EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 5, 2020, 1088-1096 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Special Issue Dedicated to Professor Hari M. Srivastava On the Occasion of his 80th Birthday Generalized Nörlund and Nörlund-type Means of Sequences of Fuzzy Numbers Pradosh Kumar Pattanaik1, Susanta Kumar Paikray2, Bidu Bhusan Jena2,∗ 1 Gandhi Institute of Engineering and Technology University, Gunupur 765022, Odisha, India 2 Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, Odisha, India Abstract. In this article we study some properties of generalized Nörlund and Nörlund-type means of sequences of fuzzy real numbers. We establish necessary and sufficient conditions for our purposed methods to transform convergent sequences of fuzzy real numbers into convergent sequences of fuzzy real numbers which also preserve the limit. Finally, we establish some results showing the connection between the generalized Nörlund and Nörlund-type limits and the usual limits under slow oscillation of sequences of fuzzy real numbers. 2020 Mathematics Subject Classifications: 40A05, 40G05, 03E72 Key Words and Phrases: Generalized Nörlund mean, Generalized Riesz mean, Fuzzy real numbers, Slow oscillation 1. Introduction Let D be the set of all closed and bounded intervals on the real line R. For X,Y ∈ D, we define d(X,Y ) = max(|a1 − b1|, |a2 − b2|), ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3680 Email addresses: pradoshmunna@gmail.com (P. K. Pattanaik), skpaikray math@vssut.ac.in (S. K. Paikray), bidumath.05@gmail.com (B. B. Jena) https://www.ejpam.com 1088 c© 2020 EJPAM All rights reserved. P. K. Pattanaik, S. K. Paikray, B. B. Jena / Eur. J. Pure Appl. Math, 13 (5) (2020), 1088-1096 1089 where X = [a1, a2], Y = [b1, b2]. It is known that (D, d) is a metric space which is also complete. A fuzzy real number X is a fuzzy set on R, and is a mapping X : R → I (= [0, 1]) associating each real number r with its grade of membership X(r). Recalling some basic terminologies, a fuzzy real number X is called convex if, X(r) ≥ X(s)∧X(t) = min(X(s), X(t)), where s < r < t. A fuzzy real number X is called normal if, there exists r0 ∈ R such that X(r0) = 1. Further, if for every ε > 0, X−1([0, a+ ε]), for all a ∈ I (is open in the usual topology of R) then X is called upper semi-continuous. Let R(I) denotes the set of all convex, upper semi continuous and normal fuzzy num- bers, and let Xα (0 < α ≤ 1) be the α level set of X, which is defined by Xα = {r ∈ R : X(r) ≥ α}. Also, for α = 0, it is closure of the strong 0-cut. Note that, the set of all numbers R can be embedded in R(I). For each t ∈ R, t ∈ R(I) is defined by t̄(r) =  1, if r = t, 0, if r 6= t. Let d̄ : R(I)× R(I)→ R be defined by d(X,Y ) = sup 0≤α≤1 d(Xα, Y α). Also, d defines a metric on R(I). It is trivial that (R(I), d) is a metric space, and is complete. Here, 0 and 1 are the additive identity and multiplicative identity respectively. The preliminary idea of fuzzy set theory was introduced and studied by Zadeh [18] in the year 1965. Gradually this theory has entered into many diversified areas of sci- ence and technology. In particular, mathematicians and researchers working on sequence spaces preferred to use fuzzy sequences because of its wide applications. The scope of such theory has been studied in the different areas of (for instance) fuzzy logic, fuzzy graph theory, fuzzy topological spaces, fuzzy differential equations, fuzzy mathematical program- ming, and so on. In this article we study the characterization of generalized Nörlund and Nörlund-type (Riesz) means of sequences of fuzzy real numbers. 2. Preliminaries and Definitions Let (pn) and (qn) be two sequences of non-negative real numbers which are not all zero, that is, Pn = n∑ u=1 pu, n ∈ N P. K. Pattanaik, S. K. Paikray, B. B. Jena / Eur. J. Pure Appl. Math, 13 (5) (2020), 1088-1096 1090 and Qn = n∑ u=0 qu, n ∈ N. Let us consider Rn = n∑ u=0 puqu and R ′ n = n∑ u=0 pn−uqu Definition 1. A sequence (Xn) of fuzzy real numbers is generalized Nörlund (N, pn, qn) summable to l if, d ( 1 R′n n∑ u=0 pn−uquXu, l ) → 0 as n→∞. Definition 2. A sequence (Xn) of real numbers is generalized Nörlund-type (N, pn, qn) summable or generalized Riesz (R, pn, qn) summable to l if, d ( 1 Rn n∑ u=1 puquXu, l ) → 0 as n→∞. Definition 3. A sequence (Xn) of fuzzy real numbers is slowly oscillating if, d(Xn, Xm)→ 0 as n,m→∞ with 1 ≤ n m → 1. Equivalently, we can say that; a sequence of fuzzy real numbers (Xn) is oscillating slowly iff for every ε > 0, there exists δ = δ(ε) > 0 and n0(ε) ∈ N such that d(Xn, Xm) < ε whenever 1 ≤ ( n m ) < 1 + δ and m,n ≥ n0(ε). Several summability methods have been defined for different fuzzy numbers valued sequences. The Cesàro summability of order one for sequences of fuzzy real numbers was studied by Altın et al. [1] in the year 2010. In the year 2017, Yavuz [17] defined a Euler summability method of sequences of fuzzy numbers and a proved Tauberian theorem. Dealing with statistical summability of sequences of fuzzy numbers, in 2016, Talo and Bal [13] studied some results based on Nörlund-type means. Recently, some works on Nörlund and Riesz means have been studied by Srivastava et al. [9], [10], [11], and [12] based on statistical convergence. Very recently, Jena et al. [6] studied the Cesàro summability of double sequences of fuzzy real numbers and proved Tauberian theorems on that basis. Also, Das et al. [2] used statistical (C, 1)(E,µ) product summability mean for sequences of fuzzy numbers to prove a fuzzy Korovkin-type approximation theorem. For more studies in this direction one may refer to [3], [4], [5], [7], [8], [14], [15] and [16]. Motivated essentially by the above mentioned works, we investigate here the characterization of generalized Nörlund and Nörlund-type (Riesz) means of sequences of fuzzy real numbers. We establish necessary and sufficient conditions for our purposed methods to transform convergent sequences of fuzzy real numbers into convergent sequences of fuzzy real numbers which also preserve the limit. Moreover, we establish some results demonstrating the connection between the generalized Nörlund and Nörlund-type limit and the usual limit under slow oscillation of sequences of fuzzy real numbers. P. K. Pattanaik, S. K. Paikray, B. B. Jena / Eur. J. Pure Appl. Math, 13 (5) (2020), 1088-1096 1091 3. Main Theorem The objective of this paper to prove the following theorem. Theorem 1. The method (N, pn, qn) is regular if and only if pn−kqk R′n → 0 as n→∞. Proof. Let (Xn) be any sequence of fuzzy real numbers which is convergent to L. That is, lim n→∞ Xn = L; then for given ε > 0 there exists a positive integer n0 for which d(Xn, L) < ε for n ≥ n0 and d(Xn, L) < H, for all n ∈ N. Let pn−kqk R′n → 0 as n → ∞, then for ε > 0 there exists n1 ∈ N such that pn−kqk R′n < ( ε 2Hmax(n0,n1) ) for n > n1. Let n2 = max(n0, n1). Then for all n ≥ n2, we have d ( 1 R′n n∑ i=1 pn−i+1qiXi, L ) ≤ d ( 1 R′n n2∑ i=1 pn−i+1qiXi, L ) + d ( 1 R′n n∑ i=n2+1 pn−i+1qiXi, L ) = d ( 1 R′n (pnq0X0 + pn−1q1X1 + ...+ pn−n2+1qn2−1Xn2−1), L ) + d ( 1 R′n (pn−n2qn2Xn2 + ...+ p0qnXn), L ) ≤ pnq0 R′n d(X0, L) + pn−1q1 R′n d(X1, L) + ...+ pn−n2+1qn2−1 R′n d(Xn2−1, L) + pn−n2qn2 R′n d(Xn2 , L) + ...+ p0qn R′n d(Xn, L) ≤ ε 2Hn2 H + ε 2Hn2 H + ...+ ε 2Hn2 H + pn−n2qn2 R′n ε 2 + ...+ p0qn R′n ε 2 ≤ ε 2n2 + ε 2n2 + ...+ ε 2n2 + pn−n2qn2 + ...+ p0qn R′n ε 2 < ε 2 + ε 2 = ε. Conversely, let (N, pn, qn) be a regular method. Consider the sequence ek = (0, 0, ..., 1, 0, 0...) = Xn where 1 appears at the kth place. Also, we have Xn → 0 as n→∞. Thus, d ( n∑ k=1 pn−k+1qk R′n ēk, 0̄ ) = pn−kqn R′n → 0 (n→∞). Theorem 2. The method (R, pn, qn) is regular if and only if pnqn Rn → 0 as n→∞. P. K. Pattanaik, S. K. Paikray, B. B. Jena / Eur. J. Pure Appl. Math, 13 (5) (2020), 1088-1096 1092 Proof. Let (Xn) be any sequence of fuzzy real numbers which is convergent to L. That is, limn→∞Xn = L; then for given ε > 0 there exists a positive integer n0 for which d(Xn, L) < ε, for all n ≥ n0 and d(Xn, L) < H for all n ∈ N. Let pnqn Rn → 0 as n→∞, then there exists n1 ∈ N such that pnqn Rn < ε 2Hmax(n0,n1) for all n > n1. Let n2 = max(n0, n1). Then for all n ≥ n2, we have d ( 1 Rn n∑ i=0 piqiXi, L ) ≤ d ( 1 Rn n2∑ i=0 piqiXi, L ) + d ( 1 Rn n∑ n2+1 piqiXi, L ) ≤ d ( 1 Rn (p0q0X0 + p1q1X1 + ...+ pn2qn2Xn2), L ) + d ( 1 Rn (pn2+1qn2+1Xn2+1 + ...+ pnqnXn), L ) ≤ p0q0 Rn d(X0, L) + p1q1 Rn d(X1, L) + ...+ pn2qn2 Rn d(Xn2 , L) + pn2+1qn2+1 Rn d(Xn2+1, L) + ...+ pnqn Rn d(Xn, L) ≤ p0q0 Rn H + p1q1 Rn H + ...+ pn2qn2 Rn H + pn2+1qn2+1 Rn ε 2 + ...+ pnqn Rn ε 2 ≤ ε 2Hn2 H + ε 2Hn2 H + ...+ ε 2Hn2 H + pn2 qn−n2 + ....+ pnq0 Rn ε 2 < ε 2 + ε 2 = ε. Conversely, let (R, pn, qn) be regular. Consider the sequence ek = (0, 0, ....., 1, 0, ...) = Xn, where 1 appears at the kth place. Also, we have Xn → 0 as n→∞. Thus, d ( n∑ k=1 pnqn Rn ēk, 0̄ ) = pnqn Rn → 0 as n→∞. Theorem 3. If (Xn) is (N, pn, qn) summable to L in R(I) and slowly oscillating then it is convergent to L in R(I). Proof. Without loss of generality we may assume that L = 0. Suppose that limn→∞ d(Xn, 0) > 0. Then there exists α > 0 and a subsequence Xni of (Xn) such that d(Xni , 0) ≥ α for all i ∈ N. 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 and d(Xn, Xm) < α 2 . P. K. Pattanaik, S. K. Paikray, B. B. Jena / Eur. J. Pure Appl. Math, 13 (5) (2020), 1088-1096 1093 Moreover, (Xn) being (N, pn, qn)- summable to 0, that means, (σn) is convergent to 0 in (R(I), d) with σn = 1 R′n n∑ k=1 pn−kqkXk; thus for all mi ≥ ni, σmi − R ′ ni R′mi σni = 1 R′mi mi∑ k=1 pmi−kqkXk − R ′ ni R′mi 1 R′ni ni∑ k=1 pni−kqkXk = 1 R′mi mi∑ k=ni+1 pmi−kqkXk. Clearly, ni ≥ g1 and ni ≤ m ≤ mi = [(1 + δ)ni], where [x] denote the integral part of x, we have d(0, Xm) ≥ d(0, Xni)− d(Xni , Xm) ≥ α− α 2 . Again, d(σmi , σni) + d ( σni , R ′ ni R′mi σni ) ≥ d ( σmi , R ′ ni R′mi σni ) ≥ d  1 R′mi mi∑ k=ni+1 pmi−kqkXk, 0̄  ≥ d ( pmi−kqmi − pni−kqni R′mi Xni , 0 ) − d  mi∑ k=ni+1 pmi−kqkXk − pni−kqniXni R′mi , 0̄  ≥ pmi−kqmi − pni−kqni R′mi d(Xni , 0) − mi∑ k=ni+1 d ( pmi−kqkXk − pni−kqniXni R′mi , 0 ) = pmi−kqmi − pni−kqni R′mi d(Xni , 0) − mi∑ k=ni+1 1 R′mi d (pmi−kqkXk, pni−kqniXni) ≥ pmi−kqmi − pni−kqni R′mi d(Xni , 0) − pmi−kqmi − pni−kqni R′mi d(Xk, Xni) REFERENCES 1094 ≥ pmi−kqmi − pni−kqni R′mi α− pmi−kqmi − pni−kqni R′mi α 2 = pmi−kqmi − pni−kqni R′mi ( α− α 2 ) ≥ pmi−kqmi − pni−kqni R′mi ( δ 1 + δ ) ≥ 0. Thus, for all mi ≥ ni ≥ gi, d(σmi , σni) + d ( σni , R ′ ni R′mi σni ) ≥ d ( σmi , R ′ ni R′mi σni ) α 2 ( δ 1 + δ ) . Consequently, 0 = lim d ( σni , R ′ ni R′mi σni ) ≥ α 2 ( δ 1 + δ ) > 0 which contradicts that (Xn) converges in R(I). Therefore, (Xn) is convergent to L in R(I). This completes the proof of the theorem. Theorem 4. If (Xn) is (N̄ , pn, qn) summable to L in R(I) and slowly oscillating then it is convergent to L in R(I). Proof. The proof can be followed in the similar lines from the proof of Theorem 3. Acknowledgements The authors would like to keep the record of the 80th birthday of Prof. H. M. Srivas- tava for his tremendous contribution to many significant developments in Mathematical research. Also, the authors express their heartfelt thanks to the editors and anonymous referees for their most valuable comments and constructive suggestions which leads to the improvement of the earlier version of the manuscript. Funding: This research received no external funding. Conflicts of Interest: The authors declare that they have no conflicts of interest. References [1] Y. Altin, M. Mursaleen, and H. Altinok. Statistical summability (C,1) for sequences of fuzzy real numbers and a Tauberian theorem. J. Intell. Fuzzy Syst., 21:379–384, 2010. [2] A. A. Das, S. K. Paikray, T. Pradhan, and H. Dutta. 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