The New Generalized Difference Sequence Space χ 2 over p – Metric Spaces Defined by Musielak Orlicz Function Associated with a Sequence of Multipliers

In the present paper, we introduce new sequence spaces by using Musielak-Orlicz function and a generalized B η -difference operator on p–metric space. Some topological properties and inclusion relations are also examined. Citation: Deepmala N, Subramanian N, Mishra VN (2016) The New Generalized Difference Sequence Space χ2 over p–Metric Spaces Defined by Musielak Orlicz Function Associated with a Sequence of Multipliers. J Appl Computat Math 5: 331. doi: 10.4172/2168-9679.1000331


Introduction
Throughout w, χ and L 2 denote the classes x of all, gai and analytic scalar valued single sequences, respectively. We write w 2 for the set of all complex sequences (x mn ) where m,n ∈ , the set of positive integers. Then w 2 is a linear space under the coordinate wise addition and scalar multiplication. For some approximations results in Musielak-Orlicz-Sobolev spaces and some applications to nonlinear partial differential equations see equation 22. The growing interest in this field is strongly stimulated by the treatment of recent problems in elasticity, fluid dynamics, calculus of variations, and differential equations.
Let (x mn ) be a double sequence of real or complex numbers. Then the series  Let M and Φ be mutually complementary Orlicz functions. Then, we have: (i) For all u,y ≥ 0, uy ≤ M (u) + Φ (y), (Young's inequality) [16] (2) (ii) For all u ≥ 0, (iii) For all u ≥ 0, and 0 < λ < 1, Lindenstrauss and Tzafriri [16] used the idea of Orlicz function to construct Orlicz sequence space The space  M with the norm becomes a Banach space which is called an Orlicz sequence space. For M (t) = t p (1≤p<∞), the spaces  M coincide with the classical sequence space  p .
A sequence f = (f mn ) of Orlicz function is called a Musielak-Orlicz function. A sequence g = (g mn ) defined by gmn(v) = sup{|v|u -fmn(u): u ≥ 0}, m,n = 1,2,… is called the complementary function of a Musielak-Orlicz function f. For a given Musielak Orlicz function f, the Musielak-Orlicz sequence space t f is defined by: We consider t f equipped with the Luxemburg metric The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [17] as follows Here c,c 0 and  ∞ denote the classes of convergent, null and bounded sclar valued single sequences respectively. The spaces c(∆),c 0 (∆),l ∞ (∆) 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: The generalized difference double notion has the following representation: and also this generalized difference double notion has the following binomial representation: Let η = (η mn ) be a sequence of nonzero scalars. Then, for a sequence space E, the multiplier sequence space E η , associated with the multiplier sequence η, is defined as: The notion of sequence spaces associated with multiplier sequences was introduced by. Later on this notion was studied from different aspects by Tripathy and Sen [18], Tripathy and Hazarika [19] and many others [20].
Let η=(η mn ) be a sequence of nonzero scalars. Then, for a sequence space E, the multiplier sequence space E η , associated with the multiplier sequence η, is defined as:

Definition and Preliminaries
Let n ∈  and X be a real vector space of dimension w, where n ≤ w. A real valued function   is called the p product metric of the Cartesian product of n metric spaces is the p norm of the n-vector of the norms of the n subspaces.
A trivial example of p product metric of n metric space is the p norm space is X =  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 pmetric space is said to be p-Banach metric space.
Let X be a linear metric space. A function w: X is called paranorm, if: (4) If (σ mn ) is a sequence of scalars with σ mn → σ as m,n→∞ and (x mn ) is a sequence of vectors with w(x mn -x) → 0 as m,n→∞, then w(σ mn x mnσx) → 0 as m,n→∞.
A paranorm w for which w(x) = 0 implies x = 0 is called total paranorm and the pair (X,w) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm by (Willansky, 1984). η=(ϕ rs ) a nondecreasing sequence of positive reals tending to infinity and ϕ 11 =1 and ϕ r+1,s+1 ≤ϕ rs +1.
The generalized de la Vallee-Pussin means is defined by:

∑ ∑
Where I rs = [rs-λ rs +1,rs]. For the set of sequences that are strongly summable to zero, strongly summable and strongly bounded by the de la Vallee-Poussin method.
The notion of λ-double gai and double analytic sequences as follows: Let = mn m n λ λ ∞ be a strictly increasing sequences of positive real numbers tending to infinity, that is: 0 < λ 00 < λ 11 < …. and λ mn → ∞ as m,n → ∞ and said that a sequence x = (x mn ) ∈ w 2 is λ-convergent to 0, called a the in the ordinary sense of convergence, then: This implies that: be a p-metric space, q = (q mn ) be double analytic sequence of strictly positive real numbers. By w 2 (p -X) we denote the space of all sequences defined over: The following inequality will be used throughout the paper. If  be an p-metric space and let s(w 2 -x) denote the space of X-valued sequences. Let q = (q mn ) be any bounded sequence of positive real numbers and f = (f mn ) be a Musielak-Orlicz function. We define the following sequence spaces in this paper: If we take f mn (x) = x, we get: If we take q = (q mn ) = 1, we get: In the present paper we plan to study some topological properties and inclusion relation between the above defined sequence spaces.
which we shall discuss in this paper.

Theorem 1
Let f = (f mn ) be a Musielak-Orlicz function, q = (q mn ) be a double analytic sequence of strictly positive real numbers, the sequence spaces

Proof
It is routine verification. Therefore the proof is omitted.

Theorem 2
Let f = (f mn ) be a Musielak-Orlicz function, q = (q mn ) be a double analytic sequence of strictly positive real numbers, the sequence space.

g x y g x g y + ≤ +
Finally, to prove that the scalar multiplication is continuous. Let λ be any complex number. By definition, If the sequence (g mn ) satisfies uniform ∆ 2 -condition, then:

Proof
Let the sequence (f mn ) satisfies uniform ∆ 2 -condition, we get: Since the sequence (f mn ) satisfies uniform ∆ 2 -condition, then: Thus: This gives that: (ii) Similarly, one can prove that: if the sequence (g mn ) satisfies uniform ∆ 2 -condition.

Proposition 1
If 0<q mn <p mn <∞ for each m and n, then:

Proof
The proof is standard, so we omit it.
The proof is standard, so we omit it.

Proposition 3
The proof is easy so we omit it.

Proposition 4
For any sequence of Musielak Orlicz functions f = (f mn ) and q = (q mn ) be double analytic sequence of strictly positive real numbers. Then The proof is easy so we omit it.

Proposition 5
The sequence space

Proposition 6
The sequence space

Proof
The proof follows from Proposition 5.

Proposition 7
If f = (f mn ) be any Musielak Orlicz function. Then

Proof
It is easy to prove so we omit.

Proposition 9
The sequence space is not monotone.

Proof
The proof follows from Proposition 9.
A sequence x = (x mn ) is said to be ϕ-statistically convergent or s ϕ -statistically convergent to 0 if for every ε > 0,