$\mathcal{I}$ and $\mathcal{I}^{\ast}$ convergence of multiple sequences of fuzzy numbers
Bol. Soc. Paran. Mat.
c SPM –ISSN-2175-1188 on line
SPM: www.spm.uem.br/bspm
(3s.) v. 33 2 (2015): 69–78.
ISSN-00378712 in press
doi:10.5269/bspm.v33i2.22008
I and I∗ convergence of multiple sequences of fuzzy numbers
Pankaj Kumar, Vijay Kumar and S. S. Bhatia
abstract: Recently, the concept of statistical convergence for multiple sequences
of fuzzy numbers has been studied by Kumar et al.. This motivate us to extend
the idea of I-convergence to sequences of fuzzy numbers of multiplicity greater than
two.
Key Words: Fuzzy number sequences; multiple sequences; ideal convergence.
Contents
1 Introduction
69
2 Background and Preliminaries
70
3 I2 -convergence
71
4 I∗2 -convergence
74
5 I2 -Cauchy and I∗2 -Cauchy sequences of fuzzy numbers
75
6 Multiple Sequences of Fuzzy Numbers
77
1. Introduction
R. P. Agnew [1] studied the summability theory of multiple sequences and
obtained certain theorems for multiple sequences which have already been proved
by the author himself for double sequences. Móricz [7] continued with the study of
multiple sequences and gave some remarks on the notion of regular convergence of
multiple series. In 2003 [8], the author extended statistical convergence from single
to multiple real sequences and obtained some results for real double sequences.
Very recently, Kumar et al. [18] studied the concept of statistical convergence for
multiple sequences of fuzzy numbers.
The notion of statistical convergence of sequence of numbers was introduced
by Fast [3] and Schoenberg [24] independently in 1951 and discussed by [4,9,19]
and many others. Nuray and Savaş [14] first introduced statistical convergence
of sequences of fuzzy numbers. After their pioneer work, many authors have been
made their contribution to study different generalizations of statistical convergence
for sequences of fuzzy numbers(see [6,15,20,21,22] etc.).
Kostyrko, Salat and Wilczynski [5] defined I-convergence for single sequences
which is a natural generalization of statistical convergence. Tripathy and Tripathy
2000 Mathematics Subject Classification: 40A05, 40C05
69
Typeset by BSP
style.
M
c Soc. Paran. de Mat.
70
Pankaj Kumar, Vijay Kumar and S. S. Bhatia
[25] introduced the concept of I-convergence and I-Cauchy for double sequences
and proved some properties related to the solidity, symmetricity, completeness and
denseness. In past years, I-convergence has also become an interesting area of
research for sequences of fuzzy numbers. The credit goes to Kumar et al. [17] who
first defined I-convergence for sequences of fuzzy numbers. For an extensive view
of this subject, we refer [2,5,10,11,12,13,16,23].
Continuing our work [18], in present paper, we study the concept of ideal convergence for sequences of fuzzy numbers having multiplicity greater than or equal
to two.
2. Background and Preliminaries
Let C(R)= {A ⊂ R: A compact and convex}. The space C(R) has a linear
structure induced by the operation A + B = {a + b : a ∈ A, b ∈ B} and λA = {λa :
a ∈ A} for A, B ∈ C(R) and λ ∈ R. The Hausdroff distance between A and B is
defined as
δ ∞ (A, B) = max{sup inf k a − b k, sup inf k a − b k}
a∈A b∈B
b∈B a∈A
It is well known that (C(R), δ ∞ ) is a complete (not separable) metric space.
Definition 2.1. A fuzzy number is a function X from R to [0,1], which satisfying
the following conditions
(i) X is normal, i.e., there exists and x0 ∈ R such that X(x0 ) = 1;
(ii) X is fuzzy convex, i.e., for any x, y ∈ R and
λ ∈ [0, 1], X(λx + (1 − λ)y) ≥ min{X(x), X(y)};
(iii) X is upper semi-continuous;
(iv) the closure of the set {x ∈ R : X(x) > 0}, denoted by X 0 , is compact.
The properties (i)-(iv) imply that for each α ∈ (0, 1], the α-level set,
α
X α = {x ∈ R : X(x) ≥ α} = [X α , X ]
is a non-empty compact convex subset of R. Let L(R) denotes the set of all fuzzy
numbers. The linear structure of L(R) induces an addition X + Y and a scalar
multiplication λX in terms of α-level sets by
[X + Y ]α = [X]α + [Y ]α and [λX]α = λ[X]α
(x, y ∈ L(R), λ ∈ R)
for each α ∈ [0, 1].
Define for each 1 ≤ q < ∞
Z 1
1
δ ∞ (X α , Y α )q dα ) q
dq (X, Y ) = (
0
α
α
and δ ∞ = sup0≤α≤1 δ ∞ (X , Y ). Clearly d∞ (X, Y ) = limq→∞ dq (X, Y ) with
dq ≤ dr if q ≤ r. Moreover, dq is a complete, separable and locally compact metric
space [1].
Throughout the paper, d will denote dq with 1 ≤ q < ∞.
We now quote the following definitions which will be needed in the sequel.
I and I∗ convergence of multiple sequences of fuzzy numbers
71
Definition 2.2. A double sequence X = (Xnk ) of fuzzy numbers is said to be
convergent to a fuzzy number X0 if for each ǫ > 0 there exist a positive integer m
such that
d(Xnk , X0 ) < ǫ for every n, k ≥ m.
The fuzzy number X0 is called the limit of the sequence (Xnk ) and we write
limn,k→∞ Xnk = X0 .
Definition 2.3. A double sequence X = (Xnk ) of fuzzy numbers is said to be
Cauchy sequence if for each ǫ > 0 there exists a positive integer n0 such that
d(Xnk , XN K ) < ǫ
for every n ≥ N ≥ n0 , k ≥ K ≥ n0 .
Definition 2.4. A double sequence X = (Xnk ) of fuzzy numbers is said to be
bounded if there exists a positive number M such that
d(Xnk , e
0) < M for all n, k.
2
Let l∞
denote the set of all bounded triple sequences of fuzzy numbers.
Definition 2.5. If X is a non-empty set. A family of sets I ⊂ P (X) is called an
ideal in X if and only if
(i) φ ∈ I;
(ii) for each A, B ∈ I we have A ∪ B ∈ I;
(iii) for each A ∈ I and B ⊂ A we have B ∈ I.
Definition 2.6. Let X is a non-empty set. A non-empty family of sets F ⊂ P (X)
is called a filter on X if and only 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 6= φ and X ∈
/ I. It immediately follows that
I ⊂ P (X) is a non-trivial ideal if and only if the class F = F (I) = X − A : A ⊂ I
is a filter on X. The filter F = F (I) is called the filter associated with the ideal I.
Definition 2.7. A non-trivial ideal I ⊂ P (X) is called an admissible ideal in X if
and only if it contains all singletons i.e., if it contains {x : x ∈ X}.
Throughout this paper, N2 denotes the usual product set N × N.
3. I2 -convergence
In this section, we shall state and prove our results only for double sequences.
Our methods can readily be applied to sequences of fuzzy numbers of any multiplicity.
Definition 3.1. Let I2 ⊂ P (N2 ) be a non-trivial ideal in N2 . A double sequence
X = (Xij ) of fuzzy numbers is said to be I2 -convergent to some fuzzy number X0 ,
in symbol: I2 − lim Xij = X0 , if for each ǫ > 0,
{(i, j) ∈ N2 : d(Xij , X0 ) ≥ ǫ} ∈ I2 .
72
Pankaj Kumar, Vijay Kumar and S. S. Bhatia
We shall denote the set of all I2 -convergent double sequences of fuzzy numbers by
I2 .
Theorem 3.2. If a double sequence X = (Xij ) of fuzzy numbers is I2 -convergent
to some limit, then it must be unique.
Theorem 3.3. Let X = (Xij ) and Y = (Yij ) be two double sequences of fuzzy
numbers, then
(i) If X = (Xij ) is convergent to X0 , then (Xij ) is I2 -convergent to X0 .
(ii) If X = (...truncated)
