Regularly ideal invariant convergence of double sequences

Abstract

In this paper, we introduce the notions of regularly invariant convergence, regularly strongly invariant convergence, regularly p-strongly invariant convergence, regularly $(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergence, regularly $(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-convergence, regularly $(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2} )$-Cauchy double sequence, regularly $(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-Cauchy double sequence and investigate the relationship among them.

1 Introduction and background

Throughout the paper, $\mathbb{N}$ and $\mathbb{R}$ denote the set of all positive integers and the set of all real numbers, respectively. The concept of convergence of real sequences was extended to statistical convergence independently by Fast [18] and Schoenberg [42]. This concept was extended to the double sequences by Mursaleen and Edely [29].

The idea of $\mathcal{I}$-convergence was introduced by Kostyrko et al. [19] as a generalization of statistical convergence. Das et al. [5] introduced the concept of $\mathcal{I}$-convergence of double sequences in a metric space and studied some properties of this convergence. Tripathy and Tripathy [45] studied $\mathcal{I}$-convergent and regularly $\mathcal{I}$-convergent double sequences. Dündar and Altay [11] introduced $\mathcal{I}_{2}$-convergence and regularly $\mathcal{I}$-convergence of double sequences. Also, Dündar [7] introduced regularly $\mathcal{I}$-convergence and regularly $\mathcal{I}$-Cauchy double sequences of functions. Recently, Dündar and Akın [8] introduced the notions of $R(\mathcal{I}_{W_{2}},\mathcal{I}_{W})$-convergence, $R(\mathcal{I}_{W_{2}}^{*},\mathcal{I}_{W}^{*})$-convergence, $R(\mathcal{I}_{W_{2}},\mathcal{I}_{W})$-Cauchy, and $R(\mathcal{I}_{W_{2}}^{*},\mathcal{I}_{W}^{*})$-Cauchy double sequence of sets and investigated the relationship among them. A lot of development has been made in this area after the works of [6, 9, 12, 13, 15, 20, 22, 23, 26, 28, 31–34, 36, 43].

Several authors have studied invariant convergent sequences (see, [4, 24, 25, 30, 35, 38–41, 44]). Recently, the concepts of σ-uniform density of the set $A\subseteq \mathbb{N}$, $\mathcal{I}_{\sigma }$-convergence and $\mathcal{I}^{*}_{\sigma }$-convergence of sequences of real numbers were defined by Nuray et al. [35]. The concept of σ-convergence of double sequences was studied by Çakan et al. [4], and the concept of σ-uniform density of $A\subseteq \mathbb{N}\times \mathbb{N}$ was defined by Tortop and Dündar [44]. Dündar et al. [16] studied ideal invariant convergence of double sequences and some properties.

Now, we recall the basic definitions and concepts (see [1–5, 9–15, 17, 19–21, 27, 34, 36, 37, 43–46]).

A double sequence $x=(x_{kj})_{k,j \in \mathbb{N}}$ of real numbers is said to be convergent to $L \in \mathbb{R}$ in Pringsheim’s sense if, for any $\varepsilon >0$, there exists $N_{\varepsilon }\in \mathbb{N}$ such that $\vert x_{kj} -L \vert < \varepsilon$, whenever $k,j> N_{\varepsilon }$. In this case, we write $P-\lim_{k,j \rightarrow \infty } x_{kj} = \lim_{k,j \rightarrow \infty } x_{kj} = L$.

A family of sets $\mathcal{I}\subseteq 2^{\mathbb{N}}$ is called an ideal if and only if

$(i)$ $\emptyset \in \mathcal{I}$, $(\mathit{ii})$ For each $A,B\in \mathcal{I}$, we have $A\cup B\in \mathcal{I}$, $(\mathit{iii})$ For each $A\in \mathcal{I}$ and each $B\subseteq A$, we have $B\in \mathcal{I}$.

An ideal is called nontrivial if $\mathbb{N}\notin \mathcal{I}$, and nontrivial ideal is called admissible if $\{ n \} \in \mathcal{I}$ for each $n\in \mathbb{N}$.

Throughout the paper we take $\mathcal{I}$ as an admissible ideal in $\mathbb{N}$.

A family of sets $\mathcal{F}\subseteq 2^{\mathbb{N}}$ is called a filter if and only if

$(i)$ $\emptyset \notin \mathcal{F}$, $(\mathit{ii})$ For each $A,B\in \mathcal{F}$, we have $A\cap B\in \mathcal{F}$, $(\mathit{iii})$ For each $A\in \mathcal{F}$ and each $B\supseteq A$, we have $B\in \mathcal{F}$.

For any ideal there is a filter $\mathcal{F}(\mathcal{I})$ corresponding to $\mathcal{I}$, given by

$$ \mathcal{F}(\mathcal{I})= \bigl\{ M\subset \mathbb{N}:(\exists A\in \mathcal{I})\ (M=\mathbb{N}\backslash A) \bigr\} . $$

An admissible ideal $\mathcal{I} \subset 2^{\mathbb{N}}$ is said to satisfy the property (AP) if, for every countable family of mutually disjoint sets $\{A_{1},A_{2}, \ldots \}$ belonging to $\mathcal{I}$, there exists a countable family of sets $\{B_{1},B_{2}, \ldots \}$ such that $A_{j} \Delta B_{j}$ is a finite set for $j \in \mathbb{N}$ and $B = \bigcup_{j=1}^{\infty }B_{j} \in \mathcal{I}$.

A nontrivial ideal $\mathcal{I}_{2}$ of $\mathbb{N} \times \mathbb{N}$ is called strongly admissible ideal if $\{i\}\times \mathbb{N}$ and $\mathbb{N}\times \{i\}$ belong to $\mathcal{I}_{2}$ for each $i \in N$.

It is evident that a strongly admissible ideal is admissible also.

Throughout the paper we take $\mathcal{I}_{2}$ as a strongly admissible ideal in $\mathbb{N} \times \mathbb{N}$.

$\mathcal{I}_{2}^{0} = \{A \subset \mathbb{N}\times \mathbb{N}: ( \exists m(A)\in \mathbb{N})\ (i,j \geq m(A) \Rightarrow (i,j) \notin A) \}$. Then $\mathcal{I}_{2}^{0}$ is a strongly admissible ideal and clearly an ideal $\mathcal{I}_{2}$ is strongly admissible if and only if $\mathcal{I}_{2}^{0} \subset \mathcal{I}_{2}$.

An admissible ideal $\mathcal{I}_{2} \subset 2^{\mathbb{N}\times \mathbb{N}}$ satisfies the property (AP2) if, for every countable family of mutually disjoint sets $\{E_{1},E_{2}, \ldots \}$ belonging to $\mathcal{I}_{2}$, there exists a countable family of sets $\{F_{1},F_{2}, \ldots \}$ such that $E_{j} \Delta F_{j} \in \mathcal{I}_{2}^{0}$, i.e., $E_{j} \Delta F_{j}$ is included in the finite union of rows and columns in $\mathbb{N}\times \mathbb{N}$ for each $j \in \mathbb{N}$ and $F = \bigcup_{j=1}^{\infty }F_{j} \in \mathcal{I}_{2}$ (hence $F_{j}\in \mathcal{I}_{2}$ for each $j \in \mathbb{N}$).

Let σ be a mapping of the positive integers into themselves. A continuous linear functional ϕ on $\ell _{\infty }$, the space of real bounded sequences, is said to be an invariant mean or a σ-mean if it satisfies following conditions:

• 1 $\phi (x)\geq 0$, when the sequence $x=(x_{n})$ has $x_{n}\geq 0$ for all n,

• 2 $\phi (e)=1$, where $e=(1,1,1,\ldots)$, and

• 3 $\phi (x_{\sigma (n)})=\phi (x_{n})$ for all $x\in \ell _{\infty }$.

The mappings σ are assumed to be one-to-one and such that $\sigma ^{m}(n)\neq n$ for all positive integers n and m, where $\sigma ^{m}(n)$ denotes the mth iterate of the mapping σ at n. Thus, ϕ extends the limit functional on c, the space of convergent sequences, in the sense that $\phi (x)=\lim x$ for all $x\in c$.

In the case σ is translation mappings $\sigma (n)=n+1$, the σ-mean is often called a Banach limit and the space $V_{\sigma }$, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences ĉ.

It can be shown that

$$ V_{\sigma }= \Biggl\{ x=(x_{n})\in \ell _{\infty }: \lim _{m\rightarrow \infty }\frac{1}{m}\sum_{k=1}^{m}x_{\sigma ^{k}(n)}=L \text{, uniformly in } n \Biggr\} . $$

Let $A\subseteq \mathbb{N}$ and

$$ s_{m}=\min_{n} \bigl\vert A\cap \bigl\{ \sigma (n),\sigma ^{2}(n),\ldots, \sigma ^{m}(n) \bigr\} \bigr\vert $$

and

$$ S_{m}=\max_{n} \bigl\vert A\cap \bigl\{ \sigma (n),\sigma ^{2}(n),\ldots, \sigma ^{m}(n) \bigr\} \bigr\vert . $$

If the limits $\underline{V}(A)=\lim_{m\rightarrow \infty }\frac{s_{m}}{m}$, $\overline{V}(A)=\lim_{m\rightarrow \infty }\frac{S_{m}}{m}$ exist, then they are called a lower and upper σ-uniform density of the set A, respectively. If $\underline{V}(A)=\overline{V}(A)$, then $V(A)=\underline{V}(A)=\overline{V}(A)$ is called σ-uniform density of A.

Denote by $\mathcal{I}_{\sigma }$ the class of all $A\subseteq \mathbb{N}$ with $V(A)=0$.

Let $\mathcal{I}_{\sigma }\subset 2^{\mathbb{N}}$ be an admissible ideal. A sequence $x=(x_{k})$ is said to be $\mathcal{I}_{\sigma }$-convergent to the number L if, for every $\varepsilon >0$, $A_{\varepsilon }=\{k: \vert x_{k}-L \vert \geq \varepsilon \}\in \mathcal{I}_{\sigma }$, that is, $V(A_{\varepsilon })=0$. In this case, we write $\mathcal{I}_{\sigma }-\lim_{k}=L$.

Let $\mathcal{I}_{\sigma }\subset 2^{\mathbb{N}}$ be an admissible ideal. A sequence $x=(x_{k})$ is said to be $\mathcal{I}^{*}_{\sigma }$-convergent to the number L if there exists a set $M=\{m_{1}< m_{2}<\cdots \}\in \mathcal{F}(\mathcal{I}_{\sigma })$ such that $\lim_{k\rightarrow \infty }x_{m_{k}}=L$. In this case, we write $\mathcal{I}^{*}_{\sigma }-\lim_{k}=L$.

Let $A\subseteq \mathbb{N}\times \mathbb{N}$ and

$$ s_{mn}=\min_{k,j} \bigl\vert A\cap \bigl\{ \bigl( \sigma (k),\sigma (j) \bigr), \bigl(\sigma ^{2}(k),\sigma ^{2}(j) \bigr),\ldots, \bigl(\sigma ^{m}(k), \sigma ^{n}(j) \bigr) \bigr\} \bigr\vert $$

and

$$ S_{mn}=\max_{k,j} \bigl\vert A\cap \bigl\{ \bigl( \sigma (k),\sigma (j) \bigr), \bigl(\sigma ^{2}(k),\sigma ^{2}(j) \bigr),\ldots, \bigl(\sigma ^{m}(k), \sigma ^{n}(j) \bigr) \bigr\} \bigr\vert . $$

If the limits exist, $\underline{V_{2}}(A)=\lim_{m,n\rightarrow \infty } \frac{s_{mn}}{mn}$, $\overline{V_{2}}(A)=\lim_{m,n \rightarrow \infty }\frac{S_{mn}}{mn}$ exist, then they are called a lower and an upper σ-uniform density of the set A, respectively. If $\underline{V_{2}}(A)=\overline{V_{2}}(A)$, then $V_{2}(A)=\underline{V_{2}}(A)=\overline{V_{2}}(A)$ is called the σ-uniform density of A.

Denote by $\mathcal{I}_{2}^{\sigma }$ the class of all $A\subseteq \mathbb{N}\times \mathbb{N}$ with $V_{2}(A)=0$.

Throughout the paper we let $\mathcal{I}^{\sigma }_{2}\subset 2^{\mathbb{N}\times \mathbb{N}}$ be a strongly admissible ideal.

A double sequence $x=(x_{kj})$ is said to be $\mathcal{I}_{2}$-invariant convergent or $\mathcal{I}_{2}^{\sigma }$-convergent to L if, for every $\varepsilon >0$, $A(\varepsilon )= \{(k,j): \vert x_{kj}-L \vert \geq \varepsilon \}\in \mathcal{I}^{\sigma }_{2}$, that is, $V_{2} (A(\varepsilon ) )=0$. In this case, we write $\mathcal{I}_{2}^{\sigma }-\lim x=L$ or $x_{kj}\rightarrow L(\mathcal{I}^{\sigma }_{2})$.

A double sequence $(x_{kj})$ is $\mathcal{I}^{*}_{2}$-invariant convergent or $\mathcal{I}^{{\sigma }*}_{2}$-convergent to L if and only if there exists a set $M_{2}\in \mathcal{F}(\mathcal{I}^{\sigma }_{2})$ ($\mathbb{N}\times \mathbb{N}\backslash M_{2}=H\in \mathcal{I}^{\sigma }_{2}$) such that, for $(k,j)\in M_{2}$, $\lim_{k,j\rightarrow \infty }x_{kj}=L$. In this case, we write $\mathcal{I}^{{\sigma }*}_{2}-\lim_{k,j\rightarrow \infty }x_{kj}= L$ or $x_{kj} \rightarrow L(\mathcal{I}^{{\sigma }*}_{2})$.

A double sequence $(x_{kj})$ is said to be $\mathcal{I}_{2}$-invariant Cauchy or $\mathcal{I}^{\sigma }_{2}$-Cauchy sequence if, for every $\varepsilon >0$, there exist numbers $r=r(\varepsilon )$, $s=s(\varepsilon )\in \mathbb{N}$ such that $A(\varepsilon )= \{(k,j): \vert x_{kj}-x_{rs} \vert \geq \varepsilon \} \in \mathcal{I}_{2}^{\sigma }$, that is, $V_{2} (A(\varepsilon ) )=0$.

A double sequence $(x_{kj})$ is $\mathcal{I}^{*}_{2}$-invariant Cauchy or $\mathcal{I}^{{\sigma }*}_{2}$-Cauchy sequence if there exists a set $M_{2}\in \mathcal{F}(\mathcal{I}^{\sigma }_{2})$ (i.e., $\mathbb{N}\times \mathbb{N} \backslash M_{2} =H \in \mathcal{I}^{ \sigma }_{2}$) such that, for every $(k,j),(r,s)\in M_{2}$, $\lim_{k,j,r,s\rightarrow \infty } \vert x_{kj}-x_{rs} \vert =0$.

A double sequence $x = (x_{kj})$ is said to be regularly $(\mathcal{I}_{2},\mathcal{I})$-convergent ($r(\mathcal{I}_{2}, \mathcal{I})$-convergent) if it is $\mathcal{I}_{2}$-convergent in Pringsheim’s sense and for every $\varepsilon >0$ the following hold:

$$\begin{aligned}& \bigl\{ k\in \mathbb{N}: \vert x_{kj} - L_{j} \vert \geq \varepsilon \bigr\} \in \mathcal{I}\quad \text{for some } L_{j} \in X \text{ and each } j \in \mathbb{N}, \\& \bigl\{ j\in \mathbb{N}: \vert x_{kj} - M_{k} \vert \geq \varepsilon \bigr\} \in \mathcal{I} \quad \text{for some } M_{k} \in X \text{ and each } k \in \mathbb{N}. \end{aligned}$$

A double sequence $x = (x_{kj})$ is said to be $r(\mathcal{I}_{2}^{*},\mathcal{I}^{*})$-convergent if there exist the sets $M \in \mathcal{F}(\mathcal{I}_{2})$, $M_{1} \in \mathcal{F}(\mathcal{I})$, and $M_{2} \in \mathcal{F}(\mathcal{I})$ (i.e., $\mathbb{N}\times \mathbb{N} \setminus M \in \mathcal{I}_{2}$, $\mathbb{N} \setminus M_{1} \in \mathcal{I}$ and $\mathbb{N} \setminus M_{2} \in \mathcal{I}$) such that the limits

$$ \mathop{\lim_{k,j\rightarrow \infty }}_{(k,j)\in M}x_{kj},\quad\quad \mathop{ \lim_{k\rightarrow \infty }}_{k\in M_{1}}x_{kj}, \quad ( j \in \mathbb{N}) \quad \text{and}\quad \mathop{\lim_{j\rightarrow \infty }}_{j\in M_{2}}x_{kj} \quad ( k \in \mathbb{N}) $$

exist. Note that if $x = (x_{kj})$ is regularly convergent to L, then the limits $\lim_{k\rightarrow \infty }\lim_{j\rightarrow \infty }x_{kj}$ and $\lim_{j\rightarrow \infty }\lim_{k\rightarrow \infty }x_{kj}$ exist and are equal to L.

A double sequence $x = (x_{kj})$ is said to be regularly $(\mathcal{I}_{2},\mathcal{I})$-Cauchy ($r(\mathcal{I}_{2}, \mathcal{I})$-Cauchy) if it is $\mathcal{I}_{2}$-Cauchy in Pringsheim’s sense and for every $\varepsilon >0$ there exist $m_{j}=m_{j}(\varepsilon )$, $n_{k}=n_{k}(\varepsilon ) \in \mathbb{N}$ such that the following hold:

$$\begin{aligned}& A_{1}(\varepsilon )= \bigl\{ k\in \mathbb{N}: \vert x_{kj}-x_{m_{j}j} \vert \geq \varepsilon \bigr\} \in \mathcal{I} \quad (j\in \mathbb{N}), \\& A_{2}(\varepsilon )= \bigl\{ j\in \mathbb{N}: \vert x_{kj}-x_{kn_{k}} \vert \geq \varepsilon \bigr\} \in \mathcal{I} \quad (k\in \mathbb{N}). \end{aligned}$$

A double sequence $x = (x_{kj})$ is said to be regularly $(\mathcal{I}_{2}^{*},\mathcal{I}^{*})$-Cauchy ($r(\mathcal{I}_{2}^{*}, \mathcal{I}^{*})$-Cauchy) if there exist the sets $M \in \mathcal{F}(\mathcal{I}_{2})$, $M_{1} \in \mathcal{F}(\mathcal{I})$, and $M_{2} \in \mathcal{F}(\mathcal{I})$ (i.e., $\mathbb{N}\times \mathbb{N} \setminus M \in \mathcal{I}_{2}$, $\mathbb{N} \setminus M_{1} \in \mathcal{I}$, and $\mathbb{N} \setminus M_{2} \in \mathcal{I}$) and for every $\varepsilon >0$ there exist $N=N(\varepsilon ) \in \mathbb{N}$, $s=s(\varepsilon )$, $t=t(\varepsilon )$, $m_{j}=m_{j}(\varepsilon )$, $n_{k}=n_{k}(\varepsilon )\in \mathbb{N}$ such that whenever $k,j, m_{j},n_{k} >N$, we have

$$\begin{aligned}& \vert x_{kj}-x_{st} \vert < \varepsilon \quad \bigl( \text{for }(k,j),(s,t) \in M, k,j,s,t > N \bigr), \\& \vert x_{kj},x_{m_{j}j} \vert < \varepsilon \quad (\text{for each } k\in M_{1} \text{ and each } j\in \mathbb{N} ), \\& \vert x_{kj},x_{kn_{k}} \vert < \varepsilon \quad (\text{for each } j\in M_{2} \text{ and each } k\in \mathbb{N} ). \end{aligned}$$

Lemma 1

([16])

Suppose that $x=(x_{kj})$ is a bounded double sequence. If $x=(x_{kj})$ is $\mathcal{I}^{\sigma }_{2}$-convergent to L, then $x=(x_{kj})$ is invariant convergent to L.

Lemma 2

([16])

Let $0< p<\infty$.

• (i) If $x_{kj}\rightarrow L ([V_{\sigma }^{2}]_{p} )$, then $x_{kj}\rightarrow L (\mathcal{I}^{\sigma }_{2} )$.

• (ii) If $(x_{kj})\in \ell _{\infty }^{2}$ and $x_{kj}\rightarrow L (\mathcal{I}^{\sigma }_{2} )$, then $x_{kj}\rightarrow L ([V_{\sigma }^{2}]_{p} )$.

• (iii) If $(x_{kj})\in \ell _{\infty }^{2}$, then $x_{kj} \rightarrow L (\mathcal{I}^{\sigma }_{2} )$ if and only if $x_{kj}\rightarrow L ([V_{\sigma }^{2}]_{p} )$.

Lemma 3

([16])

If a double sequence $(x_{kj})$ is $\mathcal{I}^{{\sigma }*}_{2}$-convergent to L, then this sequence is $\mathcal{I}^{\sigma }_{2}$-convergent to L.

Lemma 4

([16])

Let $\mathcal{I}^{\sigma }_{2}$ have the property $(\mathit{AP}2)$. If $(x_{kj})$ is $\mathcal{I}^{\sigma }_{2}$-convergent to L, then $(x_{kj})$ is $\mathcal{I}^{{\sigma }*}_{2}$-convergent to L.

Lemma 5

([16])

If a double sequence $(x_{kj})$ is $\mathcal{I}^{\sigma }_{2}$-convergent, then $(x_{kj})$ is an $\mathcal{I}^{{\sigma }}_{2}$-Cauchy double sequence.

Lemma 6

([16])

If a double sequence $(x_{kj})$ is $\mathcal{I}^{{\sigma }*}_{2}$-Cauchy, then this sequence is $\mathcal{I}^{\sigma }_{2}$-Cauchy.

2 Main results

Now, we denote the notions of regularly invariant convergence, regularly strongly invariant convergence, regularly p-strongly invariant convergence, regularly $(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergence, regularly $(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-convergence, regularly $(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2} )$-Cauchy double sequence, regularly $(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-Cauchy double sequence and investigate the relationship among them.

Definition 2.1

A double sequence $x=(x_{kj})$ is said to be regularly invariant convergent ($r(\sigma ,\sigma _{2})$-convergent) if it is invariant convergent in Pringsheim’s sense and the following limits hold:

$$\begin{aligned} \lim_{m\rightarrow \infty }\frac{1}{m}\sum_{k=0}^{m}x_{ \sigma ^{k}(s),\sigma ^{j}(t)}=L_{j}, \quad \text{uniformly in } s, \end{aligned}$$

for some $L_{j}\in X$ and each $j\in \mathbb{N}$, and

$$\begin{aligned} \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{j=0}^{n}x_{ \sigma ^{k}(s),\sigma ^{j}(t)}= M_{k}, \quad \text{uniformly in } t, \end{aligned}$$

for some $M_{k}\in X$ and each $k\in \mathbb{N}$. Note that if $x=(x_{kj})$ is $r(\sigma ,\sigma _{2})$-convergent to L, the following limits hold:

$$\begin{aligned} \lim_{m\rightarrow \infty }\lim_{n\rightarrow \infty } \frac{1}{mn} \sum _{k=0}^{m}\sum_{j=0}^{n}x_{\sigma ^{k}(s),\sigma ^{j}(t)}=L, \quad \text{uniformly in } s,t, \end{aligned}$$

and

$$\begin{aligned} \lim_{n\rightarrow \infty }\lim_{m\rightarrow \infty } \frac{1}{mn} \sum _{j=0}^{n}\sum_{k=0}^{m}x_{\sigma ^{k}(s),\sigma ^{j}(t)}=L, \quad \text{uniformly in } s,t. \end{aligned}$$

In this case, we write

$$ r(\sigma ,\sigma _{2})-\lim_{m,n\rightarrow \infty }\sum _{k=0}^{m} \sum_{j=0}^{n}x_{\sigma ^{k}(s),\sigma ^{j}(t)}=L \quad \text{or}\quad x_{kj} \overset{r(\sigma ,\sigma _{2})}{ \longrightarrow }L, \quad \text{uniformly in } s,t. $$

Definition 2.2

A double sequence $x=(x_{kj})$ is said to be regularly strongly invariant convergent ($r[\sigma ,\sigma _{2}]$-convergent) if it is strongly invariant convergent in Pringsheim’s sense and the following limits hold:

$$\begin{aligned} \lim_{m\rightarrow \infty }\frac{1}{m}\sum_{k=0}^{m} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert =0, \quad \text{uniformly in } s, \end{aligned}$$

for some $L_{j}\in X$ and each $j\in \mathbb{N}$, and

$$\begin{aligned} \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{j=0}^{n} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-M_{k} \vert =0, \quad \text{uniformly in } t, \end{aligned}$$

for some $M_{k}\in X$ and each $k\in \mathbb{N}$.

Note that if $x=(x_{kj})$ is $r[\sigma ,\sigma _{2}]$-convergent to L, the following limits hold:

$$\begin{aligned} \lim_{m\rightarrow \infty }\lim_{n\rightarrow \infty } \frac{1}{mn} \sum _{k=0}^{m}\sum_{j=0}^{n} \vert x_{\sigma ^{k}(s), \sigma ^{j}(t)}-L \vert =0, \quad \text{uniformly in } s,t, \end{aligned}$$

and

$$\begin{aligned} \lim_{n\rightarrow \infty }\lim_{m\rightarrow \infty } \frac{1}{mn} \sum _{j=0}^{n}\sum_{k=0}^{m} \vert x_{\sigma ^{k}(s), \sigma ^{j}(t)}-L \vert =0, \quad \text{uniformly in } s,t. \end{aligned}$$

In this case, we write

$$ r[\sigma ,\sigma _{2}]-\lim_{m,n\rightarrow \infty }\sum _{k=0}^{m} \sum_{j=0}^{n} \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L \vert =0 \quad \text{or}\quad x_{kj} \overset{r[ \sigma ,\sigma _{2}]}{\longrightarrow }L, \quad \text{uniformly in } s,t. $$

Definition 2.3

Let $0< p<\infty$. A double sequence $x=(x_{kj})$ is said to be regularly p-strongly invariant convergent ($r[\sigma ,\sigma _{2}]_{p}$-convergent) if it is p-strongly invariant convergent in Pringsheim’s sense and the following limits hold:

$$\begin{aligned} \lim_{m\rightarrow \infty }\frac{1}{m}\sum_{k=0}^{m} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert ^{p}=0, \quad \text{uniformly in } s, \end{aligned}$$

for some $L_{j}\in X$ and each $j\in \mathbb{N}$, and

$$\begin{aligned} \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{j=0}^{n} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-M_{k} \vert ^{p}=0, \quad \text{uniformly in } t, \end{aligned}$$

for some $M_{k}\in X$ and each $k\in \mathbb{N}$.

Note that if $x=(x_{kj})$ is $r[\sigma ,\sigma _{2}]_{p}$-convergent to L, the following limits hold:

$$\begin{aligned} \lim_{m\rightarrow \infty }\lim_{n\rightarrow \infty } \frac{1}{mn} \sum _{k=0}^{m}\sum_{j=0}^{n} \vert x_{\sigma ^{k}(s), \sigma ^{j}(t)}-L \vert ^{p}=0, \quad \text{uniformly in } s,t, \end{aligned}$$

and

$$\begin{aligned} \lim_{n\rightarrow \infty }\lim_{m\rightarrow \infty } \frac{1}{mn} \sum _{j=0}^{n}\sum_{k=0}^{m} \vert x_{\sigma ^{k}(s), \sigma ^{j}(t)}-L \vert ^{p}=0, \quad \text{uniformly in } s,t. \end{aligned}$$

In this case, we write

$$ r[\sigma ,\sigma _{2}]_{p}-\lim_{m,n\rightarrow \infty } \sum_{k=0}^{m} \sum _{j=0}^{n} \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L \vert =0 \quad \text{or}\quad x_{kj} \overset{r[\sigma ,\sigma _{2}]_{p}}{\longrightarrow }L, \quad \text{uniformly in } s,t. $$

Definition 2.4

A double sequence $x=(x_{kj})$ is said to be regularly ideal invariant convergent ($r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent) if it is ideal invariant convergent in Pringsheim’s sense and for every $\varepsilon >0$ the following hold:

$$\begin{aligned} \bigl\{ k\in \mathbb{N}: \vert x_{kj}-L_{j} \vert \geq \varepsilon \bigr\} \in \mathcal{I}_{ \sigma } \end{aligned}$$

for some $L_{j}\in X$ and each $j\in \mathbb{N}$, and

$$\begin{aligned} \bigl\{ j\in \mathbb{N}: \vert x_{kj}-M_{k} \vert \geq \varepsilon \bigr\} \in \mathcal{I}_{ \sigma } \end{aligned}$$

for some $M_{k}\in X$ and each $k\in \mathbb{N}$.

Note that if $x=(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent to L, then we write

$$ r \bigl(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2} \bigr)- \lim x=L \quad \text{or}\quad x_{kj} \overset{r \bigl( \mathcal{I}_{\sigma }, \mathcal{I}^{\sigma }_{2} \bigr)}{ \longrightarrow }L. $$

Theorem 2.1

Suppose that $x=(x_{kj})$ is a bounded double sequence. If $x=(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent, then $x=(x_{kj})$ is $r(\sigma ,\sigma _{2})$-convergent.

Proof

Let $x=(x_{kj})$ be a bounded double sequence and $x=(x_{kj})$ be regularly ideal invariant convergent to L. Then $x=(x_{kj})$ is ideal invariant convergent in Pringsheim’s sense and for every $\varepsilon >0$ the following hold:

$$\begin{aligned} \bigl\{ k\in \mathbb{N}: \vert x_{kj}-L_{j} \vert \geq \varepsilon \bigr\} \in I_{\sigma } \end{aligned}$$

for some $L_{j}\in X$ and each $j\in \mathbb{N}$, and

$$\begin{aligned} \bigl\{ j\in \mathbb{N}: \vert x_{kj}-M_{k} \vert \geq \varepsilon \bigr\} \in I_{\sigma } \end{aligned}$$

for some $M_{k}\in X$ and each $k\in \mathbb{N}$. Since $x=(x_{kj})$ is ideal invariant convergent in Pringsheim’s sense, then by Lemma 1$x=(x_{kj})$ is invariant convergent to L.

Now, let $\varepsilon >0$. We estimate

$$ u(m,s)= \Biggl\vert \frac{1}{m}\sum_{k=0}^{m}x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \Biggr\vert , \quad \text{uniformly in } s, $$

for some $L_{j}\in X$ and each $j\in \mathbb{N}$. Then we have

$$ u(m,s)\leq u^{1}(m,s)+ u^{2}(m,s), $$

where

$$ u^{1}(m,s)=\frac{1}{m}\sum^{m}_{ \substack{k=0 \\ \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert \geq \varepsilon }} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert $$

and

$$ u^{2}(m,s)=\frac{1}{m}\sum^{m}_{ \substack{k=0 \\ \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert < \varepsilon }} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert , \quad \text{uniformly in } s, $$

for some $L_{j}\in X$ and each $j\in \mathbb{N}$. Therefore, we have $u^{2}(m,s)<\varepsilon$ for every $s=1,2,\ldots$ . The boundedness of $(x_{kj})$ implies that there exists $K>0$ such that

$$ \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert \leq K, \quad (k,s=1,2,\ldots ), $$

then this implies that

$$\begin{aligned} u^{1}(m,s)&\leq \frac{K}{m} \bigl\vert \bigl\{ 1\leq k \leq m: \vert x_{\sigma ^{k}(s), \sigma ^{j}(t)}-L_{j} \vert \geq \varepsilon \bigr\} \bigr\vert \\ &\leq K \frac{\max_{s} \vert \{1\leq k \leq m: \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert \geq \varepsilon \} \vert }{m} \\ &=K\frac{S_{m}}{m}, \end{aligned}$$

and so $(x_{kj})$ is σ-convergent to $L_{j}$.

Similarly, we can show that $(x_{kj})$ is σ-convergent to $M_{k}$. Hence, $x=(x_{kj})$ is $r(\sigma ,\sigma _{2})$-convergent. □

Theorem 2.2

Let $0< p<\infty$.

• (i) If $(x_{kj})$ is $r[\sigma ,\sigma _{2}]_{p}$-convergent, then $(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent.

• (ii) If $(x_{kj})\in \ell _{\infty }^{2}$ and $(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent, then $(x_{kj})$ is $r[\sigma ,\sigma _{2}]_{p}$-convergent.

• (iii) If $(x_{kj})\in \ell _{\infty }^{2}$, then $(x_{kj})$ is $r[\sigma ,\sigma _{2}]_{p}$-convergent if and only if $(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$ is convergent.

Proof

$(i)$ Let $x=(x_{kj})$ be $r[\sigma ,\sigma _{2}]_{p}$-convergent. Then it is p-strongly invariant convergent in Pringsheim’s sense and the following limits hold:

$$\begin{aligned} \lim_{m\rightarrow \infty }\frac{1}{m}\sum_{k=0}^{m} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert ^{p}=0, \quad \text{uniformly in } s, \end{aligned}$$

for some $L_{j}\in X$ and each $j\in \mathbb{N}$, and

$$\begin{aligned} \lim_{n\rightarrow \infty }\frac{1}{n}\sum_{j=0}^{n} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-M_{k} \vert ^{p}=0, \quad \text{uniformly in } t, \end{aligned}$$

for some $M_{k}\in X$ and each $k\in \mathbb{N}$. Since $x=(x_{kj})$ is p-strongly invariant convergent in Pringsheim’s sense, then by Lemma 2$x=(x_{kj})$ is $\mathcal{I}^{\sigma }_{2}$-convergent.

Also, for every $\varepsilon >0$, some $L_{j}\in X$, and each $j\in \mathbb{N}$, we can write

$$\begin{aligned} \sum_{k=1}^{m} \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert ^{p} \geq & \sum^{m}_{ \substack{k=1 \\ \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert \geq \varepsilon }} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert ^{p} \\ \geq & {\varepsilon }^{p} \bigl\vert \bigl\{ k\leq m: \vert x_{\sigma ^{k}(s), \sigma ^{j}(t)}-L_{j} \vert \geq \varepsilon \bigr\} \bigr\vert \\ \geq & {\varepsilon }^{p}\max_{s} \bigl\vert \bigl\{ k\leq m: \vert x_{\sigma ^{k}(s), \sigma ^{j}(t)}-L_{j} \vert \geq \varepsilon \bigr\} \bigr\vert \end{aligned}$$

and

$$\begin{aligned} \frac{1}{m}\sum_{k=1}^{m} \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert ^{p} \geq & {\varepsilon }^{p} \frac{\max_{s} \vert \{k\leq m: \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert \geq \varepsilon \} \vert }{m} \\ =& {\varepsilon }^{p}\frac{S_{m}}{m} \end{aligned}$$

for every $s=1,2,\ldots$ . This implies $\lim_{m\rightarrow \infty }\frac{S_{m}}{m}=0$, and so $(x_{kj})$ is $\mathcal{I}_{\sigma }$-convergent to $L_{j}$.

Similarly, we can show that $(x_{kj})$ is $\mathcal{I}_{\sigma }$-convergent to $M_{k}$. Hence, $x=(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent.

$(\mathit{ii})$ Let $(x_{kj})\in \ell _{\infty }^{2}$ and $(x_{kj})$ be $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent. Then $x=(x_{kj})$ is ideal invariant convergent in Pringsheim’s sense and for every $\varepsilon >0$ the following hold:

$$\begin{aligned} \bigl\{ k\in \mathbb{N}: \vert x_{kj}-L_{j} \vert \geq \varepsilon \bigr\} \in \mathcal{I}_{ \sigma } \end{aligned}$$

for some $L_{j}\in X$ and each $j\in \mathbb{N}$, and

$$\begin{aligned} \bigl\{ j\in \mathbb{N}: \vert x_{kj}-M_{k} \vert \geq \varepsilon \bigr\} \in \mathcal{I}_{ \sigma } \end{aligned}$$

for some $M_{k}\in X$ and each $k\in \mathbb{N}$. Since $x=(x_{kj})$ is ideal invariant convergent in Pringsheim’s sense, then by Lemma 2, $x=(x_{kj})$ is p-strongly $\sigma _{2}$-convergent. Let $0< p<\infty$ and $\varepsilon >0$. Since $(x_{kj})$ is bounded, $(x_{kj})$ implies that there exists $K>0$ such that

$$ \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert \leq K \quad (j\in \mathbb{N}) $$

for all $k,s\in \mathbb{N}$. Then, for every $s=1,2,\ldots$ , we have

$$\begin{aligned} \frac{1}{m}\sum_{k=1}^{m} \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert ^{p} =& \frac{1}{m} \sum^{m}_{ \substack{k=1\\ \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert \geq \varepsilon }} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert ^{p} \\ & {} + \frac{1}{m}\sum^{m}_{ \substack{k=1 \\ \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert < \varepsilon }} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert ^{p} \\ \leq & K \frac{\max_{s} \vert \{ k\leq m: \vert x_{\sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert \geq \varepsilon \} \vert }{m} +\varepsilon ^{p} \\ \leq & K \frac{S_{m}}{m}+\varepsilon ^{p}. \end{aligned}$$

Hence, we obtain

$$ \lim_{m\rightarrow \infty }\frac{1}{m}\sum_{k=1}^{m} \vert x_{ \sigma ^{k}(s),\sigma ^{j}(t)}-L_{j} \vert ^{p}=0 $$

uniformly in s, and so $x=(x_{kj})$ is p-strongly σ-convergent to $L_{j}$.

Similarly, we show that $(x_{kj})$ is p-strongly σ-convergent to $M_{k}$. Hence, $x=(x_{kj})$ is $r[\sigma ,\sigma _{2}]_{p}$-convergent.

$(\mathit{iii})$ This is an immediate consequence of (i) and (ii). □

Definition 2.5

A double sequence $x=(x_{kj})$ is said to be regularly $(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-convergent ($r(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-convergent) if and only if there exist the sets $M \in \mathcal{F}(\mathcal{I}^{\sigma }_{2})$, $M_{1} \in \mathcal{F}(\mathcal{I}_{\sigma })$ and $M_{2}\in \mathcal{F}(\mathcal{I}_{\sigma })$ (i.e., $\mathbb{N}\times \mathbb{N} \setminus M \in \mathcal{I}^{\sigma }_{2}$, $\mathbb{N} \setminus M_{1} \in \mathcal{I}_{\sigma }$, and $\mathbb{N} \setminus M_{2} \in \mathcal{I}_{\sigma }$) such that the following limits hold:

$$\begin{aligned} \mathop{\lim_{k,j\rightarrow \infty }}_{(k,j)\in M}x_{kj}, \quad\quad \mathop{ \lim_{k\rightarrow \infty }}_{k\in M_{1}}x_{kj} \quad (j\in \mathbb{N}), \quad \text{and}\quad \mathop{\lim_{j\rightarrow \infty }}_{j\in M_{2}}x_{kj} \quad (k\in \mathbb{N}). \end{aligned}$$

Note that if $x=(x_{kj})$ is $r(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-convergent to L, then we write

$$ r \bigl(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2} \bigr)-\lim x=L \quad \text{or}\quad x_{kj} \overset{r \bigl( \mathcal{I}_{\sigma }^{*}, \mathcal{I}^{\sigma *}_{2} \bigr)}{\longrightarrow }L. $$

Theorem 2.3

If a double sequence $x=(x_{kj})$ is $r(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-convergent, then $(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent.

Proof

Let $(x_{kj})$ be $r(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-convergent. Then $(x_{kj})$ is $\mathcal{I}^{\sigma *}_{2}$-convergent, and so by Lemma 3$(x_{kj})$ is $\mathcal{I}^{\sigma }_{2}$ convergent. Also, there exist the sets $M_{1},M_{2} \in \mathcal{F}(\mathcal{I}_{\sigma })$ such that

$$ (\forall \varepsilon >0) \ (\exists k_{0} \in \mathbb{N})\ (\forall k \geq k_{0})\ (k \in M_{1})\quad \vert x_{kj}-L_{j} \vert < \varepsilon $$

for some $L_{j}$ and each $j \in \mathbb{N}$, and

$$ (\forall \varepsilon >0)\ (\exists j_{0} \in \mathbb{N})\ (\forall j \geq j_{0})\ (j \in M_{2}) \quad \vert x_{kj}-M_{k} \vert < \varepsilon $$

for some $M_{k}$ and each $k \in \mathbb{N}$. Hence, we have

$$\begin{aligned} A(\varepsilon ) =& \bigl\{ k\in \mathbb{N}: \vert x_{kj}-L_{j} \vert \geq \varepsilon \bigr\} \subset H_{1} \cup \bigl\{ 1,2, \ldots,(k_{0}-1) \bigr\} , \quad (j \in \mathbb{N}), \\ B(\varepsilon ) =& \bigl\{ j\in \mathbb{N}: \vert x_{kj}-M_{k} \vert \geq \varepsilon \bigr\} \subset H_{2} \cup \bigl\{ 1,2, \ldots,(j_{0}-1) \bigr\} ,\quad (k \in \mathbb{N}) \end{aligned}$$

for $H_{1}, H_{2} \in \mathcal{I}_{\sigma }$. Since $\mathcal{I}_{\sigma }$ is an admissible ideal, we get

$$ H_{1} \cup \bigl\{ 1,2,\ldots,(k_{0}-1) \bigr\} \in \mathcal{I}_{\sigma } \quad \text{and} \quad H_{2} \cup \bigl\{ 1,2, \ldots,(j_{0}-1) \bigr\} \in \mathcal{I}_{\sigma }, $$

and therefore $A(\varepsilon )\in \mathcal{I}_{\sigma }$ and $B(\varepsilon ) \in \mathcal{I}_{\sigma }$. This shows that the double sequence $(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent. □

Theorem 2.4

Let $\mathcal{I}_{\sigma }$ have the property $(\mathit{AP})$ and $\mathcal{I}^{\sigma }_{2}$ have the property $(\mathit{AP}2)$. If a double sequence $(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2} )$-convergent, then $(x_{kj})$ is $r(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-convergent.

Proof

Let a double sequence $(x_{kj})$ be $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2} )$-convergent. Then $(x_{kj})$ is $\mathcal{I}^{\sigma }_{2}$-convergent, and so by Lemma 4$(x_{kj})$ is $\mathcal{I}^{\sigma *}_{2}$-convergent. Also, for each $\varepsilon > 0$, we have

$$\begin{aligned} A(\varepsilon )= \bigl\{ k\in \mathbb{N}: \vert x_{kj}-L_{j} \vert \geq \varepsilon \bigr\} \in \mathcal{I}_{\sigma } \end{aligned}$$

for some $L_{j}$ and each $j \in \mathbb{N}$, and

$$\begin{aligned} B(\varepsilon )= \bigl\{ j\in \mathbb{N}: \vert x_{kj}-M_{k} \vert \geq \varepsilon \bigr\} \in \mathcal{I}_{\sigma } \end{aligned}$$

for some $M_{k}$ and each $k \in \mathbb{N}$.

Now put

$$\begin{aligned} A_{1} =& \bigl\{ k\in \mathbb{N}: \vert x_{kj}-L_{j} \vert \geq 1 \bigr\} , \\ A_{t} =& \biggl\{ k\in \mathbb{N}:\frac{1}{t}\leq \vert x_{kj}-L_{j} \vert < \frac{1}{t-1} \biggr\} \end{aligned}$$

for $t\geq 2$, some $L_{j}$, and each $j \in \mathbb{N}$. It is clear that $A_{m}\cap A_{n}=\emptyset$ for $m\neq n$ and $A_{m}\in \mathcal{I}_{\sigma }$ for each $m\in \mathbb{N}$. By the property (AP) there is a countable family of sets $\{B_{1},B_{2},\ldots \}$ in $\mathcal{I}_{\sigma }$ such that $A_{n}\bigtriangleup B_{n}$ is a finite set for each $n \in \mathbb{N}$ and $B=\bigcup_{n=1}^{\infty }B_{n} \in \mathcal{I}_{\sigma }$.

We prove that

$$\begin{aligned} \mathop{\lim_{k\rightarrow \infty }}_{k\in M}x_{kj} =L_{j}, \quad \text{some } L_{j} \text{ and each } j \in \mathbb{N}, \end{aligned}$$

for $M=\mathbb{N}\backslash B \in \mathcal{F}(\mathcal{I}_{\sigma })$. Let $\delta >0$ be given. Choose $t\in \mathbb{N}$ such that $1/t<\delta$. Then we have

$$ \bigl\{ k\in \mathbb{N}: \vert x_{kj}-L_{j} \vert \geq \delta \bigr\} \subset \bigcup_{n=1}^{t}x_{n} \quad \text{for some } L_{j} \text{ and each } j \in \mathbb{N}. $$

Since $A_{n}\bigtriangleup B_{n}$ is a finite set for $n\in \{1,2,\ldots,t\}$, there exists $k_{0} \in \mathbb{N}$ such that

$$\begin{aligned} \Biggl(\bigcup_{n=1}^{t}B_{n} \Biggr) \cap \{k:k\geq k_{0} \} = \Biggl( \bigcup _{n=1}^{t}A_{n} \Biggr) \cap \{k:k\geq k_{0}\}. \end{aligned}$$

If $k\geq k_{0}$ and $k\notin B$, then

$$ k\notin \bigcup_{n=1}^{t}B_{n} \quad \text{and so}\quad k\notin \bigcup_{n=1}^{t}A_{n}. $$

Thus, we have $\vert x_{kj}-L_{j} \vert < \frac{1}{t}< \delta$ for some $L_{j}$ and each $j \in \mathbb{N}$. This implies that

$$ \mathop{\lim_{k\rightarrow \infty }}x_{kj} =L_{j} $$

for $k\in M$. Hence, we have

$$ \mathcal{I}_{\sigma }^{*}-\lim_{k\to \infty }x_{kj}=L_{j} $$

for some $L_{j}$ and each $j \in \mathbb{N}$.

Similarly, for the set

$$ B(\varepsilon )= \bigl\{ j\in \mathbb{N}: \vert x_{kj}-M_{k} \vert \geq \varepsilon \bigr\} \in \mathcal{I}_{\sigma }, $$

we have

$$ \mathcal{I}_{\sigma }^{*}-\lim_{j\to \infty }x_{kj}=M_{k} $$

for some $M_{k}$ and each $k \in \mathbb{N}$. Hence, a double sequence $(x_{kj})$ is $r(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-convergent. □

Definition 2.6

A double sequence $(x_{kj})$ is said to be regularly $(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2} )$-Cauchy double sequence ($r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-Cauchy double sequence) if it is $\mathcal{I}_{2}^{\sigma }$-Cauchy in Pringsheim’s sense and for every $\varepsilon >0$ there exist numbers $m_{j}=m_{j}(\varepsilon )$, $n_{k}=n_{k}(\varepsilon )\in \mathbb{N}$ such that

$$\begin{aligned}& A_{1}(\varepsilon )= \bigl\{ k\in \mathbb{N}: \vert x_{kj}-x_{m_{j}j} \vert \geq \varepsilon \bigr\} \in \mathcal{I}_{2}^{\sigma } \quad (j\in \mathbb{N}), \\& A_{2}(\varepsilon )= \bigl\{ j\in \mathbb{N}: \vert x_{kj}-x_{kn_{k}} \vert \geq \varepsilon \bigr\} \in \mathcal{I}_{2}^{\sigma }\quad (k\in \mathbb{N}). \end{aligned}$$

Theorem 2.5

If a double sequence $(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent, then $(x_{kj})$ is an $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-Cauchy double sequence.

Proof

Let $(x_{kj})$ be $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-convergent. Then $(x_{kj})$ is $\mathcal{I}^{\sigma }_{2}$-convergent, and by Lemma 5, it is $\mathcal{I}^{\sigma }_{2}$-Cauchy double sequence. Also, for every $\varepsilon >0$, we have

$$\begin{aligned} A_{1} \biggl(\frac{\varepsilon }{2} \biggr)= \biggl\{ k\in \mathbb{N}: \vert x_{kj}-L_{j} \vert \geq \frac{\varepsilon }{2} \biggr\} \in \mathcal{I}_{\sigma } \end{aligned}$$

for some $L_{j}$ and each $j \in \mathbb{N}$, and

$$\begin{aligned} A_{2} \biggl(\frac{\varepsilon }{2} \biggr)= \biggl\{ j\in \mathbb{N}: \vert x_{kj}-M_{k} \vert \geq \frac{\varepsilon }{2} \biggr\} \in \mathcal{I}_{\sigma } \end{aligned}$$

for some $M_{k}$ and each $k \in \mathbb{N}$. Since $\mathcal{I}_{\sigma }$ is an admissible ideal, the sets

$$ A_{1}^{c} \biggl(\frac{\varepsilon }{2} \biggr)= \biggl\{ k\in \mathbb{N}: \vert x_{kj}-L_{j} \vert < \frac{\varepsilon }{2} \biggr\} $$

for some $L_{j}$ and each $j \in \mathbb{N}$, and

$$ A_{2}^{c} \biggl(\frac{\varepsilon }{2} \biggr)= \biggl\{ j\in \mathbb{N}: \vert x_{kj}-M_{k} \vert < \frac{\varepsilon }{2} \biggr\} $$

for some $M_{k}$ and each $k \in \mathbb{N}$ are nonempty and belong to $\mathcal{F}(\mathcal{I}_{\sigma })$. For $m_{j} \in A_{1}^{c}(\frac{\varepsilon }{2})$, ($j \in \mathbb{N}$ and $m_{j} > 0$) we have

$$\begin{aligned} \vert x_{m_{j}j}-L_{j} \vert < \frac{\varepsilon }{2} \end{aligned}$$

for some $L_{j}$ and each $j \in \mathbb{N}$. Now, for each $\varepsilon >0$, we define the set

$$ B_{1}(\varepsilon ) = \bigl\{ k\in \mathbb{N}: \vert x_{kj}-x_{m_{j}j} \vert \geq \varepsilon \bigr\} ,\quad (j\in \mathbb{N}), $$

where $m_{j}=m_{j}(\varepsilon )\in \mathbb{N}$. We must prove $B_{1}(\varepsilon )\subset A_{1}(\frac{\varepsilon }{2})$. Let $k\in B_{1}(\varepsilon )$. Then, for $m_{j} \in A_{1}^{c}(\frac{\varepsilon }{2})$, ($j \in \mathbb{N}$ and $m_{j} > 0$) we have

$$\begin{aligned} \varepsilon \leq \vert x_{kj}-x_{m_{j}j} \vert \leq & \vert x_{kj}-L_{j} \vert + \vert x_{m_{j}j}-L_{j} \vert \\ < & \vert x_{kj}-L_{j} \vert +\frac{\varepsilon }{2} \end{aligned}$$

for some $L_{j}$ and each $j \in \mathbb{N}$. This shows that $\frac{\varepsilon }{2}< \vert x_{kj}-L_{j} \vert$, and so $k \in A_{1}(\frac{\varepsilon }{2})$. Hence, we have $B_{1}(\varepsilon ) \subset A_{1}(\frac{\varepsilon }{2})$.

Similarly, for each $\varepsilon >0$ and for $n_{k} \in A_{2}^{c}(\frac{\varepsilon }{2})$ ($k \in \mathbb{N}$ and $n_{k} > 0$), we have

$$\begin{aligned} \vert x_{kn_{k}}-M_{k} \vert < \frac{\varepsilon }{2} \end{aligned}$$

for some $M_{k}$ and each $k \in \mathbb{N}$. Therefore, it can be seen that

$$ B_{2}(\varepsilon ) \subset A_{2} \biggl( \frac{\varepsilon }{2} \biggr), $$

where

$$ B_{2}(\varepsilon ) = \bigl\{ j\in \mathbb{N}: \vert x_{kj}-x_{kn_{k}} \vert \geq \varepsilon \bigr\} , $$

where $n_{k}=n_{k}(\varepsilon )\in \mathbb{N}$ and each $k \in \mathbb{N}$.

Hence, we have $B_{1}(\varepsilon )\in \mathcal{I}_{\sigma }$ and $B_{2}(\varepsilon )\in \mathcal{I}_{\sigma }$. This shows that $\{x_{kj}\}$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-Cauchy double sequence. □

Definition 2.7

A double sequence $(x_{kj})$ is regularly $(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-Cauchy double sequence ($r(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-Cauchy double sequence) if there exist the sets $M\in \mathcal{F}(\mathcal{I}^{\sigma }_{2})$, $M_{1}\in \mathcal{F}(\mathcal{I}^{\sigma })$ and $M_{2}\in \mathcal{F}(\mathcal{I}^{\sigma })$ (that is, $\mathbb{N}\times \mathbb{N} \backslash M =H \in \mathcal{I}^{\sigma }_{2}$, $\mathbb{N}\backslash M_{1}\in \mathcal{I}_{\sigma }$, and $\mathbb{N}\backslash M_{2}\in \mathcal{I}_{\sigma }$) and for every $\varepsilon >0$, there exist $N=N(\varepsilon )$, $s=s(\varepsilon )$, $t=t(\varepsilon )$, $m_{j}=m_{j}(\varepsilon )$, $n_{k}=n_{k}(\varepsilon )\in \mathbb{N}$ such that whenever $k,j,s,t,m_{j},n_{k}\geq N$, we have

$$\begin{aligned}& \vert x_{kj}-x_{st} \vert < \varepsilon \quad \bigl( \text{for } (k,j),(s,t)\in M, k,j,s,t \geq N \bigr), \\& \vert x_{kj}-x_{m_{j}j} \vert < \varepsilon \quad (\text{for each } k\in M_{1} \text{ and each } j\in \mathbb{N}), \\& \vert x_{kj}-x_{kn_{k}} \vert < \varepsilon \quad (\text{for each } j\in M_{2} \text{ and each } k\in \mathbb{N}). \end{aligned}$$

Theorem 2.6

If a double sequence $(x_{kj})$ is $r(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-Cauchy double sequence, then $(x_{kj})$ is $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-Cauchy double sequence.

Proof

Since a double sequence $(x_{kj})$ is $r(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-Cauchy, then $(x_{kj})$ is $\mathcal{I}^{\sigma *}_{2}$-Cauchy implies $\mathcal{I}^{\sigma }_{2}$-Cauchy by Lemma 6. Also, since $(x_{kj})$ is $r(\mathcal{I}_{\sigma }^{*},\mathcal{I}^{\sigma *}_{2})$-Cauchy, there exist the sets $M_{1}\in \mathcal{F}(\mathcal{I}^{\sigma })$ and $M_{2}\in \mathcal{F}(\mathcal{I}^{\sigma })$ (that is, $\mathbb{N}\backslash M_{1}\in \mathcal{I}_{\sigma }$ and $\mathbb{N}\backslash M_{2}\in \mathcal{I}_{\sigma }$), and for every $\varepsilon >0$, there exist $N=N(\varepsilon )$, $m_{j}=m_{j}(\varepsilon )$, $n_{k}=n_{k}(\varepsilon )\in \mathbb{N}$ such that we have

$$\begin{aligned}& \vert x_{kj}-x_{m_{j}j} \vert < \varepsilon \quad (\text{for each } k\in M_{1} \text{ and each } j\in \mathbb{N}), \\& \vert x_{kj}-x_{kn_{k}} \vert < \varepsilon \quad (\text{for each } j\in M_{1} \text{ and each } k\in \mathbb{N}), \end{aligned}$$

whenever $k,j,m_{j},n_{k}\geq N$. Therefore, $H_{1}=\mathbb{N}\backslash M_{1}\in \mathcal{I}_{\sigma }$ and $H_{2}=\mathbb{N}\backslash M_{2}\in \mathcal{I}_{\sigma }$ we have

$$ A_{1}(\varepsilon )= \bigl\{ k\in \mathbb{N}: \vert x_{kj}-x_{m_{j}j} \vert \geq \varepsilon \bigr\} \subset H_{1}\cup \bigl\{ 1,2,\ldots,(N-1) \bigr\} ,\quad (j\in \mathbb{N}) $$

for each $k\in M_{1}$ and

$$ A_{2}(\varepsilon )= \bigl\{ j\in \mathbb{N}: \vert x_{kj}-x_{kn_{k}} \vert \geq \varepsilon \bigr\} \subset H_{2}\cup \bigl\{ 1,2,\ldots,(N-1) \bigr\} ,\quad (k\in \mathbb{N}) $$

for each $j\in M_{2}$. Since $\mathcal{I}_{\sigma }$ is an admissible ideal, $H_{1}\cup \{1,2,3,\ldots,(N-1)\}\in \mathcal{I}_{\sigma }$ and $H_{2}\cup \{1,2,3,\ldots,(N-1)\}\in \mathcal{I}_{\sigma }$. Hence, we have $A_{1}(\varepsilon )\in \mathcal{I}_{\sigma }$ and $A_{2}(\varepsilon )\in \mathcal{I}_{\sigma }$, and so $(x_{kj})$ is an $r(\mathcal{I}_{\sigma },\mathcal{I}^{\sigma }_{2})$-Cauchy double sequence. □

3 Conclusions

We investigated the concepts of regularly invariant convergence types and regularly ideal invariant convergence and Cauchy sequence types. These concepts can also be studied for the lacunary sequence in the future.