The generalized χ2 sequence spaces over p- metric spaces defined by Musielak

In this paper, we introduce generalized χ2 sequence spaces over p- metric spaces defined by Musielak function f = (fmn) and study some topological properties. 40A05; 40C05; 40D05

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, Zeltser [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 [6] 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 BaŞar [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 is 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 is strongly Cesàro summable with respect to a modulus was introduced http://www.iaumath.com/content/7/1/39 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 Asummability with respect to a modulus, where A = a n,k is a non-negative regular matrix, and established some connections between strong A-summability, strong Asummability with respect to a modulus, and A-statistical convergence. In [17], the notion of convergence of double sequences was presented by 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 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 The vector space of all double analytic sequences will be denoted by 2 The double gai sequences will be denoted by χ 2 . Let φ = {all finite sequences}.
Consider a double sequence x = (x ij ). The (m, n) th section x [m,n] of the sequence is defined by x [m,n] = m,n i,j=0 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. A Fréchet coordinate space (FK-space or a metric space) X is said to have an 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 space, locally convex topology under which the coordinate mappings x = (x k ) → (x mn )(m, n ∈ N) are also continuous.
Let M and be mutually complementary modulus functions. Then, we have (1) For all u, y ≥ 0, uy ≤ M(u) + (y), (Young's inequality; see [21]). (2) (2) For all u ≥ 0, (3) 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 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 modulus function is called a Musielak-modulus function. A sequence g = g mn defined by is called the complementary function of a Musielakmodulus function f. For a given Musielak modulus function f, the Musielak-modulus sequence space t f and its subspace h f are defined, respectively, as follows: where I f is a convex modular defined by We consider that t f is equipped with the Luxemburg metric If X is a sequence space, we give the following definitions: (1) X = the continuous dual of X ; (6) X δ = a = (a mn ) : sup mn |a mn x mn | 1/m+n < ∞, where X α , X β , and X γ are called α− (or Köthe-Toeplitz) dual of X, β− (or generalized Köthe-Toeplitz) dual of X, γ − dual of X, and δ− dual of X, respectively. X α is defined http://www.iaumath.com/content/7/1/39 by Kantham and Gupta [21]. It is clear that X α ⊂ X β and X α ⊂ X γ , but X β ⊂ X γ does not hold since the sequence of partial sums of a double convergent series needs not to be bounded. The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [23] as follows: Here, c, c 0 , and ∞ denote the classes of convergent, null, and bounded scalar valued single sequences, respectively. The difference sequence space bv p of the classical space p is introduced and studied in the case 1 ≤ p ≤ ∞ and in the case 0 < p < 1 by Altay and BaŞar in [12]. 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

Definition and preliminaries
Let n ∈ N and X be a real vector space of dimension w, . . , d n (x n )) p on X satisfying the following four conditions: , for x 1 , x 2 , · · · x n ∈ X, y 1 , y 2 , · · · y n ∈ Y which 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 the p product metric of the n metric space is the p norm space which is X = R equipped with the following Euclidean metric in the product space: 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.
Let X be a linear metric space. A function w : A paranorm w for which w(x) = 0 implies x = 0 is called a 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 (see [24], Theorem 10.4.2, p.183). The notion of λ− double gai and double analytic sequences is as follows: Let λ = (λ mn ) ∞ m,n=0 be a strictly increasing sequence of positive real numbers tending to infinity, that is, 0 < λ 0 < λ 1 < · · · and λ mn → ∞ as m, n → ∞ and that a sequence If lim mn x mn = 0 in the ordinary sense of convergence, then This implies that it yields lim uv μ mn (x) = 0, and hence, x = (x mn ) ∈ w 2 is λ− convergent to 0. Let f = f mn be a Musielak-modulus function, , · · · , d (x n−1 )) p be a p-metric space, http://www.iaumath.com/content/7/1/39 and q = (q mn ) be double analytic sequence of strictly positive real numbers. By w 2 (p − X), we denote the space of all sequences as X, The following inequality will be used throughout the paper. If 0 ≤ q mn ≤ supq mn = H, K = max 1, 2 H−1 , then for all m, n and a mn , b mn ∈ C. Also, |a| q mn ≤ max 1, |a| H for all a ∈ C.
In the present paper, we define the following sequence spaces: If we take f mn (x) = x, we get If we take q = (q mn ) = 1 In the present paper, we plan to study some topological properties and inclusion relation between the above defined sequence spaces, χ , · · · , d (x n−1 )) ϕ p , which we shall discuss in this paper.

Main results
Theorem 1. Let f = f mn be a Musielak-modulus function and 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-modulus function and q = (q mn ) be a double analytic sequence of strictly positive real numbers; the sequence space p is a paranormed space with respect to the paranorm defined by where H = max (1, sup mn q mn < ∞).
Conversely, suppose that g(x) = 0, then Suppose that μ mn (x) = 0 for each m, n ∈ N. Then, by using Minkowski's inequality, we have Therefore, Finally, to prove that the scalar multiplication is continuous, let λ be any complex number. By definition, Then, where t = 1 |λ| . Since |λ| q mn ≤ max 1, |λ| supp mn , we have Proof. First, we observe that Therefore, Hence, Next, we show that Hence, it converges to zero. Therefore, for each m, n. Thus, (y mn ) is a p-metric paranormed space of double analytic sequence and, hence, an p-metric double analytic sequence. In other words. y ∈ From (6) and (7), we get · · · , d (x n−1 )) ϕ p . In other words.
Proof. We recall that λ mn (m+n)! in the (m, n)th position and zeros elsewhere, which is a p-metric of double gai sequence. Hence, x mn y mn Thus, (y mn ) is a p-metric of the double analytic sequence and an p-metric of double analytic sequence.

Theorem 5. (1) If the sequence f mn satisfies uniform
(2) If the sequence g mn satisfies uniform 2 − condition, then Proof. Let the sequence f mn satisfies uniform 2 − condition; we get To prove the inclusion Then, for all |x mn a mn | < ∞.
Since the sequence f mn satisfies the uniform 2 − condition and then ϕ rs y mn a mn λ mn (m+n)! < ∞. by (10). Thus, d (x n−1 )) ϕ p . This gives that We are granted with (9) and (11) that (3) Similarly, one can prove that if the sequence g mn satisfies the uniform 2 − condition.

Proposition 1.
If 0 < q mn < p mn < ∞ for each m and m, then This implies that for sufficiently large value of m and n. Since f mn s are nondecreasing, we get , · · · , d (x n−1 )) ϕ p .
Proof. The proof follows from Proposition 9.