ZWEIER I-CONVERGENT DOUBLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION

Abstract

In this article we introduce the zweier double sequence spaces $_2Z^I$(M), $_2Z_0^I$(M) and $_2Z_{\infty}^I$(M) using the Orlicz function M. We study the algebraic properties and inclusion relations on these spaces.

1. Introduction

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

the space of all double sequences real or complex.

Let ℓ∞, c and c0 denote the Banach spaces of bounded, convergent and null sequences respectively normed by

At the initial stage the notion of I-convergence was introduced by Kostyrko,Šalát and Wilczyński [1]. Later on it was studied by Šalát, Tripathy and Ziman[2], Demirci [3] and many others. I-convergence is a generalization of Statistical Convergence.

Now we have a list of some basic definitions used in the paper .

Definition 1.1 ([4,5]). Let X be a non empty set. Then a family of sets I⊆2X(2X denoting the power set of X) is said to be an ideal in X if

For more details see [6,7,8,9,10]. An Ideal I⊆ 2X is called non-trivial if I≠ 2X. A non-trivial ideal I⊆ 2X is called admissible if {{x} : x ∈ X} ⊆I.

A non-trivial ideal I is maximal if there cannot exist any non-trivial ideal J≠I containing I as a subset.

For each ideal I, there is a filter £(I) corresponding to I. i.e

Definition 1.2. A double sequence of complex numbers is defined as a function x : ℕ × ℕ → ℂ. We denote a double sequence as (xij), where the two subscripts run through the sequence of natural numbers independent of each other. A number a ∈ ℂ is called a double limit of a double sequence (xij) if for every ϵ > 0 there exists some N = N(ϵ) ∈ N such that

Definition 1.3 ([12]). A double sequence (xij) ∈ ω is said to be I-convergent to a number L if for every ϵ > 0,

In this case we write I − lim xij = L.

Definition 1.4 ([12]). A double sequence (xij) ∈ ω is said to be I-null if L = 0. In this case we write

Definition 1.5 ([12]). A double sequence (xij) ∈ ω is said to be I-Cauchy if for every ϵ > 0 there exist numbers m = m(ϵ), n= n(ϵ) such that

Definition 1.6 ([12]). A double sequence (xij) ∈ ω is said to be I-bounded if there exists M > 0 such that

Definition 1.7 ([12]). A double sequence space E is said to be solid or normal if (xij) ∈ E implies (αijxij) ∈ E for all sequence of scalars (αij) with |αij| < 1 for all i,j ∈ ℕ.

Definition 1.8 ([12]). A double sequence space E is said to be monotone if it contains the canonical preimages of its stepspaces.

Definition 1.9 ([12]). A double sequence space E is said to be convergence free if (yij) ∈ E whenever (xij) ∈ E and xij = 0 implies yij = 0.

Definition 1.10 ([12]). A double sequence space E is said to be a sequence algebra if (xij.yij) ∈ E whenever (xij), (yij) ∈ E.

Definition 1.11 ([12]). A double sequence space E is said to be symmetric if (xij) ∈ E implies (xπ(ij)) ∈ E, where π is a permutation on

Any linear subspace of ω, is called a sequence space.

A sequence space λ with linear topology is called a K-space provided each of maps pi → ¢defined by pi(x) = xi is continuous for all i ∈

A K-space λ is called an FK-space provided λ is a complete linear metric space.

An FK-space whose topology is normable is called a BK-space.

Let λ and μ be two sequence spaces and A = (ank) be an infinite matrix of real or complex numbers ank, where n, k ∈ Then we say that A defines a matrix mapping from λ to μ, and we denote it by writing A : λ → μ.

If for every sequence x = (xk) ∈ λ the sequence Ax = {(Ax)n}, the A transform of x is in μ, where

By (λ : μ), we denote the class of matrices A such that A : λ → μ.

Thus, A ∈ (λ : μ) if and only if the series on the right side of (1) converges for each n ∈ and every x ∈ λ. (see[14]).

The approach of constructing the new sequence spaces by means of the matrix domain of a particular limitation method have been recently studied by Başar and Altay[15], Malkowsky[16], Ng and Lee[17] and Wang[18], Başar, Altay and Mursaleen[19].

Şengönül[20] defined the sequence y = (yi) which is frequently used as the Zp transform of the sequence x = (xi) i.e,

where x−1 = 0, p ≠ 1, 1 < p < ∞ and Zp denotes the matrix Zp = (zik) defined by

Following Basar and Altay [15], Şengönül[20] introduced the Zweier sequence spaces Z and Z0 as follows

An Orlicz function is a function M : [0,∞) → ]undefined[0,∞), which is continuous, non-decreasing and convex with M(0) = 0,M(x) > 0 for x > 0 and M(x) → ∞ as x → ∞.(see]undefined[21,22]).

Lindenstrauss and Tzafriri[22] used the idea of Orlicz functions to construct the sequence space

The space ℓM is a Banach space with the norm

The space ℓM is closely related to the space ℓp which is an Orlicz sequence space with M(x) = xp for 1 ≤ p < ∞ (c.f [23],[24],[25]).

The following Lemmas will be used for establishing some results of this article.

Lemma 1.12 ([24]). A sequence space E is solid implies that E is monotone.

Lemma 1.13 ([26,27,28]). Let K ∈ £(I) and M ⊆ N. If M ∉ I, thenM⋂K ∉ I.

Lemma 1.14 ([26,27,28]). If I ⊂ 2N and M ⊆ N. If M ∉ I, then M ∩ K ∉ I.

Recently Vakeel.A.Khan et. al.[29] introduced and studied the following classes of sequence spaces.

We also denote by

and

 

2. Main results

In this article we introduce the following classes of zweier I-Convergent double sequence spaces defined by the Orlicz function.

Also we denote by

and

Throughout the article, for the sake of convenience, we will denote by Zp(xk) = x′,Zp(yk) = y′,Zp(zk) = z′ for x, y, z ∈ ω.

Theorem 2.1. For any Orlicz function M, the classes of sequences 2ZI (M), are linear spaces.

Proof. We shall prove the result for the space 2ZI (M). The proof for the other spaces will follow similarly. Let (xij), (yij) ∈ 2ZI (M) and let α, β be scalars. Then there exists positive numbers ρ1 and ρ2 such that

That is for a given ϵ > 0, we have

Let ρ3 = max{2|α|ρ1, 2|β|ρ2}. Since M is non-decreasing and convex function, we have

Now, by (1) and (2),

Therefore (αxij + βyij) ∈ 2ZI (M). Hence 2ZI (M) is a linear space.

Theorem 2.2. The spaces are Banach spaces normed by

Proof. Proof of this result is easy in view of the existing techniques and therefore is omitted.

Theorem 2.3. Let M1 and M2 be Orlicz functions that satisfy the △2-condition. Then

(a) X(M2) ⊆ X(M1.M2);

(b) X(M1) ∩ X(M2) ⊆ X(M1 + M2) For

Proof. (a) Let Then there exists ρ > 0 such that

Let ϵ > 0 and choose δ with 0 < δ < 1 such that M1(t) < ϵ for 0 ≤ t ≤ δ.

Write and consider for all we have

We have

For (yij) > δ, we have

Since M1 is non-decreasing and convex, it follows that

Since M1 satisfies the △2-condition, we have

Hence

From (3), (4) and (5), we have Thus

The other cases can be proved similarly.

(b) Let Then there exists ρ > 0 such that

The rest of the proof follows from the following equality

Theorem 2.4. The spaces are solid and monotone.

Proof. We shall prove the result for the result can be proved similarly. Let Then there exists ρ > 0 such that

Let (αij) be a sequence of scalars with |αij | ≤ 1 for all Then the result follows from (6) and the following inequality for all

By Lemma 1.12, a sequence space E is solid implies that E is monotone.

We have the space is monotone.

Theorem 2.5. The spaces are neither solid nor mono-tone in general.

Proof. Here we give a counter example. Let I = Iδ and M(x) = x2 for all x ∈ [0,∞). Consider the K-step space XK(M) of X(M) defined as follows,

let (xij) ∈ X(M) and let (yij) ∈ XK(M) be such that

Consider the sequence (xij) defined by xij = 1 for all

Then (xij) ∈ 2ZI (M) but its K-stepspace preimage does not belong to 2ZI (M).

Thus 2ZI (M) is not monotone. Hence 2ZI (M) is not solid.

Theorem 2.6. The spaces and 2ZI (M) are not convergence free in general.

Proof. Here we give a counter example. Let I = If and M(x) = x3 for all x ∈ [0,∞). Consider the sequence (xij) and (yij) defined by

Then (xij) ∈ 2ZI (M) and but (yij) ∉ 2ZI (M) and

Hence the spaces 2ZI (M) and are not convergence free.

Theorem 2.7. The spaces and 2ZI (M) are sequence algebras.

Proof. We prove that is a sequence algebra. For the space 2ZI (M), the result can be proved similarly. Let Then

Let ρ = ρ1.ρ2 > 0. Then we can show that

Thus Hence is a sequence algebra.

Theorem 2.8. If I is not maximal and I ≠ If, then the spaces 2ZI (M) and are not symmetric.

Proof. Let A ∈ I be infinite and M(x) = x for all x = (xij). If

Then by lemma 1.14. Let be such that K ∉ I and Let ϕ : K → A and be bijections, then the map defined by

is a permutation on but (xπ(i)π(j)) ∉ 2ZI (M) and

Hence and 2ZI (M) are not symmetric.