https://doi.org/10.26637/MJM0701/0011
Rough
I-statistical convergence of double
sequences in normed linear spaces
Prasanta Malik
1* and Argha Ghosh
2Abstract
The notion ofI-statistical convergence of a double sequence was first introduced by Belen et. al.[2] and the
notion of rough convergence of a sequence was first introduced by Phu [19]. In this paper we introduce and study
the notion of roughI-statistical convergence of double sequences in normed linear spaces. We also introduce
the notion of roughI-statistical limit set of a double sequence and discuss about some topological properties of
this set.
Keywords
Double sequence, I-statistical convergence, rough I-statistical convergence, rough I-statistical limit set,
I-statistical boundedness.
AMS Subject Classification 40A05, 40B99.
1,2Department of Mathematics, University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India.
*Corresponding author:1[email protected];2[email protected]
Article History: Received06February2018; Accepted12November2018 c2019 MJM.
Contents
1 Introduction. . . 55
2 Basic Definitions and Notations. . . 56
3 Main Results. . . 57
References. . . 60
1. Introduction
The notion of convergence of real double sequences was first introduced by Pringsheim [21]. A double sequencex= {xjk}j,k∈Nof real numbers is said to converge to a real number l, if for anyε>0, there existsm∈Nsuch that for all j,k≥m
|xjk−l|<ε.
In this case we write lim j→∞
k→∞
xjk=l. This notion of convergence
of real double sequences has been extended to statistical con-vergence by Mursaleen et. al. [17] ( also by Moricz [16] who introduced it for multiple sequences) using double natural density ofN×N. A subsetK⊂N×Nis said to have natural
densityd(K)if
d(K) =lim m→∞
n→∞
|K(m,n)| m.n ,
whereK(m,n) ={(j,k)∈N×N: j≤m,k≤n;(j,k)∈K}
and|K(m,n)|denotes number of elements of the setK(m,n).
A double sequencex={xjk}j,k∈Nof real numbers is said to be statistically convergent toξ ∈R, if for anyε>0, we haved(A(ε)) =0, whereA(ε) ={(j,k)∈N×N:kxjk−ξk≥
ε}.
More investigation and applications of statistical conver-gence of double sequences can be found in [3,23] and many others.
The notion of statistical convergence of double sequences has been further generalized toI-convergence of double se-quences by Das et. al. [4] using ideals inN×N. For more
details one can see [3,5,22] etc.
Recently Belen et. al. [2] introduced the notion of ideal statistical convergence of double sequences, which is a new generalization of the notions of statistical convergence and usual convergence. More investigation and applications on this notion can be found in [2,24].
by Malik et. al.[12]. Ifx=
xjk j,k∈Nbe a double sequence in some normed linear space(X,k.k)andrbe a non negative real number, thenxis said to ber-convergent toξ∈Xif for anyε>0, there existsm∈Nsuch that for all j,k≥m
|xjk−ξ|<r+ε.
Further this notion of rough convergence of double se-quence has been extended to rough statistical convergence of double sequence by Malik et. al.[13] using double natural density of the subsets ofN×Nin a similar way as the notion
of convergence of double sequence in Pringsheim sense was generalized to statistical convergence of double sequences. Further the notion of rough statistical convergence of double sequences was generalized to roughI-convergence of dou-ble sequences by Dunder et. al. [8]. So it is quite natural to think , if the new notion of I-statistical convergence of double sequences can be introduced in the theory of rough convergence.
In this paper we introduced and study the notion of rough I-statistical convergence of double sequences in a normed linear space(X,k.k)which naturally extends both the notions of rough convergence as well as rough statistical convergence of double sequences in a new way. We also define the set of all roughI-statistical limits of a double sequence and investigate some topological properties of this set. We mention that our results presented in this paper for double sequences are I-statistical analogues of those in Phu’s [19] paper.
2. Basic Definitions and Notations
Definition 2.1. [13] Let x={xjk}j,k∈Nbe a double sequence
in a normed linear space(X,k.k)and r be a non negative real number. x is said to be r- statistically convergent toξ, denoted by x−→r−st ξ, if for anyε>0 we have d(A(ε)) =0, where A(ε) ={(j,k)∈N×N: kxjk−ξ k≥r+ε}. In this caseξ is called the r-statistical limit of x.
Definition 2.2. A class I of subsets of a nonempty set X is said to be an ideal in X provided
(i)φ∈I.
(ii) A,B∈I implies AS
B∈I. (iii) A∈I,B⊂A implies B∈I. I is called a nontrivial ideal if X∈/I.
Definition 2.3. A non empty class F of subsets of a nonempty set X is said to be a filter in X provided
(i)φ∈/F.
(ii) A,B∈F implies AT
B∈F. (iii) A∈F,A⊂B implies B∈F.
If I is a nontrivial ideal in X , X6=φ, then the class
F(I) ={M⊂X:M=X\A for some A∈I}
is a filter on X , called the filter associated with I.
Definition 2.4. A nontrivial ideal I in X is called admissible if{x} ∈I for each x∈X .
Definition 2.5. A nontrivial ideal I onN×Nis called strongly
admissible if{i} ×NandN× {i}belong to I for each i∈N.
Clearly every strongly admissible ideal is admissible. Throughout the paper we takeIas a strongly admissible ideal inN×N.
Definition 2.6. [14] Let x={xjk}j,k∈Nbe a double sequence in a normed linear space(X,k.k)and r be a non negative real number. Then x is said to be rough I-convergent with
roughness degree r or r-I-convergent toξ, denoted by x−→r-I ξ, if for anyε>0we have{(j,k)∈N×N:kxjk−ξk≥r+ε} ∈
I. In this caseξ is called limit of x and x is called r-I-convergent toξ with r as roughness degree.
Now we give the definition ofI-asymptotic density of a subset ofN×N.
Definition 2.7. A subset K⊂N×Nis said to have I-asymptotic density dI(K)if
dI(K) =I−lim
m→∞
n→∞
|K(m,n)| m.n ,
where K(m,n) ={(j,k)∈N×N: j≤m,k≤n;(j,k)∈K}
and|K(m,n)|denotes number of elements of the set K(m,n).
Definition 2.8. [2]A double sequence x={xjk}j,k∈Nof real
numbers is I-statistically convergent to L, and we write xI−st→ L, provided that for anyε>0andδ>0
(m,n): mn1
(j,k):xjk−L
≥ε,j≤m,k≤n ≥δ ∈I.
Definition 2.9. Let x={xjk}j,k∈Nbe a double sequence in
a normed linear space(X,k.k)and r be a non negative real number. Then x is said to be rough I-statistically convergent toξ with roughness degree r or r-I-statistically convergent to ξ if for anyε>0andδ>0
{(m,n)∈N×N:mn1 |{(j,k):j≤m,k≤n;kxjk−ξ k≥
r+ε}| ≥δ} ∈I.
In this case ξ is called the r-I-statistical limit of x=
{xjk}j,k∈Nand we denote it byx
r-I-st
−→ξ.
Remark 2.10. Note that if I is the ideal I0={A⊂N×N:
∃m(A)∈Nsuch that i,j≥m(A)⇒(i,j)∈/A}, then rough I-statistical convergence coincide with rough statistical con-vergence.
From the above definition it is clear that the rough I -statistical limit of a double sequence is not unique. So we consider the set of roughI-statistical limits of a double se-quence xand we use the notationI-st-LIMr
x to denote the set of all r-I-statistical limits of a double sequence x. We say that a double sequencexisr-I-statistically convergent if I-st-LIMrx6=φ.
Throughout the paperX denotes a normed linear space
(X,k.k)andxdenotes the double sequence{xjk}j,k∈NinX.
3. Main Results
In this section we discuss some basic properties of rough I-statistical convergence of double sequences.
Theorem 3.1. Let x={xjk}j,k∈Nbe a double sequence in X and r≥0. Then diam (I-st-LIMrx)≤2r. In particular if x is I-statistically convergent toξ, then I-st-LIMrx=Br(ξ)(=
{y∈X:ky−ξk ≤r})and so diam (I-st-LIMrx) =2r.
Proof. If possible let, diam (I-st-LIMrx)>2r. Then there exist y,z∈I-st-LIMrxsuch thatky−zk>2r. Now chooseε>0 so thatε<ky−zk2 −r. LetA={(j,k)∈N×N:kxjk−yk ≥
r+ε}andB={(j,k)∈N×N:kxjk−zk ≥r+ε}. Then
1
mn|{(j,k): j≤m;k≤n,(j,k)∈A∪B}| ≤
1
mn|{(j,k): j≤
m;k≤n,(j,k)∈A}|+mn1|{(j,k): j≤m;k≤n,(j,k)∈B}|,
and so by the property ofI-convergence I- lim
m→∞
n→∞
1
mn|{(j,k): j≤m;k≤n,(j,k)∈A∪B}| ≤
I- lim m→∞
n→∞
1
mn|{(j,k): j≤m;k≤n,(j,k)∈A}|+
1
mn|{(j,k): j≤
m;k≤n,(j,k)∈B}|=0. Thus{(m,n)∈N×N:mn1|{(j,k):
j≤m;k≤n,(j,k)∈A∪B}| ≥δ} ∈I for allδ >0. Let K={(m,n)∈N×N:mn1|{(j,k): j≤m;k≤n,(j,k)∈A∪
B}| ≥ 1
2}. ClearlyK∈I, so choose(m0,n0)∈N×N\K. Then m01n0|{(j,k): j≤m0;k≤n0,(j,k)∈A∪B}|<12. So
1
m0n0|{(j,k): j≤m0;k≤n0,(j,k)∈/A∪B}| ≥1−
1 2=
1 2i.e.,
{(j,k):(j,k)∈/ A∪B} is a nonempty set. Take(j0,k0)∈
N×Nsuch that (j0,k0)∈/ A∪B. Then (j0,k0)∈Ac∩Bc and hencekxj0k0−yk<r+ε andkxj0k0−zk<r+ε. So
ky−zk ≤ kxj0k0−yk+kxj0k0−zk ≤2(r+ε)<ky−zk, which is absurd. Therefore diam (I-st-LIMrx)≤2r.
IfI-st- limx=ξ, then we proceed as follows. Letε>0 andδ >0 be given. ThenA={(m,n)∈N×N: mn1|{(j,k):
j≤m;k≤n,kxjk−ξk ≥ε}| ≥δ} ∈I. Then for(m,n)∈/A we havemn1|{(j,k): j≤m;k≤n,kxjk−ξk ≥ε}<δ, i.e.,
1
mn|{(j,k): j≤m;k≤n,kxjk−ξk<ε} ≥1−δ.
Now for eachy∈Br(ξ)(={y∈X:ky−ξk ≤r})we have
kxjk−yk ≤ kxjk−ξk+kξ−yk ≤ kxjk−ξk+r.
Let Bmn=(j,k): j≤m;k≤n,kxjk−ξk<ε . Then for
(j,k)∈Bmnwe havekxjk−yk<r+ε. HenceBmn⊂ {(j,k):
j≤m;k≤n,kxjk−yk<r+ε}. This implies,|Bmnmn|≤mn1|{(j,k): j≤m;k≤n,kxjk−yk<r+ε}|i.e., mn1|{(j,k): j≤m;k≤ n,kxjk−yk<r+ε}| ≥1−δ. Thus for all(m,n)∈/A,mn1 |(j,k):
j≤m;k≤n,kxjk−yk ≥r+ε}|<1−(1−δ). Hence we have
{(m,n)∈N×N: mn1 |{(j,k): j≤m;k≤n,kxjk−yk ≥r+
ε}| ≥δ} ⊂A. SinceA∈I, so{(m,n)∈N×N:mn1 |{(j,k):
j≤m;k≤n,kxjk−yk ≥r+ε}| ≥δ} ∈I. This shows that
y∈I-st-LIMxr. ThereforeI-st-LIMr
x⊃Br(ξ).
Conversely, lety∈I-st-LIMxr. If possible , letky−ξk>r. Take ε=ky−ξ2k−r. Let K1={(j,k)∈N×N:kxjk−yk ≥
r+ε}andK2={(j,k)∈N×N:kxjk−ξk ≥ε}. Then
1
mn|{(j,k):j≤m;k≤n,(j,k)∈K1∪K2}| ≤
1
mn|{(j,k): j≤
m;k≤n,(j,k)∈K1}|+mn1|{(j,k):j≤m;k≤n,(j,k)∈K2}|,
and so by the property ofI-convergence I- lim
m→∞
n→∞
1
mn|{(j,k):j≤m;k≤n,(j,k)∈K1∪K2}| ≤
I- lim m→∞
n→∞
1
mn|{(j,k):j≤m;k≤n,(j,k)∈K1}|+
1
mn|{(j,k):j≤
m;k≤n,(j,k)∈K2}|=0.
LetK={(m,n)∈N×N:mn1 |{(j,k):j≤m;k≤n,(j,k)∈
K1∪K2}| ≥12}.
ClearlyK∈I and we choose(m0,n0)∈N×N\K. Then
1
m0n0|{(j,k): j≤m0;k≤n0,(j,k)∈K1∪K2}|<
1 2. So
1
m0n0|{(j,k): j≤m0;k≤n0,(j,k)∈/K1∪K2}| ≥1−
1 2=
1 2
i.e.,{(j,k):(j,k)∈/K1∪K2}is a nonempty set. We choose
(j0,k0)∈N×Nsuch that(j0,k0)∈/K1∪K2. Then(j0,k0)∈ K1c∩K2cand hencekxj0k0−yk<r+εandkxj0k0−ξk<ε. So
ky−ξk ≤ kxj0k0−yk+kxj0k0−ξk<r+2ε=ky−ξk,
which is absurd. Thereforeky−ξk ≤rand so y∈Br(ξ). Consequently we haveI-st-LIMrx=Br(ξ)and this completes the proof.
Definition 3.2. A double sequence x=
xjk j,k∈N is said to be I- statistically bounded if there exists a positive num-ber T such that for anyδ >0the set A={(m,n)∈N×N:
1
mn|{(j,k):j≤m;k≤n,kxjkk ≥T}| ≥δ} ∈I.
The next result provides a relationship between bounded-ness and roughI-statistical convergence of double sequences.
Theorem 3.3. If a double sequence x={xjk}j,k∈Nis bounded then there exists r≥0such that I-st-LIMxr6=/0.
Proof. Let x={xjk}j,k∈N be a bounded double sequence. Then there exists a positive real numberTsuch thatkxjkk<T for all(j,k)∈N×N. Letε>0 be given. Then
(j,k):xjk−0
≥T+ε =/0.
Therefore 0∈I-st-LIMT
Note 3.4. The converse of the above theorem is not true. For example, let us consider the double sequence x={xjk}j,k∈N inRdefined by
xjk = jk ,if j and k are squares
= 3 ,otherwise.
Then I-st-LIMx0={3} 6=/0but the double sequence x is un-bounded.
We now show that the converse of Theorem 3.3 is true if the double sequence isI-statistically bounded.
Theorem 3.5. A double sequence x=xjk j,k∈N in X is
I-statistically bounded if and only if there exists a non negative real number r such that I-st-LIMxr6=/0.
Proof. Letx=xjk k∈Nbe an I-statistically bounded dou-ble sequence in X. Then there exists a positive real num-berT such that forδ >0 we have{(m,n): mn1|{(j,k): j≤
m;k≤n,kxjkk ≥T}| ≥δ} ∈I. LetA={(j,k):kxjkk ≥T}. ThenI- lim
m,n→∞ 1
mn|{(j,k): j≤m;k≤n,(j,k)∈A}|=0. Let
r0=sup{kxjkk:(j,k)∈Ac}. Then 0∈I-st-LIMr
0
x. Hence
I-st-LIMrx6=/0 forr=r0.
Conversely, letI-st-LIMxr6=/0 for somer≥0. Letξ ∈ I-st-LIMrx. Chooseε=kξk. Then for eachδ >0,{(m,n)∈
N×N:mn1|{(j,k): j≤m;k≤n,kxjk−ξk ≥r+ε}| ≥δ} ∈
I. Now taking T =r+2kξk, we have {(m,n)∈N×N: 1
mn|{(j,k): j≤m;k≤n,kxjkk ≥T}| ≥δ} ∈I. Therefore
xisI-statistically bounded.
Now let{jp}p∈Nand{kq}q∈Nbe two strictly increasing sequences of natural numbers. If x={xjk}j,k∈N is a dou-ble sequence in(X,k.k), then we define{xjpkq}p,q∈Nas a subsequence ofx.
Definition 3.6. A subsequence x0={xjpkq}p,q∈Nof a double sequence x={xjk}j,k∈Nis called a I-dense subsequence if dI({(jp,kq);p,q∈N}) =1.
In ([12], Theorem 3.4), Malik et. al. have already shown that ifx0={xjpkq}p,q∈Nis a subsequence ofx={xjk}j,k∈N, then LIMxr ⊂LIMr
x0. But this result is not true for rough
I-statistical convergence. To show this we consider the fol-lowing example.
Example 3.7. Consider the double sequence x={xjk}j,k∈N inRdefined by
xjk = jk ,if j and k both are cubes
= 0 ,otherwise.
Then x0={xjpkq}p,q∈N is a subsequence of x, where jp=
p3,p∈Nand kq=q3,q∈Nand for all r≥0, I-st-LIMxr=
[−r,r]but I-st-LIMxr0=/0.
We now presentI-statistical analogue of Theorem 3.4 [12] in the following form.
Theorem 3.8. If x0={xjpkq}p,q∈Nis a I-dense subsequence of x={xjk}j,k∈N, then I-st-LIMxr⊂I-st-LIMxr0.
Proof. Letξ∈I-st-LIMxr. Then for anyε>0,dI(A(ε)) =0, where A(ε) ={(j,k)∈N×N;kxjk−ξ k≥r+ε}. Then dI(Ac(ε)) =1.
Sincex0={xjpkq}p,q∈N is a I-dense subsequence of x, so dI(K) =1, whereK={(jp,kq);p,q∈N}. ThendI(Ac(ε)∩
K) =1.
Let A0(ε) ={(p,q)∈N×N;kxjpkq−ξ k≥r+ε}. Now
{(p,q);kxjpkq−ξ k<r+ε} ⊃A
c(
ε)∩K. Therefore
dI(A0(ε))c=1 and sodI(A0(ε)) =0
which impliesξ ∈I-st-LIMxr0.
Theorem 3.9. Let x=xjk j,k∈N be double sequence and
r≥0be a real number. Then the r-I-statistical limit set of the double sequence x i.e., the set I-st-LIMxris closed.
Proof. IfI-st-LIMxr=/0, then nothing to prove.
Let us assume thatI-st-LIMxr6= /0. Now consider a double sequence{yjk}j,k∈N inI-st-LIMxr with lim
j→∞
k→∞
yjk=y. Choose
ε>0 andδ >0. Then there existsiε
2 ∈Nsuch thatkyjk−
yk<ε
2 for all j>iε2 andk>iε2. Let j0>iε2 andk0>iε2. Thenyj0k0∈I-st-LIMxr. Consequently, we haveA={(m,n)∈
N×N: mn1 |{(j,k): j≤m;k≤n,kxjk−yj0k0k ≥r+ ε 2}| ≥ δ} ∈I. ClearlyM=N×N\Ais nonempty, choose(m,n)∈
M. We have
1
mn|{(j,k): j≤m;k≤n,kxjk−yj0k0k ≥r+ ε 2}|<δ.
⇒ 1
mn|{(j,k): j≤m;k≤n,kxjk−yj0k0k<r+ ε
2}| ≥1−δ.
PutBmn={(j,k):j≤m;k≤n,kxjk−yj0k0k<r+ ε
2}. Choose
(j,k)∈Bmn. Then
kxjk−yk ≤ kxjk−yj0k0k+kyj0k0−yk<r+
ε 2+
ε
2 =r+ε.
HenceBmn⊂ {(j,k): j≤m;k≤n,kxjk−yk<r+ε}, which implies
1−δ≤|Bmnmn|≤mn1 {(j,k): j≤m;k≤n,kxjk−yk<r+ε}
.
Thereforemn1 |{(j,k): j≤m;k≤n,kxjk−yk ≥r+ε}|<1−
(1−δ) =δ. Thus we have
{(m,n): mn1|{(j,k): j≤m;k≤n,kxjk−yk ≥r+ε}| ≥ δ} ⊂A∈I.
Theorem 3.10. Let x=
xjk j,k∈Nbe double sequence and r≥0 be a real number. Then the r-I-statistical limit set I-st-LIMrxof the double sequence x is a convex set.
Proof. Lety0,y1∈I-st-LIMxr. Letε>0 be given. Let
A0={(j,k)∈N×N:kxjk−y0k ≥r+ε},
A1={(j,k)∈N×N:kxjk−y1k ≥r+ε}.
Then by same approach as in the proof of theorem 3.1 for δ>0 we have{(m,n):mn1|{(j,k):j≤m;k≤n,(j,k)∈A0∪ A1}| ≥δ} ∈I. Choose 0<δ1<1 such that 0<1−δ1<δ. LetA={(m,n):mn1|{(j,k):j≤m;k≤n,(j,k)∈A0∪A1}| ≥ 1−δ1}. ThenA∈I. Now for all(m,n)∈/Awe have
1
mn|{(j,k): j≤m;k≤n,(j,k)∈A0∪A1}|<1−δ1
⇒ 1
mn|{(j,k):j≤m;k≤n,(j,k)∈/A0∪A1}| ≥
{1−(1−δ1)}=δ1.
Therefore{(j,k):(j,k)∈/A0∪A1}is a nonempty set. Let us take(j0,k0)∈A0c∩A1cand 0≤λ≤1. Then
kxj0k0−[(1−λ)y0+λy1]k
=k(1−λ)xj0k0+λxj0k0−[(1−λ)y0+λy1]k
≤(1−λ)kxj0k0−y0k+λkxj0k0−y1k <(1−λ)(r+ε) +λ(r+ε) =r+ε.
LetB={(j,k)∈N×N:kxjk−[(1−λ)y0+λy1]k ≥r+ ε}. Then clearly,A0c∩A1c⊂Bc. So for(m,n)∈/A,
δ1≤mn1|{(j,k): j≤m;k≤n,(j,k)∈/A0∪A1} ≤
1
mn|{(j,k): j≤m;k≤n,(j,k)∈/B}.
⇒ 1
mn|{(j,k):j≤m;k≤n,(j,k)∈B}|<1−δ1<δ.
ThusAc⊂ {(m,n): mn1|{(j,k): j≤m;k≤n,(j,k)∈B}|<
δ}. Since Ac∈F(I), so {(m,n): mn1 |{(j,k): j≤m;k≤
n,(j,k)∈B}|<δ} ∈F(I)and so{(m,n):mn1 |{(j,k): j≤
m;k≤n,(j,k)∈B}| ≥δ} ∈I. This completes the proof.
Theorem 3.11. Let r>0. Then a double sequence x=
xjk j,k∈N is r-I-statistically convergent to ξ if and only if there exists a double sequence y={yjk}j,k∈N such that I-st-limy=ξ andkxjk−yjkk≤r for all(j,k)∈N×N.
Proof. Lety={yjk}j,k∈Nbe a double sequence inX, which isI-statistically convergent toξ andkxjk−yjkk ≤rfor all
(j,k)∈N×N. Then for anyε>0 and δ >0 the setA= {(m,n)∈N×N:mn1|{(j,k): j≤m;k≤n,kyjk−ξk ≥ε} ≥
δ} ∈I. Let(m,n)∈/A. Then we have
1
mn|{(j,k): j≤m;k≤n,kyjk−ξk ≥ε}|<δ
⇒ 1
mn|{(j,k): j≤m;k≤n,kyjk−ξk<ε}| ≥1−δ.
LetBmn={(j,k): j≤m;k≤n,kyjk−ξk<ε}. Then for
(j,k)∈Bmn, we have
kxjk−ξk ≤ kxjk−yjkk+kyjk−ξk<r+ε.
Therefore,
Bmn⊂ {(j,k):j≤m;k≤n,kxjk−ξk<r+ε}.
⇒|Bmn|
mn ≤ 1
mn|{(j,k): j≤m;k≤n,kxjk−ξk<r+ε}|.
⇒ 1
mn|{(j,k): j≤m;k≤n,kxjk−ξk<r+ε}| ≥1−δ.
⇒ 1
mn|{(j,k): j≤m;k≤n,kxjk−ξk ≥r+ε}|<
{1−(1−δ)}=δ.
Thus,{(m,n)∈N×N: mn1|{(j,k): j≤m;k≤n,kxjk−
ξk ≥r+ε}| ≥δ} ⊂Aand sinceA∈I, we have{(m,n)∈
N×N:mn1 |{(j,k):j≤m;k≤n,kxjk−ξk ≥r+ε}| ≥δ} ∈I. Hencexr−→-I-stξ.
Conversely, suppose thatxr−→-I-stξ. Then forε>0 and δ>0,
A={(m,n)∈N×N: mn1|{(j,k): j≤m;k≤n,kxjk−ξk ≥
r+ε}| ≥δ} ∈I.
Let(m,n)∈/A. Then
1
mn|{(j,k): j≤m;k≤n,kxjk−ξk ≥r+ε}|<δ.
⇒ 1
mn|{(j,k): j≤m;k≤n,kxjk−ξk<r+ε}| ≥1−δ.
LetBmn={(j,k): j≤m;k≤n,kxjk−ξk<r+ε}. Now we define a double sequencey={yjk}j,k∈Nas follows,
yjk=
(
ξ, ifkxjk−ξk ≤r
xjk+r ξ−xjk
kxjk−ξk, otherwise.
Then
kyjk−ξk =
(
0, ifkxjk−ξk ≤r
kxjk−ξ+r ξ−xjk
kxjk−ξkk, otherwise.
=
0, ifkxjk−ξk ≤r
kxjk−ξk −r, otherwise.
Let(j,k)∈Bmn. Then
kyjk−ξk=0, if kxjk−ξk ≤r
and
kyjk−ξk<ε, if r<kxjk−ξk<r+ε.
ThenBmn⊂ {(j,k): j≤m;k≤n,kyjk−ξk<ε}. This im-plies
|Bmn|
mn ≤
1
Hence,
1
mn|{(j,k): j≤m;k≤n,kyjk−ξk<ε}| ≥1−δ
⇒ 1
mn|{(j,k): j≤m;k≤n,kyjk−ξk ≥ε}|<1−(1−δ) =
δ.
Thus{(m,n)∈N×N:mn1|{(j,k): j≤m;k≤n,kyjk−ξk ≥
ε}| ≥δ} ⊂A. SinceA∈I, we have
{(m,n)∈N×N:mn1 |{(j,k):j≤m;k≤n,kyjk−ξk ≥
ε}| ≥δ} ∈I.
SoI-st- limy=ξ.
Definition 3.12. A pointλ ∈X is said to be an I-statistical cluster point of a double sequence x=xjk j,k∈Nin X if for
anyε>0
dI({(j,k):kxjk−λk<ε})6=0
where dI(A) =I−lim
m→∞
n→∞
1
mn|{(j,k):j≤m;k≤n,(j,k)∈A}if
exists.
The set of I-statistical cluster point ofxis denoted by
ΛSx(I).
Lemma 3.13. [15] Let x=xjk j,k∈N∈Rnbe I-statistically
bounded double sequence. Then for everyε>0the set
(j,k):d(ΛxS(I),xjk)≥ε
has I-asymptotic density zero, where
d(ΛS
x(I),xjk) =in fy∈ΛSx(I) y−xjk
,
the distance from xjkto the setΛSx(I).
Theorem 3.14. Let x={xjk}j,k∈Nbe a double sequence in X . For any arbitrary c∈ΛSx(I), we havekξ−ck ≤r for all ξ ∈I-st-LIMxr.
Proof. Assume that there exists a pointc∈ΛSx(I)andξ ∈ I-st-LIMrxsuch thatkξ−ck>r. Letε=kξ−ck−r3 . Then
{(j,k):kxjk−ξk ≥r+ε} ⊃ {(j,k):kxjk−ck<ε}.(3.1)
Since c∈ΛSx(I) we have dI({(j,k):kxjk−ck<ε})6=0. Hence by (3.1) we havedI({(j,k):kxjk−ck ≥r+ε})6=0, which contradicts thatξ∈I-st-LIMxr. Hencekξ−ck ≤r.
Theorem 3.15. Let(Rn,k.k)be a strictly convex space and
x={xjk}j,k∈Nbe a double sequence in this space. For any r>0, let y1,y2∈I-st-LIMxrwithky1−y2k=2r. Then x is I-statistically convergent to12(y1+y2).
Proof. Lety3be an arbitraryI-statistical cluster point ofx. Now sincey1,y2∈I-st-LIMxr, so by Theorem 3.14,
ky1−y3k≤rand ky2−y3k≤r.
Then 2r=ky1−y2k≤ky1−y3k+ky3−y2k≤2r. Therefore
ky1−y3k=ky2−y3k=r. Now
1
2(y1−y2) = 1
2[(y3−y1) + (y2−y3)] (3.2)
Sinceky1−y2k=2r, so k 12(y2−y1)k=r. Again since the space is strictly convex so by (3.2) we get,12(y2−y1) = y3−y1=y2−y3. Thus y3= 12(y1+y2) is the unique I -statistical cluster point of the double sequence x. Again by the given conditionI-st-LIMxr 6= /0 and so by Theorem 3.5, xis I-statistically bounded. Since y3 is the unique I -statistical cluster point of theI-statistically bounded double sequencex, so by lemma 3.13,xisI-statistically convergent toy3=12(y1+y2).
Theorem 3.16. Let x={xjk}j,k∈Nbe a double sequence in X .
(i) If c∈ΛSx(I), then I-st-LIMxr⊂Br(c).
(ii) I-st-LIMxr= T c∈ΛSx(I)
Br(c) ={ξ ∈X:ΛSx(I)⊂Br(ξ)}.
Proof. (i)Letc∈ΛSx(I). Then by Theorem 3.14, for allξ ∈ I-st-LIMxr,kξ−ck ≤rand hence the result follows.
(ii) By (i) it is clear thatI-st-LIMrx⊂ T c∈ΛSx(I)
Br(c). Now
for allc∈ΛSx(I)andy∈ T
c∈ΛSx(I)
Br(c)we haveky−ck ≤r.
Then clearly T c∈ΛSx(I)
Br(c)⊂ {ξ ∈X:ΛSx(I)⊂Br(ξ)}.
Now, lety∈/I-st-LIMxr. Then there exists anε>0 such thatdI(A)6=0, where A={(m,n)∈N×N:kxmn−yk ≥
r+ε}. This implies the existence of anI-statistical cluster pointcof the sequencexwithky−ck ≥r+ε. This gives
ΛSx(I)*Br(y)and soy∈ {/ ξ ∈X :ΛSx(I)⊂Br(ξ)}. Hence
{ξ ∈X :ΛSx(I)⊂Br(ξ)} ⊂I-st-LIMxr. This completes the proof.
Acknowledgment
The second author is grateful to University Grants Com-mission, India for financial support under UGC NET-JRF scheme during the preparation of this paper.
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