1. Introduction

The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by Fast [1] and Schoenberg [2]. There has been an effort to introduce several generalizations and variants of statistical convergence in different spaces. One such very important generalization of this notion was introduced by Kostyrko et al. [3] by using an ideal I of subsets of the set of natural numbers, which they called I-convergence. After that the idea of I-convergence for double sequence was introduced by Das et al. [4] in 2008.

˜ ° Throughout <sup>a</sup> double sequence is defined by <sup>x</sup> <sup>¼</sup> xij and we denote <sup>2</sup><sup>ω</sup> showing the space of all real or complex double sequences.

Let X be a nonempty set then a family I ⊂ 2<sup>X</sup> is said to be an ideal in X if Ø ∈I, I is additive, i.e., for all A, B ∈I ) A ∪ B ∈I and I is hereditary, i.e., for all <sup>A</sup> <sup>∈</sup>I, <sup>B</sup> <sup>⊆</sup> <sup>A</sup> ) <sup>B</sup> <sup>∈</sup>I. <sup>A</sup> nonempty family of sets <sup>F</sup> <sup>⊂</sup>2<sup>X</sup> is said to be <sup>a</sup> filter on <sup>X</sup> if for all A, B∈ F implies A ∩ B ∈ F and for all A ∈ F with A ⊆ B implies B ∈ F. An ideal <sup>I</sup> <sup>⊂</sup>2<sup>X</sup> is said to be nontrivial if <sup>I</sup> 6¼ <sup>2</sup><sup>X</sup>, this non trivial ideal is said to be admissible if I ⊇ ff gx : x∈Xg and is said to be maximal if there cannot exist any nontrivial ideal J 6¼ I containing I as a subset. For each ideal I there is a filter Fð ÞI called as filter associate with ideal I, that is

$$\mathcal{F}(I) = \{ K \subseteq X : K^{\epsilon} \in I \}, \quad \text{where} \quad K^{\epsilon} = X \backslash K. \tag{1}$$

� � � � � � � � A double sequence x ¼ xij ∈2ω is said to be I-convergent [5–8] to a number L if for every ϵ>0, we have ð Þ i; j ∈ N � N : jxij � Lj≥ϵ ∈I: In this case, we write I � lim xij ¼ L: A double sequence x ¼ xij ∈2ω is said to be I-Cauchy if for every ϵ>0 there exists numbers m ¼ mð Þϵ , n ¼ nð Þϵ such that ð Þ i; j ∈ N � N : jxij � xmnj≥ϵ ∈I:

A continuous linear functional ϕ on l<sup>∞</sup> is said to be an invariant mean [9, 10] or σ-mean if and only if:

1. ϕð Þ x ≥ 0 where the sequence x ¼ ð Þ xk has xk ≥ 0 for all k, 2. ϕð Þ¼ e 1 where e ¼ f1; 1; 1; 1; …g,

� � 3. ϕ x<sup>σ</sup>ð Þ <sup>n</sup> ¼ ϕð Þ x for all x∈l∞,

where σ be an injective mapping of the set of the positive integers into itself having no finite orbits.

� � If <sup>x</sup> <sup>¼</sup> ð Þ xk , write Tx ¼ ðTxkÞ ¼ <sup>x</sup>σð Þ<sup>k</sup> , so we have

$$V\_{\sigma} = \left\{ \mathbf{x} = (\mathbf{x}\_k) : \lim\_{m \to \infty} t\_{m,k}(\mathbf{x}) = L \text{ uniformly in } k, L = \sigma - \lim \mathbf{x} \right\} \tag{2}$$

where m ≥ 0, k > 0:

$$t\_{m,k}(\mathbf{x}) = \frac{\mathbf{x}\_k + \mathbf{x}\_{\sigma(k)} + \dots + \mathbf{x}\_{\sigma^n(k)}}{m + 1} \quad \text{and} \ t\_{-1,k} = \mathbf{0},\tag{3}$$

where <sup>σ</sup><sup>m</sup>ð Þ<sup>k</sup> denote the <sup>m</sup>th-iterate of <sup>σ</sup>ð Þ<sup>k</sup> at k. In this case <sup>σ</sup> is the translation mapping, that is, σð Þ¼ k k þ 1, σ� mean is called a Banach limit [11] and Vσ, the set of bounded sequences of all whose invariant means are equal, is the set of almost convergent sequences. The special case of (3) in which σð Þ¼ k k þ 1 was given by Lorentz [12] and the general result can be proved in a similar way. It is familiar that a Banach limit extends the limit functional on c in the sense that

$$\phi(x) = \lim x, \quad \text{for all } x \in \mathfrak{c}. \tag{4}$$

Definition 1.1 A sequence x∈l<sup>∞</sup> is of σ-bounded variation if and only if:

(i) ∑∣ϕm,kð Þ x ∣ converges uniformly in k,

(ii) lim<sup>m</sup>!<sup>∞</sup> tm,kð Þ x , which must exist, should take the same value for all k.

We denote by BVσ, the space of all sequences of σ-bounded variation:

$$BV\_{\sigma} = \left\{ \mathbf{x} \in l\_{\infty} : \sum\_{m} |\phi\_{m,k}(\mathbf{x})| < \infty, \text{ uniformly in } k \right\}.$$

is a Banach space normed by

$$\|\|\mathbf{x}\|\| = \sup\_{k} \sum\_{m=0}^{\infty} |\phi\_{m,k}(\mathbf{x})|. \tag{5}$$

A function M : ½0; ∞Þ ! ½0; ∞Þ is said to be an Orlicz function [13, 14] if it satisfies the following conditions:

(i) M is continuous, convex and non-decreasing,

(ii) Mð Þ¼ 0 0, Mð Þ x >0 and M xð Þ! ∞ as x ! ∞:

Remark 1.1 If the convexity of an Orlicz function is replaced by Mðx þ yÞ ≤ M xð Þþ Mð Þy , then this function is called Modulus function [15–17]. If M is an Orlicz function, then MðλXÞ ≤ λM xð Þ for all λ with 0 < λ < 1: An Orlicz

function M is said to satisfy Δ2-condition for all values of u if there exists a constant K > 0 such that Mð Þ Lu ≤ KLMð Þ u for all values of L > 1 [18].

Definition 1.2 A double sequence space X is said to be:

� � � � � � [i] solid or normal if xij ∈ X implies that αijxij ∈X for all sequence of scalars αij with ∣αij∣ < 1 for all ð Þ i; j ∈ N � N.

� � � � [ii] symmetric if x<sup>π</sup> <sup>i</sup>;<sup>j</sup> ∈X whenever xij ∈X, where πð Þ i; j is a permutation on N � N. ð Þ

� � � � � � [iii] sequence algebra if xijyij ∈ E whenever xij , yij ∈ X:

� � � � [iv] convergence free if yij ∈X whenever xij ∈ X and xij ¼ 0 implies yij ¼ 0, for all ð Þ i; j ∈ N � N.

�� � Definition 1.3 Let K ¼ ni; kj : ð Þ i; j : n<sup>1</sup> < n<sup>2</sup> < n<sup>3</sup> < :… and k<sup>1</sup> < k<sup>2</sup> < k<sup>3</sup> < :…g ⊆ N � N and X be a double sequence space. A K-step space of X is a sequence space

$$
\lambda\_k^E = \left\{ \left( a\_{i\vec{\jmath}} \mathfrak{x}\_{\vec{\imath}\vec{\jmath}} \right) \, : \, (\mathfrak{x}\_{\vec{\imath}\vec{\jmath}}) \in X \right\} .
$$

� � A canonical preimage of a sequence xnikj ∈X is a sequence ðbnkÞ∈X defined as follows:

$$b\_{nk} = \begin{cases} a\_{nk}, & \text{for } n, \ k \in K \\ 0, & \text{otherwise}. \end{cases}$$

A sequence space X is said to be monotone if it contains the canonical preimages of all its step spaces.

� � � � The following subspaces l pð Þ, l∞ð Þ <sup>p</sup> , cð Þ <sup>p</sup> and <sup>c</sup>0ð Þ <sup>p</sup> where <sup>p</sup> <sup>¼</sup> pk is <sup>a</sup> sequence of positive real numbers. These subspaces were first introduced and discussed by Maddox [16]. The following inequalities will be used throughout the section. Let p ¼ pij be a double sequence of positive real numbers [19]. For any complex λ with 0 < pij ≤ supij pij ¼ G < ∞, we have

$$|\lambda|^{p\_{\vec{\eta}}} \le \max\left(\mathbf{1}, |\lambda|^G\right).$$

� � � � � � n o <sup>1</sup> Let <sup>D</sup> <sup>¼</sup> max <sup>1</sup>; <sup>2</sup><sup>G</sup>� and <sup>H</sup> <sup>¼</sup> max <sup>1</sup>;supij pij , then for the sequences aij and bij in the complex plane, we have

$$\left| a\_{\vec{v}} + b\_{\vec{v}} \right|^{p\_{\vec{v}}} \le C \left( \left| a\_{\vec{v}} \right|^{p\_{\vec{v}}} + \left| b\_{\vec{v}} \right|^{p\_{\vec{v}}} \right).$$
