ON SOME GENERALIZED I-CONVERGENT DOUBLE SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION

Abstract

In this article we introduce the sequence spaces $_2c^I_0(f,p)$, $_2c^I(f,p)$ and $_2l^I_{\infty}(f,p)$ for a modulus function f, where $p=(p_k)$ is a sequence of positive reals and study some of the properties of these spaces.

1. Introduction

The notion of I-Convergence is a generalization of the concept of statistical convergence which was first introduced by H.Fast [5] and later on studied by various mathematicians like J.A.Fridy [6,7], Kostyrko, Salat and Wilezynski [19], Salat, Tripathy, Ziman [29] and Demirci [3].

Also a double sequence is a double infinite array of elements xkl ∈ ℝ for all k, l ∈ ℕ (see [14,15]). The initial works on double sequences is found in Bromwich [1], Basarir and Solancan [2] and many others. Throughout this article a double sequence is denoted by x = (xij).

Next we discuss some preliminaries about I-convergence (see [12],[30]).

Let X be a non empty set. Then a family of sets I⊆ 2X (power set of X) is said to be an ideal if I is additive i.e A,B∈I ⇒A∪ B∈I and hereditary i.e A∈I, B⊆A⇒B∈I.

A non-empty family of sets £(I) ⊆ 2X is said to be filter on X if and only if Φ ∉ £(I), for A,B∈ £(I) we have A∩B∈ £(I) and for each A ∈ £(I)and A⊆B implies B∈ £(I). 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 £(I) = {K ⊆ N : Kc ∈ I},where Kc = N-K.

Definition 1.1. A double sequence (xij) ∈ ω is said to be I-convergent to a number L if for every ϵ > 0. {(i, j) ∈ × : |xij − L| ≥ ϵ} ∈ I. In this case we write I-lim xij = L. (see [17])

The space 2cI of all I-convergent sequences to L is given by

Definition 1.2. A sequence (xij) ∈ ω is said to be I-null if L = 0. In this case we write I-lim xij = 0.

Definition 1.3. A sequence (xij) ∈ ω is said to be I-cauchy if for every ϵ > 0 there exists a number m = m(ϵ) and n = n(ϵ) such that

Definition 1.4. A sequence (xij) ∈ ω is said to be I-bounded if there exists M > 0 such that {(i, j) ∈ × : |xij| > M}.

Definition 1.5. Let (xij), (yij) be two sequences. We say that (xij) = (yij) for almost all (i,j) relative to I (a.a.k.r.I), if {(i, j) ∈ × : xij ≠ yij} ∈ I

Definition 1.6. For any set E of sequences the space of multipliers of E, denoted by M(E) is given by

Definition 1.7. A map ħ defined on a domain D ⊂ X i.e ħ : D ⊂ X → is said to satisfy Lipschitz condition if |ħ(x)−ħ(y)| ≤ K|x−y| where K is known as the Lipschitz constant.The class of K-Lipschitz functions defined on D is denoted by ħ ∈ (D,K).

Definition 1.8. A convergence field of I-convergence is a set

The convergence field F(I) is a closed linear subspace of l∞ with respect to the supremum norm, F(I) = l∞ ∩ 2cI (See[23]).

Define a function ħ : F(I) → such that ħ (x) = I − lim x, for all x ∈ F(I), then the function ħ : F(I) → is a Lipschitz function ([11,4,13]).

Definition 1.9. The concept of paranorm is closely related to linear metric spaces [16]. It is a generalization of that of absolute value.

Let X be a linear space. A function g : X → R is called paranorm, if for all x, y, z ∈ X,

(PI) g(x) = 0 if x = θ, (P2) g(−x) = g(x), (P3) g(x + y) ≤ g(x + g(y),

(P4) If (λn) is a sequence of scalars with λn → λ (n → ∞) and xn, a ∈ X with xn → a (n → ∞) , in the sense that g(xn − a) → 0 (n → ∞) , in the sense that g (λnxn − λa) → 0 (n → ∞).

A paranorm g for which g(x) = 0 implies x = θ is called a total paranorm on X, and the pair (X, g) is called a totally paranormed space.(See[23]). The idea of modulus was structured in 1953 by Nakano.(See[24]). A function f : [0,∞)→[0,∞) is called a modulus if

(1) f(t) = 0 if and only if t = 0, (2) f(t+u)≤ f(t)+ f(u) for all t,u≥0,

(3) f is increasing and (4) f is continuous from the right at zero.

Ruckle in [25,26,27] used the idea of a modulus function f to construct the sequence space

This space is an FK space ,and Ruckle proved that the intersection of all such X(f) spaces is ϕ, the space of all finite sequences.

The space X(f) is closely related to the space l1 which is an X(f) space with f(x) = x for all real x ≥ 0. Thus Ruckle proved that,for any modulus f

Where

The space X(f) is a Banach space with respect to the norm

Spaces of the type X(f) are a special case of the spaces structured by B.Gramsch in[10]. From the point of view of local convexity, spaces of the type X(f) are quite interesting.

Symmetric sequence spaces, which are locally convex have been frequently studied by D.J.H Garling [8,9], G.Köthe [18].

The following subspaces of ω were first introduced and discussed by Maddox [22,23].

where p = (pk) is a sequence of striclty positive real numbers.

After then Lascarides[20,21] defined the following sequence spaces

where for all k ∈ .

We need the following lemmas in order to establish some results of this article.

Lemma 1.10. Then the following conditions are equivalent.(See[18]).

(a) H < ∞ and h > 0. (b) c0(p) = c0 or l∞(p) = l∞. (c) l∞{p} = l∞(p). (d) c0{p} = c0(p). (e) l{p} = l(p).

Lemma 1.11. Let K∈ £(I) and M⊆N. If M∉I, then M∩K ∉ I.(See[29,30]).

Lemma 1.12.If I ⊂ 2X and M⊆X. If M ∉ I, then M∩K ∉ I.(See[29,30]).

Throughout the article represent the bounded , I-convergent, I-null, bounded I-convergent and bounded I-null sequence spaces respectively. In this article we introduce the following classes of sequence spaces.

Also we write

 

2. Main results

Theorem 2.1. Let (pij) ∈ 2l∞. are linear spaces.

Proof. Let (xij), (yij) ∈2 cI (f, p) and α, β be two scalars. Then for a given ϵ > 0.

We have

where

be such that Then

Thus Hence (αxij + βyij) ∈ 2cI (f, p). Therefore 2cI (f, p) is a linear space. The rest of the result follows similarly.

Theorem 2.2. Let (pij) ∈ 2l∞. Then 2mI (f, p) and are paranormed spaces, paranormed by

Proof. Let x = (xij), y = (yij) ∈ 2mI (f, p).

(1) Clearly, g(x) = 0 if and only if x = 0.

(2) g(x) = g(-x) is obvious.

(3) Since using Minkowski’s inequality and the definition of f we have

(4) Now for any complex λ we have (λij) such that λij → λ, (i, j → ∞).

Let xij ∈ 2mI (f, p) such that f(|xij − L|pij) ≥ ϵ.

Therefore,

Hence g(λijxij-λL) ≤ g(λijxij) + g(λL) = λijg(xij) + λg(L) as (i, j → ∞).

Hence 2mI (f, p) is a paranormed space. The rest of the result follows similarly.

Theorem 2.3. A sequence x = (xij) ∈ 2mI (f, p) I-converges if and only if for every ϵ > 0 there exists Nϵ ∈ × where Nϵ = (m, n), m and n depending upon ϵ such that

Proof. Suppose that L = I − lim x. Then

Fixing some Nϵ ∈ Bϵ, we get

which holds for all (i, j) ∈ Bϵ. Hence

Conversely, suppose that

That is {(i, j) ∈ × : (|xij − xNϵ|pij) < ϵ} ∈ 2mI (f, p) for all ϵ > 0. Then the set

Let Jϵ = [xNϵ − ϵ, xNϵ + ϵ]. If we fix an ϵ > 0 then we have Cϵ ∈ 2mI (f, p) as well as This implies that

that is

that is

where the diam of J denotes the length of interval J. In this way, by induction we get the sequence of closed intervals

with the property that and {(i, j) ∈ × : xij ∈ Ik} ∈ 2mI (f, p) for (k=1,2,3,4,......).

Then there exists a ξ ∈ ∩Ik where (i, j) ∈ × such that ξ = I − limx. So that f(ξ) = I − lim f(x), that is L = I − lim f(x).

Theorem 2.4. and I be an admissible ideal. Then the following are equivalent.

(a) (xij) ∈ 2cI (f, p);

(b) there exists (yij) ∈ 2c(f, p) such that xij = yij, for a.a.k.r.I;

(c) there exists (yij) ∈ 2c(f, p) and such that xij = yij + zij for all (i, j) ∈ × and {(i, j) ∈ × : f(|yij − L|pij) ≥ ϵ} ∈ I;

(d) there exists a subset J × K where J = {j1, j2, ...} and K = {k1 < k2....} of × such that

Proof. (a) implies (b)

Let (xij) ∈ 2cI (f, p). Then there exists L ∈ that

Let (mt) and (nt) be increasing sequences with mt and nt ∈ such that

Define a sequence (yij) as

For mt < k ≤ mt+1, t ∈ .

Then (yij) ∈ 2c(f, p) and form the following inclusion

We get xij = yij, for a.a.k.r.I.

(b) implies (c)

For (xij) ∈ 2cI (f, p). Then there exists (yij) ∈ 2c (f, p) such that xij = yij, for a.a.k.r.I. Let K = {(i, j) ∈ × : xij ≠ yij}, then (i, j) ∈ I.

Define a sequence (zij) as

Then and yij ∈ 2c(f, p).

(c) implies (d)

Let P1 = {(i, j) ∈ × : f(|xij |pij) ≥ ϵ} ∈ I and

Then we have

(d) implies (a)

Let K = {(i1, j1) < (i2, j2) < (i3, j3) < ...} ∈ £(I) and Then for an ϵ > 0, and Lemma 1.10, we have

Thus (xij) ∈ 2cI (f, p).

Theorem 2.5. Let (pij) and (qij) be a sequence of positive real numbers. Then where Kc ⊆ × such that K ∈ I.

Proof. Let Then there exists β > 0 such that pij > βqij, for all sufficiently large (i, j) ∈ K.

Since for a given ϵ > 0, we have

Let G0 = Kc ∪ B0 Then G0 ∈ I. Then for all sufficiently large (i, j) ∈ G0,

Therefore

Corollary 2.6. Let (pij) and (qij) be two sequences of positive real numbers. Then where Kc ⊆ × such that K ∈ I.

Theorem 2.7. Let (pij) and (qij) be two sequences of positive real numbers. Then where K ⊆ × such that Kc ∈ I.

Proof. On combining Theorem 2.5 and 2.6 we get the required result.

Theorem 2.8. Then the following results are equivalent. (a) H < ∞ and h > 0.

Proof. Suppose that H < ∞ and h > 0,then the inequalities min{1, sh} ≤ spij ≤ max{1, sH} hold for any s > 0 and for all (i, j) ∈ × . Therefore the equivalence of (a) and (b) is obvious.

Theorem 2.9. Let f be a modulus function. Then (see [13]).

Proof. Let (xij) ∈ 2cI (f, p). Then there exists L ∈ that

We have

Taking supremum over (i, j) both sides we get and the inclusion is obvious. Hence and the inclusions are proper.

Theorem 2.10. then for any modulus f, we have where the inclusion may be proper.

Proof. Let This implies that for some K > 0 and all (i, j). Therefore x = (xij) ∈ 2mI (f, p) implies

which gives To show that the inclusion may be proper, consider the case when for all (i, j). Take aij = (i × j) for all (i, j).

Therefore x ∈ 2mI (f, p) implies

Thus in this case a = (aij) ∈ M( 2mI (f, p)) while

Theorem 2.11. The function ħ : 2mI (f, p) → is the Lipschitz function,where 2mI (f, p) = 2cI (f, p) ∩ 2l∞(f, p), and hence uniformly continuous.

Proof. Let x, y ∈ 2mI (f, p),x ≠ y. Then the sets

Here

Thus the sets,

Hence also B = Bx ∩ By ∈ 2mI (f, p), so that B ≠ ϕ. Now taking (i, j) in B such that

Thus ħ is a Lipschitz function. For the result can be proved similarly.

Theorem 2.12. If x, y ∈ 2mI (f, p),then (x.y) ∈ 2mI (f, p) and ħ xy) = ħ(x)ħ(y).

Proof. For ϵ > 0

Now,

As 2mI (f, p) ⊆ l∞(f, p),there exists an M ∈ such that |xij|pij < M and |ħ(y)|pk < M. Using eqn(2) we get

For all (i, j) ∈ Bx ∩ By ∈ 2mI (f.p). Hence (x.y) ∈ mI (f, p) and ħ(xy) = ħ(x)ħ(y). For the result can be proved similarly.