we study some properties of the Nörlund and Riesz mean of sequences of fuzzy real numbers. We establish necessary and sufficient conditions for the Nörlund and Riesz means to transform convergent sequences of fuzzy numbers into convergent sequences of fuzzy numbers with limit preserving. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction The concept of fuzzy set was introduced by L.A. Zadeh in 1965. The potential of the notion of fuzzy set was realized by different scientific groups and many researchers were motivated for further investigation and its applications. It has been applied for the studies in all the branches of science, where mathematics has applications. Workers on sequence spaces have also applied the notion and introduced sequences of fuzzy real numbers and studied their different properties. The concept of fuzziness has been applied in various fields like Cybernetics, Artificial Intelligence, Expert System and Fuzzy Control, Pattern Recognition, Operations Research, Decision Making, Image Analysis, Projectiles, Probability Theory, Agriculture, Weather forecasting, etc. The notion of fuzzy set theory has been applied for investigating different classes of sequences. We have listed some of the most recent papers in the list of references. Choudhary and Tripathy [1], Pattaraintakorn, Naruedomkul and Palasit [2], Tripathy and Baruah [3], Tripathy and Borgohain [4], Tripathy and Dutta [5], Tripathy and Sarma [6] are a few to be named, who have applied the notion of fuzzy for their investigations. Some of them have investigated different classes of sequences of fuzzy numbers. Therefore we were motivated to investigate the characterization of matrix classes transforming one class of sequences of fuzzy numbers into another class of sequences of fuzzy numbers. Let D denote the set of all closed and bounded intervals X = [a1 , a2 ] on the real line R. For X , Y ∈ D, we define d(X , Y ) = max(|a1 − b1 |, |a2 − b2 |), where X = [a1 , a2 ], 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 and is a mapping X : R → I (=[0, 1]) associating each real number t with its grade of membership X (t ). A fuzzy real number X is called convex if X (t ) ≥ X (s) ∧ X (r ) = min(X (s), X (r )), where s < t < r. If there exists t0 ∈ R, such that X (t0 ) = 1, then the fuzzy real number X is called normal. A fuzzy real number X is said to be upper-semicontinuous if for each ε > 0, X −1 ([0, a + ε)), for all a ∈ I is open in the usual topology of R.

∗

Corresponding author. Tel.: +91 9435819617; fax: +91 361 2740659. E-mail addresses: [email protected], [email protected] (B.C. Tripathy), [email protected] (A. Baruah).

0893-9659/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2010.02.006

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B.C. Tripathy, A. Baruah / Applied Mathematics Letters 23 (2010) 651–655

The set of all upper-semicontinuous, normal, convex fuzzy number is denoted by R(I ). The α -level set of a fuzzy real number X , for 0 < α ≤ 1 denoted by X α is defined as X α = {t ∈ R : X (t ) ≥ α}; for α = 0, it is the closure of the strong 0-cut (that is, the closure of the set {t ∈ R : X (t ) > 0}). Throughout the article α means α ∈ (0, 1] unless otherwise stated. The set R of all real numbers can be embedded in R(I ). For each r ∈ R, r ∈ R(I ) is defined by r (t ) =

1, 0,



if t = r , if t 6= r .

Let d : R(I ) × R(I ) → R be defined by d(X , Y ) = sup d(X α , Y α ). 0≤α≤1

Then d defines a metric on R(I ). It is well known that (R(I ), d) is a complete metric space. The additive identity and multiplicative identity in R(I ) are denoted by 0 and 1 respectively. 2. Definition and preliminaries Let (pn ) be a sequence of non-negative real numbers which are not all zero and Pn = p1 + p2 + p3 + · · · + pn ,

for all n ∈ N .

Definition 2.1. A sequence (Xn ) of fuzzy real numbers is said to be summable by Nörlund mean (N, pn ) to L, if d

n 1 X

P n i =1

! pn−i+1 Xi , L

→ 0,

as n → ∞.

Definition 2.2. A sequence (Xn ) of fuzzy real numbers is said to be summable by Riesz mean (R, pn ) to L, if d

n 1 X

P n i =1

! p i Xi , L

→ 0,

as n → ∞.

Definition 2.3. A sequence (Xn ) of fuzzy real numbers is said to be slowly oscillating if d(Xm , Xn ) → 0, as m, n → ∞ with → 1. 1≤ m n The above definition is equivalent to the following statement. A sequence (Xn ) of fuzzy real number is slowly oscillating if and only if for each ε > 0 we can find δ = δ(ε) > 0 and n0 (ε) ∈ N such that d(Xm , Xn ) < ε whenever 1 ≤ m < 1 + δ and m, n ≥ n0 (ε). n Recently some works on Nörlund and Riesz mean has been published by Altin, Et and Tripathy [7] applying Orlicz functions. Subrahmanyam [8] has studied the Cesàro transform of sequences of fuzzy real numbers. 3. Main results Theorem 3.1. The method (N, pn ) is regular if and only if

pn Pn

→ 0 as n → ∞.

Proof (Sufficiency). Let (Xn ) be any convergent sequence of fuzzy real numbers. Let limn→∞ Xn = L. Then for a given ε > 0 p there exists a positive integer n0 such that d(Xn , L) < ε for n ≥ n0 and d(Xn , L) < H for all n ∈ N. Let Pn → 0, as n → ∞, then for ε > 0 there exists n1 ∈ N such that have d

n 1 X

P n i =1

! pn−i+1 Xi , L

n2 1 X

≤d

Pn i=1

 =d ≤ ≤ ≤ <

1 Pn

pn Pn pn Pn

+

n

ε

for n > n1 . Let n2 = max(n0 , n1 ). Then for all n ≥ n2 , we

2H max(n0 ,n1 )

! pn−i+1 Xi , L

1

+d

n X

Pn i=n +1 2

! pn−i+1 Xi , L

   1 (pn X1 + · · · + pn−n2 +1 Xn2 ), L + d (pn−n2 Xn2 +1 + · · · + p1 Xn ), L

H + ··· +

2Hn2

ε

<

d(X1, L) + · · · +

ε

2

pn Pn

ε

= ε.

Pn

pn−n2 +1

H + ··· +

2

pn−n2 +1

Pn

ε 2Hn2

H+

Pn pn−n2 p1 d(Xn2 , L) + d(Xn2 +1 , L) + · · · + d(Xn , L) Pn Pn pn−n2 ε p1 ε Pn

 H+

2

pn−n2 Pn

+ ··· +

+ ··· +

Pn 2 p1 Pn



ε 2

B.C. Tripathy, A. Baruah / Applied Mathematics Letters 23 (2010) 651–655

653

Necessity: Let (N, Pn ) be a regular method. Consider the sequence e1 = (1, 0, 0, . . .) = (Xk ). We have Xk → 0 as k → ∞. Then d

n X pn−k+1

Pn

k=1

! e1 , 0

pn

=

→ 0,

Pn

as n → ∞. 

Theorem 3.2. The method (R, pn ) is regular if and only if Pn → ∞ as n → ∞. Proof (Sufficiency). Let (Xn ) be any convergent sequence of fuzzy real numbers. Let limn→∞ Xn = L. Then for a given ε > 0 there exists positive integer n0 such that d(Xn , L) < ε for all n ≥ n0 and d(Xn , L) < H for all n ∈ N. Let Pn → ∞, as n → ∞ p then there exists n1 ∈ N such that Pk < 2H maxε(n ,n ) for all n > n1 . Let n2 = max(n0 , n1 ). Then for all n ≥ n2 we have n

d

n 1 X

P n i =1

! ≤d

pi Xi, L

0

n2 1 X

Pn i=1

 ≤d

1 Pn

! pi Xi, L

+d

1

n X

1

Pn i=n +1 2

! pi Xi, L

   1 (p1 X1 + · · · + pn2 Xn2 ), L + d (pn2 +1 Xn2 +1 + · · · + pn Xn ), L

Pn pn2 +1 pn ≤ d(X1, L) + · · · + d(Xn2 , L) + d(Xn2 +1 , L) + · · · + d(Xn , L) Pn Pn Pn Pn pn pn +1 ε p1 pn ε ≤ H + ··· + 2 H + 2 + ··· + Pn Pn Pn 2 Pn 2 pn2

p1

≤ <

ε 2Hn2

ε

2

+

H + ··· +

ε 2

ε 2Hn2

 H+

pn2 +1 Pn

+ ··· +

pn Pn



ε 2

= ε.

Necessity: Let (R, Pn ) be regular. Consider the sequence ek = (0, 0, . . . , 1, 0, . . .) = (Xk ), where 1 appears at the k-th place. We have Xk → 0 as k → ∞. Then d

n X pi Xi i=1

Pn

!

pk

,0 =

Pn

Thus Pn → ∞, as n → ∞.

→ 0,

as n → ∞.



Theorem 3.3. If (Xn ) is (R, Pn ) summable to L in R(I ) and is slowly oscillating then it converges to L in R(I ). Proof. Without loss of generality, we may assume that L = 0. Suppose that lim d(Xn , 0) > 0. Then there exists α > 0 and a subsequence (Xni ) of (Xn ) with d(Xni , 0) ≥ α,

for all i ∈ N .

(1)

Since (Xn ) is slowly oscillating, so (Xni ) as a subsequence of (Xn ) is also slowly oscillating. Then for a given δ > 0, there exists g0 ∈ N such that g0 ≤ n ≤ m < (1 + δ)n, d(Xm , Xn ) <

α 2

.

Since (Xn ) is (R, Pn ) summable to 0, let σn =

σmi −

Pni Pmi

σni = =

mi 1 X

Pmi k=1 1

p k Xk −

mi X

Pmi k=n +1 i

1 Pn

ni Pni 1 X

Pmi Pni k=1

Pn

k=1

pk Xk , then (σn ) converges to 0 in (R(I ), d). So for mi ≥ ni

p k Xk

pk Xk .

So for all ni ≥ g1 and ni ≤ m ≤ mi = [(1 + δ)ni ], where [x] denote the integral part of x. Now d(0, Xm ) ≥ d(0, Xni ) − d(Xni , Xm )

≥ α−

α 2

.

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B.C. Tripathy, A. Baruah / Applied Mathematics Letters 23 (2010) 651–655

Again

    Pni Pni d(σmi , σni ) + d σni , σni ≥ d σmi , σni Pmi

Pmi

≥d

Pmi k=n +1 i

 ≥d

≥

=

!

mi X

1

pmi − pni Pmi

pmi − pni Pmi pmi − pni Pmi

≥

pmi − pni

≥

pmi − pni

Pmi

pk Xk , 0 mi X pk Xk − pni Xni



Xn i , 0

−d

Pmi

k=ni +1

d(Xni , 0) −

mi X

 d

pk Xk − pni Xni Pmi

k=ni +1 mi

d(Xni , 0) − d(Xni , 0) −

α−

Pmi pmi − pni 

X

1

k=ni +1

Pmi

pmi − pni Pmi

! ,0

 ,0

d(pk Xk , pni Xni ) d(Xk , Xni )

pmi − pni α Pmi

2

α

α− 2   pmi − pni δ ≥ Pmi 1+δ ≥ 0. =

Pmi

Thus for all mi ≥ ni ≥ g1 .



d(σmi , σni ) + d σni ,



Now 0 = lim d σmi ,

Pni Pmi

Pni Pmi

σni

 σni ≥



  Pn ≥ d σmi , i σni Pmi   α δ ≥ . 2 1+δ
δ > 0 which contradict that (Xn ) converges in R(I ). 1+δ

α 2



Theorem 3.4. Let (nk ) be an increasing sequence of natural numbers and (Xn ) be a sequence of fuzzy real numbers such that n Xn = Xnk+1 for nk < n < nk+1 . If (Xn ) is (R, pn ) summable to L in R(I ), then (Xn ) converges in R(I ) to L provided lim kn+1 > 1. k

Proof. Without loss of generality, we may assume that L = 0. Let Sn = to zero as n → ∞,

1 Pn

Pn

k=1

pk Xk . Suppose that d(Xni , 0) does not tend

nk+1

Snk+1 =

X p i Xi i =1

=

=

Pnk+1

1

nk X

Pnk+1 i=1 Pnk Pnk+1

pi Xi +

Snk +

1

nk+1

X

Pnk+1 i=n +1 k

pnk+1 − pnk Pnk+1

pi Xi

Xnk+1 .

(2)



Since Snk+1 and Snk tend to 0 in R(I ), it follows from (2) that Xnk+1 tend to 0 in R(I ) as limk→∞ 1 −

nk nk+1



> 0. Thus given

ε > 0, we can find k0 ∈ N such that for all k ≥ k0 , d(Xni , 0) < ε, for n ≥ nk0 + 1, then there exist k ≥ k0 such that nk0 ≤ n ≤ nk0 +1 implies that Xn = Xnk0 +1 . So d(Xni , 0) = d(Xnk0 +1 , 0) < ε as k0 ≥ k0 . Thus Xn → 0 in R(I ).  0

Conclusions: In this article we have investigated about the Nörlund and Riesz transform of sequences of fuzzy real numbers. We have obtained necessary and sufficient conditions for the Nörlund and Riesz mean to be regular. It deals with characterization of matrix transformations between sequences of fuzzy real numbers. The idea will help the workers

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655

on sequences of fuzzy numbers to go for further study in characterization of matrix classes between different classes of sequences of fuzzy numbers. References [1] B. Choudhary, B.C. Tripathy, On fuzzy real valued `(p)F sequences, in: Proc. International Conf. 8th Joint Con. Inf. Sci. (10th International Conf. on Fuzzy Theory and Technology), Saltlake City, Utha, USA, July 21–25, 2005, pp. 184–190. [2] P. Pattaraintakorn, K. Naruedomkul, K. Palasit, A note on the roughness measure of fuzzy sets, Applied Mathematics Letters 22 (8) (2009) 1170–1173. [3] B.C. Tripathy, A. Baruah, New type of difference sequence spaces of fuzzy real numbers, Mathematical Modelling and Analysis 14 (3) (2009) 391–397. [4] B.C. Tripathy, S. Borgogain, The sequence space m(M , φ, ∆nm , p)F , Mathematical Modelling and Analysis 13 (4) (2008) 577–586. p [5] B.C. Tripathy, A.J. Dutta, On fuzzy real-valued double sequence spaces 2 `F , Mathematical and Computer Modelling 46 (9–10) (2007) 1294–1299. [6] B.C. Tripathy, B. Sarma, Sequence spaces of fuzzy real numbers defined by Orlicz functions, Mathematica Slovaca 58 (5) (2008) 621–628. [7] Y. Altin, Mikail Et, B.C. Tripathy, The sequence space |N¯ p |(M , r , q, s) on seminormed spaces, Applied Mathematics and Computation 154 (2004) 423–430. [8] P.V. Subrahmanyam, Cesàro summability of fuzzy real numbers, Journal of Analysis 7 (1999) 159–168.