Open Access

On -asymptotically lacunary statistical equivalent sequences

Advances in Difference Equations20132013:111

https://doi.org/10.1186/1687-1847-2013-111

Received: 1 February 2013

Accepted: 1 April 2013

Published: 18 April 2013

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.

Keywords

asymptotical equivalentideal convergencelacunary sequencestatistical 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 -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- lim s = L provided that for every ϵ > 0 ,
lim n 1 n { the number of  k n : | x k L | 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 ,
lim n 1 n { the number of  k < n : | x k y k L | 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
| x k y k L | M for all  k .
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 ) y implies x 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 < n < k r , r C } F ( I ) .

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 ) y implies x S L ( I ) y .
Proof If limsup r q r < , then without any loss of generality, we can assume that there exists a 0 < B < such that q r < B 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 < n < k r 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 < n < k r , r C } T 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. □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Istanbul Commerce University

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© Savaş; licensee Springer 2013

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