# Almostλ

Authors: Mahmut Işık; Mikail Et
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

The main purpose of this article is to
introduce the concepts of almostλ
r-statistical convergence and
strongly almost λ
r-convergence of order β of sequences of
fuzzy numbers with respect to an Orlicz function. We give some relations
between strongly almost λ
r-convergence and almost λ
r-statistical convergence of order β of sequences of fuzzy numbers.


---

## Body

## 1. Introduction

The concept of fuzzy set was introduced by Zadeh [1]. Matloka [2] introduced sequences of fuzzy numbers and provided that every convergent sequence of fuzzy numbers is bounded. Recently, sequences of fuzzy numbers have been discussed by Altınok et al. [3, 4], Aytar and Pehlivan [5], Çolak et al. [6, 7], Gökhan et al. [8], Nuray [9], and Talo and Başar [10].The idea of statistical convergence was introduced by Fast [11] and the notion was linked with summability theory by Alotaibi et al. [12], Connor [13], Et et al. [14, 15], Fridy [16], Işık [17], Mohiuddine et al. [18, 19], and Tripathy [20]. Recently, the notion was generalized by Gadjiev and Orhan [21], Çolak [22], and Çolak and Bektaş [23].In this paper, we study the concepts of and examine some properties of almostλ
r-statistical convergence and strongly almost λ
r-convergence of order β of sequences of fuzzy numbers. The results which we obtained in this study are much more general than those obtained by Başarır et al. [24].


## 2. Definitions and Preliminaries

A fuzzy setu on R is called a fuzzy number if it has the following properties:(i)
u is normal;
(ii)
u is fuzzy convex;
(iii)
u is upper semicontinuous;
(iv)
supp
u
=
cl
⁡
{
x
∈
R
:
u
(
x
)
>
0
} is compact, where cl denoted the closure of the enclosed set.A sequenceX
=
(
X
k
) of fuzzy numbers is a function X from the set N of all positive integers into L
(
R
), where L
(
R
) is fuzzy number space. A sequence X
=
X
k is said to be bounded if the set X
k
:
k
∈
N is bounded. A sequence X
=
X
k is said to be convergent if there exists a positive integer k
0 such that d
X
k
,
X
0
<
ɛ for k
>
k
0, for every ɛ
>
0. By w
F, l
∞
F, and c
F we denote the set of a
l
l, b
o
u
n
d
e
d, and c
o
n
v
e
r
g
e
n
t sequences of fuzzy numbers, respectively [2].An Orlicz function is a functionN
:
[
0
,
∞
)
→
[
0
,
∞
), which is continuous, nondecreasing, and convex with N
(
0
)
=
0, N
(
x
)
>
0 for x
>
0, and N
(
x
)
→
∞ as x
→
∞.The generalized de la Vallée-Pousion mean is defined byt
m
x
=
1
/
λ
m
∑
k
∈
I
m
x
k
‍, where λ
=
λ
m is a nondecreasing sequence of positive numbers such that λ
m
+
1
≤
λ
m
+
1, λ
1
=
1, and λ
m
→
∞ as m
→
∞ and I
m
=
m
-
λ
m
+
1
,
m.The spacec
^ was introduced by Lorentz [25] and Maddox [26] has defined x to be strongly almost convergent to a number L if lim
m
→
∞
1
/
m
∑
k
=
1
m
‍
x
k
+
n
-
L
=
0, uniformly in n.Definition 1.
Letλ
=
λ
m be as above, let N be an Orlicz function, let r
=
r
k be any sequence of strictly positive real numbers, and let 0
<
β
≤
1 be given. A sequence X
=
X
k is said to be almost λ
r-statistically convergent of order β to X
0, with respect to the Orlicz function N, if for every ɛ
>
0
(1)
lim
m
→
∞
1
λ
m
β
k
∈
I
m
:
N
d
t
k
n
X
,
X
0
s
r
k
≥
ɛ
=
0
,
uniformly
in
n
,
s
>
0
,where (2)
t
k
n
X
=
X
n
+
X
n
+
1
+
⋯
+
X
n
+
k
k
+
1
=
1
k
+
1
∑
i
=
0
k
‍
X
n
+
i
.The set of all almost λ
r-statistically convergent sequences of order β with respect to the Orlicz function N will be denoted by S
^
β
F
N
,
r
,
λ. In this case we write X
k
→
X
0
S
^
β
F
N
,
r
,
λ. In the special case λ
m
=
m, for all m
∈
N we will write S
^
β
F
N
,
r instead of S
^
β
F
N
,
r
,
λ and in the special case r
k
=
1 for all k
∈
N and N
x
=
x we will write S
^
β
F
λ instead of S
^
β
F
N
,
r
,
λ. If we take β
=
1, then we obtain the set of all almost λ
r-statistically convergent sequence S
^
F
N
,
r
,
λ. If r
k
=
1 for all k
∈
N then we will write S
^
β
F
N
,
λ instead of S
^
β
F
N
,
r
,
λ.Definition 2.
Letλ
=
λ
m be as above, let N be an Orlicz function, let r
=
r
k be any sequence of strictly positive real numbers, and let 0
<
β
≤
1 be given. We define the following sets: (3)
w
^
β
F
N
,
r
,
λ
=
X
∈
w
F
:
lim
m
→
∞
1
λ
m
β
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
X
0
s
r
k
=
0
,
W
W
W
W
W
I
I
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
X
0
s
r
k
uniformly
in
n
,
s
>
0
,
w
^
β
F
N
,
r
,
λ
0
=
X
∈
w
F
:
lim
m
→
∞
1
λ
m
β
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
0
¯
s
r
k
=
0
,
W
W
W
W
W
I
I
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
0
¯
s
r
k
uniformly
in
n
,
s
>
0
.
w
^
β
F
N
,
r
,
λ
∞
=
X
∈
w
F
:
sup
⁡
m
,
n
1
λ
m
β
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
0
¯
s
r
k
<
∞
,
sup
⁡
m
,
n
1
λ
m
β
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
0
¯
s
r
k
<
∞
,
s
>
0
.
IfX
∈
w
^
β
F
N
,
r
,
λ, we say that X is strongly almost λ
r-convergent of order β with respect to the Orlicz function N. In this case we write X
k
→
X
0
w
^
β
F
N
,
r
,
λ.(i)
If we taker
k
=
1 for all k
∈
N, then we get w
^
β
F
N
,
r
,
λ
=
w
^
β
F
N
,
λ, w
^
β
F
N
,
r
,
λ
0
=
w
^
β
F
N
,
λ
0, and w
^
β
F
N
,
r
,
λ
∞
=
w
^
β
F
N
,
λ
∞.
(ii)
If we takeN
x
=
x and r
k
=
1 for all k
∈
N, then we get w
^
β
F
N
,
r
,
λ
=
w
^
β
F
λ, w
^
β
F
N
,
r
,
λ
0
=
w
^
β
F
λ
0, and w
^
β
F
N
,
r
,
λ
∞
=
w
^
β
F
λ
∞.
(iii)
If we takeβ
=
1 and r
k
=
1 for all k
∈
N, then we get w
^
β
F
N
,
r
,
λ
=
w
^
F
N
,
λ, w
^
β
F
N
,
r
,
λ
0
=
w
^
F
N
,
λ
0, and w
^
β
F
N
,
r
,
λ
∞
=
w
^
F
N
,
λ
∞.


## 3. Main Results

In this section we give the main results of this paper.The proof of each of the following results is straightforward, so we state the following.Theorem 3.
Letr
=
r
k be bounded, let λ
=
λ
m be as above, and let 0
<
β
≤
1; then w
^
β
F
N
,
r
,
λ
0
⊂
w
^
β
F
N
,
r
,
λ
⊂
w
^
β
F
N
,
r
,
λ
∞.Theorem 4.
Letr
=
r
k be bounded, let λ
=
λ
m be as above, and let 0
<
β
≤
1; then w
^
β
F
N
,
r
,
λ, w
^
β
F
N
,
r
,
λ
0, w
^
β
F
N
,
r
,
λ
∞, and S
^
β
F
N
,
r
,
λ are closed under the operations of addition and scalar multiplication.Theorem 5.
Letr
=
r
k be bounded, let λ
=
λ
m be as above, let 0
<
β
≤
1, and let N
1, N
2 be Orlicz functions; then(i)
w
^
β
F
N
1
,
r
,
λ
0
∩
w
^
β
F
N
2
,
r
,
λ
0
⊂
w
^
β
F
N
1
+
N
2
,
r
,
λ
0,
(ii)
w
^
β
F
N
1
,
r
,
λ
∩
w
^
β
F
N
2
,
r
,
λ
⊂
w
^
β
F
N
1
+
N
2
,
r
,
λ,
(iii)
w
^
β
F
N
1
,
r
,
λ
∞
∩
w
^
β
F
N
2
,
r
,
λ
∞
⊂
w
^
β
F
N
1
+
N
2
,
r
,
λ
∞.Theorem 6.
Let0
<
r
k
≤
q
k and q
k
/
r
k be bounded, let λ
=
λ
m be as above, and let 0
<
β
≤
1; then w
^
β
F
N
,
q
,
λ
⊆
w
^
β
F
N
,
r
,
λ.Theorem 7.
Letr
=
r
k be bounded, let λ
=
λ
m be as above, and let 0
<
β
≤
1. If lim
inf
⁡
m
→
∞
λ
m
β
/
m
β
>
0, then S
^
β
F
N
,
p
⊂
S
^
β
F
N
,
r
,
λ.Proof.
For givenɛ
>
0 we have (4)
k
≤
m
:
N
d
t
k
n
X
,
X
0
s
r
k
≥
ɛ
⊃
k
∈
I
m
:
N
d
t
k
n
X
,
X
0
s
r
k
≥
ɛand so (5)
1
m
β
k
≤
m
:
N
d
t
k
n
X
,
X
0
s
r
k
≥
ɛ
≥
λ
m
β
m
β
1
λ
m
β
k
∈
I
m
:
N
d
t
k
n
X
,
X
0
s
r
k
≥
ɛ
.Hence we get x
∈
S
^
β
F
N
,
r
,
λ.Theorem 8.
Letr
=
r
k be bounded, let λ
=
λ
m be as above, and let 0
<
β
≤
γ
≤
1; then S
^
β
F
N
,
r
,
λ
⊆
S
^
γ
F
N
,
r
,
λ.Proof.
The following inequality yields the proof:(6)
1
λ
m
γ
k
∈
I
m
:
N
d
t
k
n
X
,
X
0
s
r
k
≥
ɛ
≤
1
λ
m
β
k
∈
I
m
:
N
d
t
k
n
X
,
X
0
s
r
k
≥
ɛ
.Theorem 9.
Letr
=
r
k be bounded, let λ
=
λ
m be as above, let 0
<
β
≤
γ
≤
1, and let N be an Orlicz function; then(i)
w
^
β
F
N
,
r
,
λ
⊆
w
^
γ
F
N
,
r
,
λ,
(ii)
w
^
β
F
N
,
r
,
λ
0
⊆
w
^
γ
F
N
,
r
,
λ
0,
(iii)
w
^
β
F
N
,
r
,
λ
∞
⊆
w
^
γ
F
N
,
r
,
λ
∞.Proof.
It is easy so it is omitted.Theorem 10.
Iflim
r
k
>
0 and X is strongly almost λ
r-convergent of order β to X
0, with respect to the Orlicz function N, then X
0 is unique.Proof.
Suppose thatX
k
→
X
0
w
^
β
F
N
,
r
,
λ, X
k
→
Y
0
w
^
β
F
N
,
r
,
λ, and lim
r
k
=
l
>
0; then there exist s
1 and s
2 such that (7)
lim
m
1
λ
m
β
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
X
0
s
1
r
k
=
0
,
lim
m
1
λ
m
β
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
Y
0
s
2
p
k
=
0
,
uniformly
in
n
.Let s
=
max
⁡
2
s
1
,
2
s
2. Then we have (8)
lim
m
1
λ
m
β
∑
k
∈
I
m
‍
N
d
X
0
,
Y
0
s
r
k
≤
G
λ
m
β
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
X
0
s
1
r
k
+
G
λ
m
β
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
Y
0
s
2
r
k
⟶
0
,
m
⟶
∞
,where sup
⁡
k
r
k
=
H and G
=
max
⁡
1
,
2
H
-
1. Thus (9)
lim
m
1
λ
m
β
∑
k
∈
I
m
‍
N
d
X
0
,
Y
0
s
r
k
=
0
.Also, since clearly (10)
lim
k
N
d
X
0
,
Y
0
s
r
k
=
N
d
X
0
,
Y
0
s
l
,therefore N
d
X
0
,
Y
0
/
s
l
=
0. Hence X
0
=
Y
0.Theorem 11.
Letλ
=
λ
m be as above; then w
^
F
N
,
λ
∞
=
m
^
F
N
∞ and w
^
F
λ
∞
=
l
∞
F, where (11)
m
^
F
N
∞
=
X
∈
w
F
:
sup
⁡
k
,
n
N
d
t
k
n
X
,
0
¯
s
<
∞
,
N
d
t
k
n
X
,
0
¯
s
f
o
r
s
o
m
e
s
>
0
.Proof.
LetX
∈
w
^
F
N
,
λ
∞. Then we can find a constant K
1
>
0 such that (12)
1
λ
1
N
d
t
k
n
X
,
0
¯
s
≤
1
λ
m
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
0
¯
s
≤
K
1
for
each
k
,
n
.Hence X
∈
m
^
F
N
∞.
Conversely, ifX
∈
m
^
F
N
∞ we can find a constant K
2
>
0 such that N
d
t
k
n
X
,
0
¯
/
s
≤
K
2 for each k
,
n and so (13)
1
λ
m
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
0
¯
s

≤

K
2
λ
m
∑
k
∈
I
m
‍
1
≤
K
2
for
each
k
,
n
.Thus X
∈
w
^
F
N
,
λ
∞. Similar proof can be made for the other.
In Theorems12 and 13, we will assume that the sequence r
=
r
k is bounded and 0
<
h
=
inf
⁡
k
r
k
≤
r
k
≤
sup
⁡
k
p
k
=
H
<
∞. From now on, ∑
1
‍ will be denoting the sum over k
∈
I
m with d
t
k
n
X
,
X
0
≥
ɛ and ∑
2
‍ will be denoting the sum over k
∈
I
m with d
t
k
n
X
,
X
0
<
ɛ.Theorem 12.
Letλ
=
λ
m be as above and let β
,
γ
∈
(
0,1
] such that 0
<
β
≤
γ
≤
1; then w
^
β
F
N
,
r
,
λ
⊂
S
^
γ
F
λ.Proof.
One hasX
∈
w
^
β
F
N
,
r
,
λ; then we have (14)
1
λ
m
β
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
X
0
s
r
k
≥
1
λ
m
γ
∑
1
‍
N
d
t
k
n
X
,
X
0
s
r
k
≥
1
λ
m
γ
∑
1
‍
min
⁡
N
σ
h
,
N
σ
H
,
σ
=
ɛ
s
≥
1
λ
m
γ
k
∈
I
m
:
d
t
k
n
X
,
X
0
≥
ɛ
·
min
⁡
N
σ
h
,
N
σ
H
.Hence X
∈
S
^
γ
F
λ.Theorem 13.
Letλ
=
λ
m be as above and let β
,
γ
∈
(
0,1
] such that 0
<
β
≤
γ
≤
1, X
∈
l
∞
F, and lim
⁡
m
→
∞
⁡
λ
m
/
λ
m
β
=
1; then S
^
β
F
λ
⊂
w
^
γ
F
N
,
r
,
λ.Proof.
Suppose thatX
∈
l
∞
F and X
k
→
X
0
S
^
γ
F
λ. Since X is bounded, there exists a constant B
>
0 such that d
t
k
n
X
,
X
0
≤
B. Let ɛ
>
0, and then we have (15)
1
λ
m
γ
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
X
0
s
r
k
≤
1
λ
m
β
∑
1
‍
N
d
t
k
n
X
,
X
0
s
r
k
+
1
λ
m
β
∑
2
‍
N
d
t
k
n
X
,
X
0
s
r
k
≤
1
λ
m
β
∑
1
‍
max
⁡
N
B
s
h
,
N
B
s
H
+
1
λ
m
β
∑
2
‍
N
ɛ
s
r
k
≤
max
⁡
N
C
h
,
N
C
H
·
1
λ
n
β
k
∈
I
m
:
d
t
k
n
X
,
X
0
≥
ɛ
+
λ
m
λ
m
β
max
⁡
N
δ
h
,
N
δ
H
,
B
s
=
C
,
ɛ
s
=
δ
.Hence X
∈
w
^
γ
F
N
,
r
,
λ.
Letλ
=
λ
m and μ
=
μ
m be two sequences such that I
m
⊂
J
m
λ
m
≤
μ
m for all m
∈
N, where I
m
=
m
-
λ
m
+
1
,
m and J
m
=
m
-
μ
m
+
1
,
m. Suppose also that the parameters β and γ are fixed real numbers such that 0
<
β
≤
γ
≤
1. We will now give the following inclusion relations.Theorem 14.
Letλ
=
λ
m and μ
=
μ
m be defined as above, let N be an Orlicz function, and let β
,
γ
∈
(
0,1
] such that 0
<
β
≤
γ
≤
1. Then the following assertions hold true.(i)
If(16)
lim
inf
⁡
m
→
∞
λ
m
β
μ
m
γ
>
0
thenS
^
γ
F
N
,
μ
⊆
S
^
β
F
N
,
λ.
(ii)
If(17)
lim
n
→
∞
μ
m
λ
m
γ
=
1
thenS
^
β
F
N
,
λ
⊆
S
^
γ
F
N
,
μ.Proof.
(i) It is easy so it is omitted.
(ii) Suppose thatX
∈
S
^
β
F
N
,
λ and (17) is satisfied. Since I
m
⊂
J
m, we may write (18)
1
μ
m
γ
k
∈
J
m
:
N
d
t
k
n
X
,
X
0
s
≥
ɛ
=
1
μ
m
γ
m
-
μ
m
+
1
<
k
N
d
t
k
n
X
,
X
0
s
≤
m
-
λ
m
:
N
d
t
k
n
X
,
X
0
s
≥
ɛ
≤
m
-
λ
m
:
N
d
t
k
n
X
,
X
0
s
≥
ɛ
+
1
μ
m
γ
k
∈
I
m
:
N
d
t
k
n
X
,
X
0
s
≥
ɛ
≤
μ
m
-
λ
m
γ
λ
m
γ
+
1
λ
m
β
k
∈
I
m
:
N
d
t
k
n
X
,
X
0
s
≥
ɛ
≤
μ
m
λ
m
γ
-
1
+
1
λ
m
β
k
∈
I
m
:
N
d
t
k
n
X
,
X
0
s
≥
ɛfor all m
∈
N. Hence X
∈
S
^
γ
F
N
,
μ.Theorem 15.
Letλ
=
λ
m and μ
=
μ
m be defined as above, let N be an Orlicz function, and let β
,
γ
∈
(
0,1
] such that 0
<
β
≤
γ
≤
1. Then the following assertions hold true.(i)
If (16) holds, then w
^
γ
F
N
,
μ
⊂
w
^
β
F
N
,
λ,
(ii)
Let condition (17) be satisfied and X
∈
l
∞
F; then w
^
β
F
N
,
λ
⊂
w
^
γ
F
N
,
μ.Proof.
It is easy so it is omitted.Theorem 16.
Letλ
=
λ
m and μ
=
μ
m be defined as above, let N be an Orlicz function, and let β
,
γ
∈
(
0,1
] such that 0
<
β
≤
γ
≤
1. Then(i)
if (16) holds, then w
^
γ
F
N
,
μ
⊂
S
^
β
F
N
,
λ;
(ii)
let condition (17) be satisfied and X
∈
l
∞
F; then S
^
β
F
N
,
λ
⊂
w
^
γ
F
N
,
μ.Proof.
(i) It is easy so it is omitted.
(ii) Suppose thatX
∈
S
^
β
F
N
,
λ and let X
=
X
k be bounded; then there exists a constant B
>
0 such that d
t
k
n
X
,
X
0
≤
B. Then we may write (19)
1
μ
m
γ
∑
k
∈
I
m
‍
N
d
t
k
n
X
,
X
0
s
=
1
μ
m
γ
∑
k
∈
J
m
-
I
m
N
d
t
k
n
X
,
X
0
s
+
1
μ
m
γ
∑
k
∈
I
m
N
d
t
k
n
X
,
X
0
s
≤
μ
m
-
λ
m
γ
μ
m
γ
B
+
1
μ
m
γ
∑
k
∈
I
m
N
d
t
k
n
X
,
X
0
s
=
μ
m
μ
m
γ
-
1
B
+
1
μ
m
γ
∑
1
N
d
t
k
n
X
,
X
0
s
+
1
μ
m
γ
∑
2
N
d
t
k
n
X
,
X
0
s
≤
μ
m
λ
m
β
-
1
B
+
B
λ
m
β
k
∈
I
m
:
N
d
t
k
n
X
,
X
0
s
≥
ɛ
+
μ
m
λ
m
γ
ɛfor all n
∈
N. Hence we obtain X
∈
w
^
γ
F
N
,
μ.



---

*Source: 451050-2015-10-01.xml*