The Multi Ideal Convergence of Difference Strongly of χ 2 in P-Metric Spaces Defined by Modulus

The aim of this paper is to introduce multi and study a new concept of the χ2 space via ideal convergence of difference operator defined by modulus. Some topological properties of the resulting sequence spaces are also examined.


Introduction
Throughout w, χ and Λ denote the classes of all, gai and analytic scalar valued single sequences, respectively.We write w 2 for the set of all complex double sequences (x mn ), where m, n ∈N, the set of positive integers.Then, w 2 is a linear space under the coordinate wise addition and scalar multiplication.
Let (x mn ) be a double sequence of real or complex numbers.Then the series  The vector space of all double entire sequences are usually denoted by Γ 2 .Let the set of sequences with this property be denoted by Λ 2 and Γ 2 is a metric space with the metric Consider a double sequence x = (x mn ).The (m, n) th section x [m,n] of the sequence is defined by for all m, n ∈N, with 1 in the (m, n) th position and zero otherwise.
A sequence f = (f mn ) of modulus function is called a Musielak-modulus function.A sequence g = (g mn ) defined by is called the complementary function of a Musielak-modulus function f.For a given Musielak modulus function f, the Musielak-modulus sequence space t f is defined by We consider t f equipped with the Luxemburg metric space, (i.e.))Let (X i , d i ), i ∈ I be a family of metric spaces such that each two elements of the family are disjoint.Denote then the pair (X, d) is a Luxemburg metric space.The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz(1981)as follows Z(Δ) = {x = (x k ) ∈ w : (Δ x k ) ∈ Z}, for Z = c, c 0 and ℓ ∞ , where Δ x k = x k − x k+1 for all k ∈N.
Here c, c 0 and ℓ ∞ denote the classes of convergent, null and bounded sclar valued single sequences respectively.The difference sequence space bv p of the classical space ℓ p is introduced and studied in the case 1 ≤ p ≤∞ by Başar and Altay and in the case 0 < p < 1.The spaces c(Δ), c 0 (Δ), ℓ ∞ (Δ) and bv p are Banach spaces normed by Later on the notion was further investigated by many others.We now introduce the following difference double sequence spaces defined by Z(Δ) = {x = (x mn ) ∈ w 2 : (Δ x mn ) ∈ Z }, where Z = Λ 2 , χ 2 and Δ x mn = (x mn − x mn+1 ) − (x m+1n − x m+1n+1 ) = x mn − x mn+1 − x m+1n + x m+1n+1 for all m, n ∈N.The generalized difference double notion has the following representation: Δ m x mn = Δ m−1 x mn −Δ m−1 x mn+1 −Δ m−1 x m+1n + Δ m−1 x m+1n+1 , and also this generalized difference double notion has the following binomial representation:

Definitions and Preliminaries
Let Δ m X be a non empty set.A non-void class is called admissible if and only if {{x} : x ∈Δ m X} ⊂ I.
A double sequence space E is said to be solid or normal if (α mn Δ m x mn ) ∈ E, whenever (Δ m x mn ) ∈ E and for all double sequences α = (α mn ) of scalars with |α mn | ≤ 1. for all m, n ∈N.
A trivial example of p-product metric of n-metric space is the p-norm space is X = R equipped with the following Euclidean metric in the product space is the p-norm: If every Cauchy sequence in X converges to some L ∈ X, then X is said to be complete with respect to the p-metric.Any complete p-metric space is said to be p-Banach metric space.

Main Results
In this section we introduce the notion of different types of I-convergent double sequences.This generalizes and unifies different notions of convergence for χ 2 .We shall denote the ideal of 2 N × N by I 2 .
Let I 2 be an ideal of 2 N × N , f be an modulus function.Let u and v be two non-negative integers and μ = (μ mn ) be a sequence of non-zero reals.Then for a sequence η = (η mn ) be a double analytic sequence of strictly positive real numbers and ( ) be an p-product of n metric spaces is the p norm of the n-vector of the norms of the n subspaces.Further -valued sequence space.Now, we define the following sequence spaces:

=
The following well-known inequality will be used in this study: 0 ≤ inf mn η mn = H 0 ≤η mn ≤ sup mn = H <∞, D = max(1, 2 Since ||(d 1 (x 1 , 0), ..., d n (x n , 0))|| p be an p-product of n metric spaces is the p norm of the n-vector of the norms of the n subspaces and f is an modulus function, the following inequality holds: From the above inequality we get This completes the proof.
Proof: g rs (θ) = 0 and g rs (−x) = g rs (x) are easy to prove, so we omit them.
Let us take x, y ∈ and g rs (x + y) = g rs (x) + g rs (y).

Now, let λ, λ u
mn → where , λ u mn λ∈C and as u →∞.We have to prove that Hence by our assumption the right hand side tends to zero as u, m and n →∞.This completes the proof.
The proof can be established using standard technique.
The following result is well known.Theorem 3.6 Let f, f 1 and f 2 be modulus functions.Then we have

Let
for all m, n = 1, 2, ... .Hence from above inequality and using continuity of f, we must have x = (x mn ) is said to be double analytic if of all double analytic sequences are usually denoted by Λ 2 .A sequence x = (x mn ) is called double {x mn } and y = {y mn } in Γ 2 .Let φ = {finite sequences}.

Lemma 3 . 4
If a sequence space E is solid, then it is monotone.Theorem 3.5 The class of sequence η not solid and hence not monotone.Proof:It is routine verification.Therefore we omit the proof.
prove the result for other cases.