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The purpose of this paper is to introduce certain new sequence spaces using ideal convergence and an Orlicz function in 2-normed spaces and examine some of their properties.. Introductio

Volume 2010, Article ID 482392, 8 pages

doi:10.1155/2010/482392

Research Article

On Some New Sequence Spaces in

2-Normed Spaces Using Ideal Convergence and

an Orlicz Function

E Savas¸

Department of Mathematics, Istanbul Ticaret University, ¨ Usk ¨udar, 34672 Istanbul, Turkey

Correspondence should be addressed to E Savas¸,ekremsavas@yahoo.com

Received 25 July 2010; Accepted 17 August 2010

Academic Editor: Radu Precup

Copyrightq 2010 E Savas¸ 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

The purpose of this paper is to introduce certain new sequence spaces using ideal convergence and

an Orlicz function in 2-normed spaces and examine some of their properties

1 Introduction

The notion of ideal convergence was introduced first by Kostyrko et al.1 as a generalization

of statistical convergence which was further studied in topological spaces 2 More applications of ideals can be seen in3,4

The concept of 2-normed space was initially introduced by G¨ahler5 as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authorssee, 6,7 Recently, a lot of activities have started to study summability, sequence spaces and related topics in these nonlinear spacessee, 8 10

Recall in11 that an Orlicz function M : 0, ∞ → 0, ∞ is continuous, convex, nondecreasing function such that M0  0 and Mx > 0 for x > 0, and Mx → ∞ as

x → ∞

Subsequently Orlicz function was used to define sequence spaces by Parashar and Choudhary12 and others

If convexity of Orlicz function, M is replaced by Mx  y ≤ Mx  My, then this

function is called Modulus function, which was presented and discussed by Ruckle13 and Maddox14

Note that if M is an Orlicz function then Mλx ≤ λMx for all λ with 0 < λ < 1.

LetX, · be a normed space Recall that a sequence x nn∈Nof elements of X is called

to be statistically convergent to x ∈ X if the set Aε  {n ∈ N : x n − x ≥ ε} has natural density zero for each ε > 0.

A familyI ⊂ 2Y of subsets a nonempty set Y is said to be an ideal in Y ifi ∅ ∈ I; ii

A, B ∈ I imply A ∪ B ∈ I; iii A ∈ I, B ⊂ A imply B ∈ I, while an admissible ideal I of Y

further satisfies{x} ∈ I for each x ∈ Y, 9,10

GivenI ⊂ 2N is a nontrivial ideal inN The sequence x nn∈N in X is said to be

I-convergent to x ∈ X, if for each ε > 0 the set Aε  {n ∈ N : x n − x ≥ ε} belongs to I,

1,3

Let X be a real vector space of dimension d, where 2 ≤ d < ∞ A 2-norm on X is a

function·, · : X × X → R which satisfies i x, y  0 if and only if x and y are linearly

dependent,ii x, y  y, x, iii αx, y  |α|x, y, α ∈ R, and iv x, y  z ≤ x, y 

x, z The pair X, ·, · is then called a 2-normed space 6

Recall thatX, ·, · is a 2-Banach space if every Cauchy sequence in X is convergent

to some x in X.

Quite recently Savas¸15 defined some sequence spaces by using Orlicz function and ideals in 2-normed spaces

In this paper, we continue to study certain new sequence spaces by using Orlicz function and ideals in 2-normed spaces In this context it should be noted that though sequence spaces have been studied before they have not been studied in nonlinear structures like 2-normed spaces and their ideals were not used

2 Main Results

LetΛ  λ n  be a nondecreasing sequence of positive numbers tending to ∞ such that λ n1≥

λ n  1, λ1  0 and let I be an admissible ideal of N, let M be an Orlicz function, and let

X, ·, · be a 2-normed space Further, let p  p k be a bounded sequence of positive real

numbers By S2 − X we denote the space of all sequences defined over X, ·, · Now, we

define the following sequence spaces:

W I

λ, M, p, , ·, 





x ∈ S2 − X : ∀ε > 0



n∈ N : 1

λ n



k ∈I n



M

x k − L

ρ , z



 p k

≥ ε

∈ I for some ρ > 0, L ∈ X and each z ∈ X

,

W0I

λ, M, p, , ·, 





x ∈ S2 − X : ∀ε > 0



n∈ N : 1

λ n



k ∈I n



M

x k

ρ , z



 p k ≥ ε

∈ I for some ρ > 0, and each z ∈ X

,

λ, M, p, , ·, 





x ∈ S2 − X : ∃K > 0 s.t sup

n∈N

1

λ n



k ∈I n



M

x k

ρ , z



 p k ≤ K for some ρ > 0, and each z ∈ X

,

W∞I

λ, M, p, , ·, 





x ∈ S2 − X : ∃K > 0 s.t.



n∈ N : 1

λ n



k ∈I n



M

x k

ρ , z



 p k ≥ K

∈ I for some ρ > 0, and each z ∈ X

,

2.1

where I n  n − λ n  1, n.

The following well-known inequality 16, page 190 will be used in the study

If 0≤ p k ≤ sup p k  H, D max1, 2 H−1 2.2

then

|a k  b k|p k ≤ D|a k|p k  |b k|p k

2.3

for all k and a k , b k ∈ C Also |a| p k ≤ max1, |a| H  for all a ∈ C.

Theorem 2.1 W I λ, M, p, , ·, , W I

0λ, M, p, , ·, , and W I

∞λ, M, p, , ·,  are linear spaces.

Proof We will prove the assertion for W0I λ, M, p, , ·,  only and the others can be proved similarly Assume that x, y ∈ W I

0λ, M, , ·,  and α, β ∈ R, so



n∈ N : 1

λ n



k ∈I n



M

x k

ρ1, z



 p k ≥ ε

∈ I for some ρ1> 0,



n∈ N : 1

λ n



k ∈I



M

x k

ρ2, z



 p k ≥ ε

∈ I for some ρ2> 0.

2.4

Since, ·,  is a 2-norm, and M is an Orlicz function the following inequality holds:

1

λ n



k ∈I n



M









αx k  βy k





|α|ρ1βρ2, z





p k

≤ D1

λ n



k ∈I n



|α|



|α|ρ1βρ2M

x k

ρ1

, z

 p k

 D1

λ n



k ∈I n

 β



|α|ρ1βρ2M

y k

ρ2

, z

 p k

≤ DF 1

λ n



k ∈I n



M

x k

ρ1, z



 p k  DF 1

λ n



k ∈I n



M

y k

ρ2, z



 p k ,

2.5

where

F  max

⎡

⎣1,



|α|



|α|ρ1βρ2

H

,

 β



|α|ρ1βρ2

H⎤

⎦. 2.6

From the above inequality, we get



n∈ N : 1

λ n



k ∈I n



M









αx k  βy k





|α|ρ1βρ2, z





p k

≥ ε

⊆



n ∈ N : DF 1

λ n



k ∈I n



M

x k

ρ1, z

 p k≥ ε

2

∪



n ∈ N : DF 1

λ n



k ∈I n



M

y k

ρ2, z

 p k ≥ ε

2

.

2.7

Two sets on the right hand side belong to I and this completes the proof.

It is also easy to see that the space W∞λ, M, p, , ·,  is also a linear space and we now

have the following

paranorm defined by

g n x  inf

⎧

⎨

⎩ρ p n /H : ρ > 0 s.t.

 sup

n

1

λ n



k ∈I n



M

x k

ρ , z



 p k1/H

≤ 1, ∀z ∈ X

⎫

⎬

⎭. 2.8

Proof That g n θ  0 and g n −x  gx are easy to prove So we omit them.

iii Let us take x  x k  and y  y k  in W∞λ, M, p, , ·,  Let

A x 



ρ > 0 : sup

n

1

λ n



k ∈I n



M

x k

ρ , z



 p k ≤ 1, ∀z ∈ X

,

A

y





ρ > 0 : sup

n

1

λn



k ∈I n



M

y k

ρ , z



 p k ≤ 1, ∀z ∈ X

.

2.9

Let ρ1∈ Ax and ρ2∈ Ay, then if ρ  ρ1 ρ2, then, we have

sup

n

1

λ n



n ∈I n

M







x k  y k



ρ , z









≤ ρ1

ρ1 ρ2

sup

n

1

λ n



k ∈I n

M

x k

ρ1

, z



 ρ2

ρ1 ρ2

sup

n

1

λ n



k ∈I n

M

y k

ρ2, z



 .

2.10

Thus, supn 1/λ nn ∈I n M x k  y k /ρ1 ρ2, z p k ≤ 1 and

g n



x  y≤ inf 

ρ1 ρ2

p n /H

: ρ1∈ Ax, ρ2∈ Ay!

≤ inf ρ p n /H

1 : ρ1∈ Ax! inf ρ p n /H

2 : ρ2∈ Ay!

 g n x  g n



y

.

2.11

iv Finally using the same technique of Theorem 2 of Savas¸ 15 it can be easily seen that scalar multiplication is continuous This completes the proof

Corollary 2.3 It should be noted that for a fixed F ∈ I the space

W∞Fλ, M, p, , ·, 





x ∈ S2 − X : ∃K > 0 s.t sup

n ∈N−F

1

λ n



k ∈I n



M

x k

ρ , z



 p k ≤ K

for some ρ > 0, and each z ∈ X

,

2.12

which is a subspace of the space W I

∞λ, M, p, , ·,  is a paranormed space with the paranorms g n for

n / ∈ F and g F  infn ∈N−F g n

i W I

0λ, M1, p, , ·,  ⊆ W I

0λ, M ◦ M1, p, , ·,  provided p k  is such that H0 inf p k >

0.

ii W I

0λ, M1, p, , ·,  ∩ W I

0λ, M2, p, , ·,  ⊆ W I

0λ, M1 M2, p, , ·, .

Proof i For given ε > 0, first choose ε0 > 0 such that max {ε H

0 , ε H0

0 } < ε Now using the continuity of M choose 0 < δ < 1 such that 0 < t < δ ⇒ Mt < ε0 Let x k ∈

W0λ, M1, p, , ·,  Now from the definition

A δ 



n∈ N : 1

λ n



n ∈I n



M1



x k

ρ , z



 p k ≥ δ H

∈ I. 2.13

Thus if n / ∈ Aδ then

1

λ n



n ∈I n



M1

x k

ρ , z



 p k

< δ H , 2.14

that is,



n ∈I n



M1



x k

ρ , z



 p k < λ n δ H , 2.15

that is,



M1

x k

ρ , z



 p k < δ H , ∀k ∈ I n , 2.16 that is,

M1



x k

ρ , z



 < δ, ∀k ∈ I n 2.17

Hence from above using the continuity of M we must have

M



M1



x k

ρ , z



 < ε0, ∀k ∈ I n , 2.18 which consequently implies that



k ∈I n



M



M1

x k

ρ , z



 p k

< λ nmax ε H0 , ε H0

0

!

< λ n ε, 2.19

that is,

1

λ n



k ∈I



M



M1

x k

ρ , z



 p k < ε. 2.20

This shows that



n∈ N : 1

λ n



k ∈I n



M



M1

x k

ρ , z



 p k ≥ ε

⊂ Aδ 2.21

and so belongs to I This proves the result.

ii Let x k  ∈ W I

0M1, p, , ·,  ∩ W I

0M2, p, , ·, , then the fact

1

λ n



M1 M2



x k

ρ , z



 p k ≤ D1

λ n



M1



x k

ρ , z



 p k  D 1

λ n



M2



x k

ρ , z



 p k 2.22 gives us the result

Definition 2.5 Let X be a sequence space Then X is called solid if α k x k  ∈ X whenever

x k  ∈ X for all sequences α k  of scalars with |α k | ≤ 1 for all k ∈ N.

0λ, M, p, , ·, , W I

∞λ, M, p, , ·,  are solid.

Proof We give the proof for W I

0λ, M, p, , ·,  only Let x k  ∈ W I

0λ, M, p, , ·,  and let α k

be a sequence of scalars such that|α k | ≤ 1 for all k ∈ N Then we have



n∈ N : 1

λ n



k ∈I n



M

α k x k

ρ , z



 p k ≥ ε

⊆



n∈ N : C

λ n



k ∈I n



M

x k

ρ , z



 p k ≥ ε

∈ I,

2.23

where C maxk {1, |α k|H } Hence α k x k  ∈ W I

0λ, M, p, , ·,  for all sequences of scalars α k with|α k | ≤ 1 for all k ∈ N whenever x k  ∈ W I

0λ, M, p, , ·, .

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