σ-Convergent difference sequence spaces of second order defined by Orlicz function

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J. Math. Comput. Sci. 7 (2017), No. 2, 237-248 ISSN: 1927-5307

σ-CONVERGENT DIFFERENCE SEQUENCE SPACES OF SECOND ORDER

DEFINED BY ORLICZ FUNCTION

KHALID EBADULLAH

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

Copyright c2017 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,42),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 and difference sequences.

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 [12-13].

E-mail address: khalidebadullah@gmail.com

Received September 4, 2016; Published March 1, 2017

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`(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}_,

c₀(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[13])

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(xn−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→∞.

Lindenstrauss and Tzafriri[10] 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 satisfy4₂condition for all values ofxif there exists a constant

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

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of scalars(αk)with|αk|<1 for allk∈N.

For Orlicz function and related results see([2],[8],[17]).

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= (xk):

∞

∑

m=1

t_m,n(x) =Luniformly in n, L =σ−limx},

where form≥0,n>0.

t_m_,_n(x) =xk+xσ(k)+...+xσm(k)

m+1 ,andt−1,n=0.

whereσm(k)denotes the mthiterate ofσ at n. In particular, ifσ is the translation, aσ-mean is

often called a Banach limit andV_σ reduces to f, the set of almost convergent sequences.

Subsequently the spaces of invariant mean and Orlicz function have been studied by various authors. See( [1],[11],[15],[16],[19]).

The idea of Difference sequence sets

X₄={x= (x_k)∈ω :4x= (xk−xk+1)∈X},

whereX =`_∞, c orc0was introduced by Kizmaz [9].

Kizmaz [9] defined the sequence spaces,

`∞(4) ={x= (xk)∈ω :(4xk)∈`∞},

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c₀(4) ={x= (x_k)∈ω :(4xk)∈c0},

where4x= (x_k−x_k₊₁). These are Banach spaces with the norm

||x||₄=|x1|+||4x||∞.

After then Mikael [14] defined the sequence spaces,

`∞(42) ={x= (xk)∈ω :(42xk)∈`∞},

c(42) ={x= (x_k)∈ω :(42xk)∈c},

c₀(42) ={x= (x_k)∈ω :(42x_k)∈c0},

and showed that these are Banach spaces with norm

||x||4=|x1|+|x2|+||42x||∞.

For difference sequences see([3-5],[8],[9]).

Recently Ebadullah[6] introduced and studied the sequence space

V_σ(M,p,r) ={x= (xk):

∞

∑

m=1

1

mr[M(

|tm,n(x)|

ρ )]

pm_<_∞_{uniformly in n,}_ρ_>₀_}_.

WhereMis an Orlicz function, p= (pm)is any sequence of strictly positive real numbers and

r≥0 .

After then Ebadullah[7] introduced the sequence space

V_σ(M,p,r,4) ={x= (xk):

∞

∑

m=1

1

mr[M(

|tm,n(4x)|

ρ )]

pm_<_∞_{uniformly in n,}_ρ _>₀_}_.

and discussed the following sequence spaces ;

ForM(x) =xwe get

V_σ(p,r,4) ={x= (xk):

∞

∑

m=1

1

mr|tm,n(4x)|

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Forp_m=1, for all m, we get

V_σ(M,r,4) ={x= (xk):

∞

∑

m=1

1

mr[M(

|tm,n(4x)|

ρ )]<∞uniformly in n,ρ>0}

For r = 0 we get

V_σ(M,p,4) ={x= (xk):

∞

∑

m=1

[M(|tm,n(4x)|

ρ )]

pm_<_∞_{uniformly in n,}_ρ _>₀_}

ForM(x) =xand r=0 we get

V_σ(p,4) ={x= (x_k):

∞

∑

m=1

|tm,n(4x)|pm<∞uniformly in n,ρ >0}

Forp_k=1, for all m and r=0, we get

V_σ(M,4) ={x= (x_k):

∞

∑

m=1

[M(|tm,n(4x)|

ρ )]<∞uniformly in n,ρ >0}

ForM(x) =x, p_m=1, for all m and r=0, we get

V_σ(4x) ={x= (x_k):

∞

∑

m=1

|t_m,n(4x)|<∞uniformly in n}.

2. Main results

In this article we introduce the sequence space

V_σ(M,p,r,42) ={x= (xk):

∞

∑

m=1

1

mr[M(

|tm,n(42x)|

ρ )]

pm_<_∞_{uniformly in n,}_ρ_>₀_}_.

WhereMis an Orlicz function, p= (p_m)is any sequence of strictly positive real numbers and

r≥0 .

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ForM(x) =xwe get

V_σ(p,r,42) ={x= (x_k):

∞

∑

m=1

1

mr|tm,n(4

2_x)|pm_<

∞uniformly in n}

Forp_m=1, for all m, we get

V_σ(M,r,42) ={x= (xk):

∞

∑

m=1

1

mr[M(

|tm,n(42x)|

ρ )]<∞uniformly in n,ρ >0}

For r = 0 we get

V_σ(M,p,42) ={x= (xk):

∞

∑

m=1

[M(|tm,n(4 2_x)|

ρ )]

pm_<_∞_{uniformly in n,}_ρ_>₀_}

ForM(x) =xand r=0 we get

V_σ(p,42) ={x= (x_k):

∞

∑

m=1

|tm,n(42x)|pm<∞uniformly in n,ρ>0}

Forp_k=1, for all m and r=0, we get

V_σ(M,42) ={x= (x_k):

∞

∑

m=1

[M(|tm,n(4 2_x)|

ρ )]<∞uniformly in n,ρ>0}

ForM(x) =x, p_m=1, for all m and r=0, we get

V_σ(42x) ={x= (xk):

∞

∑

m=1

|tm,n(42x)|<∞uniformly in n}.

Theorem 2.1.The sequence spaceV_σ(M,p,r,42₎_{is a linear space over the field C of complex}

numbers.

Proof.Letx,y∈V_σ(M,p,r,42₎_and

α,β ∈Cthen there exists positive numbersρ1andρ2such

that

∞

∑

m=1

1

mr[M(

|tm,n(42x)|

ρ1

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and

∞

∑

m=1

1

mr[M(

|tm,n(42y)|

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(42x) +βtm,n(42y)|

ρ3

)]pm

≤

∞

∑

m=1

1

mr[M(

|αtm,n(42x)|

ρ3

+|βtm,n(4 2_y)|

ρ3

)]pm

≤

∞

∑

m=1

1

mr 1 2[M(

tm,n(42x)

ρ1

) +M(tm,n(4 2_y)

ρ2

)]<∞

uniformly in n.

This proves thatV_σ(M,p,r,42₎_{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,42₎_{is a paranormed space with}

g(x) = inf

n≥1{ρ pn H _:(

∞

∑

m=1

1

mr[M(

|tm,n(42x)|

ρ )]

pm₎_H1 _≤₁_, _{uniformly in n}_}

where H = max(1, suppm).

Proof.It is clear thatg(42_{x) =}_g(−42_x)_.

SinceM(0) =0,we get inf{ρpmH }= 0, forx=0

Now forα=β=1, we get

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

g(l42x) = inf

n≥1{ρ pn

H _:₍

∞

∑

m=1

1

mr[M(

|tm,n(l42x)|

ρ )]

pm₎H1 ≤₁_, _{uniformly in n}}

g(l42x) = inf

n≥1{(|l|s) pn H _:₍

∞

∑

m=1

1

mr[M(

|tm,n(l42x)|

(|l|s) )]

wheres= _|ρ_l_|.

Since|l|pm_≤_max(1,_|l|H_{), we have}

g(l42x)≤max(1,|l|H)inf n≥1{s

pn H _:₍

∞

∑

m=1

1

mr[M(

|tm,n(42x)|

(|l|s) )]

g(42lx)≤max(1,|l|H_)g(42_x)

Thereforeg(42_x)_{converges to zero when}_g(42_x)_{converges to zero in}_V

σ(M,p,r,42).

Now letxbe fixed element inVσ(M,p,r,42). There existsρ >0 such that

g(42x) = inf

n≥1{ρ pn

H _:(

∞

∑

m=1

1

mr[M(

|tm,n(42x)|

ρ )]

pm₎_H1 _≤₁_, _{uniformly in n}_}

. Now

g(l42x) =inf

n≥1{ρ pn H _:(

∞

∑

m=1

1

mr[M(

|tm,n(l42x)|

ρ )]

pm₎_H1 _≤₁_, _{uniformly in n}_{} →}_{0 as}_l_→₀_.

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Theorem 2.3.Suppose that 0<p_m<t_m<∞for eachm∈N andr>0. Then

(a)V_σ(M,p,42₎_⊆_V

σ(M,t,42).

(b)V_σ(M,42₎_⊆_V

σ(M,r,42)

Proof.(a) Suppose thatx∈V_σ(M,p,42₎_.

This implies that[M(|ti,n(42x)| ρ )]

pm₎_≤₁

for sufficiently large value of i, sayi≥m0for some fixedm0∈N.

SinceMis non decreasing, we have

∞

∑

m=m0

[M(|ti,n(4 2_x)|

ρ )]

tm_≤

∞

∑

m=m0

[M(|ti,n(4 2_x)|

ρ )]

pm_<_∞_.

Hencex∈V_σ(M,t,42).

(b) The proof is trivial.

Corollary 2.4.0< p_m≤1 for each m, thenV_σ(M,p,42)⊆V_σ(M,42)

If p_m≥1 for all m , thenV_σ(M,42₎_⊆_V

σ(M,p,42).

Theorem 2.5.The sequence spaceV_σ(M,p,r,42₎_{is solid.}

Proof.Letx∈V_σ(M,p,r,42₎_{. This implies that}

∞

∑

m=1

1

mr[M(

|tm,n(42x)|

ρ )]

pm_<

∞.

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(42x)|

ρ )]

pm_≤

∞

∑

m=1

1

mr[M(

|ti,n(42x)|

ρ )]

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Henceαx∈Vσ(M,p,r,42)for all sequence of scalars(αm)with|αm| ≤1 for allm∈N when-everx∈V_σ(M,p,r,42₎_.

Corollary 2.6.The sequence spaceV_σ(M,p,r,42₎_{is monotone.}

Theorem 2.7.LetM1,M2be Orlicz function satisfying42condition and r,r₁,r₂≥0.Then we have

(a) Ifr>1 thenV_σ(M1,p,r,42)⊆Vσ(M0M1,p,r,42),

(b)V_σ(M1,p,r,42)∩Vσ(M2,p,r,42)⊆Vσ(M1+M2,p,r,42),

(c) Ifr1≤r2thenVσ(M,p,r1,42)⊆Vσ(M,p,r2,42).

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

I₁={m∈N:M₁(|tm,n(4 2_x)|

ρ )≤δ for someρ>0},

I₂={m∈N:M₁(|tm,n(4 2_x)|

ρ )>δ for someρ>0},

when

M₁(|tm,n(4 2_x)|

ρ )>δ

we get

M(M1(

|tm,n(42x)|

ρ ))≤ {

2M(1) δ }M1(

|tm,n(42x)|

ρ )

Hence forx∈V_σ(M1,p,r,42)andr>1

∞

∑

m=1

1

mr[M0M1(

|tm,n(42x)|

ρ )]

pm₌

∑

m∈I1

1

mr[M0M1(

|tm,n(42x)|

ρ )]

pm₊

∑

m∈I2

1

mr[M0M1(

|tm,n(42x)|

ρ )]

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∞

∑

m=1

1

mr[M0M1(

|tm,n(42x)|

ρ )]

pm_≤_max(

εh,εH)

∞

∑

m=1

1

mr +max({

2M1 δ }

h_,_{2M1

δ }

H₎

where 0<h=infp_m≤p_m≤H=sup m

p_m<∞

Conflict of Interests

The author declares that there is no conflict of interests.

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