EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 5, 2020, 1131-1148
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

Applications of Lacunary Sequences to develop Fuzzy
Sequence Spaces for Ideal Convergence and Orlicz

Function

Kuldip Raj1, S. A. Mohiuddine2,3,∗

1 School of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, J&K, India
2 Department of General Required Courses, Mathematics, Faculty of Applied Studies, King
Abdulaziz University, Jeddah 21589, Saudi Arabia
3 Operator Theory and Applications Research Group, Department of Mathematics, King
Abdulaziz University, Jeddah 21589, Saudi Arabia

Abstract. In the present paper, we introduce and study ideal convergence of some fuzzy sequence
spaces via lacunary sequence, infinite matrix and Orlicz function. We study some topological and
algebraic properties of these spaces. We also make an effort to show that these spaces are normal
as well as monotone. Further, it is very interesting to show that if I is not maximal ideal then
these spaces are not symmetric.

2020 Mathematics Subject Classifications: 46A45, 40A05, 03E72

Key Words and Phrases: Lacunary sequence, ideal convergence, Orlicz function, sequence of
fuzzy numbers, difference sequence

1. Introduction and preliminaries

The concept of ordinary convergence of a sequence of fuzzy numbers was introduced
by Matloka [18] and proved some basic theorems for sequences of fuzzy numbers. Later
on Nanda [28] introduced sequences of fuzzy numbers and studied that the set of all
convergent sequences of fuzzy numbers forms a complete metric space. Recently, Nuray

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3708

Email addresses: kuldipraj68@gmail.com (K. Raj),
mohiuddine@gmail.com (S. A. Mohiuddine)

https://www.ejpam.com 1131 c© 2020 EJPAM All rights reserved.



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1132

and Savaş [30] studied statistical convergence and statistically 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. Initially the idea of I-convergence was introduced
by Kostyrko et al. [15]. A lot of developments have been made in this area, one may refer
to the articles (see [1–4, 10, 11, 16, 19, 23, 37]).

Let X be a non empty set. Then a family of sets I ⊆ 2X (power set of X) is said to
be an ideal if I is additive i.e U1, U2 ∈ I ⇒ U1 ∪ U2 ∈ I and U1 ∈ I, U2 ⊆ U1 ⇒ U2 ∈ I.
A non empty family of sets G ⊆ 2X is said to be filter on X if and only if Φ /∈ G, for
U1, U2 ∈ G we have U1 ∩ U2 ∈ G and for each U1 ∈ G and U1 ⊆ U2 implies U2 ∈ G. An
ideal I ⊆ 2X is called non trivial if I 6= 2X . A non-trivial ideal I ⊆ 2X is called admissible
if {{x} : x ∈ X} ⊆ I. A non-trivial ideal is maximal if there cannot exist any non-trivial
ideal J 6= I containing I as a subset.

A fuzzy number u is a fuzzy set [42] on the real axis, i.e., a mapping u : R → [0, 1]
which satisfies the following conditions:

(i) u is normal, i.e., there exist an x0 ∈ R such that u(x0) = 1;

(ii) u is fuzzy convex, i.e., for x, y ∈ R and 0 ≤ λ ≤ 1, u(λx+(1−λ)y) ≥ min[u(x), u(y)];

(iii) u is upper semi-continuous;

(iv) the closure of the set supp(u) is compact, where supp(u) = {x ∈ R : u(x) > 0} and
it is denoted by [u]0.

Let L(R) denotes the set of all fuzzy numbers. The α-level set of a fuzzy real number u,
for 0 < α ≤ 1 denoted by uα is defined as [u]α = {x ∈ R : u(x) ≥ α}, for α = 0 it is the
closure of the strong 0 cut (i.e. closure of the set {t ∈ R : u(t) > 0}).
For each r ∈ R, r ∈ L(R) is defined by

r̄(t) =

{
1, if t = r;
0, if t 6= 0.

Define a map d : L(R)× L(R)→ R by

d(x, y) = sup
α∈[0,1]

{max{|uα1 − vα1 |, |uα2 − vα2 |}},

where uα = [uα1 , u
α
2 ] and vα = [vα1 , v

α
2 ]. In this case, (L(R), d) is a complete metric space.

The additive identity and multiplicative identity in L(R) are denoted by 0 and 1, respec-
tively.

An Orlicz function M : [0,∞) → [0,∞) is convex, continuous and non-decreasing
function which also satisfy M(0) = 0, M(x) > 0 for x > 0 and M(x) → ∞ as x → ∞. If
convexity of Orlicz function is replaced by M(x+ y) ≤ M(x) +M(y), then this function
is called the modulus function and characterized by Nakano [27] and followed by Ruckle
[33] and others. An Orlicz function M is said to satisfy ∆2-condition for all values of u,



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1133

if there exists R > 0 such that M(2u) ≤ RM(u), u ≥ 0. Lindenstrauss and Tzafriri [17]
used the idea of Orlicz function to define the following sequence space

`M =
{
x ∈ w :

∞∑
k=1

M
( |xk|
ρ

)
<∞, for some ρ > 0

}
which is called as an Orlicz sequence space. The space `M is a Banach space with the
norm

||x|| = inf
{
ρ > 0 :

∞∑
k=1

M
( |xk|
ρ

)
≤ 1
}
.

An increasing non-negative integer sequence θ = (ir) with i0 = 0 and hr = (ir−ir−1)→
∞ as r → ∞ is known as lacunary sequence. The intervals determined by θ are denoted
by Ir = (ir−1, ir] and the ratio ir/ir−1 will be denoted by qr. Freedman et al. [7] defined
the space Nθ in the following way:

Nθ =
{
x = (xk) : lim

r→∞

1

hr

∑
k∈Ir

|xk − L| = 0 for some L
}
.

Fridy and Orhan [8] defined and studied the idea of lacunary statistical for sequence of real
number. Nuray [29] and Mursaleen and Mohiuddine [25] defined this notion, respectively,
for sequences of fuzzy numbers and in the setting of intuitionistic fuzzy normed space.
Most recently, Mohiuddine and Alamri [20] defined the notion of weighted lacunary equi-
statistical convergence and, as an application, proved some approximation theorems.

In [14] Kızmaz introduced the notion of difference sequence spaces and studied `∞(∆),
c(∆) and c0(∆) which has been recently used to define statistical convergence [12, 21].
Further this notion was generalized by Et and Çolak [6] by introducing the spaces `∞(∆m),
c(∆m) and c0(∆m). Later on, another type of generalization of the difference sequence
spaces is due to Tripathy and Esi [39] who studied the spaces `∞(∆ν), c(∆ν) and c0(∆ν).
Recently, Esi et al. [5] and Tripathy et al. [40] have introduced a new type of generalized
difference operators and unified those as follows: Let ν, m be non-negative integers, then
for Z a given sequence space, we have

Z(∆m
ν ) = {x = (xk) ∈ w : (∆m

ν xk) ∈ Z}

for Z = c, c0 and `∞ where ∆m
ν x = (∆m

ν xk) = (∆m−1
ν xk −∆m−1

ν xk+1) and ∆0
νxk = xk for

all k ∈ N, which is equivalent to the following binomial representation

∆m
ν xk =

m∑
i=0

(−1)i
(
m
i

)
xk+νi.

Taking ν = 1, we get the spaces `∞(∆m), c(∆m) and c0(∆m) studied by Et and Çolak [6].
Taking m = ν = 1, we get the spaces `∞(∆), c(∆) and c0(∆) introduced and studied by
Kızmaz [14]. For more details about sequence spaces (see [9, 31, 32, 34, 36, 38, 41]) and
references therein.



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1134

Let λ and η be two sequence spaces and A = (ank) be an infinite matrix of real or
complex numbers ank, where n, k ∈ N. Then we say that A defines a matrix mapping
from λ into η if for every sequence x = (xk)

∞
k=0 ∈ λ, the sequence Ax = {An(x)}∞n=0, the

A-transform of x, is in η, where

An(x) =

∞∑
k=0

ankxk (n ∈ N). (1)

By (λ, η), we denote the class of all matrices A such that A : λ → η. Thus, A ∈ (λ, η) if
and only if the series on the right-hand side of (1.1) converges for each n ∈ N and every
x ∈ λ.

The matrix domain λA of an infinite matrix A in a sequence space λ is defined by

λA = {x = (xk) : Ax ∈ λ}. (2)

The approach constructing a new sequence space by means of the matrix domain of a
particular limitation method has recently been employed by several authors (see [35]).

Kumar and Kumar [16] defined the notion of ideal (or, I-) convergence for sequence
of fuzzy numbers and recently studied by Mursaleen and Mohiuddine [26] in probabilistic
normed spaces (see also [22]).

Definition 1. A sequence X = (Xk) of fuzzy numbers is said to be I-convergent to a fuzzy
number X0, if for every ε > 0 such that

{k ∈ N : d(Xk, X0) ≥ ε} ∈ I.

The fuzzy number X0 is called I-limit of the sequence (Xk) of fuzzy numbers and we write
I- limXk = X0.

Definition 2. A sequence X = (Xk) of fuzzy numbers is said to be I-bounded if there
exists M > 0 such that

{k ∈ N : d(Xk, 0) > M} ∈ I.

Definition 3. Let θ = (kr) be lacunary sequence. Then a sequence (Xk) of fuzzy numbers

is said to be lacunary I-convergent if for every ε > 0 such that
{
r ∈ N : 1

hr

∑
k∈Ir

d(Xk, X) ≥

ε
}
∈ I. We write Iθ- limXk = X.

Definition 4. Let EF be denote the sequence space of fuzzy numbers. Then EF is said
to be solid (or normal) if (Yk) ∈ EF whenever (Xk) ∈ EF and d(Yk, 0) ≤ d(Xk, 0) for all
k ∈ N.

Example 1. (i) If we take I = IF = {A ⊆ N : A is a finite subset }. Then IF is a
nontrival admissible ideal of N and the corresponding convergence coincide with the usual
convergence.
(ii) If we take I = Iδ = {A ⊆ N : δ(A) = 0}. where δ(A) denote the asymptotic density of
the set A. Then Iδ is a non-trival admissible ideal of N and the corresponding convergence
coincide with the statistical convergence.



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1135

Lemma 1. [24] If d is a translation invariant metric. Then
(i) d(X + Y, 0) ≤ d(X, 0) + d(Y, 0),
(ii) d(λX, 0) ≤ |λ|d(X, 0), |λ| > 1.

Lemma 2. A sequence space EF is normal implies EF is monotone. (For the crisp set
case, one may refer to Kamthan and Gupta [13]).

Lemma 3. [15] If I ⊂ 2N is a maximal ideal then for each A ∈ N, we have either A ∈ I
or N \A ∈ I.

2. Some fuzzy sequence spaces

Throughout the paper wF denote the class of all fuzzy real-valued sequences. By N
and R we denote the set of natural and real numbers respectively. Let I be an admissible
ideal of N and θ = (ir) be lacunary sequence. Suppose p = (pk) is a bounded sequence
of positive real numbers, u = (uk) be a sequence of nonzero, nonnegative real numbers,
A = (ank) an infinite matrix and M = (Mk) be a sequence of Orlicz functions. In this
paper, we define the following sequence spaces as follows:

w
I(F )
θ [A,M, p, u,∆m

v ]

=

{
(xk) ∈ wF : ∀ε > 0,

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk
≥ ε

}
∈ I,

for some ρ > 0, s ≥ 0 and X0 ∈ L(R)

}
,

w
I(F )
θ [A,M, p, u,∆m

v ]0

=

{
(xk) ∈ wF : ∀ε > 0,

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ

)]pk
≥ ε

}
∈ I,

for some ρ > 0 and s ≥ 0

}
,

wFθ [A,M, p, u,∆m
v ]∞

=

{
(xk) ∈ wF : sup

n,r

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ

)]pk
<∞, for some ρ > 0

and s ≥ 0

}
,



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1136

and

w
I(F )
θ [A,M, p, u,∆m

v ]∞

=

{
(xk) ∈ wF : ∃ K > 0 such that

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk

≥ K

}
∈ I, for some ρ > 0 and s ≥ 0

}
.

Example 2. Let Xk(l) = 1 for k = 2q, q = 1, 2, 3.....

otherwise, Xk(l) =


k
3 (l − 2) + 1 for l ∈

[
2k−3

2 , 2
]
,

−k
3 (l − 2) + 1 for l ∈

[
2, 2k+3

2

]
.

For instance take m = v = 1, then the α−level sets of (Xk) and (∆Xk) are

[Xk]
α =

{
[1, 1] k = 2q,
[2− 3

k (1− α), 2 + 3
k (1− α)] otherwise.

and

[∆Xk]
α =


[−1− 3

k (1− α),−1 + 3
k (1− α)] k = 2q,

[1− 3
k (1− α), 1 + 3

k (1− α)] k + 1 = 2q.
[( 3
k + 3

k+1)(α− 1), ( 3
k + 3

k+1)(1− α)] otherwise.

Let A = (C, 1), the Cesàro matrix, M(x) = x, s = 0, u = (uk) = 1, p = (pk) = 1,
for all k ∈ N, ρ = 1 and θ = 2r, we have

sup
n

∑
k∈Ir

ank

[
Mk

(
d(uk∆

m
v Xk, 0)

ρ

)]pk
<∞

Thus, (Xk) ∈ wFθ [A,M, p, u,∆m
v ]∞ but (Xk) is not an Ideal convergent.

Let us consider a few special cases of the above sequence spaces:

(i) If Mk(x) = x for all k ∈ N, then we have

w
I(F )
θ [A,M, p, u,∆m

v ] = w
I(F )
θ [A, p, u,∆m

v ], w
I(F )
θ [A,M, p, u,∆m

v ]0 = w
I(F )
θ [A, p,

u,∆m
v ]0, w

F
θ [A,M, p, u,∆m

v ]∞ = wFθ [A, p, u,∆m
v ]∞ and w

I(F )
θ [A,M, p, u,∆m

v ]∞

= w
I(F )
θ [A, p, u,∆m

v ]∞.

(ii) If p = (pk) = 1, for all k, then we have

w
I(F )
θ [A,M, p, u,∆m

v ] = w
I(F )
θ [A,M, u,∆m

v ], w
I(F )
θ [A,M, p, u,∆m

v ]0 = w
I(F )
θ [A,

M, u,∆m
v ]0, w

F
θ [A,M, p, u,∆m

v ]∞ = wFθ [A,M, u,∆m
v ]∞ and w

I(F )
θ [A,M, p, u,∆m

v ]∞

= w
I(F )
θ [A,M, u,∆m

v ]∞.



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1137

(iii) If we take A = (C, 1), i.e., the Cesàro matrix, then the above classes of sequences

are denoted by w
I(F )
θ [w,M, p, u,∆m

v ], w
I(F )
θ [w,M, p, u,∆m

v ]0, w
F
θ [w,M, p, u,∆m

v ]∞
and
w
I(F )
θ [w,M, p, u,∆m

v ]∞ respectively.

(iv) If we take A = (ank) a de la Vallée-Poussin mean, i.e.,

ank =

{ 1
λn
, if k ∈ In = [n− λn + 1, n];

0, otherwise.

where (λn) is a non-decreasing sequence of positive numbers tending to ∞ and
λn+1 ≤ λn + 1, λ1 = 1, then the above classes of sequences are denoted by

w
I(F )
λ [M, p, u,∆m

v ],

w
I(F )
λ [M, p, u,∆m

v ]0, w
F
λ [M, p, u,∆m

v ]∞ and w
I(F )
λ [M, p, u,∆m

v ]∞ respectively.

(v) If I = IF then we obtain

wFθ [A,M, p, u,∆m
v ]

=

{
(xk) ∈ wF : lim

n,r→∞

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk
= 0, for some

ρ > 0 and s ≥ 0, X0 ∈ L(R)

}
,

wFθ [A,M, p, u,∆m
v ]0

=

{
(xk) ∈ wF : lim

n,r→∞

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk
= 0, for some

ρ > 0 and s ≥ 0

}
,

wFθ [A,M, p, u,∆m
v ]∞

=

{
(xk) ∈ wF : lim

n,r→∞

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ

)]pk
<∞, for some

ρ > 0 and s ≥ 0

}
.



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1138

(vi) If I = Iδ is an admissible ideal of N, then

w
I(F )
θ [A,M, p, u,∆m

v ]

=

{
(xk) ∈ wF : ∀ε > 0,

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk
≥ ε

}

∈ Iδ, for some ρ > 0, s ≥ 0 and X0 ∈ L(R)

}
,

w
I(F )
θ [A,M, p, u,∆m

v ]0

=

{
(xk) ∈ wF : ∀ε > 0,

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ

)]pk
≥ ε

}

∈ Iδ, for some ρ > 0 and s ≥ 0

}
,

and

w
I(F )
θ [A,M, p, u,∆m

v ]∞

=

{
(xk) ∈ wF : ∃ K > 0 such that

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk

≥ K

}
∈ Iδ, for some ρ > 0 and s ≥ 0

}
.

The following inequality will be used throughout the paper. Let p = (pk) be a sequence
of positive real numbers with 0 < pk ≤ supk pk = H and let D = max

{
1, 2H−1

}
. Then,

for the factorable sequences (ak) and (bk) in the complex plane, we have

|ak + bk|pk ≤ D(|ak|pk + |bk|pk). (3)

Also |ak|pk ≤ max
{

1, |a|H
}

for all a ∈ C.
The main purpose of this paper is to introduced and study some lacunary I-convergent

sequence spaces of fuzzy numbers by using an infinite matrix and a sequence of Orlicz
functions in more general setting. We also make an effort to study some properties like
linearity, paranorm, solidity and some interesting inclusion relations between the spaces

w
I(F )
θ [A,M, p, u,∆m

v ], w
I(F )
θ [A,M, p, u,∆m

v ]0, wFθ [A,M, p, u,∆m
v ]∞ and w

I(F )
θ [A,M, p, u,

∆m
v ]∞.



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1139

3. Main Results

In the current section we study some topological properties and some inclusion relations
between the sequence spaces which we have defined above.

Theorem 1. Let M = (Mk) be a sequence of Orlicz functions, p = (pk) be a bounded se-
quence of positive real numbers and u = (uk) be a sequence of strictly positive real numbers.

Then the spaces w
I(F )
θ [A,M, p, u,∆m

v ], w
I(F )
θ [A,M, p, u,∆m

v ]0 and w
I(F )
θ [A,M, p, u,∆m

v ]∞
are linear spaces over the complex field C.

Proof. We shall prove the result for the space w
I(F )
θ [A,M, p, u,∆m

n ]0 only and others
can be proved in the similar way. Let X = (Xk) and Y = (Yk) be two elements in

w
I(F )
θ [A,M, p, u,∆m

n ]0. Then there exists ρ1 > 0 and ρ2 > 0 such that

A ε
2

=

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ1

)]pk
≥ ε

2

}
∈ I

and

B ε
2

=

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Yk, 0)

ρ2

)]pk
≥ ε

2

}
∈ I.

Let α and β be two scalars. Then by using the inequality (3) and continuity of the function
M = (Mk), we have

1
hr

∑
k∈Ir

ank

[
k−sMk

(
d(αuk∆

m
v Xk + βuk∆

m
v Yk, 0)

|α|ρ1 + |β|ρ2

)]pk

≤ D
1

hr

∑
k∈Ir

ank

[
|α|

|α|ρ1 + |β|ρ2
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ1

)]pk

+ D
1

hr

∑
k∈Ir

ank

[
|β|

|α|ρ1 + |β|ρ2
k−sMk

(
d(uk∆

m
v Yk, 0)

ρ2

)]pk

≤ DK
1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ1

)]pk

+ DK
1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Yk, 0)

ρ2

)]pk
,

where K = max

{
1,

(
|α|

|α|ρ1+|β|ρ2

)H
,

(
|β|

|α|ρ1+|β|ρ2

)H}
.

From the above relation we obtain the following:



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1140{
n, r ∈ N : 1

hr

∑
k∈Ir ank

[
k−sMk

(
d(αuk∆m

v Xk+βuk∆m
v Yk,0)

|α|ρ1+|β|ρ2

)]pk
≥ ε

}

⊆

{
n, r ∈ N : DK

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ1

)]pk
≥ ε

2

}

∪

{
n, r ∈ N : DK

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Yk, 0)

ρ2

)]pk
≥ ε

2

}
∈ I.

This completes the proof.

Theorem 2. Let M = (Mk) be a sequence of Orlicz functions, p = (pk) be a bounded se-
quence of positive real numbers and u = (uk) be a sequence of strictly positive real numbers.

Then the spaces w
I(F )
θ [A,M, p, u,∆m

v ], w
I(F )
θ [A,M, p, u,∆m

v ]0 and w
I(F )
θ [A,M, p, u,∆m

v ]∞
are paranormed spaces with the paranorm g∆ defined by

g∆(X) = inf

{
(ρ)

pn
H :

(
1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ

)]pk) 1
H

≤ 1, for some ρ > 0

and s ≥ 0 , n = 1, 2, .... r ∈ N

}
where H = max{1, sup

k
pk}.

Proof. Clearly, g∆(−X) = g∆(X) and g∆(θ) = 0. Let X = (Xk) and Y = (Yk) be two

elements in w
I(F )
θ [A,M, p, u,∆m

v ]0. Then for every ρ > 0 we write

A1 =

{
ρ > 0 :

(
1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ

)]pk) 1
H

≤ 1

}

and

A2 =

{
ρ > 0 :

(
1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Yk, 0)

ρ

)]pk) 1
H

≤ 1

}
.

Let ρ1 ∈ A1 and ρ2 ∈ A2. If ρ = ρ1 + ρ2, then we get the following(
1
hr

∑
k∈Ir ank

[
k−sMk

(
d(uk∆m

v (Xk+Yk),0))
ρ

)])

≤ ρ1

ρ1 + ρ2

(
1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ1

)])



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1141

+
ρ2

ρ1 + ρ2

(
1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Yk, 0)

ρ2

)])
.

Thus, we have

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v (Xk + Yk), 0)

ρ

)]pk
≤ 1

and

g∆(X + Y ) = inf{(ρ1 + ρ2)
pn
H : ρ1 ∈ A1, ρ2 ∈ A2}

≤ inf{(ρ1)
pn
H : ρ1 ∈ A1}+ inf{(ρ2)

pn
H : ρ2 ∈ A2}

= g∆(X) + g∆(Y ).

Let tmk → t, where tmk , t ∈ C, and let g∆(Xm
k − Xk) → 0 as m → ∞. To prove that

g∆(tmk X
m
k − tXk)→ 0 as m→∞. Let tk → t, where tk, t ∈ C, and g∆(Xm

k −Xk)→ 0 as
m→∞. We have

A3 =

{
ρk > 0 :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρk

)]pk
≤ 1

}

and

A4 =

{
ρ′k > 0 :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Yk, 0)

ρ′k

)]pk
≤ 1

}
.

If ρk ∈ A3 and ρ′k ∈ A4 then by inequality (3) and continuity of the function M = (Mk),
we have that

k−sMk

(
d(uk∆m

v (tmXm
k −tX,0))

|tm−t|ρk+|t|ρ′k

)

≤ k−sMk

(
d(uk∆

m
v (tmXm

k − tXk), 0)

|tm − t|ρk + |t|ρ′k

)
+ k−sMk

(
d(uk∆

m
v (tXk − tX, 0))

|tm − t|ρk + |t|ρ′k

)

≤ |tm − t|ρk
|tm − t|ρk + |t|ρ′k

k−sMk

(
d(uk∆

m
v X

m
k , 0)

ρk

)

+
|t|ρ′k

|tm − t|ρk + |t|ρ′k
k−sMk

(
d(uk∆

m
v (Xm

k −Xk), 0)

ρ′k

)
.

From the above inequality it follows that

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v (tmXm

k − tX), 0)

|tm − t|ρk + |t|ρ′k

)]pk
≤ 1



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1142

and consequently,

g∆(tmk Xk + tX) = inf{(|tmk − t|ρk + |t|ρ′k})
pn
H : ρk ∈ A3, ρ

′
k ∈ A4}

≤ |tmk − t|ρ
pn
H
k inf{(ρk)

pn
H : ρk ∈ A3}+ |t|ρ′k inf{(ρ′k)

pn
H : ρ′k ∈ A4}

≤ max{|t|, |t|
pn
H }g∆(Xm

k −Xk).

Note that g∆(Xm
k ) ≤ g∆(Xm) + g∆(Xm

k −Xm), for all k ∈ N. Hence, by our assumption
the right hand tends to 0 as m→∞. This completes the proof.

Theorem 3. Let M = (Mk) be a sequence of Orlicz functions, p = (pk) be a bounded
sequence of positive real numbers,

(i) Let 0 < inf pk ≤ pk ≤ 1. Then

w
I(F )
θ [A,M, p, u,∆m

v ] ⊆ wI(F )
θ [A,M, u,∆m

v ], w
I(F )
θ [A,M, p, u,∆m

v ]0 ⊆ wI(F )
θ [A,M,

u,∆m
v ]0.

(ii) Let 1 ≤ pk ≤ sup pk <∞. Then

w
I(F )
θ [A,M, u,∆m

v ] ⊆ wI(F )
θ [A,M, p, u,∆m

v ], w
I(F )
θ [A,M, u,∆m

v ]0 ⊆ wI(F )
θ [A,M, p,

u,∆m
v ]0.

Proof. (i) Let X = (Xk) be an element in w
I(F )
θ [A,M, p, u,∆m

v ]. Since 0 < inf pk ≤
pk ≤ 1 we have

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]
≤ 1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk
.

Therefore,{
n, r ∈ N : 1

hr

∑
k∈Ir ank

[
k−sMk

(
d(uk∆m

v Xk,X0)
ρ

)]
≥ ε

}

⊆

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk
≥ ε

}
∈ I.

The other part can be proved in the same way.

(ii) Let X = (Xk) be an element in w
I(F )
θ [A,M, u,∆m

v ]. Since 1 ≤ pk ≤ sup pk <∞. Then
for each 0 < ε < 1 there exists a positive integer n0 such that

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]
≤ ε < 1 for all n ≥ n0.

This implies that

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk
≤ 1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]
.



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1143

Therefore, we have{
n, r ∈ N : 1

hr

∑
k∈Ir ank

[
k−sMk

(
d(uk∆m

v Xk,X0)
ρ

)]pk
≥ ε

}

⊆

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]
≥ ε

}
∈ I.

The other part can be proved in the similar way. This completes the proof.

Theorem 4. Let X = (Xk) be a sequence of Fuzzy numbers, M = (Mk) be a sequence of
Orlicz functions, p = (pk) be a bounded sequence of positive real numbers and u = (uk) be
a sequence of strictly positive real numbers. Then

w
I(F )
θ [A,M, p, u,∆m

v ]0 ⊂ wI(F )
θ [A,M, p, u,∆m

v ] ⊂ wFθ [A,M, p, u,∆m
v ]∞.

Proof. The inclusion w
I(F )
θ [A,M, p, u,∆m

v ]0 ⊂ w
I(F )
θ [A,M, p, u,∆m

v ] is obvious. Let

X = (Xk) ∈ w
I(F )
θ [A,M, p, u,∆m

v ]. Then there is some fuzzy number X0, such that

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk
≥ ε.

Now, by inequality (3), we have

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ

)]pk
≤ D 1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk

+ D
1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(X0, 0)

ρ

)]pk
.

This implies that X = (Xk) ∈ wFθ [A,M, p, u,∆m
v ]∞. This completes the proof.

Theorem 5. Let M = (Mk) and S = (Sk) be a sequence of Orlicz functions. Then

w
I(F )
θ [A,M, p, u,∆m

v ] ∩ wI(F )
θ [A,S, p, u,∆m

v ] ⊂ wI(F )
θ [A,M+ S, p, u,∆m

v ].

Proof. Let X = (Xk) ∈ w
I(F )
θ [A,M, p, u,∆m

v ]∩wI(F )
θ [A,S, p, u,∆m

v ] using the inequal-
ity (3), we have

1
hr

∑
k∈Ir ank

[
k−s(Mk + Sk)

(
d(uk∆m

v Xk,X0)
ρ

)]pk

=
1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)
+ k−sSk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1144

≤ D

{
1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk

+
1

hr

∞∑
k∈Ir

ank

[
k−sSk

(
d(uk∆

m
v Xk, X0)

ρ

)]pk}
.

Thus, X = (Xk) ∈ w
I(F )
θ [A,M+ S, p, u,∆m

v ]. This completes the proof.

Theorem 6. The sequence spaces w
I(F )
θ [A,M, p, u,∆m

v ]0 and w
I(F )
θ [A,M, p, u,∆m

v ]∞ are
normal as well as monotone.

Proof. We give the proof of the theorem for w
I(F )
θ [A,M, p, u,∆m

v ]0 only. Let X =

(Xk) ∈ w
I(F )
θ [A,M, p, u,∆m

v ]0 and Y = (Yk) be such that d(Yk, 0) ≤ d(Xk, 0) for all
k ∈ N. Then for given ε > 0 we have

B =

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Xk, 0)

ρ

)]pk
≥ ε

}
∈ I,

again the set

B1 =

{
n, r ∈ N :

1

hr

∑
k∈Ir

ank

[
k−sMk

(
d(uk∆

m
v Yk, 0)

ρ

)]pk
≥ ε

}
⊆ B.

Hence, B1 ∈ I and so Y = (Yk) ∈ w
I(F )
θ [A,M, p, u,∆m

v ]0. Thus, the space w
I(F )
θ [A,M,

p, u,∆m
v ]0 is normal. Also, from the Lemma 2, it follows that w

I(F )
θ [A,M, p, u,∆m

v ]0 is
monotone. This completes the proof.

Theorem 7. If I is not maximal ideal then the space w
I(F )
θ [A,M, p, u,∆m

v ] is neither
normal nor monotone.

Example 3. Let us consider a sequence of fuzzy numbers

Xk(l) =



1+l
2 − 1 ≤ l ≤ 1,

3−l
2 1 ≤ l ≤ 3,

0 otherwise.

If m = 0, then ∆m
v Xk = 1. Let A = (C, 1), the Cesàro matrix, M(x) = x, u =

(uk) = 1, s = 0, p = (pk) = 1, for all k ∈ N, ρ = 1 and θ = 2r then we have

(Xk) ∈ w
I(F )
θ [A,M, p, u,∆m

v ]. Since I is not maximal by Lemma 3, their exist a subset K
of N such that K /∈ I and N−K /∈ I. Let us define sequence Y = (Yk) by

Yk =

{
Xk k ∈ K
0 otherwise.



K. Raj, S. A. Mohiuddine / Eur. J. Pure Appl. Math, 13 (5) (2020), 1131-1148 1145

Then, (Yk) belongs to the canonical pre image of the k-step spaces of w
I(F )
θ [A,M, p, u,∆m

v ].

But Yk /∈ wI(F )
θ [A,M, p, u,∆m

v ]. Hence, w
I(F )
θ [A,M, p, u,∆m

v ] is not monotone. There-

fore, by Lemma 2, w
I(F )
θ [A,M, p, u,∆m

v ] is not normal.

Theorem 8. If I is neither maximal nor I = IF then the spaces w
I(F )
θ [A,M, p, u,∆m

v ]

and w
I(F )
θ [A,M, p, u,∆m

v ]0 are not symmetric.

Example 4. Let us consider a sequence of fuzzy numbers

Xk(l) =


l − 2k + 1 l ∈ [2k − 1, 2k],
−l + 2k + 1 l ∈ [2k, 2k + 1],
0 otherwise.

If m = 1, v = 1, then ∆m
v Xk = ∆Xk. Let A = (C, 1), the Cesàro matrix, M(x) = x2,

u = (uk) = 1, s = 0, I = Iδ, p = (pk) = 1, for all k ∈ N and θ = 2r. Thus, we have
(Xk) ∈ wI(F )[A,M, p, u,∆m

v ]. But the rearrangement Y = (Yk) of the sequence space (Xk)
is defined as

Yk = {X1, X4, X2, X9, X3, X16, X5, X25, X6, ...}

This implies that (Yk) ∈ w
I(F )
θ [A,M, p, u,∆m

v ]. Hence, w
I(F )
θ [A,M, p, u,∆m

v ] is not sym-

metric. Similarly, w
I(F )
θ [A,M, p, u,∆m

v ]0 is not symmetric.

Theorem 9. The spaces w
I(F )
θ [A,M, p, u,∆m

v ] and w
I(F )
θ [A,M, p, u,∆m

v ]0 are not con-
vergent free in general.

Example 5. Let us consider a sequence of fuzzy numbers

Xk(l) =



1+l
2 − 1 ≤ l ≤ 1,

3−l
2 1 ≤ l ≤ 3,

0 otherwise.

If m = 0, then ∆m
v Xk = 1. Let A = (C, 1), the Cesàro matrix, M(x) = x, u =

(uk) = 1, s = 0, p = (pk) = 1, for all k ∈ N and ρ = 1 then we have (Xk) ∈
wI(F )[A,M, p, u,∆m

v ]. Let Yk(l) = 1
k for all k ∈ N. Then (Yk) ∈ w

I(F )
θ [A,M, p, u,∆m

v ].

But Xk = 0 does not imply Yk = 0. Hence, w
I(F )
θ [A,M, p, u,∆m

v ] is not convergent free.

Similarly, w
I(F )
θ [A,M, p, u,∆m

v ]0 is not convergent free.

Acknowledgements

The authors would like to thank the referees for their invaluable comments and cor-
rections which led to the improvement of the manuscript.



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