Generalized Difference Lacunary I-convergent spaces

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Spaces of Generalized difference Lacunary I-convergent

sequences

Adem Kilic¸man1and Stuti Borgohain2

1_{Department of Mathematics and Institute for Mathematical Research University of Putra Malaysia, Serdang, Selangor 43400, Malaysia} 2_{Department of Mathematics, Indian Institute of Technology, Bombay, Powai:400076, Mumbai, Maharashtra, India.}

Received: 20 January 2015, Revised: 8 April 2015, Accepted: 19 May 2015 Published online: 17 June 2015

Abstract: In this research article, we studied some new generalized difference strongly summable lacunaryI-convergentn-normed sequence spaces related toℓpspaces which are defiend by Orlicz functions. Some results involved with these spaces are also investigated

and studied. We also give some relations related to these sequence spaces.

Keywords: Lacunary sequence,n-norm, Orlicz function; Difference operator; Ideal Convergence; de la Vall `ee Poussin mean.

1 Introduction

The concept of the crisp set sequence spacem(φ) was initiated by W.L.C. Sargent which was later on studied and investigated from the sequence space point of view by many other mathematicians. Recently, some researchers worked on some matrix classes characterized by takingm(φ)as one member.

Kostyrko [13] introduced the concept of Ideal convergence as a generalization form of statistical convergence.

A lacunary sequence is defined as an increasing integer sequenceθ= (kr)such thatk0=0 andhr=kr−kr−1→∞as r→∞.

For any lacunary sequenceθ= (kr), the spaceNθis defined as ,

Nθ= {

(xk): lim r→∞h

−1

r

∑

k∈Jr

|xk−L|=0,for someL

}

.

The spaceNθ is aBKspace with the norm,

∥(xk)∥θ=sup

r

h−_r1

_∑

k∈Jr

|xk|.

Note: Throughout this paper, the intervals determined byθ will be denoted byJr= (kr−1,kr]and the ratio _k_rk₋r₁ will be

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By an Orlicz function, we mean a functionM:[0,∞)→[0,∞), which is continuous, non-decreasing and convex with M(0) =0,M(x)>0, forx>0 andM(x)→∞, asx→∞.

A sequencex∈ℓ∞is said to be almost convergent if all of its Banach limits coincide. Let ˆcdenote the space of all almost convergent sequences. Lorentz[9] introduced the following sequence space as,

ˆ

c={x∈ℓ∞: lim

m tm,n(x)exists uniformly in n}

where

tm,n(x) =

xn+xn+1+...+xm+n

m+1 .

The notion of difference sequence spaces of crisp sets are defined asZ(∆) ={x= (xk):(∆xk)∈Z},forZ=ℓ∞,candc0, where∆x= (∆xk) = (xk−xk+1), for allk∈Nand later on which was generalized by many other recent mathematicians.

These spaces are Banach spaces, normed by,(see Kizmaz [10])

∥x∥∆=|x1|+sup

k

|∆xk|.

2 Definitions and Preliminaries

Letn∈NandXbe a real vector space. A real valued function onXnsatisfying the following four properties: 1.∥(z1,z2, ...zn)∥n=0 if and only ifz1,z2, ...znare linearly dependent;

2.∥(z1,z2, ...zn)∥nis invariant under permutation;

3.∥(z1,z2, .., .zn−1,αzn)∥n=|α|∥z1,z2, ...zn∥n, for allα∈R;

4.∥(z1,z2, ...zn−1,x+y)∥n≤ ∥(z1,z2, ...zn−1,x)∥n+∥(z1,z2, ...zn−1,y)∥n;

is called ann-norm onXand the pair(X,∥, ., .∥n)is called ann-normed space.

The spacem(φ)is defined as,

m(φ) =

{

(xk)∈w:∥x∥m(φ)= sup

s≥1,σ∈℘s 1 φs_k

∑

_∈_σ

|xk|<∞

}

.

The idea of Orlicz function is used to construct the sequence space, (see Lindenstrauss and Tzafriri [11])

ℓM=

{

(xk)∈w:

∞

∑

k=1 M

(

|x|

ρ )

<∞, for someρ>0 }

which becomes a Banach space, called as Orlicz sequence space, with the following norm,

∥x∥=inf {

ρ>0 : ∞

∑

k=1 M

(_|_x

k|

ρ )

≤1 }

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For a lacunary sequenceθ= (kr), a sequence(xk)is said to be lacunaryI-convergent if for everyε>0 such that,

{

r∈N:h−_r1

∑

k∈Jr

|xk−x| ≥ε

}

∈I.

We writeIθ−limxk=x.

In this article, we define some new generalized difference lacunaryI-convergent sequence spaces in n-normed spaces related toℓp-space by using Orlicz function. We will also introduce and examine certain new sequence spacIes using the

above tools.

3 Main Results

Letu= (uk)be a sequence of real numbers such thatuk>0 for allk, and sup k

uk<∞. Also, letIbe an admissible ideal of

NandMbe an orlicz function. In this article, we have introduced the following sequence space as,

(m(M,φ,∆qp,u,θ)I,∥, ...∥n)

=

{

x:∀ε>0 (

r∈N:_h1 r

( sup

∑

_∈_σ

( M

(

∥∆pqtkm(x)−L,z1,z2, ...zn−1∥n

ρ

))uk)

≥ε )

∈I }

.

for someρ>0,∀z1,z2, ...zn−1∈X.

Particular cases: If we takeuk=1, for allk, we have,

(m(M,φ,∆qp,θ)I,∥, ...∥n)

=

{

x:∀ε>0 (

r∈N:_h1 r

( sup

∑

_∈_σ

M (_∥_∆q

ptkm(x)−L,z1,z2, ...zn−1∥n

ρ

))

≥ε )

∈I }

.

for someρ>0,∀z1,z2, ...zn−1∈X.

Now, if we considerM(x) =x, then we can easily obtain:

(m(φ,∆qp,u,θ)I,∥, ...∥n)

=

{

x:∀ε>0 (

r∈N:_h1 r

( sup

∑

_∈_σ

(∥∆q

ptkm(x)−L,z1,z2, ...zn−1∥n)uk

)

≥ε, uniformly inm )

∈I }

.

Ifx∈(m(M,φ,∆pq,u,θ)I,∥, ...∥n)with

{ 1

hr

( sup

∑

_∈_σ

( M

(_∥_∆q

ptkm(x)−L,z1,z2, ...zn−1∥n

ρ

))uk)

≥ε }

∈Iasn→∞, uniformly inm, then we write

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The following well known inequality will be used later.

If 0≤uk≤supuk=HandC=max(1,2H−1), then

|ak+bk|uk≤C{|ak|uk+|bk|uk}, (1)

for allkandak,bk∈C.

Theorem 1. Let lim

k→∞infuk>0. Then, xk→L implies xk →L∈(m(M,φ,∆

q

p,u,θ)I,∥, ..∥n). Let lim

k→∞uk=u >0. If xk→L∈(m(M,φ,∆pq,u,θ)I,∥, ..∥n), thenLis unique.

Proof: Letxk→L.

By the definition of Orlicz function, we have, for allε>0,

{ 1 hr

( sup

∑

_∈_σ

M (_∥_∆q

ptkm(x)−L,z1,z2, ..zn−1∥n

ρ

))

≥ε }

∈I.

Since lim

k infuk>0, it follows that,

{ 1 hr

( sup

∑

_∈_σ

( M

(

∥∆qptkm(x)−L,z1, ..zn−1∥n

ρ

))uk)

≥ε }

∈I

And consequently,xk→L∈(m(M,φ,∆pq,u,θ)I,∥, ....∥n).

Let lim

k sk=s>0. Suppose that,

xk→L1∈(m(M,φ,∆qp,u,θ)I,∥, ..∥n)

xk→L2∈(m(M,φ,∆qp,u,θ)I,∥, ..∥n)

and(∥L1−L2,z1,z2, ...zn−1∥n)uk=a>0.

Now, using the definitions of inequality and Orlicz function, we have,

1 hr

{ sup

∑

_∈_σ

( M

(

|L1−L2,z1, ...zn−1∥n

ρ

))uk}

≤C

hr

{ sup

∑

_∈_σ

( M

(_|_∆q

ptkm(x)−L1,z1, ...zn−1∥n

ρ

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+C

hr

{ sup

∑

_∈_σ

( M

(_|_∆q

ptkm(x)−L2,z1, ...zn−1∥n

ρ

))uk}

Since,

{

r∈N: 1 hr

( sup

∑

_∈_σ

( M

(_∥_∆q

ptkm(x)−L1,z1, ..zn−1∥n

ρ

))uk)

≥ε }

∈I,

and

{

r∈N: 1 hr

( sup

∑

_∈_σ

( M

(_∥_∆q

ptkm(x)−L2,z1, ..zn−1∥n

ρ

))uk)

≥ε }

∈I,

Hence,

{

r∈N: 1 hr

( sup

∑

_∈_σ

( M

(

∥L1−L2,z1, ..zn−1∥n

ρ

))uk)

≥ε }

∈I (2)

Further,

M (_∥_L

1−L2,z1,z2, ..zn−1∥n

ρ

)uk

→M (_a

ρ )u

ask→∞,and therefore,

1 hr

{ sup

∑

_∈_σ

M (_∥_L

1−L2,z1, ..zn−1∥n

ρ

)uk} =M

(_a

ρ )u

. (3)

From the above equations ((3) and (4)), it follows thatM(_ρa)=0 and by the definition of an Orlicz function, we have a=0. HenceL1=L2and this completes the proof.

Theorem 2.

1. Let 0<infuk≤uk≤1. Then,m(M,φ,∆qp,u,θ)I⊂m(M,φ,∆pq,θ)I

2. Let 0<uk≤supuk<∞. Then,m(M,φ,∆qp,θ)I⊂m(M,φ,∆pq,u,θ)I.

Theorem 3.The inclusionm(M,φ,∆pq−1,u,θ)I⊂m(M,φ,∆pq,u,θ)Iis strict.

In general,m(M,φ,∆i

p,u,θ)I⊂m(M,φ,∆ q

P,u,θ)Ifor alli=1,2,3, ...p−1 and the inclusion is strict.

Theorem 4.m(M,φ,∆pq,u,θ)Iis a complete linear topological space,with paranormg, wheregis defined by,

g(x) =

pq

∑

m=1

∥tkm(x)−L,z1,z2, ..zn−1∥n

+inf {

ρukH _: 1

hr

{ sup

∑

_∈_σ

( M

(

∥∆pqtkm(x)−L,z1,z2, ..zn−1∥n

ρ

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whereH=max(1,(sup

k

uk)).

Proposition 5. m(M,φ,∆qp,u,θ)I⊆m(M,ψ,∆pq,u,θ)Iif and only if sup s≥1

(_φ

s

ψs

)

<∞, for 0<p<∞.

Proof: First, suppose that sup

s≥1 (

φs

ψs

)

=K<∞, then we have,φs≤Kψs.

Now, if(xk)∈m(M,φ,∆pq,u,θ)I, then

1

hr

{ sup

∑

_∈_σ

( M

(_∥_∆q

ρ

))uk

≥ε }

∈I

⇒_h1_r

{ sup

s≥1,σ∈℘s 1 Kψs_k

∑

_∈_σ

( M

(_∥_∆q

ρ

))uk

≥ε }

∈I i.e(xk)∈m(M,ψ,∆qp,u,θ)I.

Hence,m(M,φ,∆pq,u,θ)I⊆m(M,ψ,∆qp,u,θ)I.

Conversely, suppose that m(M,φ,∆pq,u,θ)I ⊆m(M,ψ,∆qp,u,θ)I. We should prove that sup s≥1

(_φ

s

ψs

)

=sup

s≥1

(ηs)<∞.

Suppose that sup

s≥1

(ηs) = ∞. Then there exists a subsequence (ηsi) of (ηs) such that lim

i→∞(ηsi) = ∞. Then for (xk)∈m(M,φ,∆pq,u,θ)I, we have,

1

hr

{ sup

s≥1,σ∈℘s 1 ψs_k

∑

_∈_σ

( M

(_∥_∆q

ρ

))uk}

≥ 1

hr

{ sup

s≥1,σ∈℘s

(_η

si

φsi

)

∑

k∈σ (

M (_∥_∆q

ρ

))uk} =∞

which implies that(xk)∈/m(M,φ,∆pq,u,θ)I, a contradiction. This completes the proof.

Corollary 6.m(M,φ,∆qp,u,θ)I=m(M,ψ,∆pq,u,θ)I, if and only if sup s≥1

(ηs)<∞and sup s≥1

(ηs−1)<∞, whereηs=_ψφs_s, for

0<p<∞.

References

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