The Generalized Non-absolute type of sequence spaces

In this paper we introduce the notion of $\lambda_{mn}-\chi^{2}$ and $\Lambda^{2}$ sequences. Further, we introduce the spaces $\left[\chi^{2q\lambda}_{f\mu },\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)}$ and $\left[\Lambda^{2q\lambda}_{f\mu },\left\|\left(d\left(x_{1},0\right),d\left(x_{2},0\right),\cdots, d\left(x_{n-1},0\right)\right)\right\|_{p}\right]^{\textit{I}\left(F\right)},$ which are of non-absolute type and we prove that these spaces are linearly isomorphic to the spaces $\chi^{2}$ and $\Lambda^{2},$ respectively. Moreover, we establish some inclusion relations between these spaces.


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 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. Some initial works on double sequence spaces is found in Bromwich [1]. Later on, they were investigated by Hardy [2], Moricz [3], Moricz and Rhoades [4], Basarir and Solankan [5], Tripathy [6], Turkmenoglu [7], and many others.
2010 Mathematics Subject Classification. 40A05; 40C05; 46A45; 03E72; 46B20. Received: Revised: C bp (t) := C p (t) M u (t) and C 0bp (t) = C 0p (t) M u (t); where t = (t mn ) is the sequence of strictly positive reals t mn for all m, n ∈ N and p − lim m,n→∞ denotes the limit in the Pringsheim's sense. In the case t mn = 1 for all m, n ∈ N; M u (t) , C p (t) , C 0p (t) , L u (t) , C bp (t) and C 0bp (t) reduce to the sets M u , C p , C 0p , L u , C bp and C 0bp , respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. Gökhan and Colak [8,9] have proved that M u (t) and C p (t) , C bp (t) are complete paranormed spaces of double sequences and gave the α−, β−, γ− duals of the spaces M u (t) and C bp (t) .
Quite recently, in her PhD thesis, Zelter [10] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [11] and Tripathy have independently introduced the statistical convergence and Cauchy for double sequences and given the relation between statistical convergent and strongly Cesàro summable double sequences. Altay and Basar [12] have defined the spaces BS, BS (t) , CS p , CS bp , CS r and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces M u , M u (t) , C p , C bp , C r and L u , respectively, and also examined some properties of those sequence spaces and determined the α− duals of the spaces BS, BV, CS bp and the β (ϑ) − duals of the spaces CS bp and CS r of double series. Basar and Sever [13] have introduced the Banach space L q of double sequences corresponding to the well-known space q of single sequences and examined some properties of the space L q . Quite recently Subramanian and Misra [14] have studied the space χ 2 M (p, q, u) of double sequences and gave some inclusion relations.
The class of sequences which are strongly Cesàro summable with respect to a modulus was introduced by Maddox [15] as an extension of the definition of strongly Cesàro summable sequences. Connor [16] further extended this definition to a definition of strong A− summability with respect to a modulus where A = (a n,k ) is a nonnegative regular matrix and established some connections between strong A− summability, strong A− summability with respect to a modulus, and A− statistical convergence. In [17] the notion of convergence of double sequences was presented by A. Pringsheim. Also, in [18]- [19], and [20] the four dimensional matrix transformation (Ax) k, = ∞ m=1 ∞ n=1 a mn k x mn was studied extensively by Robison and Hamilton. We need the following inequality in the sequel of the paper. For a, b, ≥ 0 and 0 < p < 1, we have The double series ∞ m,n=1 x mn is called convergent if and only if the double sequence (s mn ) is convergent, where s mn = m,n i,j=1 x ij (m, n ∈ N).
A sequence x = (x mn )is said to be double analytic if sup mn |x mn | 1/m+n < ∞. The vector space of all double analytic sequences will be denoted by Λ 2 . A sequence The double gai sequences will be denoted by x ij ij for all m, n ∈ N ; where ij denotes the double sequence whose only non zero term is a 1 (i+j)! in the (i, j) th place for each i, j ∈ N.
An FK-space(or a metric space)X is said to have AK property if ( mn ) is a Schauder basis for X. Or equivalently x [m,n] → x.
An FDK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings x = (x k ) → (x mn )(m, n ∈ N) are also continuous. Let M and Φ are mutually complementary modulus functions. Then, we have: (iii) For all u ≥ 0, and 0 < λ < 1, Lindenstrauss and Tzafriri [22] used the idea of Orlicz function to construct Orlicz sequence space The space M with the norm , the spaces M coincide with the classical sequence space p .
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 as where M f is a convex modular defined by The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz as follows Here c, c 0 and ∞ denote the classes of convergent,null and bounded sclar valued single sequences respectively. The difference sequence space bv p of the classical space p is introduced and studied in the case 1 ≤ p ≤ ∞ by Başar and Altay and in the case 0 < p < 1 by Altay and Başar in [1]. The spaces c (∆) , c 0 (∆) , ∞ (∆) 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 B µ difference operator is equivalent to the following binomial representation: Let n ∈ N and X be a real vector 0), . . . , d n (x n , 0)) p on X satisfying the following four conditions: y 1 ), (x 2 , y 2 ), · · · (x n , y n )) := sup {d X (x 1 , x 2 , · · · x n ), d Y (y 1 , y 2 , · · · y n )} , for x 1 , x 2 , · · · x n ∈ X, y 1 , y 2 , · · · y n ∈ Y 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 = 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.

Notion of λ mn − double chi and double analytic sequences
The generalized de la Vallee-Pussin means is defined by : t rs (x) = 1 ϕrs m∈Irs n∈Irs x mn , 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.

2.1.
Lemma. Every convergent sequence is λ mn − convergent to the same ordinary limit.

2.2.
Lemma. If a λ mn − Musielak convergent sequence converges in the ordinary sense, then it must Musielak converge to the same λ mn − limit.

The spaces of λ mn − double gai and double analytic sequences
In this section we introduce the sequence space and as sets of λ mn double gai and double analytic sequences: are isomorphic to the spaces and Proof : We only consider the case χ 2 can be shown similarly.
. The linearity of T is obvious. It is trivial that x = 0 whenever T x = 0 and hence T is injective.
To show surjective we define the sequence x = {x mn (λ)} by We can say that B µ η (x) = y mn from (3.1) and . We deduce from that and T x = y. Hence T is surjective. We have for every x ∈ χ 2 .

I(F )
and are ismorphic. Similarly obtain other sequence spaces.

I(F )
. We get .
[44] B.Hazarika and A.Esi, On ideal convergent sequence spaces of fuzzy real numbers associated with multiplier sequences defined by sequence of Orlicz functions, Annals of Fuzzy Mathematics and Informatics, (in press).