On generalized difference lacunary statistical convergence in a paranormed space

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R E S E A R C H

Open Access

On generalized difference lacunary statistical

convergence in a paranormed space

Selma Altunda ˘g

*

*_{Correspondence:} [email protected] Mathematics Department, Faculty of Arts and Science, Sakarya University, Sakarya, 54187, Turkey

Abstract

In this article, we introduce the concept of

m_{-lacunary statistical convergence and}

m_{-lacunary strong convergence in a paranormed space. Also, we establish some}

connections between these concepts.

1 Introduction

In order to extend convergence of sequences, the notion of statistical convergence was introduced by Fast [] and Steinhaus [] and several generalizations and applications of this concept have been investigated by various authors [, ]. This notion was studied in normed spaces by Kolk [], in locally convex Hausdorff topological spaces by Maddox [], in topological Hausdorff groups by Çakallı [] and in probabilistic normed space by Karakuş []. Recently, Alotaibi and Alroqi [] extended this notion in paranormed spaces. In this article, we study the concept of statistical convergence from difference sequence spaces which are defined over paranormed space.

2 Preliminaries and deﬁnitions

LetKbe a subset of the set of natural numbersN. Then the asymptotic density ofK de-noted byδ(K) =limn_n|{k≤n:k∈K}|, where the vertical bars denote the cardinality of

the enclosed set in [].

A number sequencex= (xk) is said to be statistically convergent to the numberLif for

eachε> , the setK(ε) ={k≤n:|xk–L| ≥ε}has asymptotic density zero,i.e.,

lim

n



nk≤n:|xk–L| ≥ε= .

In this case we writest-limx=L. This concept was studied by [, ].

By a lacunaryθ= (kr);r= , , , . . . , wherek= , we shall mean an increasing sequence

of non-negative integers withkr–kr–→ ∞asr→ ∞. The intervals determined byθwill

be denoted byIr= (kr–,kr] andhr=kr–kr–. The ratio_kk_r–r will be denoted byqr.

The notion of diﬀerence sequence spaceX() was introduced by Kızmaz [] as follows:

X() =x= (xk) : (xk)∈X

forX=l∞,c,c, wherexk=xk–xk+for allk∈N.

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The notion of diﬀerence sequence spaces was further generalized by Et and Çolak [] as follows:

Xm=x= (xk)∈w:

mxk

∈X

forX=l∞,candc, wherem∈N,mxk=m–xk–m–xk+,xk=xk.

The sequencexis said to bem-statistically convergent to the numberLprovided that for eachε> ,

lim

n



nk≤n: m_x

k–L≥ε= .

The set of allm-statistically convergent sequences was denoted byS(m) in []. Furthermore, this notion was studied in [, ].

A paranorm is a function g :X−→Rdeﬁned on a linear spaceX such that for all

x,y,z∈X,

(i) g(x) = ifx=θ; (ii) g(–x) =g(x);

(iii) g(x+y)≤g(x) +g(y);

(iv) If(αn)is a sequence of scalars withαn−→α(n−→ ∞) andxn,a∈Xwithxn−→a

(n−→ ∞) in the sense thatg(xn–a)−→(n−→ ∞), thenαnxn−→αa

(n−→ ∞), in the sense thatg(αnxn–αa)−→(n−→ ∞).

A paranormg for whichg(x) =  impliesx=θ is called a total paranorm onXand the pair (X,g) is called a total paranormed space.

Note that each seminorm (norm)ponXis a paranorm (total) but converse need not be true.

The concept of paranorm is a generalization of absolute value []. A modulus functionf is a function from [,∞) to [,∞) such that

(i) f(x) = if and only if x= ; (ii) f(x+y)≤f(x) +f(y)for allx,y≥; (iii) f increasing;

(iv) f is continuous from at the right zero.

Since|f(x) –f(y)| ≤f(|x–y|), it follows from condition (iv) thatfis continuous on [,∞). Furthermore, we havef(nx)≤nf(x) for alln∈Nfrom condition (ii) and so

f(x) =f

nx n

≤nf

x n

.

Hence, for alln∈N, 

nf(x)≤f

x n

.

A modulus may be bounded or unbounded. For example,f(x) =xpfor  <p≤ is un-bounded, butf(x) =_+x_xis bounded. Ruckle [] used the idea of a modulus functionf to construct a class of FK spaces

L(f) = x= (xk) :

f|xk|

<∞

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In [], the notion of statistical convergence was deﬁned in a paranormed space.

Deﬁnition . A sequencex= (xk) is said to be statistically convergent to the numberL

in (X,g) if for eachε> ,

lim

n



nk≤n:g(xk–L)≥ε= .

In this case, we writeg(st)-limx=L. We denote the set of allg(st)-convergent sequences bySg[].

Deﬁnition . A sequencex= (xk) is said to be stronglyp-Cesaro summable ( <p<∞)

to the limitLin (X,g) if

lim

n



n n

j=

g(xj–L)

p

= ,

and we write it asxk−→L([C,g]p). In this case,Lis called the [C,g]p-limit ofx[].

In this article, we shall study the concept of m_{-lacunary statistical convergence,}

m_{-lacunary strong convergence and}m_{-lacunary strong convergence with respect to}

a modulus function in a paranormed space.

3 Generalized difference statistical convergence in a paranormed space

Deﬁnition . A sequencex= (xk) is said to bem-statistically convergent to the number Lin (X,g) if for eachε> ,

lim

n



nk≤n:g

mxk–L

≥ε= .

In this case, we writeSg(m)-limx=L. We denote the set of allm-statistically

conver-gent sequences in (X,g) bySg(m).

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

statistically convergent to the numberLin (X,g) if for eachε> ,

lim

r



hr

k∈Ir:g

mxk–L

≥ε= .

In this case, we writeSθ

g(m)-limx=L. We denote the set of allm-lacunary statistically

convergent sequences in (X,g) bySθ g(m).

Deﬁnition . A sequencex= (xk) is said to be strongly m-Cesaro summable to the

limitLin (X,g) if

lim

n



n n

k=

gmxk–L

= ,

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Deﬁnition . A sequencex= (xk) is said to be stronglym-lacunary strongly summable

to the limitLin (X,g) if

lim

r



hr

k∈Ir

gmxk–L

= ,

and we write it asxk−→L(Ngθ(m)). In this caseLis called theNgθ(m)-limofx.

Theorem . Letθbe a lacunary sequence and(X,g)be a paranormed space.Then

(i) Ifxk−→L(Ngθ(m)),thenxk−→L(Sgθ(m))and the inclusion is strict;

(ii) If xis am_{-bounded sequence and}_x

k−→L(Sθg(m)),thenxk−→L(Ngθ(m));

(iii) l∞_g (m)∩Sθ

g(m) =lg∞(m)∩Ngθ(m).

Proof (i) Ifε>  andxk→L(Ngθ(m)), we can write

k∈Ir

gmxk–L

≥

k∈Ir g(m_x_k–_L₎_≥_ε

gmxk–L

≥εk∈Ir:g

mxk–L

≥ε,

which yields the result.

In order to prove that the inclusionNθ

g(m)⊂Sθg(m) is proper, letθ be given and

X=Nθ

(,hr) ={x= (xk) :|  hr

k∈Irxk| 

hr →_,_r→ ∞}_{with the paranorm}_g₍_x_{) =}|x|₊

sup_r|_h

r

k∈Irxk| 

hr_{. Deﬁne} _x_{= (}_x_k_{) to be }_h_r_hr _{at the ﬁrst term in}_I

r for everyr≥, xk=hr(hr– hr–· · ·– (k– )hr) between the second term and ([

√

hr] + )th term inIr, xk=hr(hr– hr–· · ·– ([

√

hr])hr) at the ([ √

hr] + )th term inIrandxk=  otherwise.

We see that

xk=

⎧ ⎨ ⎩

hrhr,hrhr, . . . ,hr[ √

hr]hr, at the ﬁrst [ √

hr] integers inIr,

, otherwise.

and

g(xk) =

⎧ ⎨ ⎩

, , . . . , [√hr], at the ﬁrst [ √

hr] integers inIr,

, otherwise.

Note thatxis not-bounded in (X,g). We have, for everyε> ,



hr

k∈Ir:g(xk)≥ε)=

[√hr]

hr →



asr→ ∞,i.e.,xk→(Sgθ()). On the other hand,



hr

k∈Ir

g(xk) =



hr

[√hr]([ √

hr] + )

 →

 = ;

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(ii) Suppose thatxk→L(Sθg(m)) and sayg(mxk–L)≤Mfor allk. Givenε> , we get



hr

k∈Ir

gmxk–L

= 

hr

k∈Ir g(m_x

k–L)≥ε

gmxk–L

+ 

hr

k∈Ir g(m_x_k–_L_)<_ε

gmxk–L

≤M hr

k∈Ir:g

mxk–L

≥ε+ε,

from which the result follows.

(iii) This is an immediate consequence of (i) and (ii).

Corollary . If xk→L(|σ|g(m)), then xk →L(Sg(m)). If x∈lg∞(m)and if xk → L(Sg(m)),then xk→L(|σ|g(m)).

Theorem . Let θ be a lacunary sequence and (X,g) be a paranormed space, then

Sθ

g(m) =Sg(m)if and only if <lim infrqr≤lim suprqr<∞.

Proof Suppose thatlim infqr> , then there exists aδ>  such thatqr≥+δfor suﬃciently

larger, which implies that

hr kr

≥ δ

 +δ.

Ifxk→L(Sg(m)), then for everyε>  and suﬃciently larger, we have



kr

k≤kr:g

mxk–L

≥ε≥  kr

k∈Ir:g

mxk–L

≥ε

≥ δ

 +δ



hr

_k_∈_I_r:gmxk–L

≥ε,

which proves theSg(m)⊂Sθg(m).

Conversely, suppose that lim infqr = . Since θ is lacunary, we can select a

subse-quence (krj) of θ satisfying k_rj k_rj– <  +

 j and

k_rj–

kr_(j–) >j, where rj ≥ rj–+  and X =

N_θ(,_h_r) = {x= (xk) :|_h_r

k∈Irxk| 

hr →_,_r→ ∞} _{with the paranorm} _g₍_x_{) =} |x| ₊

sup_r|_h

r

k∈Irxk|  hr_.

Now deﬁne a sequence by

xk=

⎧ ⎨ ⎩

hr+k, k∈Ir(j),j= , , . . . ,

, otherwise.

We can see that

g(xk) =

⎧ ⎨ ⎩

, k∈Ir(j),j= , , . . . ,

, otherwise.

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We can see thatx∈/Nθ

g(m), butx∈σg(m). Theorem .(ii) implies thatx∈/Sgθ(m),

but it follows from Corollary . thatx∈Sg(m). HenceSg(m)⊂Sθg(m) andSg(m)⊂ Sθ

g(m) implies thatlim infqr> .

To show for any lacunary sequenceθ,Sθ

g(m)⊂Sg(m) implieslim supqr<∞, the same

technique of Lemma  of [] can be used. Now suppose thatlim supqr=∞. Consider the

same space deﬁned above and the sequence deﬁned by

xi=

⎧ ⎨ ⎩

hr+i, krj–<i≤krj–,j= , , . . . ,

, otherwise.

Then we get

g(xi) =

⎧ ⎨ ⎩

, krj–<i≤krj–,j= , , . . . ,

, otherwise.

Then x∈Nθ

g(m), butx∈/ σg(m). By Theorem .(i) we conclude thatx∈Sgθ(m),

but by Corollary . thatx∈/Sg(m). Hence,Sgθ(m)⊂Sg(m). This completes the proof.

Deﬁnition . Letfbe a modulus function. Then a sequencex= (xk) is lacunary strongly p-Cesaro summable toLwith respect tof in (X,g) if

lim

r



hr

kIr

fgmxk–L

pk

= .

In this case, we writexk→L(Ngθ(f,m,p)). If we takepk=  for allk∈N, we sayxk→ L(Nθ

g(f,m)).

Lemma . Let f be a modulus function and let <δ< .Then for each x>δ we have f(x)≤f()δ–x[].

Theorem . Let f be a modulus function and (X,g) be a paranormed space. Then Nθ

g(m)⊂Ngθ(f,m).

Proof Letx∈Nθ

g(m). Then we haveτr=_h_rk∈Irg( m_x

k–L)→ asr→ ∞for someL.

Letε>  and chooseδ with  <δ<  such thatf(u) <εforuwith ≤u≤δ. Then we can write



hr

k∈Ir

fgmxk–L

= 

hr

k∈Ir g(m_x_k–_L₎_≤_δ

fgmxk–L

+ 

hr

k∈Ir g(m_x

k–L)>δ

fgmxk–L

≤  hr

(hrδ) +



hr

f()δ–hrτr

from Lemma .. Thereforex∈Nθ

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Theorem . Let <infkpk≤pk≤supkpk<∞.Then Sθg(m) =Ngθ(f,m,p)if and only if f is bounded.

Proof Following the technique applied for establishing Theorem . of [], we can prove

the theorem.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The author would like to thank the referees for their careful reading of the manuscript and for their helpful suggestions.

Received: 14 December 2012 Accepted: 6 May 2013 Published: 20 May 2013 References

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