Asymptotically lacunary I 2 -invariant equivalence

Abstract

In this paper, the concept of asymptotically lacunary $I_{2}$ -invariant equivalence, the concepts of asymptotically lacunary σ₂-equivalence and the concept of asymptotically lacunary invariant S₂-equivalence for double sequences are defined. In addition, some relationships were obtained among these new equivalence concepts.

Keywords

Asymptotically equivalence double sequence ideal convergence invariant convergence double lacunary sequence statistical convergence

1 Introduction and background

Many authors have studied on the concepts of invariant mean and invariant convergence (see, [10 , 34]).

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

φ (x) ≥0, when the sequence x = (x_n) has x_n ≥ 0 for all n,

φ (e) =1, where e = (1, 1, 1,. . .) and

φ (x_σ(n)) = φ (x_n) forall x ∈ ℓ _∞.

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

In the case σ is translation mappings σ (n) = n + 1, the σ-mean is often called a Banach limit.

Throughout the paper $ℕ$ denotes the set of natural numbers.

The idea of $I$ -convergence was introduced by Kostyrko et al. [6] which is based on the structure of the ideal $I$ of subset of the set $ℕ$ . For more detail, see [7 , 29].

A family of sets $I \subseteq 2^{N}$ is called an ideal if and only if (i) $\emptyset \in I$ , (ii) For each $A, B \in I$ we have $A \cup B \in I$ , (iii) For each $A \in I$ and each B ⊆ A we have $B \in I .$

An ideal is called non-trivial if $ℕ \notin I$ and non-trivial ideal is called admissible if ${n} \in I$ for each $n \in ℕ$ .

Several convergence concepts for double sequences and some properties of these concepts which are noted following can be seen in [1 , 33].

A double sequence x = (x_kj) is said to be bounded if there exists an M > 0 such that |x_kj| < M for all k and j, i.e., if $sup_{k, j} | x_{kj} | < \infty$ .

The set of all bounded double sequences will be denoted by $ℓ_{\infty}^{2}$ .

Mursaleen and Edely [12] introduced the concept of statistically convergence for double sequences.

A double sequence x = (x_kj) is said to be statistically convergent to L if for every ɛ > 0 $\begin{matrix} lim_{m, n \to \infty} \frac{1}{mn} | {(k, j), k \leq m and j \\ \leq n : | x_{kj} - L | \geq ɛ} | \\ = 0 . \end{matrix}$

A double sequence θ₂ = {(k_r, j_u)} is called double lacunary sequence if there exist two increasing sequence of integers such that $\begin{matrix} k_{0} & = & 0, h_{r} = k_{r} - k_{r - 1} \to \infty as r \to \infty and \\ j_{0} & = & 0, {\bar{h}}_{u} = j_{u} - j_{u - 1} \to \infty as u \to \infty . \end{matrix}$

We use the following notations in the sequel: $\begin{matrix} k_{ru} & = & k_{r} j_{u}, h_{ru} = h_{r} {\bar{h}}_{u}, \\ I_{ru} & = & {(k, j) : k_{r - 1} < k \leq k_{r} \\ and j_{u - 1} < j \leq j_{u}} . \end{matrix}$

Throughout the paper we take θ₂ = {(k_r, j_u)} as a double lacunary sequence.

Using the double lacunary sequence concept, the concepts of lacunary statistically convergence and lacunary σ-statistically convergence for double sequences were defined by Patterson and Savaş [19] and Savaş and Patterson [27], respectively.

A double sequence x = (x_kj) is said to be lacunary statistically convergent to L if for every ɛ > 0 $\begin{matrix} lim_{r, u \to \infty} \frac{1}{h_{ru}} | {(k, j) \in I_{ru} : | x_{kj} - L | \geq ɛ} | = 0 . \end{matrix}$ It is denoted by $x_{k j} \to L ({S^{″}}_{θ})$ .

A nontrivial ideal $I_{2}$ of $ℕ \times ℕ$ is called strongly admissible ideal if ${i} \times ℕ$ and $ℕ \times {i}$ belong to $I_{2}$ for each i ∈ N.

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

Let (X, ρ) be a metric space and $I_{2}$ be a strongly admissible ideal in $ℕ \times ℕ$ . A double sequence x = (x_mn) in X is said to be $I_{2}$ -convergent to L ∈ X, if for any ɛ > 0 $\begin{matrix} A (ɛ) & = & {(m, n) \in ℕ \times ℕ : ρ (x_{mn}, L) \geq ɛ} \\ \in I_{2} . \end{matrix}$ It is denoted by $I_{2} - lim_{m, n \to \infty} x_{mn} = L .$

Recently, the definitions of some invariant convergence for double sequences were presented in a study by Ulusu et al. [33] as below:

A double sequence x = (x_kj) is said to be lacunary invariant convergent to L if $\begin{matrix} lim_{r, u \to \infty} \frac{1}{h_{ru}} \sum_{k, j \in I_{ru}} x_{σ^{k} (m), σ^{j} (n)} = L, \end{matrix}$ uniformly in m, n = 1, 2,. . . and it is denoted by $x_{kj} \to L (V_{2}^{σ θ})$ .

A double sequence x = (x_kj) is said to be strongly lacunary invariant convergent to L if $\begin{matrix} lim_{r, u \to \infty} \frac{1}{h_{ru}} \sum_{k, j \in I_{ru}} | x_{σ^{k} (m), σ^{j} (n)} - L | = 0, \end{matrix}$ uniformly in m, n and it is denoted by $x_{kj} \to L ([V_{2}^{σ θ}])$ .

A double sequence x = (x_kj) is said to be strongly p-lacunary invariant convergent (0 < p < ∞) to L if $\begin{matrix} lim_{r, u \to \infty} \frac{1}{h_{ru}} \sum_{k, j \in I_{ru}} | x_{σ^{k} (m), σ^{j} (n)} - L |^{p} = 0, \end{matrix}$ uniformly in m, n and it is denoted by $x_{kj} \to L ([V_{2}^{σ θ}]_{p})$ .

Let θ₂ = {(k_r, j_u)} be a double lacunary sequence, $A \subseteq ℕ \times ℕ$ and $\begin{matrix} s_{ru} & = & min_{m, n} | A \cap {(σ^{k} (m), σ^{j} (n)) : (k, j) \in I_{ru}} |, \\ S_{ru} & = & max_{m, n} | A \cap {(σ^{k} (m), σ^{j} (n)) : (k, j) \in I_{ru}} | . \end{matrix}$

If the following limits exist $\begin{matrix} \underline{V_{2}^{θ}} (A) = lim_{r, u \to \infty} \frac{s_{ru}}{h_{ru}} and \bar{V_{2}^{θ}} (A) = lim_{r, u \to \infty} \frac{S_{ru}}{h_{ru}}, \end{matrix}$ then they are called a lower lacunary σ-uniform density and an upper lacunary σ-uniform density of the set A, respectively.

If $\underline{V_{2}^{θ}} (A) = \bar{V_{2}^{θ}} (A)$ , then $V_{2}^{θ} (A) = \underline{V_{2}^{θ}} (A) = \bar{V_{2}^{θ}} (A)$ is called the lacunary σ-uniform density of A.

Denoted by $I_{2}^{σ θ}$ the class of all $A \subseteq ℕ \times ℕ$ with $V_{2}^{θ} (A) = 0$ .

A double sequence x = (x_kj) is said to be lacunary $I_{2}$ -invariant convergent or $I_{2}^{σ θ}$ -convergent to the L if for every ɛ > 0, the set $\begin{matrix} A_{ɛ} = {(k, j) \in I_{ru} : | x_{kj} - L | \geq ɛ} \in I_{2}^{σ θ}, \end{matrix}$ i.e., $V_{2}^{θ} (A_{ɛ}) = 0$ . It is denoted by $I_{2}^{σ θ} - lim x_{kj} = L$ or $x_{kj} \to L (I_{2}^{σ θ})$ .

Marouf [9] presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. Then, the concept of asymptotically equivalence has been developed by many other researchers (see, [3–5 , 32]).

Two nonnegative sequences x = (x_k) and y = (y_k) are said to be asymptotically equivalent if $\begin{matrix} lim_{k} \frac{x_{k}}{y_{k}} = 1 . \end{matrix}$ It is denoted by x ∼ y.

Hazarika and Kumar [4] presented some asymptotically equivalence definitions for double sequences as follows:

Two nonnegative double sequences x = (x_kl) and x = (y_kl) are said to be P-asymptotically equivalent if $\begin{matrix} P - lim_{k, l} \frac{x_{kl}}{y_{kl}} = 1, \end{matrix}$ denoted by x ∼ ^Py .

Two nonnegative double sequences x = (x_kl) and x = (y_kl) are said to be asymptotically $I_{2}$ -equivalent of multiple L provided that for every ɛ > 0 $\begin{matrix} {(k, l) \in ℕ \times ℕ : | \frac{x_{kl}}{y_{kl}} - L | \geq ɛ} \in I_{2}, \end{matrix}$ denoted by $x \sim^{I^{Litsc}} y$ and simply asymptotically $I_{2}$ -equivalent if L = 1.

2 Asymptotically lacunary $I_{2}$ -invariant equivalence

In this section, the concept of asymptotically lacunary $I_{2}$ -invariant equivalence, the concepts of asymptotically lacunary σ₂-equivalence and the concept of asymptotically lacunary invariant S₂-equivalence for double sequences are defined. In addition, some relationships were obtained among these new equivalence concepts.

2.1 Definition

Two nonnegative double sequence x = (x_kj) and y = (y_kj) are said to be asymptotically lacunary σ₂-equivalent of multiple L if $\begin{matrix} lim_{r, u \to \infty} \frac{1}{h_{ru}} \sum_{k, j \in I_{ru}} \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} = L, \end{matrix}$ uniformly in m, n. In this case, we write $x \overset{N_{2 (L)}^{σ θ}}{\sim} y$ and simply asymptotically lacunary σ₂-equivalent if L = 1.

2.2 Definition

Two nonnegative double sequences x = (x_kj) and y = (y_kj) are said to be asymptotically lacunary $I_{2}$ -invariant equivalent of multiple L if for every ɛ > 0 $\begin{matrix} A_{ɛ}^{\sim} : = {(k, j) \in I_{ru} : | \frac{x_{kj}}{y_{kj}} - L | \geq ɛ} \in I_{2}^{σ θ}, \end{matrix}$ i.e., $V_{2}^{θ} (A_{ɛ}^{\sim}) = 0$ . In this case, we write $x \overset{I_{2 (L)}^{σ θ}}{\sim} y$ and simply asymptotically lacunary $I_{2}$ -invariant equivalent if L = 1.

The set of all asymptotically lacunary $I_{2}$ -invariant equivalent of multiple L sequences will be denoted by $I_{2 (L)}^{σ θ}$ .

2.3 Theorem

Suppose that x = (x_kj) and y = (y_kj) are bounded double sequences. Then, $\begin{matrix} x \overset{I_{2 (L)}^{σ θ}}{\sim} y \Rightarrow x \overset{N_{2 (L)}^{σ θ}}{\sim} y . \end{matrix}$

Proof: Let $m, n \in ℕ$ be arbitrary and ɛ > 0. Now, we calculate $\begin{matrix} t (θ_{2}, m, n) : = | \frac{1}{h_{ru}} \sum_{k, j \in I_{ru}} \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | . \end{matrix}$

We have $\begin{matrix} t (θ_{2}, m, n) \leq t_{1} (θ_{2}, m, n) + t_{2} (θ_{2}, m, n), \end{matrix}$ where $\begin{matrix} t_{1} (θ_{2}, m, n) \\ : = \frac{1}{h_{ru}} \sum_{\begin{matrix} k, j \in I_{ru} \\ | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \geq ɛ \end{matrix}} | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \end{matrix}$ and

$\begin{matrix} t_{2} (θ_{2}, m, n) \\ : = \frac{1}{h_{ru}} \sum_{\begin{matrix} k, j \in I_{ru} \\ | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | < ɛ \end{matrix}} | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | . \end{matrix}$ For every m, n = 1, 2,. . ., we get t₂ (θ₂, m, n) < ɛ. The boundedness of x and y implies that there exists an M > 0 such that $\begin{matrix} | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \leq M, \end{matrix}$ for k, j ∈ I_ru and for every m, n. Then, this implies that $\begin{matrix} \begin{matrix} t_{1} (θ_{2}, m, n) \\ \leq \frac{M}{h_{ru}} | {(k, j) \in I_{ru} : | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \geq ɛ} | \\ \leq M \frac{{max}_{m, n} | {(k, j) \in I_{ru} : | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \geq ɛ} |}{h_{ru}} \\ = M \frac{S_{ru}}{h_{ru}}, \end{matrix} \end{matrix}$ hence $x \overset{N_{2 (L)}^{σ θ}}{\sim} y$ .

The converse of Theorem 1 does not hold. For example, x = (x_kj) and y = (y_kj) are the sequences defined by following;

$\begin{matrix} \begin{matrix} x_{kj} & : = {\begin{matrix} 2 & , & \begin{matrix} if \begin{matrix} k_{r - 1} < k < k_{r - 1} + [\sqrt{h_{r}}], \\ j_{r - 1} < j < j_{r - 1} + [\sqrt{{\bar{h}}_{u}}] and \\ k + j is an even integer . \end{matrix} \end{matrix} \\ 0 & , & if \begin{matrix} k_{r - 1} < k < k_{r - 1} + [\sqrt{h_{r}}], \\ j_{r - 1} < j < j_{r - 1} + [\sqrt{{\bar{h}}_{u}}] and \\ k + j is an odd integer . \end{matrix} \end{matrix} \\ y_{kj} & : = 1 . \end{matrix} \end{matrix}$

When σ (m) = m + 1 and σ (n) = n + 1, this sequences are asymptotically lacunary σ₂-equivalent but they are not asymptotically lacunary $I_{2}$ -invariant equivalent.

2.4 Definition

Two nonnegative double sequence x = (x_kj) and y = (y_kj) are said to be strongly asymptotically lacunary σ₂-equivalent of multiple L if $\begin{matrix} lim_{r, u \to \infty} \frac{1}{h_{ru}} \sum_{k, j \in I_{ru}} | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | = 0, \end{matrix}$ uniformly in m, n. In this case, we write $x \overset{[N_{2 (L)}^{σ θ}]}{\sim} y$ and simply strongly asymptotically lacunary σ₂-equivalent if L = 1.

2.5 Definition

Let 0< p < ∞. Two nonnegative double sequence x = (x_kj) and y = (y_kj) are said to be strongly p-asymptotically lacunary σ₂-equivalent of multiple L if $\begin{matrix} lim_{r, u \to \infty} \frac{1}{h_{ru}} \sum_{k, j \in I_{ru}} {| \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L |}^{p} = 0, \end{matrix}$ uniformly in m, n. In this case, we write $x \overset{[N_{2 (L)}^{σ θ}]_{p}}{\sim} y$ and simply strongly p-asymptotically lacunary σ₂-equivalent if L = 1.

The set of all strongly p-asymptotically lacunary σ₂-equivalent of multiple L sequences will be denoted by $[N_{2 (L)}^{σ θ}]_{p}$ .

2.6 Theorem

Assume that x = (x_kj) and y = (y_kj) are double sequences. Then, $\begin{matrix} x \overset{[N_{2 (L)}^{σ θ}]_{p}}{\sim} y \Rightarrow x \overset{I_{2 (L)}^{σ θ}}{\sim} y . \end{matrix}$

Proof: Let $x \overset{[N_{2 (L)}^{σ θ}]_{p}}{\sim} y$ and given ɛ > 0. Then, for every $m, n \in ℕ$ we have $\begin{matrix} \sum_{k, j \in I_{ru}} {| \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L |}^{p} \\ \geq \sum_{\begin{matrix} k, j \in I_{ru} \\ | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \geq ɛ \end{matrix}} {| \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L |}^{p} \\ \geq ɛ^{p} \cdot | {(k, j) \in I_{ru} : | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \geq ɛ} | \\ \geq ɛ^{p} \cdot max_{m, n} | {(k, j) \in I_{ru} : | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \geq ɛ} | \end{matrix}$

and so $\begin{matrix} \begin{matrix} \frac{1}{h_{ru}} \sum_{k, j \in I_{ru}} {| \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L |}^{p} \\ \geq ɛ^{p} \cdot \frac{{max}_{m, n} | {(k, j) \in I_{ru} : | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \geq ɛ} |}{h_{ru}} \\ = ɛ^{p} \cdot \frac{S_{ru}}{h_{ru}} \end{matrix} \end{matrix}$ and this implies that $lim_{r, u \to \infty} \frac{S_{ru}}{h_{ru}} = 0$ and so $x \overset{I_{2 (L)}^{σ θ}}{\sim} y$ .

2.7 Theorem

Suppose that x = (x_kj) and y = (y_kj) are bounded double sequences. Then, $\begin{matrix} x \overset{I_{2 (L)}^{σ θ}}{\sim} y \Rightarrow x \overset{[N_{2 (L)}^{σ θ}]_{p}}{\sim} y . \end{matrix}$

Proof: Let $x, y \in ℓ_{\infty}^{2}$ , $x \overset{I_{2 (L)}^{σ θ}}{\sim} y$ and ɛ > 0. By assumption, we have $V_{2}^{θ} (A_{ɛ}^{\sim}) = 0$ . The boundedness of x and y implies that there exists an M > 0 such that $\begin{matrix} | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \leq M, \end{matrix}$ for k, j ∈ I_ru and for every m, n. Observe that, for every $m, n \in ℕ$ we have $\begin{matrix} \frac{1}{h_{ru}} \sum_{k, j \in I_{ru}} {| \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L |}^{p} \\ = \frac{1}{h_{ru}} \sum_{\begin{matrix} k, j \in I_{ru} \\ | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \geq ɛ \end{matrix}} {| \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L |}^{p} \\ + \frac{1}{h_{ru}} \sum_{\begin{matrix} k, j \in I_{ru} \\ | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | < ɛ \end{matrix}} {| \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L |}^{p} \\ \leq M \frac{{max}_{m, n} | {(k, j) \in I_{ru} : | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \geq ɛ} |}{h_{ru}} \\ + ɛ^{p} \\ \leq M \frac{S_{ru}}{h_{ru}} + ɛ^{p} . \end{matrix}$ Hence, we obtain $\begin{matrix} lim_{r, u \to \infty} \frac{1}{h_{ru}} \sum_{k, j \in I_{ru}} {| \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L |}^{p} = 0, \end{matrix}$ uniformly in m, n.

2.8 Theorem

For every double lacunary sequence θ₂ = {(k_r, j_u)}, $\begin{matrix} I_{2 (L)}^{σ θ} \cap ℓ_{\infty}^{2} = [N_{2 (L)}^{σ θ}]_{p} \cap ℓ_{\infty}^{2} . \end{matrix}$ Proof: This is an immediate consequence of Theorem 2.6 and Theorem 2.7.

2.9 Definition

Two nonnegative double sequences x = (x_kj) and y = (y_kj) are said to be asymptotically lacunary invariant S₂-equivalent of multiple L if for every ɛ > 0 $\begin{matrix} lim_{r, u \to \infty} \frac{1}{h_{ru}} | {(k, j) \in I_{ru} : | \frac{x_{σ^{k} (m), σ^{j} (n)}}{y_{σ^{k} (m), σ^{j} (n)}} - L | \geq ɛ} | \\ = 0, \end{matrix}$ uniformly in m, n. In this case, we write $x \overset{S_{2 (L)}^{σ θ}}{\sim} y$ and simply asymptotically lacunary invariant S₂-equivalent if L = 1.

Now we shall state a theorem that gives a relationship between the concepts of asymptotically lacunary $I_{2}$ -invariant equivalence and asymptotically lacunary invariant S₂-equivalence.

2.10 Theorem

Suppose that x = (x_kj) and y = (y_kj) are double sequences. Then, $\begin{matrix} x \overset{I_{2 (L)}^{σ θ}}{\sim} y \Leftrightarrow x \overset{S_{2 (L)}^{σ θ}}{\sim} y . \end{matrix}$

Footnotes

Acknowledgements

This work is supported by the Scientific Research Project Fund of Afyon Kocatepe University under the project number 18.KARİYER.74

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U. Ulusu, F. Nuray, Lacunary I -convergence, (under review).

Asymptotically lacunary I 2 -invariant equivalence

Abstract

Keywords

1 Introduction and background

2 Asymptotically lacunary I 2 -invariant equivalence

2.1 Definition

2.2 Definition

2.3 Theorem

2.4 Definition

2.5 Definition

2.6 Theorem

2.7 Theorem

2.8 Theorem

2.9 Definition

2.10 Theorem

Footnotes

Acknowledgements

References

2 Asymptotically lacunary $I_{2}$ -invariant equivalence