Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://bmathaa.org
Volume 10 Issue 4(2018), Pages 1-13.
ON ∇2-STATISTICAL CONVERGENCE OF DOUBLE
SEQUENCES OF ORDER α IN RANDOM 2-NORMED SPACE
¨
OMER K˙IS¸ ˙I
Abstract. In this present paper, we introduce the notion of∇2-statistical
convergence of double sequences of orderα,∇2-statistical Cauchy double
se-quences of orderαin random 2-normed spaces and obtain some results. We display examples which show that our method of convergence is more general in random 2-normed space.
1. Introduction
The idea of the statistical convergence was given by Zygmund [36] in the first edition of his monograph published in Warsaw in 1935. The concept of statistical convergence was introduced by Fast [7] and Steinhaus [34] and then reintroduced by Schoenberg [31] independently. Over the years, statistical convergence has been developed in ([3], [13], [14], [21], [25], [29], [35]) and turned out very useful to resolve many convergence problems arising in Analysis.
Definition 1. ([7]) A number sequence x= (xk) is said to be statistically conver-gent to the number l if for everyε >0,
lim
n→∞ 1
n|{k≤n:|xk−l| ≥ε}|= 0.
In this case we writest−limk→∞xk =l.Statistical convergence is a natural
gener-alization of ordinary convergence. Iflimxk =l, thenst−limxk =l. The converse does not hold in general.
In literature, several interesting generalizations of statistical convergence have been appeared. One among these is λ-statistical convergence given by Mursaleen [23] with a non-decreasing sequenceλ= (λn) of positive real numbers tending to
∞such thatλn+1≤λn+ 1,λ1= 1.
The idea of λ-statistical convergence can be connected to the generalized de la Vall´ee-Poussin mean. It is defined by
tn(x) = 1
λn X
k∈In
xk
2000Mathematics Subject Classification. 40A05, 40A35.
Key words and phrases. λ-convergence; 2-norm; 2-normed space. c
2018 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted September 22, 2018. Published October 10, 2018. Communicated by F. Basar.
We thank the editor and referees for their careful reading, valuable suggestions and remarks.
whereIn= [n−λn+ 1, n].
Definition 2. ([23]) A sequence x= (xk) of numbers is said to be λ -statistical convergent to a numberl provided that for everyε >0,
lim
n→∞ 1
λn
|{k∈In :|xk−l| ≥ε}|= 0.
In this case, the number l is called λ-statistical limit of the sequence x= (xk)and we writeSλ−limk→∞xk =l.
Recently, for α∈(0,1] C¸ olak and Bekta¸s [2] generalized Definition 2 in terms ofλ-statistical convergence of orderαand obtained some analogous results.
The concept of probabilistic normed spaces was initially introduced by A. N. Sherstnev [33] in 1962. Menger [22] introduced the notion of probabilistic metric spaces in 1942. The idea of Menger [22] was to use distribution function instead of non-negative real values of a metric. In last few years these spaces are grown up rapidly and many detereministic results of linear normed spaces are obtained for probabilistic normed spaces. For a detailed study on probabilistic functional anal-ysis, we refer ([1], [17], [26], [32]). In 2005, Golet [16] used the concept of 2-norm of G¨ahler [15] and presented generalized probabilistic normed space which he called random 2-normed space. G¨urdal and Pehlivan ([37], [38]) studied statistical con-vergence in 2-normed spaces and in 2-Banach spaces. Recently, Sava¸s [39] defined and studied generalized statistical convergence in random 2-normed space. Esi and
¨
Ozdemir [6] introduced and studied the concept of generalized ∆m-statistical
con-vergence of sequences in probabilistic normed space. Esi [5], defined and studied the notion of ∇-statistical convergence and ∇-statistical Cauchy sequences using byλ-sequences in random 2-normed spaces, and proved some theorems.
The existing literature on statistical convergence and its generalizations appears to have been restricted to real or complex sequences, but in recent years these ideas have been also extended to the sequences in fuzzy normed [40] and intutionistic fuzzy normed spaces [18], [20], [27] and [28]. Several authors studied on the sets of fuzzy valued sequences and the characterization of the classes of related matrix transformations ([8], [9], [10], [11], [12]).
Let R denotes the set of reals and R+0 = [0,∞). A function f : R → R + 0
is called a distribution function if it is non-decreasing and left-continuous with inft∈Rf(t) = 0 and supt∈Rf(t) = 1. We will denote the set of all distribution
functions byD. Also, a distance distribution function is a non decreasing function
F defined on R+ = [0,∞) that satisfies F(0) = 0 and F(∞) = 1; and is left
continuous on (0,∞). LetD+denotes the set of all distance distribution functions.
A triangular norm, briefly t-norm, is a binary operation ∗ on [0,1] which is continuous, commutative, associative, non-decreasing and has 1 as neutral element, i.e., it is the continuous mapping∗: [0,1]×[0,1]→[0,1] such that for all a, b, c∈
[0,1] :
(i)a∗1 =a,
(ii)a∗b=b∗a,
(iii)c∗d≥a∗b ifc≥aandd≥b,
(iv) (a∗b)∗c=a∗(b∗c).
The∗ operationsa∗b= max{a+b−1,0},a∗b=ab, anda∗b= min{a, b}on [0,1] aret-norms.
2
Definition 3. ([15]) Let X be a real vector space of dimension d >1 (d may be infinite). A real valued functionk., .k:X2→Rsatisfying the following conditions:
(i)kx1, x2k= 0, if and only ifx1, x2 are linearly dependent.
(ii)kx1, x2k=kx2, x1k for allx1, x2∈X,
(iii)kαx1, x2k=|α| kx1, x2k, for anyα∈Rand
(iv)kx1+x2, x3k ≤ kx1, x3k+kx2, x3k
is called a 2-norm and the pair (X,k., .k) is called a 2-normed space.
Definition 4. ([16]) Let X be a real vector space of dimension d >1 (d may be infinite),τ be a triangle function(a binary operation on D+ which is associative, commutative, nondecreasing andε0 as a unit) andF:X×X → D+ (forx, y∈X,
F(x, y;t)is the value ofF(x, y)at t∈R). Then F is called a probabilistic norm
(X,F, τ)a probabilistic 2-normed space if the following conditions are satisfied:
(i) F(x, y;t) = H0(t), if x, y are linearly dependent, where H0(t) = 0 if
t≤0 andH0(t) = 1 ift >0.
(ii)F(x, y;t)6=H0(t), ifx, yare linearly dependent.
(iii)F(x, y;t) =F(y, x;t), for allx, y∈X,
(iv)F(αx, y;t) =Fx, y; t
|α|
for everyt >0,α6= 0 andx, y∈X,
(v)F(x+y, z;t)≥τ(F(x, z;t),F(y, z;t)), wherex, y, z∈X.
If (v) is replaced byF(x+y, z;t1+t2)≥ F(x, z;t1)∗ F(y, z;t2) for allx, y, z∈
X andt1, t2∈R+0 then (X,F,∗) is called a random 2-normed space.
Example 1. Let (X,k., .k) be a 2-normed space with kx, zk =|x1z2−x2z1|; x=
(x1, x2), z = (z1, z2) and a∗b = ab for all a, b∈ [0,1]. For every x, y ∈X and
t >0 we define F(x, y;t) = t+ktx,yk, then (X,F,∗)is a random 2-normed space.
Definition 5. ([24]) Let (X,F,∗)be a random 2-normed space. Then a sequence x= (xk) is said to be convergent to x0 ∈ X with respect to norm F if for every
ε > 0, t ∈(0,1) and non-zero z ∈X, there exists a positive integer k0 such that
F(xk−x0, z;ε)>1−t wheneverk≥k0. It is denoted byF-limxk=x0.
Definition 6. ([24]) Let (X,F,∗)be a random 2-normed space. Then a sequence x = (xk) is said to be statistically convergent SR2N convergent to x0 ∈ X with respect to norm F if for everyε >0, t∈(0,1) and non-zeroz∈X,
δ({k∈N:F(xk−x0, z;ε)≤1−t}) = 0.
In this case, we writeSR2N-limxk =x0.
Definition 7. ([4]) Let (X,F,∗) be a random 2-normed space. Then a sequence x = (xk) is said to be statistically convergent to l with respect to F if for every ε >0, t∈(0,1) and non-zeroz∈X,
lim
n→∞ 1
n|({k≤n:F(xk−l, z;ε)≤1−t})|= 0.
In this case, we writeSR2N-limxk =l.
Ki¸si [19] has recently defined the∇2-statistical convergence of double sequences
in random 2-normed spaces.
Throughout the paper, we consider a random 2-normed space (X, F,∗) and
Let λ = (λr) and µ = (µs) be two non-decreasing sequences of positive real numbers, each tending to∞and such thatλr+1 ≤λr+ 1,λ1= 1;µs+1≤µs+ 1,
µ1= 1. LetIr= [r−λr+ 1, r], Is= [s−µs+ 1, s] andIr,s =Ir×Is.
For any setX ⊆N×N, the number,
δλ(X) =P- lim
r,s→∞ 1
λr,s|{(k, l)∈Ir×Is: (k, l)∈X}|;
is said to beλ-density of the setX, provided the limit exists, whereλr,s=λrµs.
2. Main results
In this section, we define ∇2-statistical convergent double sequence of order
α(0 < α ≤ 1) in random 2-normed space (X,F,∗). Also, we obtain some basic properties of this notion in random 2-normed space.
Definition 8. A double sequence x= (xkl)in random 2-normed space(X,F,∗)is said to be ∇2-convergent to l ∈X of order α with respect toF if for each ε >0,
t ∈ (0,1) and for non-zero z ∈ X, there exists an positive integer n0 such that
F 1
λαr,s P
(k,l)∈Ir,s
xkl−l, z;ε !
> 1−t whenewer k, l ≥ n0. In this case we write
Fα
∇2−limk,l→∞xkl=l, andl is called theF α
∇2−limit ofx= (xkl).
Definition 9. A double sequencex= (xkl)in a random 2-normed space (X,F,∗)
is said to be ∇2-Cauchy of order αwith respect to F if for every ε >0, t∈(0,1) and for non-zeroz∈X, there exists positive integersp, q such that
F
1
λαr,s X
(k,l)∈Ir,s
(xkl−xmn), z;ε
<1−t,
wheneverk, m > p,l, n > q.
Definition 10. A double sequencex= (xkl)in a random 2-normed space(X,F,∗)
is said to be∇2-statistical convergent orS∇2-convergent tol of orderα(0< α≤1) with respect toF if for everyε >0 ,t∈(0,1)andfor non-zero z∈X such that
δ∇2
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;ε
≤1−t
= 0.
In other ways we can write
(k, l)∈Ir,s :F
lim r,s→∞
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;ε
≤1−t
= 0,
or, equivalently,
δ∇2
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;ε
>1−t
= 1,
i.e.,
S∇α
2−r,slim→∞F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;ε
2
In this case, we writeS∇α2(R2N)−limk,lxkl=l orxkl→l S∇α2(R2N)
and
S∇α2(R2N) (X) =
x= (xkl) :∃l∈R, S∇α2(R2N)−limk,l xkl =l
.
The collection of all∇2-statistically convergent double sequences of order α in
random 2-normed space is symbolized asSα
∇2(R2N) (X).
Definition 11. A double sequencex= (xkl)in a random 2-normed space(X,F,∗)
is said to be∇2-statistically Cauchy of orderαwith respect toFif for everyε >0,
t∈(0,1)and for non-zeroz∈X, there exist positive integersp, q such that for all k, m > p,l, n > q
δ∇2
(k, l)∈Ir,s :F
1
λαr,s X
(k,l)∈Ir,s
(xkl−xmn), z;ε
≤1−t
= 0,
or, equivalently,
δ∇2
(k, l)∈Ir,s :F
1
λαr,s X
(k,l)∈Ir,s
(xkl−xmn), z;ε
>1−t
= 1.
Definition 11, immediately implies the following Lemma.
Lemma 1. Let (X,F,∗) be a random 2-normed space. If x = (xkl) is a double sequence inX, then for everyε >0,t∈(0,1)and for non-zeroz∈X, the following statetements are equivalent.
(i)Sα
∇2−limk,lxkl=l.
(ii)δ∇2 (
(k, l)∈Ir,s :F λα1 r,s
P
(k,l)∈Ir,s
xkl−l, z;ε !
≤1−t )!
= 0.
(iii)δ∇2 (
(k, l)∈Ir,s:F λ1α r,s
P
(k,l)∈Ir,s
xkl−l, z;ε !
>1−t )!
= 1.
(iv)Sα
∇2−limk,l→∞F
1
λαr,s P
(k,l)∈Ir,s
xkl−l, z;ε !
= 1.
Theorem 2. Let (X,F,∗)be a random 2-normed space andα∈(0,1]be given. If x= (xkl) is a double sequence in X such that S∇α2(R2N)−limk,lxkl = l exists, then it is unique.
Proof. Suppose thatSα
∇2(R2N)−limk,lxkl =l
0, wherel6=l0. Letε >0 be given.
Chooseν >0 such that
(1−ν)∗(1−ν)>1−ε. (1)
Then, for anyt >0 and for non-zeroz∈X, we define
K1(υ, t) =
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
(xkl−l), z; t
2
≤1−ν
;
K2(υ, t) =
(k, l)∈Ir,s :F
1
λαr,s X
(k,l)∈Ir,s
(xkl−l0), z; t
2
≤1−ν
Since
S∇α
2(R2N)−limk,lxkl=l andS α
∇2(R2N)−limk,l xkl =l
0,
we have
δ∇2(K1(υ, t)) = 0 andδ∇2(K2(υ, t)) = 0 for allt >0.
LetK(υ, t) =K1(υ, t)∪K2(υ, t), then it is easy to observe thatδ∇2(K(υ, t)) = 0
which immediately implies δ∇2(K
c(υ, t)) = 1. Let k ∈ Kc(υ, t) = Kc
1(υ, t)∩
Kc
2(υ, t) . Now one can write,
F(l−l0, z;t) ≥ F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t 2
∗ F
1
λαr,s X
(k,l)∈Ir,s
xkl−l0, z;t 2
> (1−ν)∗(1−ν).
It follows by (1) that
F(l−l0, z;t)>(1−ε).
Sinceεis arbitrary, it follows thatF(l−l0, z;t) = 1, for allt >0 and non-zero
z∈X. This shows that l=l0.
Theorem 3. Let (X,F,∗) be a random 2-normed space and α∈ (0,1] be given. Let x= (xkl)andy= (ykl)be two double sequences in X.
(i) If Sα
∇2(R2N)−limk,lxkl = l and 0 6= c ∈ R, then S α
∇2(R2N)−
limk,lcxkl=cl.
(ii) If Sα
∇2(R2N)−limk,lxkl = l and S α
∇2(R2N)−limk,lykl = l
0, then
S∇α2(R2N)−limk,l(xkl+ykl) =l+l0.
Proof. The proof of the theorem is not so hard so is omitted here.
Theorem 4. Let (X,F,∗) be a random 2-normed space and α∈ (0,1] be given. If x = (xkl) be a double sequence in X such that Fα
∇2−limk,l→∞xkl = l, then Sα
∇2(R2N)−limk,lxkl =l. Hovewer the converse need not be true in general.
Proof. SinceFα
∇2−limk,l→∞xkl=l, for every ε >0,t >0 and for non-zeroz∈X
there is a positive integern0 such that
F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t
>1−ε, ∀k, l > n0.
The set
K(ε, t) =
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t
≤1−ε
has at most finitely many terms. Also, since every finite subset of N has δ∇2
-density zero, consequently we haveSα
∇2(K(ε, t)) = 0. This shows thatS α
∇2(R2N)−
limk,lxkl =l. We next give the following example which shows that the converse
need not be true.
Example 2. Let X = R2 with the 2-norm kx, zk = kx1z2−x2z1k where x =
2
where each t > 0, non-zero z ∈ X, z2 > 0. We define a sequence x= (xkl) as follows:
1
λαr,s X
(k,l)∈Ir,s
xkl =
(k, l), if n−√λn+ 1≤k≤n, m−√µm+ 1≤l≤m,
(0,0), otherwise.
Now forε >0,t∈(0,1),write
K(ε, t) =
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t
≤1−ε
,
wherel= (0,0). Then
K(ε, t) =
(k, l)∈Ir,s : t
t+ 1 λαr,s P
(k,l)∈Ir,s
xkl
≤1−ε
, θ= (0,0)
=
(
(k, l)∈Ir,s: 1
λαr,s P
(k,l)∈Ir,s
xkl ≥ tε
1−ε>0 )
={(k, l)∈Ir,s :xkl= (k, l)}
=(k, l)∈Ir,s:n−√λn+ 1≤k≤n, m−√µm+ 1≤l≤m ,
so we get
1
λαr,s
|K(ε, t)| ≤ 1 λαr,s
n
(k, l)∈Ir,s:r− p
λr+ 1≤k≤r, s− √
µs+ 1≤l≤s o
≤
√ λrs
λαr,s
Letting limit, r, sas∞, we get
δ∇2(K(ε, t)) = limr,s→∞
1
λαr,s|K(ε, t)| ≤r,slim→∞
√ λrs
λαr,s = 0.
This shows that xkl →0 Sα
∇2(R2N) (X)
.
On the other hand the sequence is notFα
∇2−convergent to zero as
F 1
λαr,s P
(k,l)∈Ir,s
xkl−l, z;t ! = t t+ 1 λαr,s P
(k,l)∈Ir,s
xkl = t t+(k+l)z2,
ifn−√λn+ 1≤k≤n,
m−√µm+ 1≤l≤m,
1, otherwise.
≤1.
Theorem 5. Let (X,F,∗) be a R2N space and0 < α≤β ≤1, then Sα
∇2(X)⊂ S∇β
Proof. If 0< α≤β ≤1, then for everyε >0 and t >0 and non-zero z ∈X, we have
1
λβr,s (
(k, l)∈Ir,s:F 1
λαr,s P
(k,l)∈Ir,s
xkl−l, z;t !
≤1−ε )
≤ 1
λαr,s (
(k, l)∈Ir,s:F 1
λαr,s P
(k,l)∈Ir,s
xkl−l, z;t !
≤1−ε )
which immediately implies the inclusion Sα∇2(X) ⊂ S∇β
2(X). We next give an
example that shows the inclusion inSα
∇2(X)⊂S β
∇2(X) is strict for someαandβ
withα < β.
Example 3. Let X = R2 with the 2-norm kx, zk = kx1z2−x2z1k where x =
(x1, x2),z = (z1, z2)and a∗b=ab for all a, b∈[0,1]. Let F(xkl, z, t) = t+ktx,zk, where each t > 0, non-zero z ∈ X, z2 > 0. We define a sequence x= (xkl) as follows:
X
(k,l)∈Ir,s
xkl=
(1,0), if k+l is even,
(0,0), ifk+l is odd.
Forε >0,t∈(0,1), if we define
K(ε, t) =
(
(k, l)∈Ir,s :F 1
λαr,s P
(k,l)∈Ir,s
xkl−θ, z;t !
≤1−ε )
, θ= (0,0)
=
(k, l)∈Ir,s: t
t+
1
λαr,s
P
(k,l)∈Ir,s
xkl−θ,z
≤1−ε
=
(
(k, l)∈Ir,s :
1
λαr,s P
(k,l)∈Ir,s
xkl−θ, z
≥ εt
1−ε >0 )
={(k, l)∈Ir,s: (xkl) = (1,0)}={(k, l)∈Ir,s:k+l is even};
then,
lim
r,s→∞ 1
λαr,s |K(ε, t)|= limr,s→∞ 1
λαr,s |{(k, l)∈Ir,s :k+l is even}| ≤r,slim→∞
q λr,s+ 1
2λαr,s = 0
forα >1. Similarly, forε >0,t∈(0,1), if we define
B(ε, t) =
(k, l)∈Ir,s :F
1
λαr,s X
(k,l)∈Ir,s
xkl−x0, z;t
≤1−ε
,x0= (1,0)
then
lim
r,s→∞ 1
λαr,s |B(ε, t)|= limr,s→∞ 1
λαr,s |{(k, l)∈Ir,s:k+l is odd}| ≤r,slim→∞
q λr,s+ 1
2λαr,s = 0
forα >1. This shows that Sα
2
Example 4. Let R2,F,∗be a R2N space as defined above. We define a sequence x= (xkl)as follows:
1
λαr,s X
(k,l)∈Ir,s
xkl=
(1,0), if r−√λr+ 1≤k≤r, s− √
λs+ 1≤l≤s,
(0,0), otherwise.
Then one can easily see S∇β
2 −limk,lxkl = 0, i.e., x ∈ S β
∇2 for
1
2 < β ≤ 1 but
x /∈Sα
∇2(X)for0 < α≤
1
2. This shows that the inclusion inS
α
∇2(X)⊂S β
∇2(X) is strict.
Theorem 6. Let (X,F,∗) be a random 2-normed space and α∈ (0,1] be given. If x = (xkl) be a sequence in X, then Sα∇2 −limk,lxkl = l if and only if there exists a subsetK ={km:k1< k2< ...} of N such thatlimr,s→∞λα1
r,s
|K| = 1and Fα
∇2−limk,l→∞xkl=l.
Proof. First suppose thatSα
∇2−limk,lxkl=l. For t >0 and non-zeroz ∈X and s∈N, we define
A(s, t) =
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t
>1−
1
s
K(s, t) =
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t
≤1−
1
s
SinceS∇α2−limk,lxkl=lit follows that
δ∇2(K(s, t)) = 0
Also, fors= 1,2,3, ...and fort >0, we observe that
A(s, t)⊃A(s+ 1, t)
and
lim
r,s→∞ 1
λαr,s|A(s, t)|= 1; i.e.,δ∇2(A(s, t)) = 1. (2)
Now, to prove the result, it is sufficient to prove that Fα
∇2−limk,l→∞xkl = l.
Suppose that for k ∈A(s, t), x= (xkl) does not convergent to l with respect to
Fα
∇2. Then, there exists someu >0 such that
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t
≤1−u
for infinitely many terms (xkl). Let
A(u, t) =
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t
>1−u
andu > 1s fors= 1,2,3.... This implies thatδ∇2(A(s, t)) = 0, which contradicts
(2) asδ∇2(A(s, t)) = 1. Hence F α
∇2−limk,l→∞xkl =l.
Conversely, suppose that there exists a subset
of Nsuch that limr,s→∞λ1α r,s
|K|= 1 and Fα
∇2−limk,l→∞xkl =l. Then for every
ε >0 andt >0 and non-zeroz∈X, there exists a positive integern0 such that
(k, l)∈Ir,s :F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t
>1−ε
for allk, l > n0. If we take
K(ε, t) =
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t
≤1−ε
then it is easy to see that
K(ε, t)⊂N×N− {n0+ 1, n0+ 2, ...}
and consequently
δ∇2(K(ε, t))≤1−1 = 0.
Hence,Sα
∇2−limk,lxkl=l.
Finally, we establish the Cauchy convergence criteria of double sequences of order
αin random 2-normed spaces.
Theorem 7. Let (X,F,∗)be a random 2-normed space andα∈(0,1]be given. A double sequence x= (xkl) is said to be∇2-statistical convergent of order αif and only if it is∇2-statistical Cauchy of order α.
Proof. Let x = (xkl) be ∇2-statistical convergent sequence of order α. Suppose
that Sα
∇2−limk,lxkl =l. Forε >0,t >0 and non-zero z∈X chooses >0 such
that (1−s)∗(1−s)>1−ε. We define
A(s, t) =
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z; t 2
≤(1−s)
;
then
Ac(s, t) =
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z; t 2
>(1−s)
.
Since S∇α
2 −limk,lxkl = l it follows that δ∇2(A(s, t)) = 0 and consequently δ∇2(A
c(s, t)) = 1. Let (u, γ)∈Ac(s, t), then
F
1
λαr,s X
(k,l)∈Ir,s
xuγ−l, z; t
2
>(1−s).
If we take
B(ε, t) =
(k, l)∈Ir,s:F
1
λαr,s X
(k,l)∈Ir,s
xkl−xuγ, z;t
≤(1−ε)
2
then to prove the first part it is sufficient to prove that B(ε, t) ⊂ A(s, t). Let (k, l)∈B(ε, t), which gives
F
1
λαr,s X
(k,l)∈Ir,s
xkl−xuγ, z;t
≤(1−ε).
Suppose (k, l)∈/A(s, t), then
F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t
>(1−s),
Also it can be easily seen that
1−ε ≥ F
1
λαr,s X
(k,l)∈Ir,s
xkl−xuγ, z;t
≥ F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z; t
2
∗ F
1
λαr,s X
(k,l)∈Ir,s
xuγ−l, z; t
2
≥ (1−s)∗(1−s)>1−ε.
This contradiction shows that B(ε, t) ⊂ A(s, t) and therefore, the theorem is proved.
Conversely, letx= (xkl) is∇2-statistical Cauchy double sequence of order
αbut not double∇2-statistical convergent of orderαwith respect toF. Now
F
1
λαr,s X
(k,l)∈Ir,s
xkl−xuγ, z;t
≥ F
1
λαr,s X
(k,l)∈Ir,s
xkl−l, z;t 2
∗ F
1
λαr,s X
(k,l)∈Ir,s
xuγ−l, z; t 2
≥ (1−s)∗(1−s)>1−ε.
sincexis not double∇2-statistical convergent. Thereforeδ∇2(B
c(t, ε)) = 0, where
B(ε, t) =
(k, l)∈Ir,s :F
1
λαr,s X
(k,l)∈Ir,s
xkl−xuγ, z;t
≤1−ε
.
and soδ∇2(B(t, ε)) = 1 , which is contradiction, since xis ∇2-statistical Cauchy
double sequence. Hence x must be ∇2-statistical Cauchy. This completes the
proof.
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¨
OMER K˙IS¸ ˙I, Faculty of Science, Depertmant of Mathematics, Bartın University, Bartın, Turkey
