On generalized sequence spaces via modulus function

R E S E A R C H

Open Access

On generalized sequence spaces via modulus

function

Ekrem Sava¸s

*

*_{Correspondence:}

[email protected] Department of Mathematics, Istanbul Commerce University, Sütlüce, Istanbul, Turkey

Abstract

In this paper, we introduce and study the concept of lacunary strongly

(A,

ϕ)-convergence with respect to a modulus function and lacunary (A,ϕ)-statistical

convergence and examine some properties of these sequence spaces. We establish some connections between lacunary strongly (A,

ϕ)-convergence and lacunary

(A,

ϕ)-statistical convergence.

MSC: Primary 40H05; secondary 40C05

Keywords: modulus function; almost convergence; lacunary sequence;

ϕ-function;

statistical convergence

1 Introduction

Letsdenote the set of all real and complex sequencesx= (xk). Byl∞andc, we denote the Banach spaces of bounded and convergent sequencesx= (xk) normed byx=supn|xn|,

respectively. A linear functionalLonl∞is said to be a Banach limit [] if it has the following properties:

() L(x)≥ifn≥(i.e.xn≥for alln),

() L(e) = , wheree= (, , . . .),

() L(Dx) =L(x), where the shift operatorDis defined byD(xn) ={xn+}.

LetBbe the set of all Banach limits onl∞. A sequencex∈∞is said to be almost

con-vergent if all Banach limits ofxcoincide. Letcˆdenote the space of almost convergent

sequences. Lorentz [] has shown that

ˆ

c=

x∈l∞:lim

m tm,n(x) exists uniformly inn

,

where

tm,n(x) =

xn+xn++xn++· · ·+xn+m

m+  .

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

space of lacunary strongly convergent sequencesNθwas defined by Freedmanet al.[] as

follows:

Nθ=

x= (xk) :lim r

 hr

k∈Ir

|xk–l|= , for somel

.

In the special case whereθ= (r_{) (see []) we haveN}

θ=w, which is defined by

w=

x= (xk) :lim n

 n

n

k=

|xk–l|= , for somel

.

Das and Mishra [] have introduced the spaceACθ of lacunary almost convergent

se-quences and the space|ACθ|of lacunary strongly almost convergent sequences as follows:

ACθ=

x= (xk) :lim r

 hr

k∈Ir

(xk+n–L) = , for someLuniformly inn

and

|ACθ|=

x= (xk) :lim r

 hr

k∈Ir

|xk+n–L|= , for someLuniformly inn

.

Ruckle used the idea of a modulus functionf to construct a class ofFKspaces,

L(f) =

x= (xk) :

∞

k= f |xk|

<∞

.

The spaceL(f) is closely related to the spacel, which is anL(f) space withf(x) =xfor all realx≥.

In , Savaş [] generalized the concept of strong almost convergence by using a mod-ulusf andp= (pk) is a sequence of strictly positive real numbers as follows:

ˆ

c(f,p)=

x:lim

n

 n

n

k=

f |xk+m–L|

pk

= , for someL, uniformly inm

and

ˆ

c(f,p)_=

x:lim

n

 n

n

k=

f |xk+m|

pk

= , uniformly inm

.

More investigations in this direction and more applications of the modulus can be found in [–].

Following Ruckle [], a modulus functionf is a function from [,∞) to [,∞) such

that

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

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

By aϕ-function we understand a continuous non-decreasing functionϕ(u) defined for

u≥ and such thatϕ() = ,ϕ(u) > , foru>  andϕ(u)→ ∞asu→ ∞.

Aϕ-functionϕis called non-weaker than aϕ-functionψif there are constantsc,b,k,l>  such thatcψ(lu)≤bϕ(ku) (for all largeu) and we writeψ≺ϕ.

Aϕ-functionϕis said to satisfy the condition () (for all largeu) if for some constant k>  there is satisfied the inequalityϕ(u)≤kϕ(u) (see [, ]).

In this paper, we introduce and study some properties of the following sequence space which is generalization of Savaş [].

2 Main results

Letϕ andf be a given ϕ-function and modulus function, respectively, and letp= (pn)

be a sequence of positive real numbers. Moreover, letA= (Ai) be the generalized three

parametric real matrix withAi= (an,k(i)) and a lacunary sequenceθ be given. Then we

define the following sequence spaces:

Nθ(A,ϕ,f,p) =

x= (xk) :lim r

 hr

n∈Ir

f

∞

k=

ank(i)ϕ |xk|

pn

= , uniformly ini

.

Ifx∈N

θ(A,ϕ,f), the sequencexis said to be lacunary strong (A,ϕ)-convergent to zero with respect to a modulusf. Whenϕ(x) =xfor allx, we obtain

Nθ(A,f,p) =

x= (xk) :lim r

 hr

n∈Ir

f

∞

k= ank(i)xk

pn

= , uniformly ini

.

If we takef(x) =x, we write

Nθ(A,ϕ,p) =

x= (xk) :lim r

 hr

n∈Ir

∞

k=

ank(i)ϕ |xk|

pn

= , uniformly ini

.

If we takepk=p, for allk, we have

Nθ(A,ϕ,f) =

x= (xk) :lim r

 hr

n∈Ir

f

∞

k=

ank(i)ϕ |xk|

p

= , uniformly ini

.

If we takeA=Iandϕ(x) =x, respectively, then we have

Nθ=

x= (xk) :lim r

 hr

k∈Ir

f |xk|

pn

= 

.

If we define the matrixA= (ank(i)) as follows:

ank(i) :=



n, ifn≥k,

, otherwise,

then we have

Nθ(C,ϕ,f,p) =

x= (xk) :lim r

 hr

n∈Ir

f

 n

n

k=

ϕ |xk|

pn

= , uniformly ini

,

ank(i) :=



n, ifi≤k≤i+n– ,

then we have

N_θ(cˆ,ϕ,f,p) =

x= (xk) :lim r

 hr

n∈Ir

f  n

i+n

k=i ϕ |xk|

pn

= , uniformly ini

.

Ifx∈Nθ(ˆc,ϕ,f), the sequencexis said to be almost lacunary strongϕ-convergent to zero

with respect to a modulusf. In the next theorem we establish inclusion relations between

w(A,ϕ,f,p) andN

θ(A,ϕ,f,p). We now have the following.

Theorem . Let f be any modulus function and let there be aϕ-functionϕand a gener-alized three parametric real matrixA;let p= (pn)be a sequence of positive real numbers

and the sequenceθbe given.If

w(A,ϕ,f,p) =

x= (xk) :lim m  m m n= f ∞ k=

ank(i)ϕ |xk|

pn

= ,uniformly in i

,

then the following relations are true:

(a) Iflim infrqr> then we havew(A,ϕ,f,p)⊆Nθ(A,ϕ,f,p). (b) Ifsup_rqr<∞,then we haveNθ(A,ϕ,f,p)⊆w(A,ϕ,f,p).

(c)  <lim infrqr≤lim suprqr<∞,then we haveNθ(A,ϕ,f,p) =w(A,ϕ,f,p).

Proof (a) Let us suppose thatx∈w(A,ϕ,f,p). There existsδ>  such thatqr>  +δfor all

r≥ and we havehr/kr≥δ/( +δ) for sufficiently larger. Then, for alli,

 kr kr n= f ∞ k=

ank(i)ϕ |xk|

pn ≥  kr

n∈Ir

f ∞ k=

ank(i)ϕ |xk|

pn

=hr kr

 hr

n∈Ir

f ∞ k=

ank(i)ϕ |xk|

pn

≥ δ

 +δ

 hr

n∈Ir

f ∞ k=

ankϕ |xk|

pn

.

Hence,x∈Nθ(A,ϕ,f,p).

(b) If lim sup_rqr<∞ then there existsM>  such thatqr<Mfor all r≥. Letx∈

N

θ(A,ϕ,f,p) andεis an arbitrary positive number, then there exists an indexjsuch that for everyj≥jand alli,

Rj=

 hj

n∈Ir

f ∞ k=

ank(i)ϕ |xk|

pn

<ε.

Thus, we can also findK>  such thatRj≤Kfor allj= , , . . . . Now letmbe any integer

withkr–≤m≤kr, then we obtain, for alli,

I=  m m n= f ∞ k=

ank(i)ϕ |xk|

pn

≤ 

kr–

kr n= f ∞ k=

ank(i)ϕ |xk|

pn

where

I=  kr–

j

j=

n∈Ij

f

∞

k=

ank(i)ϕ |xk|

pn

,

I=  kr–

m

j=j+

n∈Ij

f

∞

k=

ank(i)ϕ |xk|

pn

.

It is easy to see that

I =  kr–

j

j=

n∈Ij

f

∞

k=

ank(i)ϕ |xk|

pn

= 

kr–

n∈I

f

∞

k=

ank(i)ϕ |xk|

pn

+· · ·+

n∈Ij_

f

∞

k=

ank(i)ϕ |xk|

pn

≤ 

kr–

(hR+· · ·+hjRj)

≤ 

kr–

jkj sup

≤i≤j

Ri

≤jkj

kr– K.

Moreover, we have for alli

I =  kr–

m

j=j+

n∈Ij

f

∞

k=

ankϕ |xk|

pn

= 

kr–

m

j=j+

 hj

n∈Ij

f

∞

k=

ankϕ |xk|

pn

hj

≤ε 

kr–

m

j=j+

hj

≤ε kr

kr– =εqr<ε·M.

ThusI≤jkj

kr–K+ε·M. Finally,x∈w(A,ψ,f,p).

The proof of (c) follows from (a) and (b). This completes the proof.

Theorem . Let f,f,be modulus functions.Then we have

Nθ(A,f,ϕ,p)⊂(A,ϕ,fof,p).

Proof This can be proved by using techniques similar to those used in the theorem of

Savaş [].

Letθ be a lacunary sequence, and letA= (ank(i)) be the generalized three parametric

real matrix, the sequence x= (xk), theϕ-functionϕ(u) and a positive numberε>  be

given. We write, for alli,

Kθr(A,ϕ,ε) =

n∈Ir:

∞

k=

ank(i)ϕ |xk|

≥ε

.

The sequencexis said to be (A,ϕ)-statistically convergent to a number zero if for every

ε> 

lim

r

 hr

μ Kθr(A,ϕ,ε)

= , uniformly ini,

whereμ(Kr

θ(A,ϕ,ε)) denotes the number of elements belonging toKθr(A,ϕ,ε). We denote byS

θ(A,ϕ), the set of sequencesx= (xk) which are lacunary (A,ϕ)-statistical convergent

to zero and we write

S_θ(A,ϕ) =

x= (xk) :lim r

 hr

μ K_θr(A,ϕ,ε)= , uniformly ini

.

More investigations in this direction can be found in [–].

We now establish inclusion relations betweenN

θ(A,ϕ,f,p) andSθ(A,ϕ). In the following theorem we assume that  <h=infpn≤pn≤suppk≤H≤ ∞. Theorem . (a)If the matrix A and the sequenceθand functions f andϕare given,then

Nθ(A,ϕ,f,p)⊂Sθ(A,ϕ).

(b)If theϕ-functionϕ(u)and the matrix A are given,and if the modulus function f is bounded,then

Sθ(A,ϕ)⊂Nθ(A,ϕ,f,p).

Proof (a) Letf be a modulus function and letεbe a positive numbers. We write the fol-lowing inequalities, for alli,

 hr

n∈Ir

f

∞

k=

ank(i)ϕ |xk|

pn

= 

hr

n∈Ir

f

∞

k=

ank(i)ϕ |xk|

pn

≥ 

hr

n∈Ir

f(ε)pn

≥ 

hr

n∈Ir

min f(ε)infpn,f(ε)H

≥ 

hr

μ Kθr(A,ϕ,ε)

where

I_r=

n∈Ir:

∞

k=

ank(i)ϕ |xk|

≥ε

.

Finally, ifx∈N

θ(A,ϕ,f,p) thenx∈Sθ(A,ϕ,f).

(b) Let us suppose thatx∈Sθ(A,ϕ). If the modulus functionf is a bounded function, then there exists an integerKsuch thatf(x) <Kforx≥. Let us take

I_r=

n∈Ir:

∞

k=

ank(i)ϕ |xk|

<ε

.

Thus we have, for alli,

 hr

n∈Ir

f

∞

k=

ank(i)ϕ |xk|

pn

≤ 

hr

n∈Ir

f

∞

k=

ank(i)ϕ |xk|

pn

+ 

hr

n∈Ir

f

∞

k=

ank(i)ϕ |xk|

pn

≤ 

hr

n∈Ir

max Kh,KH+  hr

n∈Ir

f(ε)pn

≤max Kh,KH hr

μ Kθr(A,ϕ,ε)

+max f(ε)h,f(ε)H.

Taking the limit asε→, we observe thatx∈N

θ(A,ϕ,f,p).

This completes the proof.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

I would like to express my gratitude to the referee of the paper for his useful comments and suggestions towards the quality improvement of the paper. This paper was presented during the ‘2nd International Eurasian Conference on Mathematical Sciences and Applications’ held on Sarajevo, Bosnia and Herzegovina on 26th to 29th August 2013, and it was submitted for the conference proceedings.

Received: 25 September 2013 Accepted: 4 February 2014 Published:04 Mar 2014 References

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