Hindawi Publishing Corporation
International Journal of Analysis
Volume 2013, Article ID 749684, 7 pages
http://dx.doi.org/10.1155/2013/749684
Research Article
Regularly (I2, I)-Convergence and Regularly (I2, I)-Cauchy
Double Sequences of Fuzzy Numbers
Erdinç Dündar,1 Özer Talo,2 and Feyzi BaGar3
1
Department of Mathematics, Kocatepe University, Gazligöl Yolu, 03200 Afyon, Turkey
Department of Mathematics, Celal Bayar University, 45040 Manisa, Turkey
3
Department of Mathematics, Fatih University, The Hadimköy Campus, Büyükçekmece, 34500 Istanbul, Turkey
2
Correspondence should be addressed to Feyzi Başar; [email protected]
Received 1 November 2012; Accepted 28 February 2013
Academic Editor: Malte Braack
Copyright © 2013 Erdinç Dündar et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, we introduce the notions of regularly (I2 , I), (I∗2 , I∗ )-convergence and regularly (I2 , I), (I∗2 , I∗ )-Cauchy
double sequence of fuzzy numbers. Also, we study some properties of these concepts.
1. Introduction, Notations, and Definitions
The concept of ordinary convergence of a sequence of fuzzy
numbers was firstly introduced by Matloka [1] and proved
some basic theorems for sequences of fuzzy numbers. Nanda
[2] studied the sequences of fuzzy numbers and showed that
the set of all convergent sequences of fuzzy numbers forms a
complete metric space. Recently, Nuray and Savaş [3] defined
the concepts of statistical convergence and statistical Cauchy
for sequence of fuzzy numbers. They proved that a sequence
of fuzzy numbers, is statistically convergent if and only if it
is statistically Cauchy. Nuray [4] introduced Lacunary statistical convergence of sequences of fuzzy numbers whereas
Savaş [5] studied some equivalent alternative conditions for a
sequence of fuzzy numbers to be statistically Cauchy. A lot of
developments have been made in this area after the works of
[6–8].
Throughout the paper N and R denote the set of all
positive integers and the set of all real numbers, respectively.
The idea of I-convergence was introduced by Kostyrko et al.
[9] as a generalization of statistical convergence which is
based on the structure of the ideal I of subset of the set N.
Nuray and Ruckle [10] independently introduced the same
with another name generalized statistical convergence. Das
et al. [11] introduced the concept of I-convergence of double
sequences in a metric space and studied some properties
of this convergent sequences of this type. Balcerzak et al.
[12] studied on statistical convergence and ideal convergence
for sequences of functions. Komisarski [13] discussed the
pointwise I-convergence and I-convergence in measure of
sequences of functions. Mursaleen and Alotaibi investigated
the notion of ideal convergence in [14] for random 2-normed
space and construct some interesting examples. Mursaleen
and Mohiuddine defined and studied the concept of Iconvergence, I∗ -convergence, I-limit points and I-cluster
points in probabilistic normed space, in [15]. Şahiner et al.
introduced and investigated I-convergence in 2-normed
spaces, and also defined and examined some new sequence
spaces using norm, in [16]. A lot of developments have been
made in this area after the works of [17–21].
V. Kumar and K. Kumar studied the concepts of Iconvergence, I∗ -convergence, and I-Cauchy sequence for
sequences of fuzzy numbers, in [22]. Mursaleen et al. studied
the concept of ideal convergence and ideal Cauchy for double
sequences in intuitionistic fuzzy normed spaces, in [23].
Recently, Dündar and Talo have introduced the concepts
of I2 -convergence, I∗2 -convergence for double sequences
of fuzzy numbers and studied their some properties and
relations, in [24]. Quite recently, Dündar and Talo have
introduced the concepts of I2 -Cauchy, I∗2 -Cauchy double
sequences of fuzzy numbers, in [25].
In this paper, we introduce the notions of regularly
(I2 , I), (I∗2 , I∗ )-convergence and regularly (I2 , I),
2
International Journal of Analysis
(I∗2 , I∗ )-Cauchy double sequence of fuzzy numbers. Also,
we study some properties of those sequences.
In a possible application we can say that if we choose
the statistical convergence of a special case of the ideal
convergence, the results in this paper can be obtained for
statistical convergence.
Now, we recall the concept of fuzzy numbers, convergence, ideal convergence of the sequences and double
sequences (see [9, 11, 21, 24–30]).
A double sequence 𝑥 = (𝑥𝑚𝑛 )𝑚,𝑛∈N of real numbers is said
to be convergent to 𝐿 ∈ R in Pringsheim’s sense, if for any
𝜀 > 0 there exists 𝑁𝜀 ∈ N such that |𝑥𝑚𝑛 − 𝐿| < 𝜀 whenever
𝑚, 𝑛 > 𝑁𝜀 . In this case we write
𝑃 − lim 𝑥𝑚𝑛 = 𝐿
𝑚,𝑛 → ∞
or
lim 𝑥
𝑚,𝑛 → ∞ 𝑚𝑛
= 𝐿.
(1)
Let 𝑋 ≠ 0. A class I of subsets of 𝑋 is said to be an ideal in 𝑋
provided that
(i) 0 ∈ I,
(ii) 𝐴, 𝐵 ∈ I implies 𝐴 ∪ 𝐵 ∈ I,
(iii) 𝐴 ∈ I, 𝐵 ⊂ 𝐴 implies 𝐵 ∈ I.
I is called a nontrivial ideal if 𝑋 ∉ I.
Let 𝑋 ≠ 0. A non-empty class F of subsets of 𝑋 is said to
be a filter in 𝑋 provided that
󵄨
󵄨
{𝑚 ∈ N : 󵄨󵄨󵄨𝑥𝑚𝑛 − 𝐿 𝑛 󵄨󵄨󵄨 ≥ 𝜀} ∈ I
(ii) 𝐴, 𝐵 ∈ F implies 𝐴 ∩ 𝐵 ∈ F,
(iii) 𝐴 ∈ F, 𝐴 ⊂ 𝐵 implies 𝐵 ∈ F.
If I is a nontrivial ideal in 𝑋, 𝑋 ≠ 0, then the class
(2)
is called a filter on 𝑋 associated with I.
A nontrivial ideal I in 𝑋 is called admissible if {𝑥} ∈ I
for each 𝑥 ∈ 𝑋.
Throughout the paper we take I as a nontrivial admissible ideal in N.
Let I ⊂ 2N be a nontrivial ideal, and let (𝑋, 𝜌) be a
metric space. A sequence (𝑥𝑛 ) of elements of 𝑋 is said to be
I-convergent to 𝐿 ∈ 𝑋, if for each 𝜀 > 0 we have 𝐴(𝜀) = {𝑛 ∈
N : 𝜌(𝑥𝑛 , 𝐿) ≥ 𝜀} ∈ I.
A sequence (𝑥𝑛 ) of elements of 𝑋 is said to be I∗ -convergent to 𝐿 ∈ 𝑋 if and only if there exists a set 𝑀 ∈ F(I)
(i.e., N \ 𝑀 ∈ I), 𝑀 = {𝑚1 < 𝑚2 < ⋅ ⋅ ⋅ < 𝑚𝑘 < ⋅ ⋅ ⋅} such that
𝜌(𝑥𝑚𝑘 , 𝐿) → 0, as 𝑘 → ∞.
Throughout the paper we take I2 as a nontrivial admissible ideal in N × N.
A nontrivial ideal I2 ⊂ 2N×N is called strongly admissible
if {𝑖} × N and N × {𝑖} belong to I2 for each 𝑖 ∈ N. It is evident
that a strongly admissible ideal is also admissible.
Let I02 = {𝐴 ⊂ N × N : (∃𝑚(𝐴) ∈ N) (𝑖, 𝑗 ≥ 𝑚(𝐴) ⇒
(𝑖, 𝑗) ∉ 𝐴)}. Then I02 is a nontrivial strongly admissible ideal
and clearly an ideal I2 is strongly admissible if and only if
I02 ⊂ I2 .
Let (𝑋, 𝜌) be a linear metric space, and let I2 ⊂ 2N×N be
a strongly admissible ideal. A double sequence 𝑥 = (𝑥𝑚𝑛 ) in
(3)
for some 𝐿 𝑛 ∈ C, for each 𝑛 ∈ N and
󵄨
󵄨
{𝑛 ∈ N : 󵄨󵄨󵄨𝑥𝑚𝑛 − 𝐾𝑚 󵄨󵄨󵄨 ≥ 𝜀} ∈ I
(i) 0 ∉ F,
F (I) = {𝑀 ⊂ 𝑋 : (∃𝐴 ∈ I) (𝑀 = 𝑋 \ 𝐴)}
𝑋 is said to be I2 -convergent to 𝐿 ∈ 𝑋, if for any 𝜀 > 0 we
have 𝐴(𝜀) = {(𝑚, 𝑛) ∈ N × N : 𝜌(𝑥𝑚𝑛 , 𝐿) ≥ 𝜀} ∈ I2 and is
written I2 − lim𝑚,𝑛 → ∞ 𝑥𝑚𝑛 = 𝐿.
If I2 ⊂ 2N×N is a strongly admissible ideal, then usual
convergence implies I2 -convergence.
Let (𝑋, 𝜌) be a linear metric space, and let I2 ⊂ 2N×N be
a strongly admissible ideal. A double sequence 𝑥 = (𝑥𝑚𝑛 ) of
elements of 𝑋 is said to be I∗2 -convergent to 𝐿 ∈ 𝑋, if and
only if there exists a set 𝑀 ∈ F(I2 ) (i.e., N × N \ 𝑀 ∈ I2 )
such that lim𝑚,𝑛 → ∞ 𝑥𝑚𝑛 = 𝐿, for (𝑚, 𝑛) ∈ 𝑀 and is written
I∗2 − lim𝑚,𝑛 → ∞ 𝑥𝑚𝑛 = 𝐿.
Let (𝑋, 𝜌) be a linear metric space, and let I2 ⊂ 2N×N be
a strongly admissible ideal. A double sequence 𝑥 = (𝑥𝑚𝑛 ) of
elements of 𝑋 is said to be I2 -Cauchy, if for every 𝜀 > 0 there
exist 𝑠 = 𝑠(𝜀), 𝑡 = 𝑡(𝜀) ∈ N such that 𝐴(𝜀) = {(𝑚, 𝑛) ∈ N × N :
𝜌(𝑥𝑚𝑛 , 𝑥𝑠𝑡 ) ≥ 𝜀} ∈ I2 .
Let I2 be an ideal of N × N and I be an ideal of N,
then a double sequence 𝑥 = (𝑥𝑚𝑛 ) in C, which is the set of
complex numbers, is said to be regularly (I2 , I)-convergent
(r (I2 , I)-convergent), if it is I2 -convergent in Pringsheim’s
sense and for every 𝜀 > 0, the following statements hold:
(4)
for some 𝐾𝑚 ∈ C, for each 𝑚 ∈ N.
We say that an admissible ideal I ⊂ 2N satisfies the
property (AP), if for every countable family of mutually
disjoint sets {𝐴 1 , 𝐴 2 , . . .} belonging to I, there exists a
countable family of sets {𝐵1 , 𝐵2 , . . .} such that 𝐴 𝑗 △ 𝐵𝑗 is a
finite set for 𝑗 ∈ N and 𝐵 = ⋃∞
𝑗=1 𝐵𝑗 ∈ I. (hence 𝐵𝑗 ∈ I for
each 𝑗 ∈ N).
We say that an admissible ideal I2 ⊂ 2N×N satisfies the
property (AP2), if for every countable family of mutually
disjoint sets {𝐴 1 , 𝐴 2 , . . .} belonging to I2 , there exists a
countable family of sets {𝐵1 , 𝐵2 , . . .} such that 𝐴 𝑗 △ 𝐵𝑗 ∈ I02 ,
that is, 𝐴 𝑗 △ 𝐵𝑗 is included in the finite union of rows and
columns in N × N for each 𝑗 ∈ N and 𝐵 = ⋃∞
𝑗=1 𝐵𝑗 ∈ I2
(hence 𝐵𝑗 ∈ I2 for each 𝑗 ∈ N).
A fuzzy number is a fuzzy set on the real axis, that is, a
mapping 𝑢 : R → [0, 1] which satisfies the following four
conditions.
(i) 𝑢 is normal, that is, there exists an 𝑥0 ∈ R such that
𝑢(𝑥0 ) = 1.
(ii) 𝑢 is fuzzy convex, that is, 𝑢[𝜆𝑥 + (1 − 𝜆)𝑦] ≥
min{𝑢(𝑥), 𝑢(𝑦)} for all 𝑥, 𝑦 ∈ R and for all 𝜆 ∈ [0, 1].
(iii) 𝑢 is upper semicontinuous.
(iv) The set [𝑢]0 := {𝑥 ∈ R : 𝑢(𝑥) > 0} is compact, (cf.
Zadeh [31]), where {𝑥 ∈ R : 𝑢(𝑥) > 0} denotes the
closure of the set {𝑥 ∈ R : 𝑢(𝑥) > 0} in the usual
topology of R.
International Journal of Analysis
3
We denote the set of all fuzzy numbers on R by 𝐸1 and called
it as the space of fuzzy numbers. 𝛼-level set [𝑢]𝛼 of 𝑢 ∈ 𝐸1 is
defined by
{{𝑡 ∈ R : 𝑥 (𝑡) ≥ 𝛼} ,
[𝑢]𝛼 := {
{𝑡 ∈ R : 𝑥(𝑡) > 𝛼},
{
(0 < 𝛼 ≤ 1) ,
(𝛼 = 0) .
(5)
The set [𝑢]𝛼 is closed, bounded, and nonempty interval for
each 𝛼 ∈ [0, 1] which is defined by [𝑢]𝛼 := [𝑢− (𝛼), 𝑢+ (𝛼)]. R
can be embedded in 𝐸1 , since each 𝑟 ∈ R can be regarded as
a fuzzy number 𝑟 defined by
1, (𝑥 = 𝑟) ,
𝑟 (𝑥) := {
0, (𝑥 ≠ 𝑟) .
(6)
Theorem 1 (see [27]). Let [𝑢]𝛼 = [𝑢− (𝛼), 𝑢+ (𝛼)] for 𝑢 ∈ 𝐸1
and for each 𝛼 ∈ [0, 1]. Then the following statements hold.
(i) 𝑢− is a bounded and nondecreasing left continuous
function on (0, 1].
(ii) 𝑢+ is a bounded and nonincreasing left continuous
function on (0, 1].
(iii) The functions 𝑢− and 𝑢+ are right continuous at the
point 𝛼 = 0.
(iv) 𝑢− (1) ≤ 𝑢+ (1).
Then it can easily be observed that 𝑑 is a metric on 𝑊 and
(𝑊, 𝑑) is a complete metric space, (cf. Nanda [2]). Now, we
may define the metric 𝐷 on 𝐸1 by means of the Hausdorff
metric 𝑑 as
𝐷 (𝑢, V) := sup 𝑑 ([𝑢]𝛼 , [V]𝛼 )
𝛼∈[0,1]
󵄨 󵄨
󵄨
󵄨
:= sup max {󵄨󵄨󵄨𝑢− (𝛼) − V− (𝛼)󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑢+ (𝛼) − V+ (𝛼)󵄨󵄨󵄨} .
𝛼∈[0,1]
(10)
One can see that
󵄨 󵄨
󵄨
󵄨
𝐷 (𝑢, 0) = sup max {󵄨󵄨󵄨𝑢− (𝛼)󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑢+ (𝛼)󵄨󵄨󵄨}
𝛼∈[0,1]
(11)
󵄨
󵄨 󵄨
󵄨
= max {󵄨󵄨󵄨𝑢− (0)󵄨󵄨󵄨 , 󵄨󵄨󵄨𝑢+ (0)󵄨󵄨󵄨} .
The partial ordering relation ⪯ on 𝐸1 is defined as follows:
𝑢 ⪯ V ⇐⇒ 𝑢− (𝛼) ≤ V− (𝛼) ,
𝑢+ (𝛼) ≤ V+ (𝛼)
∀𝛼 ∈ [0, 1] .
(12)
Now, we may give the following.
Conversely, if the pair of functions 𝛼 and 𝛽 satisfies the
conditions (i)–(iv), then there exists a unique 𝑢 ∈ 𝐸1 such that
[𝑢]𝛼 := [𝑢− (𝛼), 𝑢+ (𝛼)] for each 𝛼 ∈ [0, 1]. The fuzzy number 𝑢
corresponding to the pair of functions 𝑢− and 𝑢+ is defined by
𝑢 : R → [0, 1], 𝑢(𝑥) := sup{𝛼 : 𝑢− (𝛼) ≤ 𝑥 ≤ 𝑢+ (𝛼)}.
1
Let 𝑢, V, 𝑤 ∈ 𝐸 and 𝑘 ∈ R. Then the operations addition,
scalar multiplication, and product are defined on 𝐸1 by
𝑢 + V = 𝑤 ⇐⇒ [𝑤]𝛼 = [𝑢]𝛼 + [V]𝛼 ,
𝑤+ (𝛼) = 𝑢+ (𝛼) + V+ (𝛼) ,
[𝑘𝑢]𝛼 = 𝑘[𝑢]𝛼 ,
∀𝛼 ∈ [0, 1] ,
𝑢V = 𝑤 ⇐⇒ [𝑤]𝛼 = [𝑢]𝛼 [V]𝛼 ,
∀𝛼 ∈ [0, 1] ,
Proposition 2 (see [6]). Let 𝑢, V, 𝑤, 𝑧 ∈ 𝐸1 and 𝑘 ∈ R. Then,
the following statements hold.
(i) (𝐸1 , 𝐷) is a complete metric space.
(ii) 𝐷(𝑘𝑢, 𝑘V) = |𝑘|𝐷(𝑢, V).
(iii) 𝐷(𝑢 + V, 𝑤 + V) = 𝐷(𝑢, 𝑤).
(iv) 𝐷(𝑢 + V, 𝑤 + 𝑧) ≤ 𝐷(𝑢, 𝑤) + 𝐷(V, 𝑧).
(v) |𝐷(𝑢, 0) − 𝐷(V, 0)| ≤ 𝐷(𝑢, V) ≤ 𝐷(𝑢, 0) + 𝐷(V, 0).
∀𝛼 ∈ [0, 1]
⇐⇒ 𝑤− (𝛼) = 𝑢− (𝛼) + V− (𝛼) ,
(7)
where it is immediate that
𝑤− (𝛼) = min {𝑢− (𝛼) V− (𝛼) , 𝑢− (𝛼) V+ (𝛼) , 𝑢+ (𝛼) V− (𝛼) ,
𝑢+ (𝛼) V+ (𝛼)} ,
𝑤+ (𝛼) = max {𝑢− (𝛼) V− (𝛼) , 𝑢− (𝛼) V+ (𝛼) , 𝑢+ (𝛼) V− (𝛼) ,
𝑢+ (𝛼) V+ (𝛼)} ,
(8)
for all 𝛼 ∈ [0, 1].
Let 𝑊 be the set of all closed bounded intervals 𝐴 of real
numbers with endpoints 𝐴 and 𝐴, that is, 𝐴 := [𝐴, 𝐴]. Define
the relation 𝑑 on 𝑊 by
󵄨
󵄨 󵄨
󵄨
(9)
𝑑 (𝐴, 𝐵) := max {󵄨󵄨󵄨𝐴 − 𝐵󵄨󵄨󵄨 , 󵄨󵄨󵄨󵄨𝐴 − 𝐵󵄨󵄨󵄨󵄨} .
Following Matloka [1], we give some definitions concerning the sequences of fuzzy numbers which are needed in the
text.
A sequence 𝑢 = (𝑢𝑘 ) of fuzzy numbers is a function 𝑢 from
the set N into the set 𝐸1 . The fuzzy number 𝑢𝑘 denotes the
value of the function at 𝑘 ∈ N and is called as the general term
of the sequence. By 𝑤(𝐹), we denote the set of all sequences
of fuzzy numbers
A sequence (𝑢𝑛 ) ∈ 𝑤(𝐹) is called convergent with limit
𝑢 ∈ 𝐸1 , if for every 𝜀 > 0 there exists an 𝑛0 = 𝑛0 (𝜀) ∈ N such
that 𝐷(𝑢𝑛 , 𝑢) < 𝜀 for all 𝑛 ≥ 𝑛0 .
A double sequence 𝑢 = (𝑢𝑛𝑘 ) of fuzzy real numbers is
defined by a function 𝑢 from the set N × N into the set 𝐸1 .
The fuzzy number 𝑢𝑛𝑘 denotes the value of the function at
(𝑛, 𝑘) ∈ N × N.
A double sequence 𝑢 = (𝑢𝑚𝑛 ) of fuzzy numbers is said to
be convergent in the Pringsheim’s sense or P-convergent, if for
every 𝜀 > 0 there exists 𝑘 ∈ N such that 𝐷(𝑢𝑚𝑛 , 𝑢0 ) < 𝜀 for all
𝑚, 𝑛 ≥ 𝑘 and is denoted by 𝑃 − lim𝑚,𝑛 → ∞ 𝑢𝑚𝑛 = 𝑢0 . The fuzzy
number 𝑢0 is called the Pringsheim limit of 𝑢.
4
International Journal of Analysis
Let I2 ⊂ 2N×N be a strongly admissible ideal.
2. Main Results
(i) A double sequence 𝑢 = (𝑢𝑚𝑛 ) of fuzzy numbers is said
to be I2 -convergent to a fuzzy number 𝑢0 if for any
𝜀 > 0 we have 𝐴(𝜀) = {(𝑚, 𝑛) ∈ N × N : 𝐷(𝑢𝑚𝑛 , 𝑢0 ) ≥
𝜀} ∈ I2 and is written I2 − lim𝑚,𝑛 → ∞ 𝑢𝑚𝑛 = 𝑢0 .
In this section, we study certain properties of regular convergence, regularly (I2 , I)-convergence and regularly (I2 , I)Cauchy double sequences of fuzzy numbers.
(ii) A double sequence 𝑢 = (𝑢𝑚𝑛 ) of fuzzy numbers is said
to be I∗2 -convergent to 𝑢0 ∈ 𝐸1 if there exists 𝑀 ∈
F(I2 ) (i.e., 𝐻 = N × N \ 𝑀 ∈ I2 ) such that
lim 𝑢
𝑚,𝑛 → ∞ 𝑚𝑛
(𝑚,𝑛)∈𝑀
= 𝑢0
(13)
and is written I∗2 − lim𝑚,𝑛 → ∞ 𝑢𝑚𝑛 = 𝑢0 .
(iii) A double sequence 𝑢 = (𝑢𝑚𝑛 ) of fuzzy numbers is said
to be I2 -Cauchy, if for each 𝜀 > 0, there exist 𝑠 =
𝑠(𝜀), 𝑡 = 𝑡(𝜀) ∈ N such that 𝐴(𝜀) = {(𝑚, 𝑛) ∈ N × N :
𝐷(𝑢𝑚𝑛 , 𝑢𝑠𝑡 ) ≥ 𝜀} ∈ I2 .
(iv) A double sequence 𝑢 = (𝑢𝑚𝑛 ) of fuzzy numbers is said
to be I∗2 -Cauchy if there exists a set 𝑀 ∈ F(I2 ) (i.e.,
𝐻 = N×N\𝑀 ∈ I2 ) such that for every 𝜀 > 0 and for
(𝑚, 𝑛), (𝑠, 𝑡) ∈ 𝑀, 𝑚, 𝑛, 𝑠, 𝑡 > 𝑘0 = 𝑘0 (𝜀), 𝐷(𝑢𝑚𝑛 , 𝑢𝑠𝑡 ) <
𝜀, that is,
lim
𝐷 (𝑢𝑚𝑛 , 𝑢𝑠𝑡 ) = 0.
𝑚,𝑛,𝑠,𝑡 → ∞
(𝑚,𝑛),(𝑠,𝑡)∈𝑀
(14)
Now, we begin with quoting the following five lemmas
due to Dündar and Talo [24, 25] which are needed throughout
the paper.
Lemma 3 (see [24, Theorem 3.3]). Let I2 ⊂ 2N×N be a
strongly admissible ideal, let 𝑢 = (𝑢𝑚𝑛 ) be a double sequence
of fuzzy numbers, and let 𝑢0 be a fuzzy number. Then, 𝑃 −
lim𝑚,𝑛 → ∞ 𝑢𝑚𝑛 = 𝑢0 implies I2 − lim𝑚,𝑛 → ∞ 𝑢𝑚𝑛 = 𝑢0 .
Lemma 4 (see [24, Theorem 4.2]). Let I2 ⊂ 2N×N be a
strongly admissible ideal, let 𝑢 = (𝑢𝑚𝑛 ) be a double sequence of
fuzzy numbers, and let 𝑢0 ∈ 𝐸1 . Then, I∗2 − lim𝑚,𝑛 → ∞ 𝑢𝑚𝑛 =
𝑢0 implies I2 − lim𝑚,𝑛 → ∞ 𝑢𝑚𝑛 = 𝑢0 .
Lemma 5 (see [24, Theorem 4.4]). Let I2 ⊂ 2N×N be a
strongly admissible ideal with property (AP2), let 𝑢 = (𝑢𝑚𝑛 )
be a double sequence of fuzzy numbers, and let 𝑢0 be a fuzzy
real number. Then, I2 − lim𝑚,𝑛 → ∞ 𝑢𝑚𝑛 = 𝑢0 implies I∗2 −
lim𝑚,𝑛 → ∞ 𝑢𝑚𝑛 = 𝑢0 .
Lemma 6 (see [25, Theorem 3.2]). Let I2 ⊂ 2N×N be a
strongly admissible ideal. A double sequence 𝑢 = (𝑢𝑚𝑛 ) of
fuzzy numbers is I2 -convergent if and only if it is I2 -Cauchy
sequence.
Lemma 7 (see [25, Theorem 3.4]). Let I2 ⊂ 2N×N be a strongly admissible ideal. If a double sequence 𝑢 = (𝑢𝑚𝑛 ) of fuzzy
numbers is an I∗2 -Cauchy sequence, then it is I2 -Cauchy.
Definition 8. A double sequence (𝑢𝑚𝑛 ) of fuzzy numbers
is said to be regularly convergent, if it is convergent in
Pringsheim’s sense and the limits
lim 𝑢 ,
𝑚 → ∞ 𝑚𝑛
(𝑛 ∈ N) ,
lim 𝑢 ,
𝑛 → ∞ 𝑚𝑛
(𝑚 ∈ N) ,
(15)
exist for each fixed 𝑛 ∈ N and 𝑚 ∈ N, respectively. Note that if
(𝑢𝑚𝑛 ) is regularly convergent to a fuzzy number 𝑢0 , then the
limits
lim lim 𝑢 ,
𝑛 → ∞𝑚 → ∞ 𝑚𝑛
lim lim 𝑢
𝑚 → ∞𝑛 → ∞ 𝑚𝑛
(16)
exist and are equal to 𝑢0 . In this case we write
𝑟 − lim 𝑢𝑚𝑛 = 𝑢0
𝑚,𝑛 → ∞
𝑟
or 𝑢𝑚𝑛 󳨀→ 𝑢0
as 𝑚, 𝑛 󳨀→ ∞.
(17)
Definition 9. Let I2 ⊂ 2N×N be a strongly admissible ideal,
and let I ⊂ 2N be an admissible ideal. A double sequence
(𝑢𝑚𝑛 ) of fuzzy numbers is said to be regularly (I2 , I)convergent (𝑟(I2 , I)-convergent), if it is I2 -convergent in
Pringsheim’s sense and for every 𝜀 > 0, the following
statements hold:
{𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , V𝑛 ) ≥ 𝜀} ∈ I
(18)
for some fuzzy numbers V𝑛 , for each 𝑛 ∈ N and
{𝑛 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑤𝑚 ) ≥ 𝜀} ∈ I
(19)
for some fuzzy numbers 𝑤𝑚 , for each 𝑚 ∈ N.
In the case (𝑢𝑚𝑛 ) is regularly (I2 , I)-convergent
(𝑟(I2 , I)-convergent) to a fuzzy number 𝑢0 , then the limits
I − lim𝑛 → ∞ lim𝑚 → ∞ 𝑢𝑚𝑛 and I − lim𝑚 → ∞ lim𝑛 → ∞ 𝑢𝑚𝑛
exist and are equal to 𝑢0 .
Theorem 10. Let I2 be a strongly admissible ideal of N × N,
and let I be an admissible ideal of N. If a double sequence
(𝑢𝑚𝑛 ) of fuzzy numbers is regularly convergent, then it is
𝑟(I2 , I)-convergent.
Proof. Let (𝑢𝑚𝑛 ) be regularly convergent. Then (𝑢𝑚𝑛 ) is convergent in Pringsheim’s sense and the limits lim𝑚 → ∞ 𝑢𝑚𝑛 (𝑛 ∈
N) and lim𝑛 → ∞ 𝑢𝑚𝑛 (𝑚 ∈ N) exist. By Lemma 3, (𝑢𝑚𝑛 ) is
I2 -convergent. Also, for 𝜀 > 0, there exist 𝑚 = 𝑚0 (𝜀) and
𝑛 = 𝑛0 (𝜀) such that
𝐷 (𝑢𝑚𝑛 , V𝑛 ) < 𝜀
(20)
for some fuzzy numbers V𝑛 and each fixed 𝑛 ∈ N for every
𝑚 ≥ 𝑚0 and
𝐷 (𝑢𝑚𝑛 , 𝑤𝑚 ) < 𝜀
(21)
International Journal of Analysis
5
for some fuzzy numbers 𝑤𝑚 and each fixed 𝑚 ∈ N for every
𝑛 ≥ 𝑛0 . Then, since I is admissible ideal so for 𝜀 > 0, we have
{𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , V𝑛 ) ≥ 𝜀} = {1, 2, . . . , 𝑚0 − 1} ∈ I,
{𝑛 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑤𝑚 ) ≥ 𝜀} = {1, 2, . . . , 𝑛0 − 1} ∈ I.
(22)
𝐴 (𝜀) = {𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , V𝑛 ) ≥ 𝜀} ∈ I
Hence, (𝑢𝑚𝑛 ) is 𝑟(I2 , I)-convergent.
Definition 11. Let I2 be a strongly admissible ideal of N ×
N, and let I be an admissible ideal of N. A double sequence
(𝑢𝑚𝑛 ) of fuzzy numbers is said to be 𝑟(I∗2 , I∗ )-convergent, if
there exist the sets 𝑀 ∈ F(I2 ) (i.e., N × N \ 𝑀 ∈ I2 ), 𝑀1 ∈
F(I), and 𝑀2 ∈ F(I) (i.e., N \ 𝑀1 ∈ I and N \ 𝑀2 ∈ I)
such that the limits
lim 𝑢 ,
𝑚,𝑛 → ∞ 𝑚𝑛
(𝑚,𝑛)∈𝑀
lim 𝑢 ,
𝑚 → ∞ 𝑚𝑛
𝑚∈𝑀1
lim 𝑢
𝑛 → ∞ 𝑚𝑛
𝑛∈𝑀2
Theorem 12. Let I2 be a strongly admissible ideal of N × N,
and let I be an admissible ideal of N. If a double sequence
(𝑢𝑚𝑛 ) of fuzzy numbers is 𝑟(I∗2 , I∗ )-convergent, then it is
𝑟(I2 , I)-convergent.
Proof. Let (𝑢𝑚𝑛 ) be 𝑟(I∗2 , I∗ )-convergent. Then, it is I∗2 convergent and so, by Lemma 4, it is I2 -convergent. Also,
there exist the sets 𝑀1 , 𝑀2 ∈ F(I) such that
(∃𝑚0 ∈ N)
𝐷 (𝑢𝑚𝑛 , V𝑛 ) < 𝜀,
(∀𝑚 ≥ 𝑚0 )
(𝑚 ∈ 𝑀1 )
(𝑛 ∈ N)
(24)
for some fuzzy numbers V𝑛 and
(∀𝜀 > 0)
(∃𝑛0 ∈ N)
𝐷 (𝑢𝑚𝑛 , 𝑤𝑚 ) < 𝜀,
(∀𝑛 ≥ 𝑛0 )
(𝑛 ∈ 𝑀2 )
(𝑚 ∈ N)
(25)
(29)
for some fuzzy numbers 𝑤𝑚 , for each 𝑚 ∈ N.
Now put
𝐴 1 = {𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , V𝑛 ) ≥ 1} ,
𝐴 𝑘 = {𝑚 ∈ N :
1
1
≤ 𝐷 (𝑢𝑚𝑛 , V𝑛 ) <
}
𝑘
𝑘−1
(30)
for 𝑘 ≥ 2, for some fuzzy numbers V𝑛 and for each 𝑛 ∈ N.
It is clear that 𝐴 𝑖 ∩ 𝐴 𝑗 = 0 for 𝑖 ≠ 𝑗 and 𝐴 𝑖 ∈ I for each
𝑖 ∈ N. By the property (𝐴𝑃) there is a countable family of sets
{𝐵1 , 𝐵2 , . . .} in I such that 𝐴 𝑗 △ 𝐵𝑗 is a finite set for each
𝑗 ∈ N and 𝐵 = ⋃∞
𝑗=1 𝐵𝑗 ∈ I.
We prove that
lim 𝑢
𝑚 → ∞ 𝑚𝑛
𝑚∈𝑀
= V𝑛 for some fuzzy numbers V𝑛 and for each 𝑛 ∈ N
(31)
{𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , V𝑛 ) ≥ 𝛿}
𝑘
𝐴 (𝜀) = {𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , V𝑛 ) ≥ 𝜀} ⊂ 𝐻1 ∪ {1, 2, . . . , 𝑚0 −1} ,
(𝑛 ∈ N) ,
𝐵 (𝜀) = {𝑛 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑤𝑚 ) ≥ 𝜀} ⊂ 𝐻2 ∪ {1, 2, . . . , 𝑛0 −1} ,
(𝑚 ∈ N) ,
(26)
for 𝐻1 , 𝐻2 ∈ I. Since I is admissible ideal we get
𝐻2 ∪ {1, 2, . . . , 𝑛0 − 1} ∈ I,
𝐶 (𝜀) = {𝑛 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑤𝑚 ) ≥ 𝜀} ∈ I
for 𝑀 = N \ 𝐵 ∈ F(I). Let 𝛿 > 0 be given. Choose 𝑘 ∈ N
such that 1/𝑘 < 𝛿. Then, we have
for some fuzzy numbers 𝑤𝑚 . Hence, we have
𝐻1 ∪ {1, 2, . . . , (𝑚0 − 1)} ∈ I,
(28)
for some fuzzy numbers V𝑛 , for each 𝑛 ∈ N and
(23)
exist for each fixed 𝑛 ∈ N and 𝑚 ∈ N, respectively.
(∀𝜀 > 0)
Proof. Let a double sequence (𝑢𝑚𝑛 ) of fuzzy numbers be
𝑟(I2 , I)-convergent. Then (𝑢𝑚𝑛 ) is I2 -convergent and so
(𝑢𝑚𝑛 ) is I∗2 -convergent, by Lemma 5. Also, for every 𝜀 > 0
we have
⊂ ⋃𝐴 𝑗 for some fuzzy numbers V𝑛 and for each 𝑛 ∈ N.
𝑗=1
(32)
Since 𝐴 𝑗 △ 𝐵𝑗 is a finite set for 𝑗 ∈ {1, 2, . . . , 𝑘}, there exists
𝑚0 ∈ N such that
𝑘
𝑘
𝑗=1
𝑗=1
(⋃𝐵𝑗 ) ∩ {𝑚 : 𝑚 ≥ 𝑚0 } = (⋃𝐴 𝑗 ) ∩ {𝑚 : 𝑚 ≥ 𝑚0 } .
(33)
(27)
If 𝑚 ≥ 𝑚0 and 𝑚 ∉ 𝐵 then
𝑘
𝑘
𝑗=1
𝑗=1
and therefore 𝐴(𝜀), 𝐵(𝜀) ∈ I. This shows that the double
sequence (𝑢𝑚𝑛 ) is 𝑟(I2 , I)-convergent.
𝑚 ∉ ⋃𝐵𝑗 and so 𝑚 ∉ ⋃𝐴 𝑗 .
Theorem 13. Let I2 ⊂ 2N×N be a strongly admissible ideal
with property (AP2), and let I ⊂ 2N be an admissible
ideal with property (AP). If a double sequence (𝑢𝑚𝑛 ) of fuzzy
numbers is 𝑟(I2 , I)-convergent, then (𝑢𝑚𝑛 ) is 𝑟(I∗2 , I∗ )convergent.
Thus, we have 𝐷(𝑢𝑚𝑛 , V𝑛 ) < 1/𝑘 < 𝛿 for some fuzzy numbers
V𝑛 and for each 𝑛 ∈ N. This implies that
lim 𝑢
𝑚 → ∞ 𝑚𝑛
𝑚∈𝑀
(𝑥) = V𝑛 .
(34)
(35)
6
International Journal of Analysis
Hence, we have
∗
I − lim 𝑢𝑚𝑛 = V𝑛
𝑚→∞
(36)
for some fuzzy numbers V𝑛 and for each 𝑛 ∈ N.
Similarly, for the set 𝐶(𝜀) = {𝑛 ∈ N : 𝐷(𝑢𝑚𝑛 , 𝑤𝑚 ) ≥ 𝜀} ∈
I, we have
I∗ − lim 𝑢𝑚𝑛 = 𝑤𝑚
𝑛→∞
(37)
for some fuzzy numbers 𝑤𝑚 and for each 𝑚 ∈ N. Hence,
a double sequence (𝑢𝑚𝑛 ) of fuzzy numbers is 𝑟(I∗2 , I∗ )convergent.
Now, we give the definitions of 𝑟(I2 , I)-Cauchy sequence and 𝑟(I∗2 , I∗ )-Cauchy sequence.
Definition 14. Let I2 be a strongly admissible ideal of
N × N, and let I be an admissible ideal of N. A double
sequence (𝑢𝑚𝑛 ) of fuzzy numbers is said to be regularly
(I2 , I)-Cauchy (𝑟(I2 , I)-Cauchy), if it is I2 -Cauchy in
Pringsheim’s sense and for every 𝜀 > 0 there exist 𝑘𝑛 = 𝑘𝑛 (𝜀) ∈
N and 𝑙𝑚 = 𝑙𝑚 (𝜀) ∈ N such that the following statements hold:
𝐴 1 (𝜀) = {𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑢𝑘𝑛 𝑛 ) ≥ 𝜀} ∈ I,
(𝑛 ∈ N) ,
𝐴 2 (𝜀) = {𝑛 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑢𝑚𝑙𝑚 ) ≥ 𝜀} ∈ I,
(𝑚 ∈ N) .
(38)
A double sequence (𝑢𝑚𝑛 ) is said to be regularly (I∗2 , I∗ )Cauchy (𝑟(I∗2 , I∗ )-Cauchy), if there exist the sets 𝑀 ∈
F(I2 ), 𝑀1 ∈ F(I), and 𝑀2 ∈ F(I) (i.e., N × N \ 𝑀 ∈ I2 ,
N \ 𝑀1 ∈ I, and N \ 𝑀2 ∈ I) and for every 𝜀 > 0 there exist
𝑁 = 𝑁(𝜀), 𝑠 = 𝑠(𝜀), 𝑡 = 𝑡(𝜀), 𝑘𝑛 = 𝑘𝑛 (𝜀), 𝑙𝑚 = 𝑙𝑚 (𝜀) ∈ N such
that
𝐷 (𝑢𝑚𝑛 , 𝑢𝑠𝑡 ) < 𝜀 for (𝑚, 𝑛) , (𝑠, 𝑡) ∈ 𝑀,
𝐷 (𝑢𝑚𝑛 , 𝑢𝑘𝑛 𝑛 ) < 𝜀 for each 𝑚 ∈ 𝑀1 and for each 𝑛 ∈ N,
𝐷 (𝑢𝑚𝑛 , 𝑢𝑚𝑙𝑚 ) < 𝜀 for each 𝑛 ∈ 𝑀2 and for each 𝑚 ∈ N,
(39)
whenever 𝑚, 𝑛, 𝑠, 𝑡, 𝑘𝑛 , 𝑙𝑚 ≥ 𝑁.
Theorem 15. Let I2 be a strongly admissible ideal of N × N,
and let I be an admissible ideal of N. If a double sequence
(𝑢𝑚𝑛 ) of fuzzy numbers is 𝑟(I∗2 , I∗ )-Cauchy, then it is
𝑟(I2 , I)-Cauchy.
Proof. Since a double sequence (𝑢𝑚𝑛 ) of fuzzy numbers is
𝑟(I∗2 , I∗ )-Cauchy, it is I∗2 -Cauchy. We know that I∗2 Cauchy implies I2 -Cauchy by Lemma 7. Also, since the
double sequence (𝑢𝑚𝑛 ) of fuzzy numbers is 𝑟(I∗2 , I∗ )Cauchy so there exist the sets 𝑀1 , 𝑀2 ∈ F(I) and for every
𝜀 > 0 there exist 𝑘𝑛 = 𝑘𝑛 (𝜀), 𝑙𝑚 = 𝑙𝑚 (𝜀) ∈ N such that
𝐷 (𝑢𝑚𝑛 , 𝑢𝑘𝑛 𝑛 ) < 𝜀 for each 𝑚 ∈ 𝑀1 and for each 𝑛 ∈ N,
𝐷 (𝑢𝑚𝑛 , 𝑢𝑚𝑙𝑚 ) < 𝜀 for each 𝑛 ∈ 𝑀2 and for each 𝑚 ∈ N
(40)
for 𝑁 = 𝑁(𝜀) ∈ N and 𝑚, 𝑛, 𝑘𝑛 , 𝑙𝑚 ≥ 𝑁. Therefore, for 𝐻1 =
N \ 𝑀1 , 𝐻2 = N \ 𝑀2 ∈ I we have
𝐴 1 (𝜀) = {𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑢𝑘𝑛 𝑛 ) ≥ 𝜀}
⊂ 𝐻1 ∪ {1, 2, . . . , 𝑁 − 1} ,
(𝑛 ∈ N)
(41)
for 𝑚 ∈ 𝑀1 and
𝐴 2 (𝜀) = {𝑛 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑢𝑚𝑙𝑚 ) ≥ 𝜀}
⊂ 𝐻2 ∪ {1, 2, . . . , 𝑁 − 1} ,
(𝑚 ∈ N)
(42)
for 𝑛 ∈ 𝑀2 . Since I is admissible ideal
𝐻1 ∪ {1, 2, . . . , 𝑁 − 1} ∈ I,
𝐻2 ∪ {1, 2, . . . , 𝑁 − 1} ∈ I.
(43)
Hence, we have 𝐴 1 (𝜀), 𝐴 2 (𝜀) ∈ I and (𝑢𝑚𝑛 ) is 𝑟(I2 , I)Cauchy.
Theorem 16. Let I2 be a strongly admissible ideal of N × N,
and let I be an admissible ideal of N. If a double sequence
(𝑢𝑚𝑛 ) of fuzzy numbers is 𝑟(I2 , I)-convergent, then (𝑢𝑚𝑛 ) is
𝑟(I2 , I)-Cauchy sequence.
Proof. Let (𝑢𝑚𝑛 ) be a 𝑟(I2 , I)-convergent double sequence
of fuzzy numbers. Then (𝑢𝑚𝑛 ) is I2 -convergent, and by
Lemma 6, it is I2 -Cauchy sequence of fuzzy numbers. Also
for every 𝜀 > 0, we have
𝜀
𝜀
𝐴 1 ( ) = {𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , V𝑛 ) ≥ } ∈ I
2
2
(44)
for some fuzzy numbers V𝑛 , for each 𝑛 ∈ N and
𝜀
𝜀
𝐴 2 ( ) = {𝑛 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑤𝑚 ) ≥ } ∈ I
2
2
(45)
for some fuzzy numbers 𝑤𝑚 , for each 𝑚 ∈ N. Since I is
admissible ideal, the sets
𝜀
𝜀
𝐴𝑐1 ( ) = {𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , V𝑛 ) < } ,
2
2
(𝑛 ∈ N)
(46)
(𝑚 ∈ N)
(47)
for some fuzzy numbers V𝑛 and
𝜀
𝜀
𝐴𝑐2 ( ) = {𝑛 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑤𝑚 ) < } ,
2
2
for some fuzzy numbers 𝑤𝑚 , are nonempty and belong to
F(I). For 𝑘𝑛 ∈ 𝐴𝑐1 (𝜀/2), (𝑛 ∈ N and 𝑘𝑛 > 0) we have
𝐷(𝑢𝑘𝑛 𝑛 , V𝑛 ) < 𝜀/2 for some fuzzy numbers V𝑛 . Now, for 𝜀 > 0
we define the set
𝐵1 (𝜀) = {𝑚 ∈ N : 𝐷 (𝑢𝑚𝑛 , 𝑢𝑘𝑛 𝑛 ) ≥ 𝜀} ,
(𝑛 ∈ N) ,
(48)
where 𝑘𝑛 = 𝑘𝑛 (𝜀). Let 𝑚 ∈ 𝐵1 (𝜀). Then for 𝑘𝑛 ∈ 𝐴𝑐1 (𝜀/2),
(𝑛 ∈ N and 𝑘𝑛 > 0) we have
𝜀 ≤ 𝐷 (𝑢𝑚𝑛 , 𝑢𝑘𝑛 𝑛 ) ≤ 𝐷 (𝑢𝑚𝑛 , V𝑛 ) + 𝐷 (𝑢𝑘𝑛 𝑛 , V𝑛 )
< 𝐷 (𝑢𝑚𝑛 , V𝑛 ) +
𝜀
2
(49)
International Journal of Analysis
7
for some fuzzy numbers V𝑛 . This shows that
𝜀
𝜀
< 𝐷 (𝑢𝑚𝑛 , V𝑛 ) and so 𝑚 ∈ 𝐴 1 ( ) .
2
2
(50)
Hence, we have 𝐵1 (𝜀) ⊂ 𝐴 1 (𝜀/2).
Similarly, for 𝑙𝑚 ∈ 𝐴𝑐2 (𝜀/2) (𝑚 ∈ N and 𝑙𝑚 > 0) we have
𝜀
𝐷 (𝑢𝑚𝑙𝑚 , 𝑤𝑚 ) < ,
2
(𝑚 ∈ N)
(51)
for some fuzzy numbers 𝑤𝑚 . Therefore, it can be seen that
𝜀
𝐵2 (𝜀) = {𝑚 ∈ N : 𝐷 (𝑢𝑚𝑙𝑚 , 𝑤𝑚 ) ≥ 𝜀} ⊂ 𝐴 2 ( ) .
2
(52)
Hence, we have 𝐵1 (𝜀), 𝐵2 (𝜀) ∈ I. This shows that (𝑢𝑚𝑛 ) is
𝑟(I2 , I)-Cauchy sequence of fuzzy numbers.
Acknowledgment
The authors would like to express their pleasure to the referees
for making some useful remarks and helpful suggestions
which improved the presentation of the paper.
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