J. Math. Comput. Sci. 6 (2016), No. 6, 965-974 ISSN: 1927-5307

σ-CONVERGENT SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION

KHALID EBADULLAH

College of Science and Theoretical Studies, Saudi Electronic Universiy, Kingdom of Saudi Arabia

Copyright c2016 Khalid Ebadullah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract. In this paper, we introduce the sequence spaceVσ(M,p,r),whereMis an Orlicz function,p= (pm)is any sequence of strictly positive real numbers andr≥0 and study some of the properties and inclusion relations

that arise on the said space.

Keywords:invariant mean; paranorm; Orlicz function.

2010 AMS Subject Classification:40F05, 40C05, 46A45.

### 1. Introduction

Let N, R and C be the sets of all natural, real and complex numbers respectively. We write

ω ={x= (xk):xk∈R or C}, the space of all real or complex sequences.

Let`∞, candc0denote the Banach spaces of bounded, convergent and null sequences

respec-tively.

The following subspaces ofω were first introduced and discussed by Maddox [5-6].

Received April 20, 2016

`(p) ={x∈ω :∑

k

|x_{k}|pk_{<}∞},

`∞(p) ={x∈ω : sup

k

|xk|pk<∞},

c(p) ={x∈ω : lim

k |xk−l|

pk_{=}_{0}_{,} _{for some}_{l}∈_{C}},

c_{0}(p) ={x∈ω : lim

k |xk|

pk_{=}_{0}_{},}

where p= (p_{k})is a sequence of striclty positive real numbers.

The concept of paranorm is closely related to linear metric spaces.It is a generalization of that of absolute value.(see[5-6])

Let X be a linear space. A functiong:X −→Ris called paranorm, if for allx,y,z∈X,

(PI)g(x) =0i f x=θ,

(P2)g(−x) =g(x),

(P3)g(x+y)≤g(x) +g(y),

(P4) If(λn)is a sequence of scalars withλn→λ (n→∞)andxn,a∈X withxn→a(n→∞),
in the sense thatg(x_{n}−a)→0(n→∞), in the sense thatg(λnxn−λa)→0(n→∞).

An Orlicz function is a function M:[0,∞)→[0,∞), which is continuous, non-decreasing and

convex withM(0) =0,M(x)>0 forx>0 andM(x)→∞asx→∞.(see[2],[9],[12])

Lindenstrauss and Tzafriri[3] used the idea of Orlicz functions to construct the sequence space

`M={x∈ω :

∞

### ∑

k=1

M(|xk|

ρ )<∞,for someρ>0}

The space`M is a Banach space with the norm

||x||=inf{ρ >0 :

∞

### ∑

k=1

M(|xk| ρ )≤1}

The space`Mis closely related to the space`pwhich is an Orlicz sequence space withM(x) =xp for 1≤ p<∞.

An Orlicz functionMis said to satisfy4_{2}condition for all values of x if there exists a constant

K>0 such thatM(Lx)≤KLM(x)for all values ofL>1.

of scalars(α_{k})with|αk|<1 for allk∈N.

Letσ be an injection on the set of positive integers N into itself having no finite orbits and T be

the operator defined on`∞byT(xk) = (xσ(k)).

A positive linear functionalΦ, with||Φ||=1, is called aσ-mean or an invariant mean ifΦ(x) = Φ(T x)for allx∈`∞.

A sequencexis said to beσ-convergent, denoted byx∈Vσ, ifΦ(x)takes the same value, called

σ−limx, for allσ-meansΦ. We have
V_{σ} ={x= (x_{k}):

∞

### ∑

m=1

tm,n(x) =Luniformly in n, L =σ−limx},

where form≥0,n>0.

t_{m},n(x) =

x_{k}+x_{σ}_{(}_{k}_{)}+...+x_{σ}m_{(}_{k}_{)}

m+1 ,andt−1,n=0.

where σm(k) denotes the mth iterate of σ at n. In particular, if σ is the translation, a σ

-mean is often called a Banach limit and V_{σ} reduces to f, the set of almost convergent
se-quences.(see[1],[4],[7],[8],[10],[11])

Subsequently the spaces of invariant mean and Orlicz function have been studied by various authors.(See [1-13]).

### 2. Main results

In this article we introduce the sequence space

V_{σ}(M,p,r) ={x= (x_{k}):

∞

### ∑

m=1

1

mr[M(

|tm,n(x)|

ρ )]

pm_{<}_{∞}_{uniformly in n,}_{ρ}_{>}_{0}_{}.}

WhereMis an Orlicz function, p= (p_{m})is any sequence of strictly positive real numbers and

Now we define the sequence spaces as follows;

ForM(x) =xwe get

V_{σ}(p,r) ={x= (x_{k}):

∞

### ∑

m=1

1

mr|tm,n(x)|

pm_{<}_{∞}_{uniformly in n}_{}}

Forpm=1, for all m, we get

V_{σ}(M,r) ={x= (x_{k}):

∞

### ∑

m=1

1

mr[M(

|t_{m},n(x)|

ρ )]<∞uniformly in n,ρ >0}

For r = 0 we get

V_{σ}(M,p) ={x= (x_{k}):

∞

### ∑

m=1

[M(|tm,n(x)|

ρ )]

pm_{<}_{∞}_{uniformly in n,}_{ρ}_{>}_{0}_{}}

ForM(x) =xand r=0 we get

V_{σ}(p) ={x= (x_{k}):

∞

### ∑

m=1

|tm,n(x)|pm<∞uniformly in n,ρ>0}

Forp_{k}=1, for all m and r=0, we get

V_{σ}(M) ={x= (x_{k}):

∞

### ∑

m=1

[M(|tm,n(x)|

ρ )]<∞uniformly in n,ρ>0}

ForM(x) =x, p_{m}=1, for all m and r=0, we get

V_{σ}(x) ={x= (x_{k}):

∞

### ∑

m=1

|tm,n(x)|<∞uniformly in n}.

Theorem 2.1. The sequence spaceV_{σ}(M,p,r) is a linear space over the field C of complex
numbers.

Proof.Letx,y∈V_{σ}(M,p,r)andα,β∈Cthen there exists positive numbersρ1andρ2such that

∞

### ∑

m=1

1

mr[M(

|tm,n(x)|

ρ1

and

∞

### ∑

m=1

1

mr[M(

|tm,n(y)|

ρ2

)]pm_{<}

∞

uniformly in n.

Defineρ3= max(2|α|ρ1, 2|β|ρ2).

Since M is non decreasing and convex we have

∞

### ∑

m=1

1

mr[M(

|αtm,n(x) +βtm,n(y)|

ρ3

)]pm

≤

∞

### ∑

m=1

1

mr[M(

|αtm,n(x)|

ρ3

+|βtm,n(y)| ρ3

)]pm

≤

∞

### ∑

m=1

1

mr 1 2[M(

tm,n(x)

ρ1

) +M(tm,n(y) ρ2

)]<∞

uniformly in n.

This proves thatV_{σ}(M,p,r)is a linear space over the field C of complex numbers.

Theorem 2.2.For any Orlicz function M and a bounded sequence p= (pm)of strictly positive
real numbers,V_{σ}(M,p,r)is a paranormed space with

g(x) = inf n≥1{ρ

pn H :(

∞

### ∑

m=1

1

mr[M(

|tm,n(x)|

ρ )]

pm_{)}_{H}1 _{≤}_{1}_{,} _{uniformly in n}_{}}

where H = max(1, suppm).

Proof.It is clear thatg(x) =g(−x). SinceM(0) =0,we get

inf{ρpmH }= 0, forx=0

For the continuity of scalar multiplication letl 6=0 be any complex number. Then by the defi-nition we have

g(lx) = inf n≥1{ρ

pn H :(

∞

### ∑

m=1

1

mr[M(

|tm,n(lx)|

ρ )]

pm_{)}_{H}1 _{≤}_{1}_{,} _{uniformly in n}_{}}

g(lx) = inf n≥1{(|l|s)

pn H :(

∞

### ∑

m=1

1

mr[M(

|tm,n(lx)|

(|l|s) )]

pm_{)}_{H}1 _{≤}_{1}_{,} _{uniformly in n}_{}}

wheres= _{|}ρ_{l}_{|}.

Since|l|pm_{≤}_{max(1,}_{|l|}H_{), we have}

g(lx)≤max(1,|l|H)inf n≥1{s

pn H :(

∞

### ∑

m=1

1

mr[M(

|tm,n(x)|

(|l|s) )]

pm_{)}_{H}1 _{≤}_{1}_{,} _{uniformly in n}_{}}

g(lx)≤max(1,|l|H)g(x)

Thereforeg(lx)converges to zero wheng(x)converges to zero inV_{σ}(M,p,r).

Now letxbe fixed element inV_{σ}(M,p,r). There existsρ>0 such that

g(x) = inf n≥1{ρ

pn
H _{:}_{(}

∞

### ∑

m=1

1

mr[M(

|tm,n(x)|

ρ )]

pm_{)}H1 ≤_{1}_{,} _{uniformly in n}}

. Now

g(lx) =inf n≥1{ρ

pn
H _{:}_{(}

∞

### ∑

m=1

1

mr[M(

|t_{m},n(lx)|

ρ )]

pm_{)}H1 ≤_{1}_{,} _{uniformly in n}} →_{0 as}_{l}→_{0}_{.}

Theorem 2.3.The sequence space

V_{σ}(M,p,r) ={x= (x_{k}):

∞

### ∑

m=1

1

mr[M(

|tm,n(x)|

ρ )]

pm_{<}_{∞}_{uniformly in n,}_{ρ}_{>}_{0}_{}.}

is a Banach space with the norm

g(x) = inf n≥1{ρ

pn
H _{:}_{(}

∞

### ∑

m=1

1

mr[M(

|tm,n(x)|

ρ )]

pm_{)}H1 ≤_{1}}.

Theorem 2.4.Suppose that 0<p_{m}<t_{m}<∞for eachm∈N andr>0. Then

(a)V_{σ}(M,p)⊆V_{σ}(M,t).

(b)V_{σ}(M)⊆V_{σ}(M,r)

Proof.(a) Suppose thatx∈V_{σ}(M,p).
This implies that[M(|ti,n(x)|

ρ )]

pm_{)}_{≤}_{1}

for sufficiently large value of i, sayi≥m_{0}for some fixedm_{0}∈N.

SinceMis non decreasing, we have

∞

### ∑

m=m0

[M(|ti,n(x)|

ρ )]

tm_{≤}

∞

### ∑

m=m0

[M(|ti,n(x)|

ρ )]

pm_{<}_{∞.}

Hencex∈V_{σ}(M,t).

(b) The proof is trivial.

Corollary 2.5.0< pm≤1 for each m, thenVσ(M,p)⊆Vσ(M)

If p_{m}≥1 for all m , thenV_{σ}(M)⊆V_{σ}(M,p).

Theorem 2.6.The sequence spaceV_{σ}(M,p,r)is solid.

Proof.Letx∈V_{σ}(M,p,r). This implies that

∞

### ∑

m=1

1

mr[M(

|tm,n(x)|

ρ )]

Letαmbe a sequence of scalars such that|αm| ≤1 for allm∈N. Then the result follows from the following inequality.

∞

### ∑

m=1

1

mr[M(

|αmti,n(x)|

ρ )]

pm_{≤}

∞

### ∑

m=1

1

mr[M(

|ti,n(x)|

ρ )]

pm_{<}_{∞.}

Henceαx∈Vσ(M,p,r)for all sequence of scalars(αm)with|αm| ≤1 for allm∈Nwhenever

x∈V_{σ}(M,p,r).

Corollary 2.7.The sequence spaceV_{σ}(M,p,r)is monotone.

Theorem 2.8.LetM_{1},M_{2}be Orlicz function satisfying4_{2}condition and

r,r_{1},r_{2}≥0.Then we have

(a) Ifr>1 thenV_{σ}(M_{1},p,r)⊆V_{σ}(M0M_{1},p,r),
(b)V_{σ}(M_{1},p,r)∩Vσ(M2,p,r)⊆Vσ(M1+M2,p,r),

(c) Ifr1≤r2thenVσ(M,p,r1)⊆Vσ(M,p,r2).

Proof. (a) Since M is continuous at 0 from right, for ε >0 there exists 0<δ <1 such that

0≤c≤δ impliesM(c)<ε.

If we define

I_{1}={m∈N:M_{1}(|tm,n(x)|

ρ )≤δ for someρ >0},

I_{2}={m∈N :M_{1}(|tm,n(x)|

ρ )>δ for someρ >0},

when

we get

M(M_{1}(|tm,n(x)|
ρ ))≤ {

2M(1) δ }M1(

|tm,n(x)|

ρ )

Hence forx∈V_{σ}(M_{1},p,r)andr>1

∞

### ∑

m=1

1

mr[M0M1(

|tm,n(x)|

ρ )]

pm_{=}

### ∑

m∈I1

1

mr[M0M1(

|tm,n(x)|

ρ )]

pm_{+}

### ∑

m∈I2

1

mr[M0M1(

|tm,n(x)|

ρ )]

pm_{.}

∞

### ∑

m=1

1

mr[M0M1(

|tm,n(x)|

ρ )]

pm_{≤}_{max}_{(ε}h_{,}_{ε}H_{)}

∞

### ∑

m=1

1

mr +max({

2M_{1}

δ }

h_{,}_{{}2M1

δ }

H_{)}

where 0<h=infp_{m}≤p_{m}≤H=sup
m

p_{m}<∞

(b)The proof follows from the following inequality

1

mr[(M1+M2)(

|tm,n(x)|

ρ )]

pm _{≤}_{C} 1

mr[M1(

|tm,n(x)|

ρ )]

pm_{+}_{C} 1

mr[M2(

|tm,n(x)|

ρ )]

pm

(c)The proof is straightforward.

Corollary 2.9.LetMbe an Orlicz function satisfying42condition. Then we have

(a) Ifr>1 thenV_{σ}(p,r)⊆V_{σ}(M,p,r),
(b)V_{σ}(M,p)⊆V_{σ}(M,p,r),

(c)V_{σ}(p)⊆V_{σ}(p,r),
(d)V_{σ}(M)⊆V_{σ}(M,r).

Conflict of Interests

The author declares that there is no conflict of interests. REFERENCES

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