ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 12 Issue 1(2020), Pages 1-11.
ROUGH ∆I2-STATISTICAL CONVERGENCE OF DOUBLE SEQUENCES IN NORMED LINEAR SPACES
OMER K˙IS¨ ¸ ˙I, HATICE K ¨UBRA ¨UNAL
Abstract. In this paper, we introduce rough ∆I2-statistical convergence as an extension of rough convergence. We define the set of rough ∆I2-statistical limit points of a sequence and analyze the results with proofs.
1. Introduction
The concept of convergence of a sequence of real numbers was independently extended to statistical convergence independently by Fast [10] and Schoenberg [37].
The idea ofI-convergence was introduced by Kostyrko et al. [24] as a generalization of statistical convergence which is based on the structure of the idealI of subset of the set of natural numbers. Kostyrko et al. [25] studied the idea ofI-convergence and extremalI-limit points.
The idea of I-statistical convergence was introduced by Sava¸s and Das [27] as an extension of ideal convergence. Later on it was further investigated by Debnath and Debnath [11], Et et al. [12], Sava¸s and G¨urdal [26] and many others.
The idea of rough convergence was first introduced by Phu [32] in finite-dimensional normed spaces. In another paper [33] related to this subject, Phu defined the rough continuity of linear operators and showed that every linear operatorf :X →Y is r -continuous at every pointx∈X under the assumption dimY <∞ and r >0, where X and Y are normed spaces. In [34], Phu extended the results given in [32] to infinite-dimensional normed spaces. Aytar [3] studied the rough statistical convergence. Also, Aytar [4] studied that the rough limit set and the core of a real sequence. Pal et al. [9] generalized the idea of rough convergence into rough statistical convergence and rough ideal convergence. Recently, rough convergence of double sequences has been introduced by Malik and Maity [14] and investigated some basic properties of this type of convergence and also studied the relation between rough convergence and Pringsheim convergence for double sequences. In [15] rough statistical convergence of double sequences in finite dimensional normed linear spaces was studied and investigated some basic properties of this type of convergence rough statistical convergence of double sequences. Recently, D¨undar and C¸ akan [5, 6, 7] introduced the notion of rough I-convergence and the set of
2000Mathematics Subject Classification. 40A05, 40A35.
Key words and phrases. Statistical convergence;I2-convergence; rough convergence.
c
2020 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e.
Submitted November 23, 2019. Published January 22, 2020.
Communicated by Feyzi Basar.
1
rough I-limit points of a sequence and studied the notions of rough convergence, I2-convergence and the sets of rough limit points and rough I2-limit points of a double sequence. Sava¸s et al. [28] introduced roughI-statistical convergence as an extension of rough convergence. Rough convergence, rough statistical convergence and ∆I-convergence for difference sequences and for double difference sequences have been studied. For details, see ([17], [18], [19], [20], [21], [22], [23]).
In view of the recent applications of ideals in the theory of convergence of se- quences, it seems very natural to extend the interesting concept of rough ∆I2- statistical convergence of double seqeunces in normed linear spaces further by using ideals which we mainly do here.
2. Definitions and notations
In this section, we recall some definitions and notations, which form the base for the present study (See [1, 2, 3, 4, 5, 6, 7, 16, 32, 33, 34]).
During the preparation of the paper, let r be a nonnegative real number and Rn denotes the realn-dimensional space with the norm k.k. Consider a sequence x= (xk)⊂X =Rn.
The sequence x= (xk) is said to ber-convergent to x∗, denoted byxk −→r x∗ provided that
∀ε >0∃iε∈N: k≥iε⇒ kxk−x∗k< r+ε.
The set
LIMrx:={x∗∈Rn:xk
−→r x∗}
is called ther-limit set of the sequencex= (xk). A sequencex= (xi) is said to be r-convergent if LIMrx6=∅. In this case, r is called the convergence degree of the sequencex= (xk). Forr= 0, we get the ordinary convergence. There are several reasons for this interest (see [32]).
A family of setsI ⊆2Nis called an ideal if and only if (i) ∅ ∈ I,
(ii) for eachA, B∈ I we haveA∪B∈ I,
(iii) for eachA∈ I and eachB⊆Awe have B∈ I.
An ideal is called non-trivial ifN∈ I/ and non-trivial ideal is called admissible if {n} ∈ I for eachn∈N.
A family of setsF ⊆2Nis a filter inNif and only if (i) ∅∈ F/ ,
(ii) for eachA, B∈ F we haveA∩B∈ F,
(iii) for eachA∈ F and eachB⊇Awe have B∈ F.
IfI ⊆2Nis a nontrivial ideal, then the family of sets F(I) ={M ⊂N:∃A∈ I:M =N\ A}
is a filter ofNand it is called the filter associated with the idealI.
Mursaleen et al. [31] definedI-statistical cluster point of real number sequence.
A real sequencex= (xk) is said to be ∆-ideal convergent tox∈Rprovided for eachε >0
{k∈N:|∆xk−x| ≥ε} ∈ I.
The set of all ∆-ideal convergent sequences is doneted bycI(∆).
A real sequencex= (xk) is said to be ∆I∗-convergent to x∈R, if there exists M ={m=mi:mi< mi+1, i∈N} such thatM ∈ F(I) and ∆−limk→∞xkm =x.
In this case, we write ∆I∗−limxk =x.
Theorem 2.1. If a I-statistically bounded sequence has one cluster point then it isI-statistically convergent.
A sequencex= (xk) is said to be roughI-convergent (r-I-convergent) tox∗with the roughness degreer, denoted byxk −→r−Ix∗ provided that{k∈N:kxk−x∗k ≥ r+ε} ∈ I for every ε >0; or equivalently, if the condition
I −lim supkxk−x∗k ≤r is satisfied. In addition, we can writexk
−→r−Ix∗iff the inequalitykxk−x∗k< r+ε holds for everyε >0 and almost allk.
A double sequencex= (xmn)(m,n)∈N×Nof real numbers is said to be bounded if there exists a positive real numberM such that|xmn|< M,for allm, n∈N. That is
kxk∞= sup
m,n
|xmn|<∞.
A double sequencex= (xmn) of real numbers is said to be convergent toL∈R in Pringsheim’s sense (shortly,p-convergent toL∈R), if for anyε >0, there exists Nε∈Nsuch that |xmn−L|< ε,wheneverm, n > Nε. In this case, we write
m,n→∞lim xmn=L.
We recall that a subsetK ofN×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}|.
Throughout the paper we consider a sequencex= (xmn) such that (xmn)∈Rn. Let x = (xmn) be a double sequence in a normed space (X,k.k) and r be a non negative real number. xis said to be r-statistically convergent toξ, denoted by xr−st−→2 ξ, if for ε > 0 we have d(A(ε)) = 0, where A(ε) = {(m, n)∈ N×N : kxmn−ξk ≥r+ε}.In this case,ξis called the r-statistical limit ofx.
A nontrivial idealI2ofN×Nis called strongly admissible if{i} ×NandN× {i}
belong toI2 for eachi∈N.
It is evident that a strongly admissible ideal is admissible also.
Throughout the paper we takeI2 as a strongly admissible ideal inN×N. Let (X, ρ) be a metric space A double sequencex= (xmn) in X is said to be I2-convergent to L ∈ X, if for any ε > 0 we have A(ε) = {(m, n) ∈ N×N : ρ(xmn, L)≥ε} ∈ I2.In this case, we say thatxisI2-convergent and we write
I2− lim
m,n→∞xmn=L.
A double sequence x= (xmn) is said to be rough convergent (r-convergent) to x∗with the roughness degree r, denoted byxmn
−→r x∗provided that
∀ε >0 ∃kε∈N:m, n≥kε⇒ kxmn−x∗k< r+ε,
or equivalently, if
lim supkxmn−x∗k ≤r.
A double sequence x = (xmn) is said to be r-I2-convergent to x∗ with the roughness degreer, denoted byxmnr−I−→2x∗ provided that
{(m, n)∈N×N:kxmn−x∗k ≥r+ε} ∈ I2, (2.1) for everyε >0; or equivalently, if the condition
I2−lim supkxmn−x∗k ≤r (2.2) is satisfied. In addition, we can writexmn
r−I2
−→ x∗ iff the inequalitykxmn−x∗k<
r+εholds for every ε >0 and almost all (m, n).
Now, we give the definition ofI2-asymptotic density ofN×N.
A subsetK⊂N×Nis said to be haveI2-asymptotic density dI2(K) if dI2(K) =I2− 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)| de- notes number of elements of the setK(m, n).
A double sequencex={xjk}of real numbers isI2-statistically convergent toε, and we writexI2→−stξ, provied that for anyε >0 andδ >0
(m, n)∈N×N: 1
mn|{(j, k) :kxjk−ξk ≥ε,j≤m, k≤n}| ≥δ
∈ I2. Letx={xjk} be a double sequence in a normed linear space (X,k.k) and r be a non negative real number. Then xis said to be roughI2-statistical convergent to ξor r-I2-statistical convergent toξif for anyε >0 andδ >0
(m, n)∈N×N: 1
mn|{(j, k) ,j≤m, k≤n:kxjk−ξk ≥r+ε}| ≥δ
∈ I2. In this case,ξis called the roughI2-statistical limit ofx={xjk}and we denote it byxr−I−→2−stξ.
3. Main results
Definition 3.1. Let (∆xkl)be a double sequence in a normed linear space(X,k.k) andrbe a nonnegative real number. Then,(∆xkl)is said to be roughI2-convergent tox∗ orr-I2-convergent to x∗ if for anyε >0
{(k, l)∈N×N:k∆xkl−x∗k ≥r+ε} ∈ I2.
In this case x∗ is called the r-I2-limit of (∆xkl)and we denote it by ∆xr−I−→2 x∗. Here ris called roughness degree.
Definition 3.2. A double sequence (∆xkl)in X is said to be roughI2-statistical convergent to x∗ or r-I2-statistical convergent to x∗, denoted by ∆x r−I−→2−st x∗, provided that
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−x∗k ≥r+ε}| ≥δ
∈ I2
for any ε >0andδ >0, or equivalently we can sayI2-stlim supk∆xkl−x∗k ≤r.
If we take r= 0, we obtain the notion∆I2-statistical convergence.
For instance assume that the sequence (∆ykl) is roughI2-statistical convergent which can not be calculated exactly, one has to do with an approximated sequence (∆xkl) satisfyingk∆xkl−∆yklk ≤rfor allk,l. Then, roughI2-statistical conver- gence of sequence (∆xkl) is not assured, but as the inclusion,
(n, m)∈N×N: nm1 |{k≤n, l≤m:k∆xkl−x∗k ≥r+ε}| ≥δ
⊆
(n, m)∈N×N: nm1 |{k≤n, l≤m:k∆ykl−x∗k ≥ε}| ≥δ holds, so the sequence (∆xkl) isr-I2-statistically convergent.
In general the roughI2-statistical limit of a sequence may not be unique for the roughness degreer >0. We define the set of all roughI2-statistical limit of (∆xkl) with
I2−st−LIMr(∆xkl) =n
x∗∈X : ∆xkl r−I−→2−stx∗o .
The double sequence (∆xkl) is said to ber-I2-statistically convergent provided I2-−st−LIMr(∆xkl) 6= ∅. It is clear that if I2-st-LIMr(∆x
k) 6= ∅ for a sequence (∆xkl), then we have
I2−st−LIMr(∆xkl) = [I2−st−lim sup (∆xkl)−r, I2−st−lim inf (∆xkl)−r]. Definition 3.3. A double sequence (∆xkl) is said to be I2-statistical bounded if there is a number K such that
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xklk> K}|> δ
∈ I2
for any δ >0.
Theorem 3.4. For a sequence (∆xkl), diam
I2−st−LIMr(∆xkl)
≤2r.
In generaldiam(I2−st−LIMr(∆xkl))has no smaller bound.
Proof. Assume thatdiam(I2−st−LIMr(∆xkl))> 2r. Then, there exist y, z ∈ I2 −st−LIMr(∆xkl) such that ky−zk > 2r. Now, we select ε > 0 so that ε < ky−zk2 −r andδ >0. Since y, z ∈ I2−st−LIMr(∆xkl), for everyε >0 and δ >0, we have
A1(ε, δ) =
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−yk ≥r+ε}| ≥δ
∈ I2
and A2(ε, δ) =
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−zk ≥r+ε}| ≥δ
∈ I2. Hence,P =N×N\(A1(ε, δ)∪A2(ε, δ))∈ F(I2). So,P 6=∅. Letp, q∈P. Then for infinitely manyk≤p,l≤q
ky−zk ≤ k∆xkl−yk+k∆xkl−zk<2 (r+ε)<2
r+ky−zk
2 −r
=ky−zk, which is contradiction. Thus,
diam
I2−st−LIMr(∆xkl)
≤2r.
To prove the second part, consider a double sequence (∆xkl) such that I2-st- lim ∆xkl = x∗. Let ε > 0 and δ > 0 then by the definition of ∆I2-statistical convergence
A(ε, δ) =
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−x∗k ≥ε}|< δ
∈ F(I2). Letp, q∈A(ε, δ) then
1
pq|{k≤p, l≤q:k∆xkl−x∗k ≥ε}|< δ i.e., for maximumk≤p,l≤q,k∆xkl−x∗k< ε.
Now, for each
y∈Br(x∗) ={y∈X :ky−x∗k ≤r}
we have
k∆xkl−yk ≤ k∆xkl−x∗k+kx∗−yk ≤ k∆xkl−x∗k+r < r+ε for maxiumk≤p∈A(ε, δ),l≤q∈A(ε, δ) i.e.,
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−yk ≥r+ε}|< δ
⊇A(ε)∈ F(I2) which shows thaty∈ I2−st−LIMr(∆xkl) and consequently,I2−st−LIMr(∆xkl) = Br(x∗). This shows that in general upper bound 2r of the diameter of the set I2−st−LIMr(∆xkl) can not be decreased any more.
Theorem 3.5. A double sequence(∆xkl)isI2-st-bounded if and only if there exists a non-negative real numberr such thatI2−st−LIMr(∆xkl)6=∅.
Proof. Let (∆xkl) beI2-st-bounded double sequence. Then, there exists a positive real numberKsuch that
P =
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xklk> K}|> δ
∈ I2. Let r= sup{k∆xklk for almostk≤t, l≤s∈M =N×N\P}. The set I −st− LIMr(∆xkl) contains the origin ofX and soI2−st−LIMr(∆xkl)6=∅.
Conversely, assume thatI2−st−LIMr(∆xkl)6=∅for somer≥0. Then, there existsx∗∈ I2−st−LIMr(∆xkl) i.e.,
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−x∗k ≥r+ε}| ≥δ
∈ I2, for eachε >0 andδ >0. This implies that (∆xkl) isI2-st-bounded.
Theorem 3.6. The setI2−st−LIMr(∆xkl)of a sequence(∆xkl)is a closed set.
Proof. IfI2−st−LIMr(∆xkl) =∅, then there is nothing to prove.
Assume that I2−st−LIMr(∆xkl) 6= ∅. Now, consider a sequence (∆ykl) in I2−st−LIMr(∆xkl) with lim
k,l→∞∆ykl =y∗. Choose ε >0. Then, there exists nε
2 ∈Nsuch that
k∆ykl−y∗k< ε 2, for allk, l≥nε
2.
Let (n0, m0)∈N×Nsuch thatyn0m0 ∈(∆ykl)⊆ I2−st−LIMr(∆xkl). So, A=
(n, m)∈N×N: 1 nm
n
k≤n, l≤m:k∆xkl−yn0m0k ≥r+ε 2
o < δ
∈ F(I2). Letp, q∈A. Then,
1 pq
n
k≤p, l≤q:k∆xkl−yn0m0k ≥r+ε 2
o < δ, i.e., for maximumk≤p, l≤q,k∆xkl−yn0m0k< r+2ε.
Choose ann0, m0> nε
2 we get
k∆xkl−y∗k ≤ k∆xkl−yn0m0k+kyn0m0−y∗k< r+ε for maximumk≤p∈A, l≤q∈A.That is
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−y∗k ≥r+ε}|< δ
⊇A∈ F(I2). Hence,y∗∈ I2−st−LIMr(∆xkl) and so,I2−st−LIMr(∆xkl) is a closed set.
Theorem 3.7. The setI2−st−LIMr(∆xkl)of a double sequence(∆xkl)is convex.
Proof. Lety0, y1∈ I2−st−LIMr(∆xkl) andε >0 andδ >0 be given. Let A0(ε, δ) =
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−y0k ≥r+ε}| ≥δ
∈ I2
and A1(ε, δ) =
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−y1k ≥r+ε}| ≥δ
∈ I2. M =N×N\(A1(ε, δ)∪A2(ε, δ))∈ F(I2) and soM must be a infinite set. Let (t, s)∈M then,d(K1) = 0, where
K1={k≤t, l≤s:k∆xkl−y0k ≥r+ε}
andd(K2) = 0, where
K2={k≤t, l≤s:k∆xkl−y1k ≥r+ε}. Now for each (k, l)∈K1c∩K2c and eachλ∈[0,1],
k∆xkl−[(1−λ)y0+λy1]k=k(1−λ) (∆xkl−y0) +λ(∆xkl−y1)k< r+ε.
Since,d(K1c∩K2c) = 1, we get 1
ts|{k≤t, l≤s:k∆xkl−[(1−λ)y0+λy1]k ≥r+ε}|< δ.
Therefore,
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−[(1−λ)y0+λy1]k ≥r+ε}|< δ
⊇M ∈ F(I2), which shows that (1−λ)y0+λy1∈ I2−st−LIMr(∆xkl) for anyλ∈[0,1]. Hence,
the setI2−st−LIMr(∆xkl) is convex.
Definition 3.8. An elementc∈X is calledI2-statistical cluster point of a sequence (∆xkl)if for eachε >0 andδ >0 the set
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−ck ≥ε}|< δ
∈ I/ 2. The set of allI2-statistical cluster points of(∆xkl)will be denoted byI2-S Γ(∆xkl)
. Theorem 3.9. Let (∆xkl) be a double sequence, then for an arbitrary c ∈ I2- S Γ(∆xkl)
,kx∗−ck ≤r for allx∗ ∈ I2−st−LIMr(∆xkl).
Proof. If possible suppose that there exists c ∈ ∆I2-S(Γx) and x∗ ∈ I2−st− LIMr(∆xkl) such thatkx∗−ck> r. Letε= kx∗−ck−r2 .Then, we have
M =
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−ck ≥ε}|< δ
∈ I/ 2.
Now, we define the set K=
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−x∗k ≥r+ε}| ≥δ
. Lett, s∈M, i.e.,
1
ts|{k≤t, l≤s:k∆xkl−ck ≥ε}|< δ.
Hence, for maximumk≤t, l≤s,kxkl−ck ≤ε.Now
k∆xkl−x∗k ≥ kx∗−ck − k∆xkl−ck> r+ε,
for all k ≤ t ∈ M, l ≤ s ∈ M. Therefore, K ⊇ M implies that K /∈ I2, which contradicts the fact that x∗ ∈ I2−st−LIMr(∆xkl). Thus, kx∗−ck ≤ r for all x∗∈ I2−st−LIMr(∆xkl) andc∈ I2-S Γ(∆xkl)
.
Theorem 3.10. A double sequence(∆xkl)is roughI2-statistical convergent to x∗
if and only if I2−st−LIMr(∆xkl) =Br(x∗).
Proof. The necessary part of the theorem is already proved in the 2nd part of Theorem 2. For the sufficiency, letI2−st−LIMr(∆xkl) = Br(x∗)6= ∅. Thus, the sequence (∆xkl) is I2-statistically bounded. Suppose that (∆xkl) has another
∆I2-statistical cluster pointx0∗ different from x∗.The point x∗=x∗+ r
kx∗−x0∗k(x∗−x0∗) satisfies,kx∗−x0∗k> r.Since,x0∗∈ I2−S Γ(∆xkl)
, byTheorem 6,x∗∈ I/ 2−st− LIMr(∆xkl). But this is impossible as
kx∗−x0∗k=r andI2−st−LIMr(∆xkl) =Br(x∗).
Therefore x∗ is the uniqueI2-statistical cluster point of (∆xkl). Also, (∆xkl) is I2-bounded. So, byTheorem 1, (∆xkl) is roughI2-statistical convergent tox∗. Theorem 3.11. Letr >0. Then, a double sequence(∆xkl)is roughI2-statistical convergent to x∗ if and only if there exists a double sequence (∆ykl) such that I2−st−LIMr(∆ykl) =x∗ andk∆xkl−∆yklk ≤r, for all(k, l)∈N×N.
Proof. Necessity: Let ∆xr−I−→2−stx∗. Then, we have,
I2−stlim supk∆xkl−x∗k ≤r. (1) Now we define,
∆ykl:=
x∗, if k∆xkl−x∗k ≤r,
∆xkl+rk∆xx∗−∆xkl
kl−x∗k, otherwise.
Then,
k∆ykl−x∗k=
0, if k∆xkl−x∗k ≤r, k∆xkl−x∗k −r, otherwise
(2) by the definition ofykl we havek∆xkl−∆yklk ≤r, for all (k, l)∈N×N. Now by (1) and (2) we get,
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆ykl−x∗k ≥ε}| ≥δ
∈ I2
which implies thatI2−st−LIMr(∆ykl) =x∗.
Sufficiency: Assume that the given condition holds. For any ε > 0 and δ > 0 the set
M(ε, δ) =
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆ykl−x∗k ≥ε}|< δ
∈ I2
andk∆xkl−∆yklk ≤r, for all (k, l)∈N×N. Therefore,
k∆xkl−x∗k=k∆xkl−∆yklk+k∆ykl−x∗k< r+ε, for maximumk≤t∈Mc(ε, δ), l≤s∈Mc(ε, δ).This shows that
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−x∗k ≥r+ε}|< δ
⊇Mc(ε, δ)∈ F(I2)
and so,r-I2-stlim ∆xkl=x∗.
Corollary 3.12. If(X,k.k)is a strictly convex space,(∆xkl)is a double sequence in X and there exists y1, y2 ∈ I2−st−LIMr(∆xkl) such thatky1−y2k = 2r, then this sequence is roughI2-statistically convergent to y1+y2 2.
The proof is straightforward and so is omitted.
Theorem 3.13. (i)If c∈Γ(∆xkl)(I2), then I2−st−LIMr(∆xkl)⊆Br(c).
(ii)I2-st-LIMr(∆xkl)= T
c∈Γ(∆xkl)(I2)
Br(c) =
x∗∈Rn: Γ(∆xkl)(I2)⊆Br(x∗) . Proof. (i) If x∗ ∈ I2−st−LIMr(∆xkl) andc ∈Γ(∆xkl)(I2), then kx∗−ck ≤ r.
Hence the result follows.
(ii) By (i) we can write
I2−st−LIMr(∆xkl)⊆ \
c∈Γ(∆xkl)(I2)
Br(c).
Assume that y ∈ T
c∈Γ(∆xkl)(I2)
Br(c). We haveky−ck ≤r for allc ∈Γ(∆xkl)(I2) and so
Γ(∆xkl)(I2)⊆Br(x∗).
Then, clearly
\
c∈Γ(∆xkl)(I2)
Br(c) =
x∗∈Rn : Γ(∆xkl)(I2)⊆Br(x∗) .
If possible lety /∈ I2−st−LIMr(∆xkl). Then, there exists anε >0 such that K=
(n, m)∈N×N: 1
nm|{k≤n, l≤m:k∆xkl−yk ≥r+ε}|< δ
∈ I,/ which implies the existence of an I2-cluster point c of the sequence (∆xkl) with ky−ck ≥r+ε. Hence
Γ(∆xkl)(I2)⊆Br(y) and y /∈
x∗∈Rn : Γ(∆xkl)(I2)⊆Br(x∗) . Finally the fact thaty∈ I2−st−LIMr(∆xkl) follows from the observation that
y∈
x∗∈Rn: Γ(∆xkl)(I2)⊆Br(x∗) .
4. Conclusion
The rough convergence have been recently studied by several authors. In view of the recent applications of ideals in the theory of convergence of sequences, it seems very natural to extend the interesting concept of rough ∆I2-statistical convergence further by using ideals which we mainly do here and investigate some properties of this new type convergence. So, we have extended some well-known results.
References
[1] M. Arslan and E. D¨undar,Rough convergence in 2-normed spaces,Bulletin of Mathematical Analysis and Applications,10(3), (2018), 1–9.
[2] M. Arslan and E. D¨undar,On Rough convergence in 2-normed spaces and some properties, Filomat,33(16), (2019), 5077–5086.
[3] S. Aytar, Rough statistical convergence, Numer. Funct. Anal. Optimiz., 29(3-4), (2008), 291–303.
[4] S. Aytar,The rough limit set and the core of a real sequence, Numer. Funct. Anal. Optimiz., 29(3-4), (2008), 283-290.
[5] E. D¨undar and C. C¸ akan,RoughI-convergence, Gulf J. Math.,2(1),(2014), 45-51.
[6] E. D¨undar and C. C¸ akan, Rough convergence of double sequences, Demonstratio Math., 47(3), (2014), 638-651.
[7] E. D¨undar,On Rough I2-convergence, Numer. Funct. Anal. Optimiz.,37(4), (2016), 480- 491.
[8] H. G¨um¨u¸s and F. Nuray, ∆m-Ideal Convergence, Sel¸cuk J. Appl. Math., 12(2), (2011), 101-110.
[9] S.K. Pal, D. Chandra and S. Dutta,Rough ideal convergence, Hacet. J. Math. Stat.,42(6), (2013), 633-640.
[10] H. Fast,Sur la convergence statistique, Colloq. Math.,2,(1951), 241–244.
[11] S. Debnath and J. Debnath, On I-statistically convergent sequence spaces defined by se- quences of Orlicz functions using matrix transformation,Proyecciones,33(3), (2014) 277- 285.
[12] M. Et, A. Alotaibi and S. A. Mohiuddine,On(∆m-I)-statistical convergence of orderα, Sci.
World J., Article ID 535419 (2014), 5 pages
[13] M. Mursaleen, O.H.H. Edely,Statistical convergence of double sequence, J. Math. Anal. Appl., 288, (2003), 223-231.
[14] P. Malik, M. Maity,On rough convergence of double sequences in normed linear spaces, Bull.
Allahabad Math. Soc.,28(1), (2013), 89-99.
[15] P. Malik, M. Maity,On rough statistical convergence of double sequences in normed linear spaces, Afrika Mat.,27, (2016), 141-148.
[16] ¨O. Ki¸si and E. D¨undar, Rough I2-lacunary statistical convergence of double sequences, J.
Inequal. Appl.,2018:23016 pages.
[17] N. Demir and H. G¨um¨u¸s,Rough convergence for difference sequences, under review.
[18] N. Demir and H. G¨um¨u¸s,Rough statistical convergence for difference sequences, under review.
[19] H. G¨um¨u¸s and N. Demir, ∆I-Rough convergence,under review.
[20] ¨O. Ki¸si and E. D¨undar,Rough∆I-statistical convergence, under review.
[21] ¨O. Ki¸si,Rough statistical convergence for double difference sequences in normed linear spaces, under review.
[22] ¨O. Ki¸si, ∆I2-Rough convergence, under review.
[23] ¨O. Ki¸si,Rough∆I2-convergence of double difference sequences, under review.
[24] P. Kostyrko, T. ˘Sal´at and W. Wilczy´nski,I-convergence, Real Anal. Exchange,26(2), (2000), 669-686.
[25] P. Kostyrko, M. Macaj, T. ˘Sal´at and M. Sleziak,I-convergence and extremal I-limit points, Math. Slovaca,55, (2005), 443-464.
[26] E. Sava¸s and M. G¨urdal, I-statistical convergence in probabilistic normed space, Sci. Bul.
Ser. A Appl. Math. Phys.,77(4), (2015), 195-204.
[27] E. Sava¸s and P. Das,A generalized statistical convergence via ideals, Appl. Math. Lett.,24, (2011), 826–830.
[28] E. Sava¸s, S. Debnath and D. Rakshit,OnI-statistically rough convergence, Publ. Inst. Math., 105(119), (2019), 145-150.
[29] M. Mohiuddine and B. Hazarika,Some classes of ideal convergent sequences and generalized difference matrix operator ,Filomat,31(6), (2017), 1827-1834.
[30] M. Mursaleen and S.A. Mohiuddine, On ideal convergence in probabilistic normed spaces, Mathematica Slovaca,62(1), (2012) 49-62.
[31] M. Mursaleen, S. Debnath and D. Rakshit,I-statistical limit superior and I-statistical limit inferior, Filomat,31(7), (2017), 2103-2108.
[32] H. X. Phu,Rough convergence in normed linear spaces, Numer. Funct. Anal. Optimiz.,22, (2001), 199-222.
[33] H. X. Phu,Rough continuity of linear operators,Numer. Funct. Anal. Optimiz.,23, (2002), 139-146.
[34] H. X. Phu,Rough convergence in infinite dimensional normed spaces, Numer. Funct. Anal.
Optimiz., 24, (2003) 285-301.
[35] T. ˘Sal´at, B. C. Tripathy and M. Ziman,OnI-convergence field,Italian J. Pure Appl. Math., 17, (2005), 45-54.
[36] T. ˘Sal´at,On statistically convergent sequences of real numbers, Math. Slovaca,30, (1980), 139-150.
[37] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly,66, (1959), 361-375.
Omer K˙IS¨ ¸ ˙I
Bartın University, Faculty of Science, Department of Mathematics, 74100, Bartın, Turkey
E-mail address:[email protected]
Hatice K¨ubra ¨Unal
Bartın University, Graduate School of Natural and Applied Science, 74100, Bartın, Turkey
E-mail address:[email protected]
