Applied Mathematics, 2014, 5, 2602-2611 Published Online September 2014 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2014.516248

How to cite this paper: Aiyub, M. (2014) Some Lacunary Sequence Spaces of Invariant Means Defined by Musielak-Orlicz Functions on 2-Norm Space. Applied Mathematics, 5, 2602-2611. http://dx.doi.org/10.4236/am.2014.516248

Some Lacunary Sequence Spaces of Invariant Means Defined by Musielak-Orlicz Functions on 2-Norm Space Mohammad Aiyub Department of Mathematics, University of Bahrain, Sakhir, Bahrain Email: maiyub2002@gmail.com Received 11 July 2014; revised 15 August 2014; accepted 25 August 2014

Copyright © 2014 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract The purpose of this paper is to introduce and study some sequence spaces which are defined by combining the concepts of sequences of Musielak-Orlicz functions, invariant means and lacunary convergence on 2-norm space. We establish some inclusion relations between these spaces under some conditions.

Keywords Invariant Means, Musielak-Orlicz Functions, 2-Norm Space, Lacunary Sequence

1. Introduction Let ω be the set of all sequences of real numbers ,c∞ and 0c be respectively the Banach spaces of bounded, convergent and null sequences ( )kx x= with ( )kx ∈ or the usual norm sup kkx x= , where 1,2,3,k ∈ = , the positive integers.

The idea of difference sequence spaces was first introduced by Kizmaz [1] and then the concept was gene- ralized by Et and Çolak [2]. Later on Et and Esi [3] extended the difference sequence spaces to the sequence spaces:

( ) ( ) ( ){ }: ,m mv k vX x x x X∆ = = ∆ ∈

for , X c∞= and 0c , where ( )kv v= be any fixed sequence of non zero complex numbers and ( ) ( )1 1

1 .m m mv k v k v kx x x− −

+∆ = ∆ − ∆

The generalized difference operator has the following binomial representation,

M. Aiyub

2603

( )0

1 , for all .m im

v k k i k ii

mx v x k

i + +=

∆ = − ∈

∑

The sequence spaces ( )mv ∞∆ , ( )m

v c∆ and ( )0mv c∆ are Banach spaces normed by

1.

mm

i i vi

x v x x∆ ∞

=

= + ∆∑

The concept of 2-normed space was initially introduces by Gahler [4] as an interesting linear generalization of normed linear space which was subsequently studied by many others [5] [6]). Recently a lot of activities have started to study summability, sequence spaces and related topics in these linear spaces [7] [8]).

Let X be a real vector space of dimension d , where 2 d≤ < ∞ . A 2-norme on X is a function .,. : X X× → which satisfies:

1) , 0x y = if and only if x and y are linearly dependent, 2) , ,x y y x= ,

3) , , ,x y x yα α α= ∈ ,

4) , , ,x y z x y x z+ ≤ + . The pair ( ), .,.X is called a 2-normed space. As an example of a 2-normed space we may take 2X =

being equiped with the 2-norm ,x y = the area os paralelogram spaned by the vectors x and y , which may be given explicitly by the formula

11 221 2

21 22

, E

x xx x abs

x x

=

.

Then clearly ( ), .,.X is 2-normed space. Recall that ( ), .,.X is a 2-Banach space if every cauchy sequence in X is convergent to some x X∈ .

Let σ be a mapping of the positive integers into itself. A continuous linear functional φ on ∞ is said to be an invariant mean or σ -mean if and only if

1) ( ) 0xφ ≥ , when the sequence ( )nx x= has, 0nx ≥ for all n , 2) ( ) ( )1, 1,1,1,e eφ = = , 3) ( )( ) ( )nx xσφ φ= for all .x ∞∈

If ( )kx x= , where ( ) ( )( )k kTx Tx xσ= = . It can be shown that

( ){ }: lim , uniformly inknkV x t x l nσ ∞= ∈ =

lim ,l xσ= − where

( ) ( ) ( ) ( )1 2

1n k nn n

kn

x x x xt x

kσ σ σ

+ + + +=

+

[9].

In the case σ is the translation mapping 1n n→ + , σ -mean is often called a Banach limit and Vσ the set of bounded sequences of all whose invariant means are equal is the set of almost convergent sequence [10].

By Lacunary sequence ( ) , 0,1, 2,rk rθ = = where 0 0k = we mean an increasing sequence of non neg- ative integers ( ) ( )1 r r rh k k r−= − → ∞ →∞ . The intervals determined by θ are denoted by [ ]1r r rI k k−= −

and the ratio 1

r

r

kk −

will be denoted by rq . The space of lacunary strongly convergent sequence Nθ was de-

fined by Freedman et al. [11] as follows:

( ) 1: lim 0 for some .r

i kr k Ir

N x x x lhθ →∞ =

= = − =

∑

An Orlicz function is a function [ ) [ ): 0, 0,M ∞ → ∞ which is continuous, non-decreasing and convex with ( ) ( )0 0, 0M M x= > for 0x > and ( )M x →∞ as .x →∞

M. Aiyub

2604

It is well known that if M is convex function and ( )0 0M = then ( ) ( )M x M xλ λ≤ , for all λ with 0 1.λ≤ ≤

Lindenstrauss and Tzafriri [12] used the idea of Orlicz function and defined the sequence space which was called an Orlicz sequence space M such as

( )1

: , for some 0kM k

k

xx x M ρ

ρ

∞

=

= = < ∞ >

∑

which was a Banach space with the norm

1inf 0 : 1k

k

xx Mρ

ρ

∞

=

= > ≤

∑

which was called an Orlicz sequence space. The M was closely related to the space p which was an Orlicz sequence space with ( ) ,pM t t= for 1 p≤ < ∞ . Later the Orlicz sequence spaces were investigated by Prashar and Choudhry [13], Maddox [14], Tripathy et al. [15]-[17] and many others.

A sequence of function ( )kM M= of Orlicz function is called a Musielak-Orlicz function [18] [19]. Also a Musielak-Orlicz function ( )kΦ = Φ is called complementary function of a Musielak-Orlicz function M if

( ) ( ){ }sup : 0 , for 1, 2,3, .k kt t s M s s kΦ = − ≥ =

For a given Musielak-Orlicz function M , the Musielak-Orlicz sequence space Ml and its subspaces M are defined as follow:

( ){ }: , for some 0M k Ml x x I cx cω= = ∈ < ∞ >

( ){ }: , for all 0M k Mx x I cx cω= = ∈ < ∞ >

where MI is a convex modular defined by

( ) ( ) ( )1

, .M k k k Mk

I x M x x x l∞

=

= = ∈∑

We consider Ml equipped with the Luxemburg norm

inf 0 : 1Mxx k Ik

= > ≤

or equipped with the Orlicz norm

( )( )0 1inf 1 : 0 .Mx I kx kk

= + >

The main purpose of this paper is to introduce the following sequence spaces and examine some properties of the resulting sequence spaces. Let ( )kM M= be a Musielak-Orlicz function, ( ), .,.X is called a 2-normed space. Let ( )kp p= be any sequences of positive real numbers, for all k ∈ and ( )ku u= such that

( )0 1, 2,3,ku k≠ = . Let s be any real number such that 0s ≥ . By ( )2 Xω − we denote the space of all sequences defined over ( ), .,.X . Then we define the following sequence spaces:

( )

( ) ( )( )

,

, , , , , .,.

12 : , 0, 0sup

k

r

mv

pm

kn v ksk k k

r n k Ir

M p u s

t xx x X k u M z s

h

θ

σω

ω ρρ

∞

−

∈

∆

∆ = = ∈ − < ∞ > ≥

∑

M. Aiyub

2605

( )

( ) ( )( )

, , , , , .,.

12 :: , 0 for some , 0, 0lim

k

r

mv

pm

kn v ksk k k

r k Ir

M p u s

t xx x X k u M z l s

h

θ

σω

ω ρρ

−

∈

∆

∆ = = ∈ − = > ≥

∑

( )

( ) ( )( )

0, , , , , .,.

12 :: , 0 0, 0lim

k

r

mv

pm

kn v ksk k k

r k Ir

M p u s

t xx x X k u M z s

h

θ

σω

ω ρρ

−

∈

∆

∆ = = ∈ − = > ≥

∑

Definition 1. A sequence space E is said to be solid or normal if ( )k kx Eα ∈ whenever ( )kx E∈ and for all sequences of scalar ( )kα with 1kα ≤ [20].

Definition 2. A sequence space E is said to be monotone if it contains the canonical pre-images of all its steps spaces, [20].

Definition 3. If X is a Banach space normed by . , then ( )m X∆ is also Banach space normed by

( )1

.m

mk

kx x f x

∆=

= + ∆∑

Remark 1. The following inequality will be used throughout the paper. Let ( )kp p= be a positive sequence of real numbers with 0 supk kp p G< ≤ = , ( )1max 1,2GD −= . Then for all ,k ka b ∈ for all k ∈ . We have

( ).k k kp p pk k k ka b D a b+ ≤ + (1)

2. Main Results Theorem 1. Let ( )kM M= be a Musielak-Orlicz function, ( )kp p= be a bounded sequence of positive real number and ( )rkθ = be a lacunary sequence. Then ( ), , , , , .,. ,m

vM p u sθ

σω

∞ ∆ ( ), , , , , .,. m

vM p u sθ

σω ∆

and ( )0, , , , , .,. m

vM p u sθ

σω ∆ are linear spaces over the field of complex numbers.

Proof 1. Let ( ) ( ) ( )0, , , , , , .,. m

k k vx x y y M p u sθ

σω = = ∈ ∆ and ,α β ∈C . In order to prove the result we

need to find some 3ρ such that,

( )( )3

1lim , 0, uniformly in .

k

r

pm

nk v k ksk kr k Ir

t x yu k M z n

h

α β

ρ−

→∞ ∈

∆ + = ∑

Since ( ) ( ) ( )0, , , , , , .,. m

k k vx y M p u sθ

σω ∈ ∆ , there exist positive 1 2,ρ ρ such that

( )1lim , 0 uniformly in

k

r

pm

kn v ksk kr k Ir

t xu k M z n

h ρ−

→∞ ∈

∆ =

∑

and

( )1lim , 0 uniformly in .

k

r

pm

kn v ksk kr k Ir

t xu k M z n

h ρ−

→∞ ∈

∆ =

∑

Define ( )3 1 2max 2 ,2 .ρ α ρ β ρ= Since ( )kM is non decreasing and convex

M. Aiyub

2606

( )( )

( )( ) ( )( )

( ) ( )

( )

3

3 3

1 2

1 ,

1 , ,

1 , ,

,

k

r

k

r

k

r

r

pm

nk v k ksk k

k Ir

pm m

nk v k nk v ksk k

k Ir

pm m

nk v k nk v ksk k

k Ir

mnk v ks

k kk Ir

t x yu k M z

h

t x t yu k M z z

h

t x t yu k M z z

h

t xD u k M zh

α β

ρ

α β

ρ ρ

ρ ρ

ρ

−

∈

−

∈

−

∈

−

∈

∆ +

∆ ∆ ≤ +

∆ ∆ ≤ +

∆≤

∑

∑

∑

∑( )

,

0, as , uniformly in .

k k

r

p pm

nk v ksk k

k Ir

t yD u k M zh

r n

ρ−

∈

∆ +

→ →∞

∑

So that ( ) ( ) ( )0, , , , , .,. .m

k k vx y M p u sθ

σα β ω + ∈ ∆ This completes the proof. Similarly, we can prove that

( ), , , , , .,. mvM p u sθ

σω ∆ and ( ), , , , , .,. m

vM p u sθ

σω

∞ ∆ are linear spaces.

Theorem 2. Let ( )kM M= be a Musielak-Orlicz function, ( )kp p= be a bounded sequence of positive real number and ( )rkθ = be a lacunary sequence. Then ( )0

, , , , , .,. mvM p u sθ

σω ∆ is a topological linear

space totalparanormed by

( )( )

1

1

1inf : , 1 for some , 1, 2,

k

r

r

Hpmm nk v kp H s

k k kk k Ir

t xg x x u k M z r

hρ ρ

ρ−

∆= ∈

∆ = + ≤ =

∑ ∑

Proof 2. Clearly ( ) ( )g x g x∆ ∆= − . Since ( )0 0kM = , for all k ∈ . we get ( ) 0g θ∆ = , for .x θ= Let

( )kx x= , ( ) ( )0, , , , , .,. m

k vy y M p u sθ

σω = ∈ ∆ and let us choose 1 0ρ > and 2 0ρ > such that

( )( )1

1

sup , 1 1,2,3,

k

r

pm

nk v ksr k k

r k I

t xh u k M z r

ρ− −

∈

∆ ≤ =

∑

and

( )( )1

2

sup , 1 1,2,3,

k

r

pm

nk v ksr k k

r k I

t yh u k M z r

ρ− −

∈

∆ ≤ =

∑

Let 1 2ρ ρ ρ= + , then we have

( )( )

( )( )

( )( )

1

11

1 2 1

11

1 2 2

sup ,

sup ,

sup , 1.

k

r

k

r

k

r

pm

nk v k ksr k k

r k I

pm

nk v ksr k k

r k I

pm

nk v ksr k k

r k I

t x yh u k M z

t xh u k M z

t yh u k M z

ρ

ρρ ρ ρ

ρρ ρ ρ

− −

∈

− −

∈

− −

∈

∆ +

∆ ≤ +

∆ + ≤ +

∑

∑

∑

M. Aiyub

2607

Since 0ρ > , we have

( )

( )

( )

1

1

1

11 1

1inf : , 1 for some 0, 1, 2,

1inf : , 1 for some

k

r

r

k

r

r

Hpmm

nk v k kp H sk k k k

k k Ir

Hpmm

nk v kp H sk k k

k k Ir

g x y

t x yx y u k M z r

h

t xx u k M z

h

ρ ρρ

ρρ

∆

−

= ∈

−

= ∈

+

∆ + = + + ≤ > =

∆ ≤ + ≤

∑ ∑

∑ ∑

( )

1

1

2 21 2

0, 1, 2,

1inf : , 1 for some 0, 1, 2, .k

r

r

Hpmm

nk v kp H sk k k

k k Ir

r

t yy u k M z r

h

ρ

ρ ρρ

−

= ∈

> = ∆ + + ≤ > =

∑ ∑

( ) ( ) ( ).g x y g x g y∆ ∆ ∆+ ≤ +

Finally, we prove that the scalar multiplication is continuous. Let λ be a given non zero scalar in . Then the continuity of the product follows from the following expression.

( ) ( )

( ) ( )

1

1

1

1

1inf : , 1 for some 0, 1, 2,

1inf : , 1 for some 0

k

r

r

k

r

r

Hpmm

nk v kp H sk k k

k k Ir

Hpmm p H nk v ks

k k kk k Ir

t xg x x u k M z r

h

t xx u k M z

h

λλ λ ρ ρ

ρ

λ λ ζ ζζ

−∆

= ∈

−

= ∈

∆ = + ≤ > =

∆ = + ≤ >

∑ ∑

∑ ∑

, 1, 2,r =

where 0.ρζλ

= > Since ( )supmax 1, rr ppλ λ≤ ,

( ) ( )

( )

sup

1

max 1,

1inf : , 1, for some 0, 1, 2, .

r

k

r

r

p

Hpm

nk v kp H sk k

k Ir

g x

t xu k M z r

h

λ λ

ρ ρρ

∆

−

∈

=

∆ + ≤ > =

∑

This completes the proof of this theorem. Theorem 3. Let ( )kM M= be a Musielak-Orlicz function, ( )kp p= be a bounded sequence of positive

real number and ( )rkθ = be a lacunary sequence. Then

( ) ( ) ( )0, , , , , .,. , , , , , .,. , , , , , .,. .m m m

v v vM p u s M p u s M p u sθ θ θ

σ σ σω ω ω

∞ ∆ ⊂ ∆ ⊂ ∆

Proof 3. The inclusion ( ) ( )0, , , , , .,. , , , , , .,.m m

v vM p u s M p u sθ θ

σ σω ω ∆ ⊂ ∆ is obvious. Let

( ), , , , , .,. mk vx M p u sθ

σω ∈ ∆ . Then there exists some positive number 1ρ such that

( )1

1lim , 0

k

r

pm

nk v ksk kr k Ir

t x leu k M z

h ρ−

→∞ ∈

∆ − →

∑

as r →∞ , uniformly in n . Define 12ρ ρ= . Since kM is non decreasing and convex for all k ∈ , we have

M. Aiyub

2608

( )

( )

( )

1

1

1

1

1 ,

,

,

,

max 1, ,

k

r

k

r

k

r

k

r

pm

nk v ksk k

k Ir

pm

nk v ksk k

k Ir

p

kk Ir

pm

nk v kk

k Ir

t xu k M z

h

t x leD u k M zh

D leM zh

t x leD M zh

leD M z

ρ

ρ

ρ

ρ

ρ

−

∈

−

∈

∈

∈

∆

∆ − ≤

+

∆ − ≤

+

∑

∑

∑

∑

G

where ( )sup kkG p= , ( )max 1,2 1GD = − by (1). Thus ( ), , , , , .,. m

k vx M p u sθ

σω ∈ ∆ .

Theorem 4. Let ( )kM M= be a Musielak-Orlicz functions. If ( )sup kpk kM z < ∞ for all 0z > , then

( ) ( ), , , , , .,. , , , , , .,. .m mv vM p u s M p u sθ θ

σ σω ω

∞ ∆ ⊂ ∆

Proof 4. Let ( ), , , , , .,. mk vx M p u sθ

σω ∈ ∆ by using (1), we have

( )

( )

1 ,

,

, .

k

r

k

r

k

r

pm

nk v ksk k

k Ir

pm

nk v kk

k Ir

p

kk Ir

t xu k M z

h

t x leD M zh

leD M zh

ρ

ρ

ρ

−

∈

∈

∈

∆

∆ − ≤

+

∑

∑

∑

Since ( )sup kpk M z < ∞ , we can take the ( )sup kp

k M z K= . Hence we can get ( ), , , , , .,. m

k vx M p u sθ

σω ∈ ∆ .

This complete the proof. Theorem 5. Let 1m ≥ be fixed integer. Then the following statements are equivalent: 1) ( ) ( )1, , , , , .,. , , , , , .,.m m

v vM p u s M p u sθ θ

σ σω ω

∞ ∞− ∆ ⊂ ∆ ,

2) ( ) ( )1, , , , , .,. , , , , , .,.m mv vM p u s M p u sθ θ

σ σω ω− ∆ ⊂ ∆ ,

3) ( ) ( )0 01, , , , , .,. , , , , , .,. .m mv vM p u s M p u sθ θ

σ σω ω− ∆ ⊂ ∆

Proof 5. Let ( )0 1, , , , , .,. .mk vx M p u sθ

σω − ∈ ∆ Then there exist 0ρ > such that

( )1lim , 0.

k

r

pm

nk v ksk kr k Ir

t xu k M z

h ρ−

→∞ ∈

∆ →

∑

Since kM is non decreasing and convex, we have

M. Aiyub

2609

( )

( )

( ) ( )

1 11

1 11

1 ,2

1 ,2

1 1, ,2 2

k

r

k

r

k

r r

pm

nk v ksk k

k Ir

pm m

nk v k ksk k

k Ir

pm m

nk v k nk v ks sk k k k

k I k Ir r

t xu k M z

h

t x xu k M z

h

t x t xu k M z u k M z

h h

ρ

ρ

ρ ρ

−

∈

− −+−

∈

− −+− −

∈ ∈

∆

∆ − ∆ = ∆ ∆ ≤ +

∑

∑

∑ ∑

( ) ( )1 11, , .

k

k k

r r

p

p pm m

nk v k nk v ks sk k k k

k I k Ir r

t x t xD Du k M z u k M zh hρ ρ

− −+− −

∈ ∈

∆ ∆ ≤ +

∑ ∑

Taking limr→∞ , we have

( )1 , 0,

k

r

pm

nk v ksk k

k Ir

t xu k M z

h ρ−

∈

∆ =

∑

i.e. ( )0 1, , , , , .,. .mk vx M p u sθ

σω − ∈ ∆ The rest of these cases can be proved in similar way.

Theorem 6. Let ( )kM M= and ( )kT T= be two Musielak-Orlicz functions. Then we have 1) ( ) ( ) ( ), , , , , .,. , , , , , .,. , , , , , .,. .m m m

v v vM p u s T p u s M T p u sθ θ θ

σ σ σω ω ω

∞ ∞ ∞ ∆ ∆ ⊂ + ∆

2) ( ) ( ) ( ), , , , , .,. , , , , , .,. , , , , , .,. .m m mv v vM p u s T p u s M T p u sθ θ θ

σ σ σω ω ω ∆ ∆ ⊂ + ∆

3) ( ) ( ) ( )0 0 0, , , , , .,. , , , , , .,. , , , , , .,. .m m m

v v vM p u s T p u s M T p u sθ θ θ

σ σ σω ω ω ∆ ∆ ⊂ + ∆

Proof 6. Let ( ) ( ), , , , , .,. , , , , , .,. .m mk v vx M p u s T p u sθ θ

σ σω ω

∞ ∞ ∈ ∆ ∆ Then

( ),

1sup ,

k

r

pm

nk v ksk k

r n k Ir

t xu k M z

h ρ−

∈

∆ < ∞

∑

and

( ),

1sup ,

k

r

pm

nk v ksk k

r n k Ir

t xu k T z

h ρ−

∈

∆ < ∞

∑

uniformly in n. We have

( )( ) ( ) ( )

, , ,

k k kp p pm m m

nk v k nk v k nk v kk k k k

t x t x t xM T z D M z D T z

ρ ρ ρ

∆ ∆ ∆ + ≤ +

by (1). Applying rk I∈∑ and multiplying by 1,k

r

uh

and sk − both side of this inequality, we get

( )( )

( ) ( )

1 ,

, ,

k

r

k k

r r

pm

nk v ksk k k

k Ir

p pm m

nk v k nk v ks sk k k k

k I k Ir r

t xu k M T z

h

t x t xD Du k M z u k T zh h

ρ

ρ ρ

−

∈

− −

∈ ∈

∆ +

∆ ∆ ≤ +

∑

∑ ∑

M. Aiyub

2610

uniformly in n. This completes the proof 2) and 3) can be proved similar to 1). Theorem 7. 1) The sequence spaces , , , , , .,.M p u sθ

σω

∞ and

0, , , , , .,.M p u sθ

σω are solid and hence

they are monotone. 2) The space , , , , , .,.M p u sθ

σω is not monotone and neither solid nor perfect.

Proof 7. We give the proof for 0

, , , , , .,.M p u sθ

σω . Let

0, , , , , .,.kx M p u sθ

σω ∈ and ( )kα be a

sequence of scalars such that 1kα ≤ for all k ∈ . Then we have

( ) ( )1 1, , 0k k

r r

p p

nk k k nk ks sk k k k

k I k Ir r

t x t xu k M z u k M z

h hαρ ρ

− −

∈ ∈

≤ →

∑ ∑

( )r →∞ , uniformly in n. Hence ( )0

, , , , , .,.k kx M p u sθ

σα ω ∈ for all sequence of scalars ( )kα with

1kα ≤ for all k ∈ , whenever 0

, , , , , .,.kx M p u sθ

σω ∈ . The spaces are monotone follows from the re-

mark (1).

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