# Fuzzy real valued lacunary I-convergent sequences

Article (PDF Available)inApplied Mathematics Letters 25(3):466-470 · March 2012with25 Reads
DOI: 10.1016/j.aml.2011.09.037 · Source: DBLP
Abstract
In this article, we introduce the concept of lacunary II-convergent sequence of fuzzy real numbers and study some basic properties.

### Full-text (PDF)

Available from: Bipan Hazarika, Oct 28, 2014
copy is furnished to the author for internal non-commercial research
and education use, including for instruction at the authors institution
and sharing with colleagues.
Other uses, including reproduction and distribution, or selling or
licensing copies, or posting to personal, institutional or third party
websites are prohibited.
In most cases authors are permitted to post their version of the
article (e.g. in Word or Tex form) to their personal website or
institutional repository. Authors requiring further information
regarding Elsevier’s archiving and manuscript policies are
encouraged to visit:
Author's personal copy
Applied Mathematics Letters 25 (2012) 466–470
Contents lists available at SciVerse ScienceDirect
Applied Mathematics Letters
journal homepage: www.elsevier.com/locate/aml
Fuzzy real valued lacunary I-convergent sequences
Bipan Hazarika
Department of Mathematics, Rajiv Gandhi University, Itanagar-791 112, Arunachal Pradesh, India
a r t i c l e i n f o
Article history:
Received in revised form 10 September
2011
Accepted 19 September 2011
Keywords:
Ideal
I-convergent
I-Cauchy
Fuzzy number
Lacunary sequence
a b s t r a c t
In this article, we introduce the concept of lacunary I-convergent sequence of fuzzy real
numbers and study some basic properties.
1. Introduction
The works on I-convergence of real valued sequences was initially studied by Kostyrko et al. [1]. Later on, it was further
studied by Šalàt et al. [2,3], Tripathy et al. [4] and many others.
Let S be a non-empty set. A non-empty family of sets I P(S) (power set of S) is called an ideal on S if (i) for each A,
B I, we have A B I; (ii) for each A I and B A, we have B I. Let S be a non-empty set. A family F P(S) (power
set of S) is called a filter on S if (i) φ ̸∈ F ; (ii) for each A, B F, we have A B F; (iii) for each A F and B A, we have
B F. An ideal I is called non-trivial if I ̸= φ and S ̸∈ I. It is clear that I P(S) is a non-trivial ideal if and only if the class
F = F (I) = {S A : A I} is a filter on S. The filter F (I) is called the filter associated with the ideal I. A non-trivial ideal
I P(S) is called an admissible ideal on S if and only if it contains all singletons, i.e., if it contains {{x} : x S}. A non-trivial
ideal I is maximal if there cannot exist any non-trivial ideal J ̸= I containing I as a subset (for details see [1]).
The concept of fuzzy sets was initially introduced by Zadeh [5]. Later on, sequences of fuzzy real numbers have been
discussed by Nanda [6], Nuray and Savas [7] and many others.
A lacunary sequence is an increasing integer sequence θ = (k
r
) such that k
0
= 0 and h
r
= k
r
k
r1
as r .
The intervals are determined by θ and it will be defined by J
r
= (k
r1
, k
r
] and the ratio
k
r
k
r1
will be defined by φ
r
.
Freedman et al. [8] defined the space N
θ
in the following way. For any lacunary sequence θ = (k
r
),
N
θ
=
(x
k
) : lim
r→∞
h
1
r
kI
r
|x
k
L| = 0, for some L
.
The space N
θ
is a BK space with the norm
(x
k
)
θ
= sup
r
h
1
r
kI
r
|x
k
|.
N
0
θ
denotes the subset of these sequences in N
θ
for which θ = 0, (N
0
θ
, ·
θ
) is also a BK space.
Tel.: +91 3602278512; fax: +91 360 2277881.