On some new sequence spaces in 2-normed spaces using ideal convergence and an orlicz function

Volume 2010, Article ID 482392,8pages 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 xn_n∈Nof elements of X is called to be statistically convergent to x ∈ X if the set Aε  {n ∈ N : xn− x ≥ ε} has natural density zero for each ε > 0.

A familyI ⊂ 2^Y 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 ⊂ 2^N is a nontrivial ideal inN. The sequence xn_n∈N in X is said to beI- convergent to x ∈ X, if for each ε > 0 the set Aε  {n ∈ N : xn− 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  λn be a nondecreasing sequence of positive numbers tending to ∞ such that λn1≥ λn  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  pk 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 λn



k∈In

 M

x_k− L ρ , z

 _pk

≥ ε 

∈ I

for some ρ > 0, L∈ X and each z ∈ X 

, W₀^I

λ, M, p,, ·, 





x∈ S2 − X : ∀ε > 0



n∈ N : 1 λn



k∈In

 M

x_k ρ , z

 _pk

≥ ε 

∈ I

for some ρ > 0, and each z∈ X 

,

W_∞

λ, M, p,, ·, 





x∈ S2 − X : ∃K > 0 s.t. sup

n∈N

1 λn



k∈In

 M

x_k ρ , z

 _pk

≤ K

for some ρ > 0, and each z∈ X 

, W_∞^I

λ, M, p,, ·, 





x∈ S2 − X : ∃K > 0 s.t.



n∈ N : 1 λn



k∈In

 M

x_k ρ , z

 _pk

≥ K 

∈ I

for some ρ > 0, and each z∈ X 

,

2.1

where In n − λn 1, n.

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

If 0≤ pk≤ sup pk H, D max

1, 2^H−1

2.2

then

|ak bk|^pk ≤ D

|ak|^pk |bk|^pk

2.3

for all k and ak, bk∈ C. Also |a|^pk≤ max1, |a|^H for all a ∈ C.

Theorem 2.1. W^Iλ, M, p, , ·, , W₀^Iλ, M, p, , ·, , and W_∞^I λ, M, p, , ·,  are linear spaces.

Proof. We will prove the assertion for W₀^Iλ, M, p, , ·,  only and the others can be proved similarly. Assume that x, y∈ W₀^Iλ, M, , ·,  and α, β ∈ R, so



n∈ N : 1 λ_n



k∈In

 M

xk

ρ₁, z

 _pk

≥ ε 

∈ I for some ρ1> 0,



n∈ N : 1 λ_n



k∈Ir

 M

xk

ρ₂, z

 _pk

≥ ε 

∈ I for some ρ2> 0.

2.4

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

1 λn



k∈In

 M



αx_k βyk



|α|ρ1βρ₂, z



_pk

≤ D1 λn



k∈In

 |α|

|α|ρ1βρ2

M

x_k ρ1

, z

 ^pk

 D1 λn



k∈In

 β

|α|ρ1βρ2

M

y_k ρ2

, z

 ^pk

≤ DF 1 λ_n



k∈In

 M

xk

ρ₁, z

 pk

 DF 1 λ_n



k∈In

 M

y_k ρ₂, z

 pk

,

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 λn



k∈In

 M



αx_k βyk



|α|ρ1βρ₂, z



_pk

≥ ε 

⊆



n∈ N : DF 1 λn



k∈In

 M

xk

ρ1, z

 pk

≥ ε 2

∪



n∈ N : DF 1 λn



k∈In

 M

yk

ρ2, z

 pk

≥ 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.

Theorem 2.2. For any fixed n ∈ N, W_∞λ, M, p, , ·,  is paranormed space with respect to the paranorm defined by

g_nx  inf

⎧⎨

⎩ρ^pn/H: ρ > 0 s.t.

 sup

n

1 λn



k∈In

 M

x_k ρ , z

 _pk_1/H

≤ 1, ∀z ∈ X

⎫⎬

⎭. 2.8

Proof. That g_nθ  0 and gn−x  gx are easy to prove. So we omit them.

iii Let us take x  xk and y  yk in W∞λ, M, p, , ·, . Let

Ax 



ρ > 0 : sup

n

1 λn



k∈In

 M

x_k ρ , z

 _pk

≤ 1, ∀z ∈ X 

,

A y





ρ > 0 : sup

n

1 λn



k∈In

 M

yk

ρ , z

 _pk

≤ 1, ∀z ∈ X 

.

2.9

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

sup

n

1 λn



n∈In

M



xk yk

 ρ , z





≤ ρ₁ ρ1 ρ2

sup

n

1 λn



k∈In

M

x_k ρ1

, z



 ρ₂ ρ₁ ρ2

sup

n

1 λ_n



k∈In

M

y_k ρ₂, z



.

2.10

Thus, sup_n1/λn

n∈InMxk yk/ρ1 ρ2, z^pk ≤ 1 and gn

x y

≤ inf  ρ1 ρ2

_pn/H

: ρ1∈ Ax, ρ2∈ A y!

≤ inf ρ₁^pn/H: ρ₁∈ Ax!

 inf ρ^p₂^n/H : ρ₂∈ A y!

 gnx  gn

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_∞F

λ, M, p,, ·, 





x∈ S2 − X : ∃K > 0 s.t. sup

n∈N−F

1 λn



k∈In

 M

x_k ρ , z

 _pk

≤ K

for some ρ > 0, and each z∈ X 

,

2.12

which is a subspace of the space W_∞^Iλ, M, p, , ·,  is a paranormed space with the paranorms gnfor n /∈ F and g^F  inf_n∈N−Fg_n.

Theorem 2.4. Let M,M1, M2, be Orlicz functions. Then we have

i W₀^Iλ, M1, p,, ·,  ⊆ W₀^Iλ, M ◦ M1, p,, ·,  provided pk is such that H0 inf pk>

0.

ii W₀^Iλ, M1, p, , ·,  ∩ W₀^Iλ, M2, p,, ·,  ⊆ W₀^Iλ, M1 M2, p, , ·, .

Proof. i For given ε > 0, first choose ε0 > 0 such that max{ε₀^H, ε^H₀⁰} < ε. Now using the continuity of M choose 0 < δ < 1 such that 0 < t < δ ⇒ Mt < ε0. Let xk ∈ W₀λ, M1, p,, ·, . Now from the definition

Aδ 



n∈ N : 1 λ_n



n∈In

 M1

xk

ρ, z

 pk

≥ δ^H 

∈ I. 2.13

Thus if n /∈ Aδ then

1 λ_n



n∈In

 M₁

x_k ρ , z

 _pk

< δ^H, 2.14

that is,



n∈In

 M1

xk

ρ , z

 pk

< λnδ^H, 2.15

that is,

 M₁

x_k ρ, z

 _pk

< δ^H, ∀k ∈ In, 2.16

that is,

M1

xk

ρ , z



< δ, ∀k ∈ In. 2.17

Hence from above using the continuity of M we must have

M

 M1

xk

ρ , z



< ε0, ∀k ∈ In, 2.18

which consequently implies that



k∈In

 M

 M₁

x_k ρ, z

 _pk

< λ_nmax ε^H₀ , ε^H₀⁰!

< λ_nε, 2.19

that is,

1 λ_n



k∈In

 M

 M₁

xk

ρ , z

 _pk

< ε. 2.20

This shows that



n∈ N : 1 λn



k∈In

 M

 M₁

x_k ρ , z

 _pk

≥ ε 

⊂ Aδ 2.21

and so belongs to I. This proves the result.

ii Let xk ∈ W₀^IM1, p,, ·,  ∩ W₀^IM2, p,, ·, , then the fact

1 λn



M1 M2

x_k ρ , z

 _pk

≤ D1 λn

 M1

x_k ρ, z

 _pk

 D 1 λn

 M2

x_k ρ , z

 _pk

2.22

gives us the result.

Definition 2.5. Let X be a sequence space. Then X is called solid if αkx_k ∈ X whenever

xk ∈ X for all sequences αk of scalars with |αk| ≤ 1 for all k ∈ N.

Theorem 2.6. The sequence spaces W₀^Iλ, M, p, , ·, , W_∞^Iλ, M, p, , ·,  are solid.

Proof. We give the proof for W₀^Iλ, M, p, , ·,  only. Let xk ∈ W₀^Iλ, M, p, , ·,  and let αk be a sequence of scalars such that|αk| ≤ 1 for all k ∈ N. Then we have



n∈ N : 1 λn



k∈In

 M

αkxk ρ , z

 _pk

≥ ε 

⊆



n∈ N : C λn



k∈In

 M

xk

ρ , z

 _pk

≥ ε 

∈ I,

2.23

where C maxk{1, |αk|^H}. Hence αkx_k ∈ W₀^Iλ, M, p, , ·,  for all sequences of scalars αk with|αk| ≤ 1 for all k ∈ N whenever xk ∈ W₀^Iλ, M, p, , ·, .

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