A generalization on ℐ asymptotically lacunary statistical equivalent sequences

R E S E A R C H

Open Access

A generalization on

I

-asymptotically

lacunary statistical equivalent sequences

Ekrem Sava¸s

_{and Hafize Gumu¸s}

*_{Correspondence:} [email protected] 1_{Department of Mathematics,} Istanbul Commerce University, Istanbul, Turkey

Full list of author information is available at the end of the article

Abstract

In this study we extend the concept ofI-asymptotically lacunary statistical equivalent sequences by using the sequencep= (pk) which is the sequence of positive real numbers where

θ

MSC: 40G15; 40A35

Keywords: lacunary sequence; ideal convergence; asymptotically equivalent

sequences

1 Introduction

The concept ofI-convergence was introduced by Kostyrkoet al.in a metric space []. Later it was further studied by Dems [], Das and Savaş [], Savaş [–] and many others.

I-convergence is a generalization form of statistical convergence, which was introduced by Fast (see []) and that is based on the notion of an ideal of the subset of positive inte-gersN.

Definition . A familyI⊂Nis said to be an ideal ofNif the following conditions hold:

(a) A,B∈IimpliesA∪B∈I, (b) A∈I,B⊂AimpliesB∈I.

An ideal is called non-trivial ifN∈/I, and a non-trivial ideal is called admissible if{n} ∈I for eachn∈N.

Definition . A family of setsF⊂Nis a filter inNif and only if:

(i) ∅∈/F.

(ii) For eachA,B∈F, we haveA∩B∈F.

(iii) For eachA∈Fand eachB⊇A, we haveB∈F.

Proposition . Iis a non-trivial ideal inNif and only if

F=F(I) ={M=N\A:A∈I}

is a filter inN(see[]).

Definition . A real sequencex= (xk) is said to beI-convergent toL∈Rif and only if

for eachε>  the set

Aε=k∈N:|xk–L| ≥ε

belongs toI. The numberLis called theI-limit of the sequencex(see []).

Remark  If we takeI=If ={A⊆N:Ais a finite subset}. ThenIf is a non-trivial

admis-sible ideal ofNand the corresponding convergence coincides with the usual convergence. A lacunary sequence is an increasing integer sequenceθ = (kr) such thatk=  and hr=kr–kr–→ ∞asr→ ∞. The intervals determined byθare denotedIr= (kr–,kr] and

the ratio kr

kr– is denoted byqr.

In , Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices.

Definition .[] Two nonnegative sequencesx= (xk) andy= (yk) are said to be

asymp-totically equivalent if

lim

k

xk

yk

= 

and it is denoted byx∼y.

Definition . (Fridy []) The sequence x = (xk) has statistic limit L, denoted by

st-limx=L, provided that for every> ,

lim

n



n

the number ofk≤n:|xk–L| ≥

= .

In , Patterson defined asymptotically statistical equivalent sequences by using the definition of statistical convergence as follows.

Definition .(Patterson []) Two nonnegative sequencesx= (xk) andy= (yk) are said

to be asymptotically statistical equivalent of multipleLprovided that for every> ,

lim

n



n

the number ofk<n:xk

yk

–L≥

= 

(denoted byx∼SLy), and simply asymptotically statistical equivalent ifL= .

In , Patterson and Savaş presented definitions for asymptotically lacunary statisti-cal equivalent sequences (see []).

Definition . Letθ= (kr) be a lacunary sequence, two nonnegative sequences [x] and

[y] are said to be asymptotically lacunary statistical equivalent of multipleLprovided that for everyε> 

lim

r



hr

k∈Ir:

xk

yk

–L≥ε= ,

Definition . Letθ = (kr) be a lacunary sequence, two number sequencesx= (xk) and

y= (yk) are said to be strong asymptotically lacunary equivalent of multipleLprovided

that

lim

r



hr

k∈Ir

xk

yk

–L= .

In , Savaş and Patterson gave an extension on asymptotically lacunary statistical equivalent sequences, and they investigated some relations between strongly asymptot-ically lacunary equivalent sequences and strongly Cesáro asymptotasymptot-ically equivalent se-quences. More applications of the asymptotically statistical equivalent sequences can be seen in [–].

Definition .[] Letθ= (kr) be a lacunary sequence and letp= (pk) be a sequence of

positive real numbers. Two number sequencesx= (xk) andy= (yk) are said to be strongly

asymptotically lacunary equivalent of multipleLprovided that

lim

r



hr

k∈Ir

xk

yk

–L

pk

= 

(denoted byxN L(p)

θ

∼ y) and simply strongly asymptotically lacunary equivalent ifL= .

Definition . Letp= (pk) be a sequence of positive real numbers. Two number

se-quencesx= (xk) andy= (yk) are said to be strongly Cesáro asymptotically equivalent toL

provided that

lim

n



n

n

k=

xk

yk

–L

pk

= 

(denoted byxσ∼(p)y) and simply strongly Cesáro asymptotically equivalent ifL= . The following definitions are given in [].

Definition . A sequencex= (xk) is said to beI-statistically convergent toLorS(I

)-convergent toLif, for anyε>  andδ> ,

n∈N: 

nk≤n:|xk–L| ≥ε≥δ

∈I.

In this case, we writexk→L(S(I)). The class of allI-statistically convergent sequences

will be denoted byS(I).

Definition . Letθbe a lacunary sequence. A sequencex= (xk) is said to beI-lacunary

statistically convergent toLorSθ(I)-convergent toLif, for anyε>  andδ> ,

r∈N: 

hr

k∈Ir:|xk–L| ≥ε≥δ

∈I.

In this case, we writexk→L(Sθ(I)). The class of allI-lacunary statistically convergent

Definition . Letθ be a lacunary sequence. A sequencex= (xk) is said to be strong

I-lacunary convergent toLorNθ(I)-convergent toLif, for anyε> ,

r∈N: 

hr

k∈Ir

|xk–L| ≥ε

∈I.

In this case, we writexk→L(Nθ(I)). The class of all strongI-lacunary statistically

con-vergent sequences will be denoted byNθ(I).

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

Recently, Savaş definedI-asymptotically lacunary statistical equivalent sequences by using the definitionsI-convergence and asymptotically lacunary statistical equivalent se-quences together.

Definition .[] Letθ= (kr) be a lacunary sequence; the two nonnegative sequences

x= (xk) andy= (yk) are said to beI-asymptotically lacunary statistical equivalent of

mul-tipleLprovided that for everyε>  andδ> ,

r∈N: 

hr

k∈Ir:

xk

yk

–L≥ε≥δ

∈I.

In this case we writexS L

θ(I)

∼ y.

2 Main results

In this section we shall give some new definitions and also examine some inclusion rela-tions.

Definition . Letθ = (kr) be a lacunary sequence and letp= (pk) be a sequence of

pos-itive real numbers. Two number sequencesx= (xk) andy= (yk) are said to be strongly

I-asymptotically lacunary equivalent of multipleLfor the sequencepprovided that

r∈N: 

hr

k∈Ir

xk

yk

–L

pk

≥ε

∈I.

In this situation we writexN L(p)

θ (I)

∼ y.

If we takepk=pfor allk∈N, we writex N_θLp(I)

∼ yinstead ofxN L(p)

θ (I)

∼ y.

Definition . Letθ= (kr) be a lacunary sequence and letp= (pk) be a sequence of

pos-itive real numbers. Two number sequencesx= (xk) andy= (yk) are said to be strongly

CesároI-asymptotically equivalent of multipleLprovided that

n∈N: 

n

n

k=

xk

yk

–L

pk

≥ε ∈I

(denoted byxσ

(p)_(I)

Theorem . Letθ= (kr)be a lacunary sequence.Then:

(a) IfxN Lp

θ (I)

∼ y,thenxS L

θ(I)

∼ y;

(b) Ifx,y∈l∞andx S_θL(I)

∼ y,thenxN Lp

θ (I)

∼ y;

(c) (xS L

θ(I)

∼ y)∩l∞= (x N_θLp(I)

∼ y)∩l∞.

Proof (a) LetxN Lp

θ (I)

∼ yandε>  be given. Then

k∈Ir

xk yk –L p ≥

k∈Ir&|xk_yk–L|≥ε

xk

yk

–L

p

≥εp

k∈Ir:

xk

yk

–L≥ε

and so 

εp_h r

k∈Ir

xk yk –L p ≥  hr k∈Ir:

xk

yk

–L≥ε.

Then, for anyδ> ,

r∈N: 

hr

k∈Ir:

xk

yk

–L≥ε≥δ

⊆

r∈N: 

hr

k∈Ir

xk

yk

–L

p

≥εpδ

∈I.

ThereforexS L

θ(I)

∼ y.

(b) Letxandybe bounded sequences andxS L

θ(I)

∼ y. Then there is anMsuch that|xk

yk–L| ≤ Mfor allk. For eachε> ,



hr

k∈Ir

xk yk –L p =  hr

k∈Ir&|xk_yk–L|≥ε

xk yk –L p +  hr

k∈Ir&|xk_yk–L|<ε

xk yk –L p ≤  hr Mp

k∈Ir:

xk

yk

–L≥ε+ 

hr εp

k∈Ir:

xk

yk

–L<ε

≤Mp

hr

k∈Ir:

xk

yk

–L≥ε+εp.

Then, for anyδ> ,

r∈N: 

hr

k∈Ir

xk yk –L p ≥ε ⊆

r∈N: 

hr

k∈Ir:

xk

yk

–L≥ε≥ ε p

Mp

∈I.

ThereforexN Lp

θ (I)

∼ y.

Theorem . Letθ= (kr)be a lacunary sequence,infkpk=h andsupkp=H.Then

xN L(p)

θ (I)

∼ y implies xS L

θ(I)

∼ y.

Proof Assume thatxN L(p)

θ (I)

∼ yandε> . Then 

hr

k∈Ir

xk yk –L pk =  hr

k∈Ir&|xk_yk–L|≥ε

xk yk –L pk +  hr

k∈Ir&|xk_yk–L|<ε

xk yk –L pk ≥  hr

k∈Ir&|xk_yk–L|≥ε

xk yk –L pk ≥  hr

k∈Ir&|xk_yk–L|≥ε

(ε)pk

≥ 

hr

k∈Ir&|xk_yk–L|≥ε

min(ε)h, (ε)H

≥ 

hr

min(ε)h, (ε)H

k∈Ir:

xk

yk

–L≥ε

and

r∈N: 

hr

k∈Ir:

xk

yk

–L≥ε≥δ

⊆

r∈N: 

hr

k∈Ir

xk

yk

–L

pk

≥δmin(ε)h, (ε)H∈I.

Thus we havexS L

θ(I)

∼ y.

Theorem . Let x and y be bounded sequences,infkpk=h andsupkp=H.Then

xS L

θ(I)

∼ y implies xN L(p)

θ (I)

∼ y.

Proof Suppose thatxandyare bounded andε> . Then there is an integerKsuch that

|xk

yk –L| ≤Kfor allk,



hr

k∈Ir

xk yk –L pk =  hr

k∈Ir&|xk_yk–L|≥ε

xk yk –L pk +  hr

k∈Ir&|xk_yk–L|<ε

xk yk –L pk ≤  hr k∈Ir:

xk

yk

–L≥ε



maxKh,KH

+ 

hr

k∈Ir:

xk

yk

–L<ε 

max(ε)pk



≤maxKh,KH hr

k∈Ir:

xk

yk

–L≥ε



+max{ε

h_,εH_}

and

r∈N: 

hr

k∈Ir

xk

yk

–L

pk

≥ε

⊆

r∈N: 

hr

k∈Ir:

xk

yk

–L≥ε



≥ε–max{εh,εH}

max{Kh_,KH_}

∈I.

Thus we havexN L(p)

θ (I)

∼ y.

Theorem . Letθ= (kr)be a lacunary sequence withlim infrqr> ,then

xσ

(p)_(I)

∼ y implies xN L(p)

θ (I)

∼ y.

Proof Iflim infrqr> , then there existsδ>  such thatqr≥ +δfor allr≥. Sincehr=

kr–kr–, we have k_hr_r ≤+δδ and

kr– hr ≤



δ. Letε>  and define the set

S=

kr∈N:



kr kr

k=

xk

yk

–L

pk

<ε .

We can easily say thatS∈F(I), which is the filter of the idealI,



hr

k∈Ir

xk

yk

–L

pk

= 

hr kr

k=

xk

yk

–L

pk

– 

hr kr–

k=

xk

yk

–L

pk

= kr

hr



kr kr

k=

xk

yk

–L

pk

–kr–

hr



kr–

kr–

k=

xk

yk

–L

pk

≤

 +δ δ

ε–

δε

for eachkr∈S. Chooseη= (+_δδ)ε–_δε. Therefore,

r∈N: 

hr

k∈Ir

xk

yk

–L

pk

<η

∈F(I)

and it completes the proof.

For the next result, we assume that the lacunary sequenceθsatisfies the condition that for any setC∈F(I),{n:kr–<n<kr,r∈C} ∈F(I).

Theorem . Letθ= (kr)be a lacunary sequence withlim supqr<∞,then

xN L(p)

θ (I)

∼ y implies xσ

(p)_(I)

Proof Iflim supqr<∞, then there existsB>  such thatqr<Bfor allr≥. Letx N_θL(p)(I)

∼ y

and define the setsTandRsuch that

T=

r∈N: 

hr

k∈Ir

xk

yk

–L

pk

<ε

and

R=

n∈N: 

n

n

k=

xk

yk

–L

pk

<ε .

Let

Aj=



hj

k∈Ij

xk

yk

–L

pk

<ε

for allj∈T. It is obvious thatT∈F(I). Choosenis any integer withkr–<n<kr, where

r∈T, 

n

n

k=

xk

yk

–L

pk

≤ 

kr–

kr

k=

xk

yk

–L

pk

= 

kr–

k∈I

xk

yk

–L

pk

+

k∈I

xk

yk

–L

pk

+· · ·+

k∈Ir

xk

yk

–L

pk

= k

kr–



h

k∈I

xk

yk

–L

pk

+k–k

kr–



h

k∈I

xk

yk

–L

pk

+· · ·

+kr–kr–

kr–



hr

k∈Ir

xk

yk

–L

pk

= k

kr– A+

k–k kr–

A+· · ·+

kr–kr– kr–

Ar

≤sup

j∈T

Aj

_k

r

kr–

<εB.

Chooseε=ε_B and in view of the fact that

{n:kr–<n<kr,r∈T} ⊂R, whereT∈F(I),

it follows from our assumption onθthat the setRalso belongs toF(I) and this completes the proof of the theorem.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors completed the paper together. Both authors read and approved the final manuscript.

Author details

Acknowledgements

Dedicated to Professor Hari M Srivastava.

We would like to thanks the referees for their valuable comments.

Received: 24 November 2012 Accepted: 4 May 2013 Published: 30 May 2013

References

1. Kostyrko, P, ˘Salát, T, Wilezy ´nski, W:I-Convergence. Real Anal. Exch.26(2), 669-686 (2000/2001) 2. Dems, K: OnI-Cauchy sequences. Real Anal. Exch.30, 123-128 (2004-2005)

3. Das, P, Sava¸s, E, Ghosal, SK: On generalizations of certain summability methods using ideals. Appl. Math. Lett.24(9), 1509-1514 (2011)

4. Sava¸s, E:m_{-Strongly summable sequences spaces in 2-normed spaces defined by ideal convergence and an Orlicz} function. Appl. Math. Comput.217, 271-276 (2010)

5. Sava¸s, E:A-Sequence spaces in 2-normed space defined by ideal convergence and an Orlicz function. Abstr. Appl. Anal.2011, Article ID 741382 (2011)

6. Sava¸s, E: On some new sequence spaces in 2-normed spaces using ideal convergence and an Orlicz function. J. Inequal. Appl.2010, Article ID 482392 (2010). doi:10.1155/2010/482392

7. Sava¸s, E, Das, P: A generalized statistical convergence via ideals. Appl. Math. Lett.24, 826-830 (2011) 8. Fast, H: Sur la convergence statistique. Colloq. Math.2, 241-244 (1951)

9. Marouf, MS: Asymptotic equivalence and summability. Int. J. Math. Math. Sci.16(4), 755-762 (1993) 10. Fridy, JA: On statistical convergence. Analysis5, 301-313 (1985)

11. Patterson, RF: On asymptotically statistical equivalent sequences. Demonstr. Math.XXXVI(1), 1-7 (2003)

12. Patterson, RF, Sava¸s, E: On asymptotically lacunary statistical equivalent sequences. Thai J. Math.4(2), 267-272 (2006) 13. Sava¸s, E: On asymptotically lacunary statistical equivalent sequences of fuzzy numbers. J. Fuzzy Math.17(3), 527-533

(2009)

14. Sava¸s, E, Patterson, RF: An extension asymptotically lacunary statistical equivalent sequences. Aligarh Bull. Math. 27(2), 109-113 (2008)

15. Sava¸s, E: On asymptoticallyλ-statistical equivalent sequences of fuzzy numbers. New Math. Nat. Comput.3(3), 301-306 (2007)

16. Sava¸s, E, Patterson, RF: Generalization of two asymptotically statistical equivalent theorems. Filomat20(2), 81-86 (2006)

17. Sava¸s, E, Patterson, RF: An extension asymptotically lacunary statistical equivalent sequences. Aligarh Bull. Math. 27(2), 109-113 (2008)

18. Sava¸s, E: OnI-asymptotically lacunary statistical equivalent sequences. Adv. Differ. Equ.2013, 111 (2013). doi:10.1186/1687-1847-2013-111

doi:10.1186/1029-242X-2013-270