R E S E A R C H
Open Access
On asymptotically double lacunary statistical
equivalent sequences in ideal context
Bipan Hazarika
1*and Vijay Kumar
2*Correspondence:
1Department of Mathematics, Rajiv
Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791112, India
Full list of author information is available at the end of the article
Abstract
An idealIis a family of subsets of positive integersN×Nwhich is closed under taking finite unions and subsets of its elements. In this paper, we present some definitions which are a natural combination of the definition of asymptotic equivalence, statistical convergence, lacunary statistical convergence, double sequences and an ideal. In addition, we also present asymptoticallyI-equivalent double sequences and study some properties of this concept.
MSC: 40A35; 40G15
Keywords: asymptotic equivalence; statistical convergence; lacunary statistical convergence; ideal; ideal convergence; double sequence
1 Introduction
Pobyvancts [] introduced the concept of asymptotically regular matrices which preserve the asymptotic equivalence of two nonnegative numbers sequences. Marouf [] presented definitions for asymptotically equivalent and asymptotic regular matrices. Patterson [] extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. Patterson and Savaş [] introduced the concept of an asymptotically lacunary statistical equivalent sequences of real numbers.
The idea of statistical convergence was formerly given under the name ‘almost conver-gence’ by Zygmund in the first edition of his celebrated monograph published in Warsaw in []. The concept was formally introduced by Steinhaus [] and Fast [], and later, it was introduced by Schoenberg [] and also independently by Buck []. A lot of devel-opments have been made in this area after the works of ˘Salát [] and Fridy []. Over the years and under different names, statistical convergence has been discussed in the theory of Fourier analysis, ergodic theory and number theory. Fridy and Orhan [] intro-duced the concept of lacunary statistical convergence. Mursaleen and Mohiuddine [] introduced the concept of lacunary statistical convergence with respect to the intuition-istic fuzzy normed space. Savaş and Patterson [, ] introduced the concept of lacunary statistical convergence for double sequences. Recently Mohiuddineet al.[] introduced statistical convergence of double sequences in locally solid Riesz spaces. For details related to lacunary statistical convergence, we refer to [, –].
Kostyrkoet al.[] introduced the notion ofI-convergence with the help of an admis-sible ideal,Idenotes the ideal of subsets ofN, which is a generalization of statistical con-vergence. Quite recently, Daset al.[] unified these two approaches to introduce new
concepts ofI-statistical convergence,I-lacunary statistical convergence and investigated some of their consequences. The notion of lacunary ideal convergence of real sequences was introduced in [, ]. Hazarika [, ] introduced the lacunary ideal convergent sequences of fuzzy real numbers and studied some basic properties of this notion. Kumar and Sharma [] studied asymptotically generalized statistical equivalent sequences using ideals. Recently Savas [] and Savas and Gumus [] studied ideal asymptotically lacu-nary statistical equivalent single sequences. For more applications of ideals, we refer to [–].
In this paper, we define asymptotically lacunary statistical equivalent double sequences using an ideal and establish some basic results regarding this notion.
2 Definitions and preliminaries
In this section, we recall some definitions and notations, which form the base for the present study.
A family of setsI⊂P(N) (power sets ofN) is called anidealif and only if for eachA,B∈I, we haveA∪B∈I, and for eachA∈Iand eachB⊂A, we haveB∈I. A non-empty family of setsF⊂P(N) is afilteronNif and only ifφ∈/F, for eachA,B∈F, we haveA∩B∈F and eachA∈Fand eachB⊃A, we haveB∈F. An idealIis called non-trivial ideal ifI=φ
andN∈/I. Clearly,I⊂P(N) is a non-trivial ideal if and only ifF=F(I) ={N–A:A∈I}is a filter onN. A non-trivial idealI⊂P(N) is calledadmissibleif and only if{{x}:x∈N} ⊂I. A non-trivial idealIismaximalif there cannot exists any non-trivial idealJ=Icontaining Ias a subset. Further details on ideals ofP(N) can be found in Kostyrkoet al.[]. Recall that a sequencex= (xk) of points inRis said to beI-convergent to a real numberif {k∈N:|xk–| ≥ε} ∈Ifor everyε> []. In this case, we writeI–limxk=.
By a lacunary sequenceθ= (kr), wherek= , we mean an increasing sequence of
non-negative integers withhr:=kr–kr–→ ∞asr→ ∞. The intervals determined byθ will
be denoted byJr:= (kr–,kr], and the ratiokrkr– will be defined byqr(see []).
The notion of statistical convergence depends on the density (asymptotic or natural) of subsets ofN. A subset ofNis said to have natural densityδ(E) if
δ(E) = lim
n→∞
n{k≤n:k∈E}exists.
Definition . A real or complex number sequencex= (xk) is said to bestatistically
con-vergenttoLif for everyε> ,
lim
n
nk≤n:|xk–L| ≥ε= .
In this case, we writeS–limx=Lorxk→L(S), andSdenotes the set of all statistically
convergent sequences.
Definition .[] A sequencex= (xk) is said to belacunary statistically convergentto
the numberLif for everyε> ,
lim
r→∞
hr
k∈Jr:|xk–L| ≥ε= .
LetSθdenote the set of all lacunary statistically convergent sequences. Ifθ= (r), thenSθ
Definition .[] LetI⊂P(N) be a non-trivial ideal. A sequence (xk) isI-statistically
convergenttoLif for eachε> andδ> ,
n∈N:
nk≤n:|xk–L| ≥ε≥δ
∈I.
In this case, we writeI(S) –limxk=L.
Definition .[] LetI⊂P(N) be a non-trivial ideal. A sequence (xk) is said to be
I-lacunary statistically convergenttoLif for eachε> andδ> ,
r∈N: hr
k∈Jr:|xk–L| ≥ε≥δ
∈I.
In this case, we writeI(Sθ) –limxk=L. Ifθ= (r), thenI(Sθ) is the same asI(S).
Definition .[] Two nonnegative sequencesx= (xk) andy= (yk) are said to be
asymp-totically equivalentif
lim
k
xk
yk
= ,
denoted byx∼y.
Definition .[] Two nonnegative sequencesx= (xk) andy= (yk) are said to be
asymp-totically statistical equivalentof multipleLprovided that for everyε> ,
lim
n
n
k≤n:xk yk
–L≥ε= ,
denoted byx∼SLyand simply asymptotically statistical equivalent ifL= .
Patterson and Savas [] defined the asymptotically lacunary statistical equivalent se-quences as follows.
Definition . Two nonnegative sequencesx= (xk) andy= (yk) are said to be
asymptot-ically lacunary statistical equivalentof multipleLprovided that for everyε> ,
lim
r
hr k∈Jr:
xk
yk
–L≥ε=
denoted byx∼SLθ yand simply asymptotically lacunary statistical equivalent ifL= . If we
takeθ= (r), then we get Definition ..
By the convergence of a double sequence, we mean the convergence in Pringsheim’s sense []. A double sequencex= (xk,l) has aPringsheimlimitL(denoted byP–limx=L)
provided that given anε> , there exists ann∈Nsuch that|xk,l–L|<ε, wheneverk,l>n.
We describe such anx= (xk,l) more briefly as‘P-convergent’. We denote the space of all
P-convergent sequences byc. The double sequencex= (x
a positive numberMsuch that|xk,l|<Mfor allkandl. We denote all bounded double
sequences byl∞.
LetK⊂N×NandK(m,n) denote the number of (i,j) inK such thati≤mandj≤n (see []). Then the lower natural density ofK is defined byδ(K) =lim infm,n→∞K(mnm,n).
In case the sequence (K(mnm,n)) has a limit in Pringsheim’s sense, then we say thatKhas a double natural densityand is defined byP–limm,n→∞Kmn(m,n) =δ(K).
For example, letK={(i,j) :i,j∈N}. Then
δ(K) =P– lim m,n→∞
K(m,n)
mn ≤P–mlim,n→∞ √
m√n mn = ,
i.e.,the setK has double natural density zero, while the set{(i, j) :i,j∈N}has double natural density .
Definition .[] A real double sequencex= (xk,l) is said to beP-statistically convergent
toprovided that for eachε> ,
P–lim
mn
mn(k,l) :k<mandl<n,|xk,l–| ≥ε= .
We denote the set of all statistical convergent double sequences byS.
The double sequenceθ=θr,s={(kr,ls)}is calleddouble lacunary sequenceif there exist
two increasing sequences of integers such that (see [])
ko= , hr=kr–kr–→ ∞ asr→ ∞
and
lo= , hs=ls–ls–→ ∞ ass→ ∞.
Notations:kr,s=krls,hr,s=hrhsandθr,sis determined by
Jr,s=
(k,l) :kr–<k≤krandls–<l≤ls
,
qr=
kr
kr–
, qs=
ls
ls–
and qr,s=qrqs.
Definition . A double sequence (xk,l) is said to bedouble lacunary convergenttoLif
P–lim
r,s
hr,s
(k,l)∈Jr,s
xk,l=L.
In this case, we writeθ¯–limk,lxk,l=L. We denoteN¯θ the set of all double lacunary
con-vergent sequences.
Definition .[] LetI⊂P(N×N) be a non-trivial ideal. A double sequence (xk,l) is
said to beI-convergenttoLif for eachε> ,
(k,l)∈N×N:|xk,l–L| ≥ε
∈I.
Throughout the paper, we denoteIas admissible ideal of subsets ofN×N, unless oth-erwise stated.
3 Asymptotically lacunary statistical equivalent double sequences using ideals In this section, we define asymptoticallyI-equivalent, asymptoticallyI-statistical equiv-alent, asymptotically I-lacunary statistical equivalent and asymptotically lacunary I -equivalent double sequences and obtain some analogous results from these new defini-tions point of views.
Definition . LetI⊂P(N×N) be a non-trivial ideal. A double sequence (xk,l) is said to
beI-statistically convergenttoLif for eachε> andδ> ,
(m,n)∈N×N:
mnk≤m;l≤n:|xk,l–L| ≥ε≥δ
∈I.
In this case, we writeI(S) –limxk,l=L.
Definition . LetI⊂P(N×N) be a non-trivial ideal. A double sequence (xk,l) is said to
beI-lacunary statistically convergenttoLif for eachε> andδ> ,
(r,s)∈N×N: hr,s
(k,l)∈Jr,s:|xk,l–L| ≥ε≥δ
∈I.
In this case, we writeI(Sθ¯) –limxk,l=L.
Definition . Two nonnegative double sequencesx= (xk,l) andy= (yk,l) are said to be
P-asymptotically equivalentif
P–lim
k,l
xk,l
yk,l
= ,
denoted byx∼Py.
Definition . Two nonnegative double sequencesx= (xk,l) andy= (yk,l) are said to be
asymptotically statistical equivalentof multipleLprovided that for everyε>
P–lim
m,n
mn
k≤m,l≤n:xk,l yk,l
–L≥ε= ,
denoted byx∼SLyand simply asymptotically statistical equivalent ifL= .
Definition . Two nonnegative double sequencesx= (xk,l) andy= (yk,l) are said to be
asymptotically lacunary statistical equivalentof multipleLprovided that for everyε> ,
P–lim
r,s
hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε=
Definition . Two non-negative double sequences (xk,l) and (yk,l) are said to be
asymp-toticallyI-equivalentof multipleLprovided that for everyε> ,
(k,l)∈N×N:xk,l yk,l
–L≥ε
∈I,
denoted by (xk,l)∼I L
(yk,l) and simply asymptoticallyI-equivalent ifL= .
Lemma . LetI⊂P(N×N)be an admissible ideal.Let(xk,l), (yk,l)be two double
se-quences and(xk,l), (yk,l)∈∞withI–limk,lxk,l= =I–limk,lyk,lsuch that(xk,l)∼I L
(yk,l).
Then there exists a sequence(zk,l)∈∞withI–limk,lzk,l= such that(xk,l)∼I L
(zk,l)∼I L
(yk,l).
Definition . Two non-negative double sequences (xk,l) and (yk,l) are said to be
asymp-toticallyI-statistically equivalentof multipleLprovided that for everyε> and for every
δ> ,
(m,n)∈N×N: mn
k≤m,l≤n:xk,l yk,l
–L≥ε≥δ
∈I,
denoted by (xk,l)∼I(S) L
(yk,l) and simply asymptoticallyI-statistical equivalent ifL= .
Definition . Two non-negative double sequences (xk,l) and (yk,l) are said to beCesaro
asymptoticallyI-equivalent(orI(σ)-equivalent) of multipleLprovided that for every δ> ,
(m,n)∈N×N:
mn
m,n
k,l=
xk,l
yk,l
–L
≥δ ∈I
denoted by (xk,l)∼I(σ) L
(yk,l) and simply asymptoticallyI(σ)-equivalent ifL= .
Definition . Two non-negative double sequences (xk,l) and (yk,l) are said to bestrongly
Cesaro asymptoticallyI-equivalent(orI(|σ|)-equivalent) of multipleLprovided that for
everyδ> ,
(m,n)∈N×N: mn
m,n
k,l= xk,l
yk,l
–L≥δ ∈I
denoted by (xk,l)∼I(|σ|) L
(yk,l) and simply strongly Cesaro asymptoticallyI(|σ|
)-equiva-lent ifL= .
Definition . Two non-negative double sequences (xk,l) and (yk,l) are said to bestrongly
asymptotically lacunary equivalentof multipleLprovided that
P–lim
r,s
hr,s
(k,l)∈Jr,s xk,l
yk,l
–L=
denoted by (xk,l)∼[Nθ¯] L
(yk,l) and simply strongly asymptotically lacunary equivalent if
Definition . Two non-negative double sequences (xk,l) and (yk,l) are said to be
asymp-toticallyI-lacunary equivalent(orI(N¯θ)-equivalent) of multipleLprovided that for every δ> ,
(r,s)∈N×N: hr,s
(k,l)∈Jr,s xk,l
yk,l
–L≥δ
∈I
denoted by (xk,l)∼I(Nθ¯) L
(yk,l) and simply asymptoticallyI(N¯θ)-equivalent ifL= .
Definition . Two non-negative double sequences (xk,l) and (yk,l) are said to be
asymp-toticallyI-lacunary statistically equivalent(orI(Sθ¯)-equivalent) of multipleLprovided that for everyε> , for everyδ> ,
(r,s)∈N×N: hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε≥δ
∈I
denoted by (xk,l)∼I(Sθ¯)L(yk,l) and simply asymptoticallyI(Sθ¯)-equivalent ifL= .
Theorem . LetI⊂P(N×N)be a non-trivial ideal.Letθ¯={(kr,ls)}be a double lacunary
sequence.If(xk,l), (yk,l)∈∞and(xk,l)∼I(S) L
(yk,l).Then(xk,l)∼I(σ) L
(yk,l).
Proof (a) Suppose that (xk,l), (yk,l)∈∞and (xk,l)∼I(S) L
(yk,l). Then we can assume that
xk,l
yk,l
–L≤M for almost allk,l.
Letε> . Then we have
mn
m,n
k,l=
xk,l
yk,l
–L
≤
mn
m,n
k,l= xk,l
yk,l
–L
≤
mn
k,l;
|xk,l yk,l–L|≥ε
xk,l
yk,l
–L+ mn
k,l;
|xk,l yk,l–L|<ε
xk,l
yk,l
–L
≤M·
mn
k≤m;l≤n:xk,l yk,l
–L≥ε+
mn·mn·ε.
Consequently, ifδ>ε> ,δandεare independent, putδ=δ–ε> , we have
(m,n)∈N×N: mn
m,n
k,l=
xk,l
yk,l
–L≥δ
⊆
(m,n)∈N×N: mn
k≤m;l≤n:xk,l yk,l
–L≥ε≥ δ
M
∈I.
This shows that (xk,l)∼I(σ) L
(yk,l).
Corollary . LetI⊂P(N×N)be a non-trivial ideal.If(xk,l), (yk,l)∈∞and(xk,l)∼I(S) L
(yk,l).Then(xk,l)∼I(|σ|) L
Theorem . LetI⊂P(N×N)be a non-trivial ideal.Letθ¯={(kr,ls)}be a double
lacu-nary sequence.Then
(a) (xk,l)∼I(Nθ¯)L(yk,l)⇒(xk,l)∼I(Sθ¯)L(yk,l). (b) I(Nθ¯)Lis a proper subset ofI(Sθ¯)L.
(c) Let(xk,l), (yk,l)∈∞and(xk,l)∼I(Sθ¯) L
(yk,l),then(xk,l)∼I(Nθ¯) L
(yk,l). (d) I(Sθ¯)L∩∞=I(N¯θ)L∩∞.
Proof (a) Letε> and (xk,l)∼I(Nθ¯) L
(yk,l). Then we can write
(k,l)∈Jr,s xk,l
yk,l
–L≥
(k,l)∈Jr,s
|xk,l yk,l–L|≥ε
xk,l
yk,l
–L
≥ε
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε
⇒
ε·hr,s
(k,l)∈Jr,s xk,l
yk,l
–L≥ hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε.
Thus, for anyδ> ,
hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε≥δ
implies that
hr,s
(k,l)∈Jr,s xk,l
yk,l
–L≥εδ.
Therefore, we have
(r,s)∈N×N: hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε≥δ
⊂
(r,s)∈N×N: hr,s
(k,l)∈Jr,s xk,l
yk,l
–L≥εδ
.
Since (xk,l)∼I(Nθ¯) L
(yk,l), so that
(r,s)∈N×N: hr,s
(k,l)∈Jr,s xk,l
yk,l
–L≥εδ
∈I,
which implies that
(r,s)∈N×N: hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε≥δ
∈I.
This shows that (xk,l)∼I(Sθ¯) L
(b) Suppose thatI(N¯θ)L⊂I(Sθ¯)L. Let (xk,l) and (yk,l) be two sequences defined as
fol-lows:
xk,l= ⎧ ⎨ ⎩
kl, ifkr–<k≤kr–+ [
√
hr],ls–<l≤ls–+ [
√
hs],r,s= , , , . . . ;
, otherwise,
and
yk,l= for allk,l∈N.
It is clear that (xk,l) /∈∞, and forε> ,
hr,s
(k,l)∈Jr,s: xk,l
yk,l
– ≥ε≤[
hr,s]
hr,s
and [
hr,s]
hr,s
→ asr,s→ ∞. (.)
This implies that
(r,s)∈N×N: hr,s
(k,l)∈Jr,s: xk,l
yk,l
– ≥ε≥δ
⊆
(r,s)∈N×N:[
hr,s]
hr,s ≥ δ
.
By virtue of last part of (.), the set on the right side is a finite set, and so it belongs toI. Consequently, we have
(r,s)∈N×N: hr,s
(k,l)∈Jr,s: xk,l
yk,l
– ≥ε≥δ
∈I.
Therefore, (xk,l)∼I(Sθ¯)
(yk,l).
On the other hand, we shall show that (xk,l)∼I(Nθ¯)
(yk,l) is not satisfied. Suppose that
(xk,l)∼I(Nθ¯)
(yk,l). Then for everyδ> , we have
(r,l)∈N×N: hr,s
(k,l)∈Jr,s xk,l
yk,l
– ≥δ
∈I. (.)
Now,
lim
r,s
hr,s
(k,l)∈Jr,s xk,l
yk,l
– = hr,s
[hr,s]([
hr,s] – )
→
asr,s→ ∞.
It follows for the particular choiceδ= that
(r,s)∈N×N: hr,s
(k,l)∈Jr,s xk,l
yk,l
– ≥
=
(r,s)∈N×N:
[hr,s]([
hr,s] – )
hr,s
≥
for somem,n∈Nwhich belongs toF asI is admissible. This contradicts (.) for the choiceδ=. Therefore, (xk,l)I(Nθ¯)
(yk,l).
(c) Suppose that (xk,l)∼I(Sθ¯)L(yk,l) and (xk,l), (yk,l)∈∞. We assume that|xk,l
yk,l –L| ≤M
and for allk,l∈N. Givenε> , we get
hr,s
(k,l)∈Jr,s xk,l
yk,l
–L= hr,s
(k,l)∈Jr,s
|xk,l yk,l–L|≥ε
xk,l
yk,l
–L+ hr,s
(k,l)∈Jr,s
|xk,l yk,l–L|<ε
xk,l
yk,l
–L
≤ M
hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε+ε.
If we put
A(ε) =
(r,s)∈N×N: hr,s
(k,l)∈Jr,s xk,l
yk,l
–L≥ε
and
B(ε) =
(r,s)∈N×N: hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε≥ ε
M
,
whereε=δ–ε> , (δandεare independent), then we haveA(ε)⊂B(ε), and soA(ε)∈I.
This shows that (xk,l)∼I(Nθ¯) L
(yk,l).
(d) It follows from (a), (b) and (c).
Theorem . LetI⊂P(N×N)be an admissible ideal.Suppose that for givenδ> and everyε> such that
(m,n)∈N×N: mn
≤k≤m– ; ≤l≤n– :xk,l yk,l
–L≥ε<δ
∈F,
then(xk,l)∼I(S) L
(yk,l).
Proof Letδ> be given. For everyε> , choosem,nsuch that
mn
≤k≤m– ; ≤l≤n– :xk,l yk,l
–L≥ε
<δ
, for allm≥m,n≥n. (.)
It is sufficient to show that there existsm,nsuch that form≥m,n≥n,
mn
≤k≤m– ; ≤l≤n– :xk,l yk,l
–L≥ε<δ
. (.)
Letm=max{m,m};n=max{n,n}. The relation (.) will be true form>m,n>n.
Ifs,tchosen fixed, then we get
≤k≤s– ; ≤l≤t– : xk,l
yk,l
Now, form>s,n>t, we have
mn
≤k≤m– ; ≤l≤n– :xk,l yk,l
–L≥ε
≤
mn
≤k≤s– ; ≤l≤t– : xk,l
yk,l
–L≥ε
+ mn
s≤k≤m– ;t≤l≤n– : xk,l
yk,l
–L≥ε
≤ M
mn+ mn
s≤k≤m– ;t≤l≤n– : xk,l
yk,l
–L≥ε≤ M
mn+
δ
.
Thus, for sufficiently largen,
mn
s≤k≤m– ;t≤l≤n– : xk,l
yk,l
–L≥ε≤ M
mn+
δ
<δ.
This established the result.
Theorem . LetI⊂P(N×N)be a non-trivial ideal.Letθ¯= (kr,ls)be a double lacunary
sequence withlim infr,sqr,s> .Then(xk,l)∼I(S) L
(yk,l)⇒(xk,l)∼I(Sθ¯) L
(yk,l).
Proof Suppose thatlim infr,sqr,s> , then there exists anα> such thatqr,s≥ +αfor
sufficiently larger,s. Then we have
hr,s
krls≥ α
+α.
If (xk,l)∼I(S L)
(yk,l), then for everyε> and for sufficiently larger,s, we have
krls
k≤kr;l≤ls: xk,l
yk,l
–L≥ε
≥
krls
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε
≥ α
+α ·
hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε.
Therefore, for anyδ> , we have
(r,s)∈N×N: hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε≥δ
⊆
(r,s)∈N×N: krls
k≤kr;l≤ls: xk,l
yk,l
–L≥ε≥ αδ
+α
∈I.
This completes the proof.
Theorem . LetI=Ifin={A⊂N×N: A is a finite set}be a non-trivial ideal,and let
¯
θ= (kr,ls)be a double lacunary sequence withlim supr,sqr,s<∞.Then(xk,l)∼I(Sθ¯) L
(yk,l)⇒
(xk,l)∼I(S) L
Proof Iflim supr,sqr,s<∞. Then there exists aK> such thatqr,s<Kfor allr,s≥. Let
(xk,l)∼I(Sθ¯) L
(yk,l). Then there existsB> andε> , we put
Mr,s=
hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε.
Since (xk,l)∼I(Sθ¯) L
(yk,l). Then for everyε> andδ> , we have
(r,s)∈N×N: hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L≥ε≥δ
=
(r,s)∈N×N:Mr,s hr,s
≥δ
∈I,
and, therefore, it is a finite set. We choose integersr,s∈Nsuch that
Mr,s
hr,s
<δ for allr>r,s>s. (.)
LetM=max{Mr,s: ≤r≤r, ≤s≤s}andm,nbe two integers with satisfyingkr–<
m≤kr,ls–<n≤ls, then we have
mn
k≤m;l≤n:xk,l yk,l
–L≥ε
≤
kr–ls–
k≤kr;l≤ls: xk,l
yk,l
–L≥ε
=
kr–ls–{
M,+M,+· · ·+Mr,s+Mrr+,s++· · ·+Mr,s}
≤ M
kr–ls–·
rs+
kr–ls–
hr+,s+
Mr+,s+
hr+,s+
+· · ·+hr,s
Mr,s
kr,s
≤ M
kr–ls–
·rs+
kr–ls–
sup
r>r,s>s
Mr,s
hr,s
{hr+,s++· · ·+hr,s}
≤ M
kr–ls–·
rs+δ
krls–krls
kr–ls–
≤ M
kr–ls–·
rs+δ·qr,s≤
M kr–ls–·
rs+δK.
This completes the proof of the theorem.
Definition . Letp∈(,∞). Two non-negative double sequences (xk,l) and (yk,l) are
said to bestrongly asymptotically lacunary p-equivalentof multipleLif
P–lim
r,s
hr,s
(k,l)∈Jr,s xk,l
yk,l
–L
p
=
denoted by (xk,l)∼[Nθ¯] L p(y
k,l) and simply strongly asymptotically lacunaryp-equivalent if
Definition . Letp∈(,∞). Two non-negative double sequences (xk,l) and (yk,l) are
said to beasymptotically lacunary p-statistically equivalentof multipleLif for everyε> ,
P–lim
r,s
hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L
p
≥ε=
denoted by (xk,l)∼ SL
¯
θp(y
k,l) and simply asymptotically lacunaryp-statistical equivalent if
L= .
Theorem . Letθ¯= (kr,ls)be a double lacunary sequence.Then
(a) (xk,l)∼[Nθ¯] L p(y
k,l)⇒(xk,l)∼ SL
¯
θp(y k,l). (b) [N¯θ]Lpis a proper subset ofSθpL¯ .
(c) Let(xk,l), (yk,l)∈∞and(xkml)∼ SL
¯
θp(y
k,l),then(xk,l)∼[Nθ¯] L p(y
k,l). (d) SθL¯
p∩
∞= [N¯θ]Lp∩∞.
The proof of the above theorem is similar to Theorem . forI=Ifin.
Definition . Letp∈(,∞). We say that two non-negative double sequences (xk,l) and
(yk,l) arestrongly asymptoticallyI-lacunary p-equivalentof multipleLif for everyε> ,
(r,s)∈N×N: hr,s
(k,l)∈Jr,s xk,l
yk,l
–L
p
≥ε
∈I
denoted by (xk,l)∼I(Nθ¯) L p(y
k,l) and simply strongly asymptoticallyI-lacunaryp-equivalent
ifL= .
Definition . Letp∈(,∞). We say that two non-negative double sequences (xk,l) and
(yk,l) areasymptotically I-lacunary p-statistically equivalent of multiple Lif for every ε> , for everyδ>
(r,s)∈N×N: hr,s
(k,l)∈Jr,s: xk,l
yk,l
–L
p
≥ε≥δ
∈I
denoted by (xk,l)∼I (Sθ¯p)L(y
k,l) and simply asymptoticallyI-lacunaryp-statistical
equiva-lent ifL= .
Theorem . LetI⊂P(N×N)be a non-trivial ideal,and letθ¯= (kr,ls)be a double
la-cunary sequence.Then
(a) (xk,l)∼I (Nθ¯p)
L
(yk,l)⇒(xk,l)∼I (Sθ¯p)
L
(yk,l). (b) I(Nθ¯p)
Lis a proper subset ofI(S¯ θp)
L.
(c) Let(xk,l), (yk,l)∈∞and(xk,l)∼I (Sθ¯p)
L
(yk,l),then(xk)∼I(Nθp) L
(yk). (d) I(Sθp¯ )L∩∞=I(N¯θp)L∩∞.
Proof The proof of the theorem follows from the proofs of the Theorems . and ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
BH drafted the manuscript. VK checked and organized the manuscript to be its final form. BH also makes the revision as the corresponding author.
Author details
1Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791112, India. 2Department of Mathematics, Haryana College of Technology and Management, Kaithal, Haryana 136027, India.
Acknowledgements
The authors thank all three reviewers for their useful comments that led to the improvement of the original manuscript.
Received: 3 April 2013 Accepted: 24 September 2013 Published:19 Nov 2013
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