Some generalized difference sequence spaces of invariant means defined by Orlicz functions

INVARIANT MEANS DEFINED BY ORLICZ FUNCTIONS

Received 23 September 2004 and in revised form 5 April 2005

We define the sequence spaces [wθ,M,p,u,∆]σ, [wθ,M,p,u,∆]0

σ, and [wθ,M,p,u,∆]∞σ which are defined by combining the concepts of Orlicz functions, invariant means, and lacunary convergence. We also study some inclusion relations and linearity properties of the above-mentioned spaces. These are generalizations of those defined and studied by Savas¸ and Rhoades in 2002 and some others before.

1. Introduction

Letl∞andcdenote the Banach spaces of bounded and convergent sequencesx=(xk),

withxk∈RorC, normed byx =supk|xk|, respectively.

A paranorm on a linear topological spaceXis a functiong:X→Rwhich satisfies the following axioms for anyx,y,x0∈Xandλ,λ0∈C:

(i)g(θ)=0, whereθ=(0, 0, 0,...), the zero sequence, (ii)g(x)=g(−x),

(iii)g(x+y)≤g(x) +g(y) (subadditivity),

(iv) the scalar multiplication is continuous, that is,

λ−→λ0, x−→x0 implyλx−→λ0x0; (1.1)

in other words,

_{λ−λ0−→0, gx}

−x0−→0 implygλx−λ0x0−→0. (1.2)

A paranormed space is a linear spaceXwith a paranormgand is written as (X,g), (see Maddox [10, page 92]).

Any functiong which satisfies all conditions (i), (ii), (iii), and (iv), together with the condition

(v)g(x)=0 if and only ifx=θ,

is called a total paranorm onX, and the pair (X,g) is called a total paranormed space, (see Maddox [10, page 92]).

Copyright©2005 Hindawi Publishing Corporation

Schaefer [20] defined theσ-convergence as follows.

Letσ be the mapping of the set of positive integers into itself. A continuous linear functionalϕonl∞is said to be an invariant mean orσ-mean if and only if

(i)ϕ(x)≥0 when the sequencex=(xn) hasxn≥0 for alln, (ii)ϕ(e)=1,

(iii)ϕ(xσ(n))=ϕ(x) for allx∈l∞.

In caseσis the translation mappingn→n+ 1, aσ-mean is often called a Banach limit andVσ, the set of bounded sequences all of whose invariant means are equal, is the set of almost convergence sequences.

A sequencex=(xk)∈l∞ is said to be almost convergent if all of its Banach limits

coincide (see Banach [1]). Let cdenote the space of all almost convergent sequences. Lorentz [8] proved that

c=x∈l∞: lim_m→∞tmn(x) exists, uniformly inn

, (1.3)

where (tmn(x)=xn+xn+1+···+xn+m)/(m+ 1).

The space [c] of strongly almost convergence sequences was introduced by Maddox [11] and Freedman et al. [4] as follows:

[c]=x∈l∞: lim_m→∞tmn|x−le|=0, uniformly inn, for somel∈R, (1.4)

wheree=(1, 1,...).

Ifx=(xk), writeTx=(Txk)=(xσ(k)). It can be shown that

Vσ=

x∈l∞: lim_m→∞tkm(x)=l, uniformly inm

, (1.5)

l=σ−limx, where

tkm(x)=xm+xσ(m)+x_kσ2(m)+···+xσk(m)

+ 1 . (1.6)

Hereσk(m) denotes thekth iterate of the mappingσatm.

Aσ-mean extends the limit functional oncin the sense thatϕ(x)=limxfor allx∈cif and only ifσhas no finite orbits; that is to say, if and only if for alln≥0,j≥0,σj(n)=n (see Mursaleen [13]).

A sequencexis said to be stronglyσ-convergent if there exists a numberlsuch that

_{xk−l∈Vσ (1.7)}

with the limit zero (see Mursaleen [12]).

Using the concept of invariant means, the following sequence spaces have been intro-duced and examined by Mursaleen et al. [14] as a generalization of Das and Sahoo [3]:

wσ=

x: lim_{n n}1 + 1

n

k=0

tkm(x−l)=0, for somel∈R, uniformly inm ,

[w]σ=

x: lim_{n n}1 + 1

n

k=0

_{tkm(x−l)=0, for somel∈R, uniformly inm ,}

wσ=

x: lim_n 1 n+ 1

n

k=0

tkm|x−l|=0, for somel∈R, uniformly inm .

(1.8)

By a lacunary sequenceθ=(kr);r=0, 1, 2,..., wherek0=0, we will mean an increas-ing sequence of nonnegative integers withkr−kr−1→ ∞. The intervals determined by θ will be denoted byIr=(kr−1,kr] and we lethr=kr−kr−1. The ratiokr/kr−1 will be denoted byqr. The space of lacunary strongly convergent sequencesNθ was defined by Freedman et al. [4] as

Nθ=

x=xk: lim_{r h}1 r

k∈Ir

_{xk−l=0, for somel . (1.9)}

The concept of lacunary strongσ-convergence was introduced by Savas¸ [18] which is a generalization of the idea of lacunary strong almost convergence due to Das and Patel [2].

If [Vσθ] denotes the set of all lacunary stronglyσ-convergent sequences, then Savas¸ [18] defined

Vθ σ=

x=xk: lim_r→∞h1 r

k∈Ir

_xσk_(n)−l=0, for somel, uniformly inn . (1.10)

Recall [5,7] that an Orlicz function is a functionM: [0,∞)→[0,∞) which is contin-uous, nondecreasing, and convex withM(0)=0,M(x)>0 forx >0, andM(x)→ ∞as x→ ∞.

If convexity ofMis replaced byM(x+y)≤M(x) +M(y), then it is called a modulus function, defined and discussed by Ruckle [16].

Lindenstrauss and Tzafriri [7] used the idea of Orlicz function to define what is called an Orlicz sequence space:

lM:=

x:

∞

k=1

Mx_ρk

<∞, for someρ >0 (1.11)

which is a Banach space with the norm

xM=inf

ρ >0 :

∞

k=1

Mx_ρk

An Orlicz functionM is said to satisfy the∆2-condition for all values ofu, if there exists a constantK >0 such that

M(2u)≤KM(u) (u≥0). (1.13)

It is easy to see that alwaysK >2. The∆2-condition is equivalent to the satisfaction of the inequality

M(lu)≤K(lu)M(u) (1.14)

for every value ofuand forl >1 (see Krasnosel’ski˘ı and Ruticki˘ı [6]).

Parashar and Choudhary [15] have introduced and examined some properties of four sequence spaces defined by using an Orlicz functionM, which generalizes the well-known Orlicz sequence spacelM and strong summable sequence spaces [C, 1,p], [C, 1,p]0, and [C, 1,p]∞. It may be noted that the spaces of strongly summable sequences were discussed

by Maddox [9].

The main object of this paper is to define and study the sequence spaces: [wθ,M,p,u,

∆]σ, [wθ,M,p,u,∆]0

σand [wθ,M,p,u,∆]∞σ, which are defined by combining the concepts of an Orlicz function, invariant mean, and lacunary convergence. We examine some lin-earity and inclusion relations of these spaces and establish the connection between lacu-nary [w]σ-convergence and lacunary [w]σ-convergence with respect to an Orlicz function which satisfies the∆2-condition.

Now, ifu=(uk) is any sequence such thatuk=0(k=1, 2,...) and for any sequence x=(xn), the difference sequence∆xis given by∆x=(∆xn)∞n=1=(xn−xn−1)∞n=1, then we

define the following sequence spaces:

wθ_,M,p,u,∆ σ=

x=xk: lim_{r h}1 r

k∈Ir

Mtkm

u∆x−le ρ

pk

=0, for somel,ρ >0, uniformly inm ,

wθ_,M,p,u,∆0 σ=

x=xk: lim_{r h}1 r

k∈Ir

Mtkm(_ρu∆x)

pk

=0, for someρ >0, uniformly inm ,

wθ_,M,p,u,∆∞

σ =

x=xk: sup r,m

1 hr

k∈Ir

M tkm(_ρu∆x)

pk

<∞for someρ >0 .

(1.15)

Ifu=e, and∆xk=xk, for allk, then these spaces reduce to those defined and studied by Savas¸ and Rhoades [19]. Also some sequence spaces are obtained by specializingθ, M, and p. For example, if u=e,∆xk=xk, andpk=1 for allk, then we get the spaces [wθ,M]σ, [wθ,M]0

Ifx∈[wθ,M]σ, we say thatxis lacunary [w]σ-convergence with respect to the Orlicz functionM.

Whenu=e,∆xk=xk, for allk, andσ(n)=n+ 1, the spaces [wθ,M,p,u,∆]σ, [wθ,M,p, u,∆]0

σ, and [wθ,M,p,u,∆]∞σ reduce to the spaces [wθ,M,p]σ, [wθ,M,p]0σ, and [wθ,M, p]∞_σ, respectively, which are defined as

wθ,M,p=

x=xk: lim_{r h}1 r

m∈Ir

Mtmn(x_ρ−l)

pk

=0, uniformly inn, for somel,ρ >0 ,

wθ,M,p₀=

x=xk: lim_{r h}1 r

m∈Ir

Mtmn(_ρx)

pk

=0, uniformly inn, for someρ >0 ,

wθ,M,p_∞=

x=xk: sup r,n

1 hr

m∈Ir

Mtmn(_ρx)

pk

<∞, for someρ >0 .

(1.16)

Ifu=e, and∆xk=xk, for allk,M(x)=x,θ=(2r), andpk=1, for allk, then [wθ,M,p, u,∆]σ=[w]σ (see Mursaleen et al. [14]) and [wθ,M,p,u,∆]=[w] (see Das and Sahoo [3]). Whenu=e,∆xk=xk, for allk,M(x)=x, andpk=1, for allk, then [wθ,M,p,u,∆]σ =[wθ]σ, [wθ,M,p,u,∆]0

σ=[w]0σ. Ifu=e,∆xk=xk, for allk, andθ=(2r), then [wθ,M,p, u,∆]σ=[w,M,p]σ, [wθ,M,p,u,∆]0

σ=[w,M,p]0σ, and [wθ,M,p,u,∆]∞σ =[w,M,p]∞σ.

2. Main results

We proved the following theorems.

Theorem2.1. For any Orlicz functionMand a bounded sequencep=(pk)of strictly posi-tive real numbers,[wθ,M,p,u,∆]σ,[wθ,M,p,u,∆]0

σ, and[wθ,M,p,u,∆]∞σ are linear spaces over the set of complex numbers.

Proof. We will prove the result only for [wθ,M,p,u,∆]0

σ. The others can be treated sim-ilarly. Letx,y∈[wθ,M,p,u,∆]0

σ andα,β∈C, the set of complex numbers. In order to prove the result, we need to find someρ3>0 such that

lim_{r h}1 r

k∈Ir

Mtkm(αu∆_ρx+βu∆y) 3

pk

Sincex,y∈[wθ,M,p,u,∆]0

σ, there exist positiveρ1andρ2such that

lim_{r h}1 r

k∈Ir

Mtkm(_ρu∆x) 1

pk =0,

lim_{r h}1 r

k∈Ir

Mtkm(_ρu∆y) 2

pk =0,

(2.2)

uniformly inm.

Defineρ3=max(2|α|ρ1, 2|β|ρ2). SinceMis nondecreasing and convex,

1 hr

k∈Ir

Mtkm(αu∆_ρx+βu∆y) 3

pk

≤_h1 r

k∈Ir 1 2pk

Mtkm(αu_ρ ∆x) 1

+Mtkm(βu_ρ ∆y) 2

pk

≤D_h1 r

k∈Ir

Mtkm(αu_ρ ∆x) 1

pk +D_h1

r

k∈Ir

Mtkm(βu_ρ ∆y) 2

pk

(2.3)

→0, asr→ ∞, uniformly inm, whereD=max(1, 2H−1_{),H=suppk, so thatαux+βuy∈} [wθ,M,p,u,∆]0

σ. This completes the proof.

Theorem2.2. For any Orlicz functionMand a bounded sequencep=(pk)of strictly posi-tive real numbers,[wθ,M,p,u,∆]0

σ is a topological linear space, totally paranormed by:

g(x)=inf

    ρpr/H:

 1

hr

k∈Ir

Mtkm(_ρx)

pk



1/H

≤1, r=1, 2,m=1, 2,...

    , (2.4)

whereH=max(1, suppk).

Proof. It is easy to see thatg(x)=g(−x). By usingTheorem 2.1, forα=β=1, we get g(x+y)≤g(x) +g(y). SinceM(0)=0, we get inf{ρpr/H_{} =0 forx=0. Conversely,} sup-poseg(x)=0, then

inf     ρ pr/H_:  1 hr

k∈Ir

Mtkm(u_ρ∆x)

pk  1/H ≤1    

=0. (2.5)

This implies that for a given>0, there exists someρ3(0< ρ3<) such that

 1

hr

k∈Ir

M tkm(_ρu∆x) 3

pk



1/H

Thus

 1

hr

k∈Ir

Mtkm(u∆x)

pk



1/H ≤

 1

hr

k∈Ir

M tkm(_ρu∆x) 3

pk



1/H ≤1,

(2.7)

for eachrandm.

Suppose thatxσi_(j)=0 for eachiand j. This implies thatt_ij(x)=0, for eachiand j. Let→0. Then

tij(x)

−→ ∞. (2.8)

It follows that

 1

hr

i∈Ir

Mtij(u∆x)

pk



1/H

−→ ∞ (2.9)

which is a contradiction.

Therefore,tij(x)=0 for eachiand j, and thusxσi_(j)=0 for eachiandj.

Showing that scalar multiplication is continuous is a standard argument, so we omit

it.

Theorem 2.3. For any Orlicz function M which satisfies the ∆2-condition, [wθ,u,

∆]σ⊆[wθ,M,u,∆].

To prove the theorem, we need the following lemma.

Lemma2.4. LetMbe an Orlicz function which satisfies the∆2-condition and let0< δ <1. Then for eachx≥δ,M(x)< Kxδ−1_{M(2)for some constantK >0.}

Proof. It follows by a straightforward calculation using the∆2-condition. Proof ofTheorem 2.3. Letx∈[wθ,u,∆]σ. Then we have

Ar=_h1 r

k∈Ir

_{tkm(u∆x−le)−→0, (2.10)}

asr→ ∞, uniformly inm, for somel.

Let>0 and chooseδwith 0< δ <1 such thatM(t)<for 0≤t≤δ. Then we can write

1 hr

k∈Ir

Mtkm(u∆x−le)=_h1 r

k∈Ir,|tkm(u∆x−le)|≤δ

Mtkm(u∆x−le)

+_h1 r

k∈Ir,|tkm(u∆x−le)|>δ

Mtkm(u∆x−le)

< h−1

r hr+h−r1Kδ−1M(2)hrAr, byLemma 2.4.

(2.11)

Theorem2.5. Letθ=(kr)be a lacunary sequence withlim infrqr>1. Then for any Or-licz functionM,[w,M,p,u,∆]σ⊂[wθ,M,p,u,∆]σ,[w,M,p,u,∆]0

σ⊂[wθ,M,p,u,∆]0σ, and [w,M,p,u,∆]∞σ ⊂[wθ,M,p,u,∆]∞σ, where

[w,M,p,u,∆]σ=

x=xk: lim_{n n}1 + 1

n

k=0

M tkm(u∆_ρx−le)

pk

=0,uniformly inm,for somel,ρ >0 ,

[w,M,p,u,∆]∞_σ =

x=xk: sup n

1 n+ 1

n

k=0

Mtkm(_ρu∆x)

pk

<∞,for someρ >0 .

(2.12)

(In casel=0, write[w,M,p,u,∆]σ=[w,M,p,u,∆]0 σ).

Proof. We will prove [w,M,p,u,∆]σ⊂[wθ,M,p,u,∆]σ only. The others can be treated similarly. It is sufficient to show that [w,M,p,u,∆]0

σ⊂[wθ,M,p,u,∆]0σ; the general in-clusion follows by linearity. Suppose lim infrqr>1, then there exists δ >0 such that qr=(kr/kr−1)≥1 +δfor allr≥1. Then forx∈[w,M,p,u,∆]0σ, we write

Ar=_h1 r

k∈Ir

Mtkm(u_ρ∆x)

pk

=_h1 r

kr

k=1

Mtkm(_ρu∆x)

pk −_h1

r kr−1

k=1

Mtkm(u∆_ρx−le)

pk

=_hkr r

 _k−1

r kr

k=1

M tkm(u_ρ∆x)

pk

₋kr−1 hr

 _k−1

r−1 kr−1

k=1

Mtkm(u_ρ∆x)

pk

_.

(2.13)

Sincehr=kr−kr−1, we have

kr hr ≤

1 +δ δ ,

kr−1 hr ≤

1

δ. (2.14)

The termsk−1

r krk=1[M(|tkm(u∆x)|/ρ)]pkandk−r−11

kr−1

k=1[M(|tkm(u∆x)|/ρ)]pkboth con-verge to zero uniformly inm, and it follows thatArconverges to zero asr→ ∞, uniformly inm, that is,x∈[wθ,M,p,u,∆]0

σ. This completes the proof.

Theorem2.6. Letθ=(kr)be a lacunary sequence withlim sup_rqr<∞. Then for any Or-licz functionM,[wθ,M,p,u,∆]σ⊂[w,M,p,u,∆]σ,[wθ,M,p,u,∆]0

σ⊂[w,M,p,u,∆]0σ, and [wθ,M,p,u,∆]∞σ ⊂[w,M,p,u,∆]∞σ.

Proof. We will prove [wθ,M,p,u,∆]σ ⊂[w,M,p,u,∆]σ only. The others can be treated similarly. It is sufficient to show that [w,M,p,u,∆]0

σ⊂[wθ,M,p,u,∆]0σ; the general inclu-sion follows by linearity. Suppose lim suprqr<∞, then there existsB >0 such thatqr< B for allr≥1. Let x∈[wθ,M,p,u,∆]0

everyj≥Land allm,

Aj=_h1 j

k∈Ij

Mtkm(_ρu∆x)

pk

<. (2.15)

We can also findK >0 such thatAj< Kfor all j=1, 2,.... Now letnbe any integer withkr−1< n≤kr, wherer > L. Then

1 n

n

k=1

Mtkm(u_ρ∆x)

pk

≤k−1 r−1

kr

k=1

Mtkm(_ρu∆x)

pk

=k−1 r−1

  

k∈I1

M tkm(_ρu∆x)

pk

+

k∈I2

Mtkm(_ρu∆x)

pk

+···+ k∈Ir

Mtkm(_ρu∆x)

pk_



= k1 kr−1k

−1 1

k∈I1

Mtkm(u_ρ∆x)

pk

+k2_k−k1 r−1

k2−k1

−1

k∈I2

Mtkm(u_ρ∆x)

pk

+···+kR_k−kR−1 r−1

kR−kR−1

−1

k∈I2

Mtkm(u_ρ∆x)

pk

+···+kr_k−kr−1 r−1

kr−kr−1

−1

k∈I2

Mtkm(_ρu∆x)

pk

= k1 kr−1A1+

k2−k1

kr−1 A2+···+

kR−kR−1 kr−1 AR

+kR+1_k −kR

r−1 AR+1+···+

kr−kr−1 kr−1 Ar

≤

sup j≥1

Aj

kR kr−1+

sup j≥RAj

kr−kR kr−1 < K

kR kr−1+B.

(2.16) Sincekr−1→ ∞asn→ ∞, it follows that

1 n

n

k=1

Mtkm(_ρu∆x)

pk

−→0, (2.17)

uniformly inm, and consequentlyx∈[w,M,p,u,∆]0

σ. This completes the proof of the

theorem.

Theorem2.7. Letθ=(kr)be a lacunary sequence with1<lim infrqr<lim suprqr<∞. Then for any Orlicz functionM,[w,M,p,u,∆]σ=[wθ,M,p,u,∆]σ.

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Ahmad H. A. Bataineh: Department of Mathematics, Al al-Bayt University, Mafraq 25113, Jordan

E-mail address:[email protected]

Laith E. Azar: Department of Mathematics, Al al-Bayt University, Mafraq 25113, Jordan