Rough I2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${I}_{2}$\end{document}-lacunary statistical convergence of double sequences

In this paper, we introduce and study the notion of rough I2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {I}_{2}$\end{document}-lacunary statistical convergence of double sequences in normed linear spaces. We also introduce the notion of rough I2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{I}_{2}$\end{document}-lacunary statistical limit set of a double sequence and discuss some properties of this set.


Introduction
Throughout the paper, N and R denote the set of all positive integers and the set of all real numbers, respectively. The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [1] and Schoenberg [2]. This concept was extended to the double sequences by Mursaleen and Edely [3]. Lacunary statistical convergence was defined by Fridy and Orhan [4]. Çakan and Altay [5] presented multidimensional analogues of the results presented by Fridy and Orhan [4].
The idea of I-convergence was introduced by Kostyrko et al. [6] as a generalization of statistical convergence which is based on the structure of the ideal I of subset of the set of natural numbers. Kostyrko et al. [7] studied the idea of I-convergence and extremal I-limit points. Das et al. [8,9] introduced the concept of I-convergence of double sequences in a metric space and studied some properties of this convergence. A lot of development have been made in area about statistical convergence, I-convergence and double sequences after the work of [1,2,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28].
The notion of lacunary ideal convergence of real sequences was introduced in [29]. Das et al. [30,31] introduced new notions, namely I-statistical convergence and I-lacunary statistical convergence by using ideal. Belen et al. [32] introduced the notion of ideal statistical convergence of double sequences, which is a new generalization of the notions of statistical convergence and usual convergence. Kumar et al. [33] introduced I-lacunary statistical convergence of double sequences. Further investigation and applications on this notion can be found in [34].
The idea of rough convergence was first introduced by Phu [35] in finite-dimensional normed spaces. In another paper [36] related to this subject, Phu defined the rough continuity of linear operators and showed that every linear operator f : X → Y is r -continuous at every point x ∈ X under the assumption dim Y < ∞ and r > 0, where X and Y are normed spaces. In [37], Phu extended the results given in [35] to infinite-dimensional normed spaces. Aytar [38] studied the rough statistical convergence. Also, Aytar [39] studied that the rough limit set and the core of a real sequence. Recently, Dündar and Çakan [11,40], Pal et al. [41] introduced the notion of rough I-convergence and the set of rough I-limit points of a sequence and studied the notion of rough convergence and the set of rough limit points of a double sequence. Further this notion of rough convergence of double sequence has been extended to rough statistical convergence of double sequence by Malik et al. [42] using double natural density of N × N in the similar way as the notion of convergence of double sequence in Pringsheim sense was generalized to statistical convergence of double sequence. Also, Dündar [43] investigated rough I 2 -convergence of double sequences. The notion of I-statistical convergence of double sequences was introduced by Malik and Ghosh [44] in the theory of rough convergence.
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 lacunary statistical convergence further by using ideals which we mainly do here.
So it is quite natural to think, if the new notion of I-lacunary statistical convergence of double sequences can be introduced in the theory of rough convergence.
Throughout the paper, let r be a nonnegative real number and R n denotes the real ndimensional space with the norm · . Consider a sequence x = (x i ) ⊂ R n .
The sequence x = (x i ) is said to be r-convergent to x * , denoted by x i The set In this case, r is called the convergence degree of the sequence x = (x i ). For r = 0, we get the ordinary convergence. There are several reasons for this interest (see [35] is satisfied. In addition, we can write x i r-I − → x * iff the inequality x ix * < r + ε holds for every ε > 0 and almost all i. A double sequence x = (x mn ) (m,n)∈N×N of real numbers is said to be bounded if there exists a positive real number M such that |x mn | < M, for all m, n ∈ N. That is A double sequence x = (x mn ) of real numbers is said to be convergent to L ∈ R in Pringsheim's sense (shortly, p-convergent to L ∈ R), if for any ε > 0, there exists N ε ∈ N such that |x mn -L| < ε, whenever m, n > N ε . In this case, we write We recall that a subset K of N × N is said to have natural density d(K) if Throughout the paper we consider a sequence x = (x mn ) such that (x mn ) ∈ R n . Let x = (x mn ) be a double sequence in a normed space (X, · ) and r be a non-negative real number. x is said to be r-statistically convergent to ξ , denoted by x It is evident that a strongly admissible ideal is admissible also. Throughout the paper we take I 2 as a strongly admissible ideal in N × N.
In this case, we say that x is I 2 -convergent and we write A double sequence x = (x mn ) is said to be rough convergent (r-convergent) to x * with the roughness degree r, denoted by x mn or equivalently, if A double sequence x = (x mn ) is said to be r-I 2 -convergent to x * with the roughness degree r, denoted by x mn for every ε > 0; or equivalently, if the condition is satisfied. In addition, we can write x mn for every ε > 0 and almost all (m, n). Now, we give the definition of I 2 -asymptotic density of N × N. → ξ , provided that for any ε > 0 and δ > 0 Let x = {x jk } be a double sequence in a normed linear space (X, · ) and r be a non-negative real number. Then x is said to be rough I 2 -statistical convergent to ξ or r-I 2 -statistical convergent to ξ if for any ε > 0 and δ > 0 In this case, ξ is called the rough I 2 -statistical limit of x = {x jk } and we denote it by A double sequence θ = θ us = {(k u , l s )} is called a double lacunary sequence if there exist two increasing sequences of integers (k u ) and (l s ) such that We will use the notation k us := k u l s , h us := h u h s and θ us is determined by and q us := q u q s .
Throughout the paper, by θ 2 = θ us = {(k u , l s )} we will denote a double lacunary sequence of positive real numbers, respectively, unless otherwise stated. A double sequence x = {x mn } of numbers is said to be I 2 -lacunary statistical convergent or S θ 2 (I 2 )-convergent to L, if for each ε > 0 and δ > 0, In this case, we write x mn → L(S θ 2 (I 2 )) or S θ 2 (I 2 )-lim m,n→∞ x mn = L.

Main results
Definition 3.1 Let x = {x jk } be a double sequence in a normed linear space (X, · ) and r be a non-negative real number. Then x is said to be rough lacunary statistical convergent to ξ or r-lacunary statistical convergent to ξ if for any ε > 0 In this case ξ is called the rough lacunary statistical limit of x = {x jk } and we denote it by Definition 3.2 Let x = {x jk } be a double sequence in a normed linear space (X, · ) and r be a non-negative real number. Then x is said to be rough I 2 -lacunary statistical convergent to ξ or r-I 2 -lacunary statistical convergent to ξ if for any ε > 0 and δ > 0 In this case, ξ is called the rough I 2 -lacunary statistical limit of x = {x jk } and we denote it

Remark 3.3 Note that if I 2 is the ideal
then rough I 2 -lacunary statistical convergence coincides with rough lacunary statistical convergence.
Here r in the above definition is called the roughness degree of the rough I 2 -lacunary statistical convergence. If r = 0, we obtain the notion of I 2 -lacunary convergence. But our main interest is when r > 0. It may happen that a double sequence x = {x jk } is not I 2lacunary statistical convergent in the usual sense, but there exists a double sequence y = {y jk }, which is I 2 -lacunary statistically convergent and satisfying the condition x jk -y jk ≤ r for all (j, k). Then x is rough I 2 -lacunary statistically convergent to the same limit.
From the above definition it is clear that the rough I 2 -lacunary statistical limit of a double sequence is not unique. So we consider the set of rough I 2 -lacunary statistical limits of a double sequence x and we use the notation I θ 2 -st-LIM r x to denote the set of all rough I 2 -lacunary statistical limits of a double sequence x. We say that a double sequence x is rough I 2 -lacunary statistically convergent if I θ 2 -st-LIM r x = ∅. Throughout the paper X denotes a normed linear space (X, · ) and x denotes the double sequence {x jk } in X. Now, we discuss some basic properties of rough I 2 -lacunary statistically convergence of double sequences.
and so by the property of I 2 -convergence Clearly H ∈ I 2 , so choose (u 0 , s 0 ) ∈ N × N \ H. Then i.e., {(j, k) ∈ J us : (j, k) / ∈ A ∪ B} is a nonempty set.
Then for (j, k) ∈ B us we have x jky < r + ε. Hence, we have This implies Thus, for all (j, k) / ∈ A, and so we have This shows that y ∈ I θ 2 -st-LIM r x . Therefore, I θ 2 -st-LIM r x ⊃ B r (ξ ). Conversely, let y ∈ I θ 2 -st-LIM r x , yξ > r and ε = y-ξ -r 2 . Now, we take and by the property of I 2 -convergence Clearly M ∈ I 2 and we choose (u 0 , s 0 ) ∈ N × N \ M. Then we have and so which is absurd. Therefore, yξ ≤ r and so y ∈ B r (ξ ). Consequently, we have Let j 0 , k 0 > i ε 2 . Then y j 0 k 0 ∈ I θ 2 -st-LIM r x . Consequently, we have Clearly M = N × N \ A is nonempty, choose (u, s) ∈ M. We have Put B us = (j, k) ∈ J us : x jky j 0 k 0 < r + ε 2 and select (j, k) ∈ B us . Then we have and so Therefore, 1 h us (j, k) ∈ J us : x jky ≥ r + ε < 1 -(1δ) = δ and so we have This shows that y ∈ I θ 2 -st-LIM r x . Hence, I θ 2 -st-LIM r x is a closed set. Proof Let y 0 , y 1 ∈ I θ 2 -st-LIM r x and ε > 0 be given. Let A 0 = (j, k) ∈ J us : x jky 0 ≥ r + ε and A 1 = (j, k) ∈ J us : x jky 1 ≥ r + ε .
Then by Theorem 3.4, for δ > 0 we have Now, we choose 0 < δ 1 < 1 such that 0 < 1δ 1 < δ and let Then A ∈ I 2 . For all (u, s) / ∈ A, we have Then clearly, A c 0 ∩ A c 1 ⊂ M c . So for (u, s) / ∈ A, we have Therefore, Since A c ∈ F(I 2 ), we have (u, s) ∈ N × N : 1 h us (j, k) ∈ J us : (j, k) ∈ M < δ ∈ F (I 2 ) and so (u, s) ∈ N × N : This completes the proof. Proof Let y = {y jk } be a double sequence in X, which is I 2 -lacunary statistically convergent to ξ and x jky jk ≤ r, for all (j, k) ∈ N × N. Then for any ε > 0 and δ > 0 Let (u, s) / ∈ A. Then we have Then, for (j, k) ∈ B us , we have x jkξ ≤ x jky jk + y jkξ < r + ε, and so Thus, we have and, since A ∈ I 2 , (u, s) ∈ N × N : 1 h us (j, k) ∈ J us : x jkξ ≥ r + ε ≥ δ ∈ I 2 .
Definition 3.8 A double sequence x = {x jk } is said to be I θ 2 -statistically bounded if there exists a positive number K such that for any δ > 0 the set The next result provides a relationship between boundedness and rough I θ 2 -statistical convergence of double sequences. Proof Let x = {x jk } be bounded double sequence. There exists a positive real number K such that x jk < K , for all (j, k) ∈ J us . Let ε > 0 be given. Then  x (I 2 ).

Theorem 3.12
For any arbitrary α ∈ S θ 2 x (I 2 ) of a double sequence x = {x jk } we have ξα ≤ r, for all ξ ∈ I θ 2 -st-LIM r x .

Conclusion
The rough convergence has recently been 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 lacunary statistical convergence further by using ideals, which we mainly do here; and we investigate some properties of this new type of convergence. So, we have extended some well-known results.