On ℐ-asymptotically lacunary statistical equivalent sequences

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Abstract

This paper presents the following definition, which is a natural combination of the definitions for asymptotically equivalent, -statistically limit and -lacunary statistical convergence. Let θ be a lacunary sequence; the two nonnegative sequences $x=( x k )$ and $y=( y k )$ are said to be -asymptotically lacunary statistical equivalent of multiple L provided that for every $ϵ>0$, and $δ>0$,

${ r ∈ N : 1 h r | { k ∈ I r : | x k y k − L | ≥ ε } | ≥ δ } ∈I$

(denoted by $x ∼ S θ L ( I ) y$) and simply -asymptotically lacunary statistical equivalent if $L=1$.

MSC:40A99, 40A05.

1 Introduction

In 1993, Marouf [1] presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. In 2003, Patterson [2] extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices.

In [3], asymptotically lacunary statistical equivalent, which is a natural combination of the definitions for asymptotically equivalent, statistical convergence and lacunary sequences. Later on, the extension asymptotically lacunary statistical equivalent sequences is presented (see [4]).

Recently, Das, Savaş and Ghosal [5] introduced new notions, namely -statistical convergence and -lacunary statistical convergence by using ideal.

In this short paper, we shall use asymptotical equivalent and lacunary sequence to introduce the concepts -asymptotically statistical equivalent and -asymptotically lacunary statistical equivalent. In addition to these definitions, natural inclusion theorems shall also be presented.

First, we introduce some definitions.

2 Definitions and notations

Definition 2.1 (Marouf [1])

Two nonnegative sequences $x=( x k )$ and $y=( y k )$ are said to be asymptotically equivalent if

$lim k x k y k =1$

(denoted by $x∼y$).

Definition 2.2 (Fridy [6])

The sequence $x=( x k )$ has statistic limit L, denoted by st-$lims=L$ provided that for every $ϵ>0$,

The next definition is natural combination of Definitions 2.1 and 2.2.

Definition 2.3 (Patterson [2])

Two nonnegative sequences $x=( x k )$ and $y=( y k )$ are said to be asymptotically statistical equivalent of multiple L provided that for every $ϵ>0$,

(denoted by $x ∼ S L y$) and simply asymptotically statistical equivalent if $L=1$.

By a lacunary $θ=( k r )$; $r=0,1,2,…$ , where $k 0 =0$, we shall mean an increasing sequence of nonnegative integers with $k r − k r − 1$ as $r→∞$. The intervals determined by θ will be denoted by $I r =( k r − 1 , k r ]$ and $h r = k r − k r − 1$. The ratio $k r k r − 1$ will be denoted by $q r$.

Definition 2.4 ([3])

Let θ be a lacunary sequence; the two nonnegative sequences $x=( x k )$ and $y=( y k )$ are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every $ϵ>0$

$lim r 1 h r | { k ∈ I r : | x k y k − L | ≥ ϵ } |=0$

(denoted by $x ∼ S θ L y$) and simply asymptotically lacunary statistical equivalent if $L=1$.

More investigations in this direction and more applications of asymptotically statistical equivalent can be found in [7, 8] where many important references can be found.

The following definitions and notions will be needed.

Definition 2.5 ([9])

A nonempty family $I⊂ 2 Y$ of subsets a nonempty set Y is said to be an ideal in Y if the following conditions hold:

1. (i)

$A,B∈I$ implies $A∪B∈I$;

2. (ii)

$A∈I$, $B⊂A$ imply $B∈I$.

Definition 2.6 ([10])

A nonempty family $F⊂ 2 N$ is said to be a filter of if the following conditions hold:

1. (i)

$∅∉F$;

2. (ii)

$A,B∈F$ implies $A∩B∈F$;

3. (iii)

$A∈F$, $B⊂A$ imply $B∈F$.

If is proper ideal of (i.e., $N∉I$), then the family of sets $F(I)={M⊂N:∃A∈I:M=N∖A}$ is a filter of . It is called the filter associated with the ideal.

Definition 2.7 ([9, 10])

A proper ideal is said to be admissible if ${n}∈I$ for each $n∈N$.

Throughout will stand for a proper admissible ideal of , and by sequence we always mean sequences of real numbers.

Definition 2.8 ([9])

Let $I⊂ 2 N$ be a proper admissible ideal in .

The sequence $( x k )$ of elements of is said to be -convergent to $L∈R$ if for each $ϵ>0$ the set $A(ϵ)={n∈N:| x n −L|≥ϵ}∈I$.

Following these results, we introduce two new notions -asymptotically lacunary statistical equivalent of multiple L and strong -asymptotically lacunary equivalent of multiple L.

The following definitions are given in [5].

Definition 2.9 A sequence $x=( x k )$ is said to be -statistically convergent to L or $S(I)$-convergent to L if, for any $ε>0$ and $δ>0$,

${ n ∈ N : 1 n | { k ≤ n : | x k − L | ≥ ε } | ≥ δ } ∈I.$

In this case, we write $x k →L(S(I))$. The class of all -statistically convergent sequences will be denoted by $S(I)$.

Definition 2.10 Let θ be a lacunary sequence. A sequence $x=( x k )$ is said to be -lacunary statistically convergent to L or $S θ (I)$-convergent to L if, for any $ε>0$ and $δ>0$,

${ r ∈ N : 1 h r | { k ∈ I r : | x k − L | ≥ ε } | ≥ δ } ∈I.$

In this case, we write $x k →L( S θ (I))$. The class of all -lacunary statistically convergent sequences will be denoted by $S θ (I)$.

Definition 2.11 Let θ be a lacunary sequence. A sequence $x=( x k )$ is said to be strong -lacunary convergent to L or $N θ (I)$-convergent to L if, for any $ε>0$

${ r ∈ N : 1 h r ∑ k ∈ I r | x k − L | ≥ ε } ∈I.$

In this case, we write $x k →L( N θ (I))$. The class of all strong -lacunary statistically convergent sequences will be denoted by $N θ (I)$.

3 New definitions

The next definitions are combination of Definitions 2.1, 2.9, 2.10 and 2.11.

Definition 3.1 Two nonnegative sequences $x=( x k )$ and $y=( y k )$ are said to be -asymptotically statistical equivalent of multiple L provided that for every $ϵ>0$ and $δ>0$,

${ n ∈ N : 1 n | { k ≤ n : | x k y k − L | ≥ ε } | ≥ δ } ∈I$

(denoted by $x ∼ S L ( I ) y$) and simply -asymptotically statistical equivalent if $L=1$.

For $I= I fin$, -asymptotically statistical equivalent of multiple L coincides with asymptotically statistical equivalent of multiple L, which is defined in [3].

Definition 3.2 Let θ be a lacunary sequence; the two nonnegative sequences $x=( x k )$ and $y=( y k )$ are said to be -asymptotically lacunary statistical equivalent of multiple L provided that for every $ϵ>0$ and $δ>0$,

${ r ∈ N : 1 h r | { k ∈ I r : | x k y k − L | ≥ ε } | ≥ δ } ∈I$

(denoted by $x ∼ S θ L ( I ) y$) and simply -asymptotically lacunary statistical equivalent if $L=1$.

For $I= I fin$, -asymptotically lacunary statistical equivalent of multiple L coincides with asymptotically lacunary statistical equivalent of multiple L, which is defined in [3].

Definition 3.3 Let θ be a lacunary sequence; two number sequences $x=( x k )$ and $y=( y k )$ are strong -asymptotically lacunary equivalent of multiple L provided that

${ r ∈ N : 1 h r ∑ k ∈ I r | x k y k − L | ≥ ε } ∈I$

(denoted by $x ∼ N θ L ( I ) y$) and strong simply -asymptotically lacunary equivalent if $L=1$.

4 Main result

In this section, we state and prove the results of this article.

Theorem 4.1 Let $θ={ k r }$ be a lacunary sequence then

1. (1)
1. (a)

If $x ∼ N θ L ( I ) y$ then $x ∼ S θ L ( I ) y$,

2. (b)

$x ∼ N θ L ( I ) y$ is a proper subset of $x ∼ S θ L ( I ) y$;

2. (2)

If $x,y∈ l ∞$ and $x ∼ S θ L ( I ) y$ then $x ∼ N θ L ( I ) y$;

3. (3)

$x ∼ S θ L ( I ) y∩ l ∞ =x ∼ N θ L ( I ) y∩ l ∞$,

where $l ∞$ denote the set of bounded sequences.

Proof Part (1a): If $ϵ>0$ and $x ∼ N θ L ( I ) y$ then

$∑ k ∈ I r | x k y k − L | ≥ ∑ k ∈ I r & | x k y k − L | ≥ ϵ | x k y k − L | ≥ ϵ | { k ∈ I r : | x k y k − L | ≥ ϵ } |$

and so

$1 ε h r ∑ k ∈ I r | x k y k −L|≥ 1 h r | { k ∈ I r : | x k y k − L | ≥ ϵ } |.$

Then, for any $δ>0$,

${ r ∈ N : 1 h r | { k ∈ I r : | x k y k − L | ≥ ϵ } | ≥ δ } ⊆ { r ∈ N : 1 h r ∑ k ∈ I r | x k y k − L | ≥ ϵ ⋅ δ } ∈I.$

Hence, we have $x ∼ S θ L ( I ) y$.

Part (1b): $x ∼ N θ L ( I ) y⊂x ∼ S θ L ( I ) y$, let $x=( x k )$ be defined as follows: $x k$ to be $1,2,…,[ h r ]$ at the first $[ h r ]$ integers in $I r$ and zero otherwise. $y k =1$ for all k. These two satisfy the following $x ∼ S θ L ( I ) y$, but the following fails $x ∼ N θ L ( I ) y$.

Part (2): Suppose $x=( x k )$ and $y=( y k )$ are in $l ∞$ and $x ∼ S θ L ( I ) y$. Then we can assume that

Given $ϵ>0$, we have

$1 h r ∑ k ∈ I r | x k y k − L | = 1 h r ∑ k ∈ I r & | x k y k − L | ≥ ϵ | x k y k − L | + 1 h r ∑ k ∈ I r & | x k y k − L | < ϵ | x k y k − L | ≤ M h r | { k ∈ I r : | x k y k − L | ≥ ϵ 2 } | + ϵ 2 .$

Consequently, we have

${ r ∈ N : 1 h r ∑ k ∈ I r | x k y k − L | ≥ ε } ⊆ { r ∈ N : 1 h r | { k ∈ I r : | x k y k − L | ≥ ϵ 2 } | ≥ ε 2 M } ∈I.$

Therefore, $x ∼ N θ L ( I ) y$.

Part (3): Follows from (1) and (2). □

Theorem 4.2 Let is an ideal and $θ={ k r }$ is a lacunary sequence with $liminf q r >1$, then

$x ∼ S L ( I ) yimpliesx ∼ S θ L ( I ) y.$

Proof Suppose first that $liminf q r >1$, then there exists a $δ>0$ such that $q r ≥1+δ$ for sufficiently large r, which implies

$h r k r ≥ δ 1 + δ .$

If $x ∼ S θ L ( I ) y$, then for every $ε>0$ and for sufficiently large r, we have

$1 k r | { k ≤ k r : | x k y k − L | ≥ ϵ } | ≥ 1 k r | { k ∈ I r : | x k y k − L | ≥ ϵ } | ≥ δ 1 + δ 1 h r | { k ∈ I r : | x k y k − L | ≥ ϵ } | .$

Then, for any $δ>0$, we get

${ r ∈ N : 1 h r | { k ∈ I r : | x k y k − L | ≥ ε } | ≥ δ } ⊆ { r ∈ N : 1 k r | { k ≤ k r : | x k y k − L | ≥ ϵ } | ≥ δ α ( 1 + α ) } ∈ I .$

This completes the proof. □

For the next result we assume that the lacunary sequence θ satisfies the condition that for any set $C∈F(I)$, $⋃{n: k r − 1 .

Theorem 4.3 Let is an ideal and $θ=( k r )$ is a lacunary sequence with $sup q r <∞$, then

$x ∼ S θ L ( I ) yimpliesx ∼ S L ( I ) y.$

Proof If $limsup r q r <∞$, then without any loss of generality, we can assume that there exists a $0 such that $q r for all $r≥1$. Suppose that $x ∼ S θ L y$ and for $ϵ,δ, δ 1 >0$ define the sets

$C= { r ∈ N : 1 h r | { k ∈ I r : | x k y k − L | ≥ ϵ } | < δ }$

and

$T= { n ∈ N : 1 n | { k ≤ n : | x k y k − L | ≥ ϵ } | < δ 1 } .$

It is obvious from our assumption that $C∈F(I)$, the filter associated with the ideal . Further observe that

$A j = 1 h j | { k ∈ I j : | x k y k − L | ≥ ϵ } | <δ$

for all $j∈C$. Let $n∈N$ be such that $k r − 1 for some $r∈C$. Now

$1 n | { k ≤ n : | x k y k − L | ≥ ϵ } | ≤ 1 k r − 1 | { k ≤ k r : | x k y k − L | ≥ ϵ } | = 1 k r − 1 | { k ∈ I 1 : | x k y k − L | ≥ ϵ } | + ⋯ + 1 k r − 1 | { k ∈ I r : | x k y k − L | ≥ ϵ } | = k 1 k r − 1 1 h 1 | { k ∈ I 1 : | x k y k − L | ≥ ϵ } | + k 2 − k 1 k r − 1 1 h 2 | { k ∈ I 2 : | x k y k − L | ≥ ϵ } | + ⋯ + k r − k r − 1 k r − 1 1 h r | { k ∈ I r : | x k y k − L | ≥ ϵ } | = k 1 k r − 1 A 1 + k 2 − k 1 k r − 1 A 2 + ⋯ + k r − k r − 1 k r − 1 A r ≤ sup j ∈ C A j ⋅ k r k r − 1 < B δ .$

Choosing $δ 1 = δ B$ and in view of the fact that $⋃{n: k r − 1 where $C∈F(I)$, it follows from our assumption on θ that the set T also belongs to $F(I)$ and this completes the proof of the theorem. □

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Correspondence to Ekrem Savaş.

Competing interests

The author declares that they have no competing interests.

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Savaş, E. On -asymptotically lacunary statistical equivalent sequences. Adv Differ Equ 2013, 111 (2013) doi:10.1186/1687-1847-2013-111