Research

# On ℐ-asymptotically lacunary statistical equivalent sequences

Ekrem Savaş

Author Affiliations

Department of Mathematics, Istanbul Commerce University, Üsküdar, Istanbul, Turkey

Advances in Difference Equations 2013, 2013:111  doi:10.1186/1687-1847-2013-111

 Received: 1 February 2013 Accepted: 1 April 2013 Published: 18 April 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This paper presents the following definition, which is a natural combination of the definitions for asymptotically equivalent, ℐ-statistically limit and ℐ-lacunary statistical convergence. Let θ be a lacunary sequence; the two nonnegative sequences and are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every , and ,

(denoted by ) and simply ℐ-asymptotically lacunary statistical equivalent if .

MSC: 40A99, 40A05.

##### Keywords:
asymptotical equivalent; ideal convergence; lacunary sequence; statistical convergence

### 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 ℐ-statistical convergence and ℐ-lacunary statistical convergence by using ideal.

In this short paper, we shall use asymptotical equivalent and lacunary sequence to introduce the concepts ℐ-asymptotically statistical equivalent and ℐ-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 and are said to be asymptotically equivalent if

(denoted by ).

Definition 2.2 (Fridy [6])

The sequence has statistic limitL, denoted by st- provided that for every ,

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

Definition 2.3 (Patterson [2])

Two nonnegative sequences and are said to be asymptotically statistical equivalent of multiple L provided that for every ,

(denoted by ) and simply asymptotically statistical equivalent if .

By a lacunary ;  , where , we shall mean an increasing sequence of nonnegative integers with as . The intervals determined by θ will be denoted by and . The ratio will be denoted by .

Definition 2.4 ([3])

Let θ be a lacunary sequence; the two nonnegative sequences and are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every

(denoted by ) and simply asymptotically lacunary statistical equivalent if .

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 of subsets a nonempty set Y is said to be an ideal in Y if the following conditions hold:

(i) implies ;

(ii) , imply .

Definition 2.6 ([10])

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

(i) ;

(ii) implies ;

(iii) , imply .

If ℐ is proper ideal of ℕ (i.e., ), then the family of sets is a filter of ℕ. It is called the filter associated with the ideal.

Definition 2.7 ([9,10])

A proper ideal ℐ is said to be admissible if for each .

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

Definition 2.8 ([9])

Let be a proper admissible ideal in ℕ.

The sequence of elements of ℝ is said to be ℐ-convergent to if for each the set .

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

The following definitions are given in [5].

Definition 2.9 A sequence is said to be ℐ-statistically convergent to L or -convergent to L if, for any and ,

In this case, we write . The class of all ℐ-statistically convergent sequences will be denoted by .

Definition 2.10 Let θ be a lacunary sequence. A sequence is said to be ℐ-lacunary statistically convergent to L or -convergent to L if, for any and ,

In this case, we write . The class of all ℐ-lacunary statistically convergent sequences will be denoted by .

Definition 2.11 Let θ be a lacunary sequence. A sequence is said to be strong ℐ-lacunary convergent to L or -convergent to L if, for any

In this case, we write . The class of all strong ℐ-lacunary statistically convergent sequences will be denoted by .

### 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 and are said to be ℐ-asymptotically statistical equivalent of multiple L provided that for every and ,

(denoted by ) and simply ℐ-asymptotically statistical equivalent if .

For , ℐ-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 and are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every and ,

(denoted by ) and simply ℐ-asymptotically lacunary statistical equivalent if .

For , ℐ-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 and are strong ℐ-asymptotically lacunary equivalent of multiple L provided that

(denoted by ) and strong simply ℐ-asymptotically lacunary equivalent if .

### 4 Main result

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

Theorem 4.1Letbe a lacunary sequence then

(1)

(a) Ifthen,

(b) is a proper subset of;

(2) Ifandthen;

(3) ,

wheredenote the set of bounded sequences.

Proof Part (1a): If and then

and so

Then, for any ,

Hence, we have .

Part (1b): , let be defined as follows: to be at the first integers in and zero otherwise. for all k. These two satisfy the following , but the following fails .

Part (2): Suppose and are in and . Then we can assume that

Given , we have

Consequently, we have

Therefore, .

Part (3): Follows from (1) and (2). □

Theorem 4.2Letis an ideal andis a lacunary sequence with, then

Proof Suppose first that , then there exists a such that for sufficiently large r, which implies

If , then for every and for sufficiently large r, we have

Then, for any , we get

This completes the proof. □

For the next result we assume that the lacunary sequence θ satisfies the condition that for any set , .

Theorem 4.3Letis an ideal andis a lacunary sequence with, then

Proof If , then without any loss of generality, we can assume that there exists a such that for all . Suppose that and for define the sets

and

It is obvious from our assumption that , the filter associated with the ideal ℐ. Further observe that

for all . Let be such that for some . Now

Choosing and in view of the fact that where , it follows from our assumption on θ that the set T also belongs to and this completes the proof of the theorem. □

### Competing interests

The author declares that they have no competing interests.

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