Skip to main content

Advertisement

On -asymptotically lacunary statistical equivalent sequences

Article metrics

  • 1189 Accesses

  • 16 Citations

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 xy).

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,

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,BI implies ABI;

  2. (ii)

    AI, BA imply BI.

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,BF implies ABF;

  3. (iii)

    AF, BA imply BF.

If is proper ideal of (i.e., NI), then the family of sets F(I)={MN:AI:M=NA} 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 nN.

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 LR if for each ϵ>0 the set A(ϵ)={nN:| 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 ) yx 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|Mfor 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 ) 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 CF(I), {n: k r 1 <n< k r ,rC}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 ) 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<B< such that q r <B for all r1. 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 CF(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 jC. Let nN be such that k r 1 <n< k r for some rC. 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 ,rC}T where CF(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. □

References

  1. 1.

    Marouf M: Asymptotic equivalence and summability. Int. J. Math. Math. Sci. 1993, 16(4):755–762. 10.1155/S0161171293000948

  2. 2.

    Patterson RF: On asymptotically statistically equivalent sequences. Demonstr. Math. 2003, 36(1):149–153.

  3. 3.

    Patterson RF, Savaş E: On asymptotically lacunary statistical equivalent sequences. Thai J. Math. 2006, 4(2):267–272.

  4. 4.

    Savaş E, Patterson RF: An extension asymptotically lacunary statistical equivalent sequences. Aligarh Bull. Math. 2008, 27(2):109–113.

  5. 5.

    Das P, Savaş E, Ghosal SK: On generalizations of certain summability methods using ideals. Appl. Math. Lett. 2011, 36: 1509–1514.

  6. 6.

    Fridy JA: On statistical convergence. Analysis 1985, 5: 301–313.

  7. 7.

    Li J: Asymptotic equivalence of sequences and summability. Int. J. Math. Math. Sci. 1997, 20(4):749–758. 10.1155/S0161171297001038

  8. 8.

    Patterson RF: Analogues of some fundamental theorems of summability theory. Int. J. Math. Math. Sci. 2000, 23(1):1–9. 10.1155/S0161171200001782

  9. 9.

    Kostyrko P, Salat T, Wilczyński W: I -Convergence. Real Anal. Exch. 2000, 26(2):669–686.

  10. 10.

    Kostyrko P, Macaj M, Salat T, Sleziak M: I -Convergence and extremal I -limit points. Math. Slovaca 2005, 55: 443–464.

Download references

Author information

Correspondence to Ekrem Savaş.

Additional information

Competing interests

The author declares that they have no competing interests.

Rights and permissions

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

Reprints and Permissions

About this article

Keywords

  • asymptotical equivalent
  • ideal convergence
  • lacunary sequence
  • statistical convergence