On multiplicative difference sequence spaces and related dual properties

The main purpose of the present article is to introduce the multiplicative difference sequence spaces of order $m$  by defining the multiplicative difference operator $\Delta_{*}^m(x_k)=x^{}_k~x^{-m}_{k+1}~x^{\binom{m}{2}}_{k+2}~x^{-\binom{m}{3}}_{k+3}~x^{\binom{m}{4}}_{k+4}\dots x^{(-1)^m}_{k+m}$ for all $m, k \in \mathbb N$. By using the concept of multiplicative linearity various topological properties are investigated  and the relations related to their dual spaces are studied via multiplicative infinite matrices.

Let ω * be the space of all positive real valued sequences defined by and any subspace of ω * is called a sequence space over the real field R + .

Ugur Kadak
Quite recently Çakmak and Başar [8] have defined the multiplicative sets ℓ * ∞ , c * , c * 0 and ℓ * 1 of sequences as follows: In particular the sets ℓ * ∞ , c * and c * 0 of sequences over R + , normed by • ∞ = sup k∈N |x k | * , it is proved that these are all Banach spaces.Also Kadak [9] have examined the spaces bs * , cs * and cs * 1 defined by Also the inverse operator can be interpreted as: For instance, x 3 k+2 , . . . .The main focus of this paper is to extend the difference sequence spaces of order m defined earlier to the multiplicative form of these spaces.Moreover, by using multiplicative difference operator ∆ m * , the duals of these spaces are investigated with respect to the multiplicative infinite matrices.

Preliminaries and definitions
In the period from 1967 till 1972, Grossman and Katz [10] introduced the non-Newtonian calculus consisting of the branches of geometric, bigeometric, quadratic and biquadratic calculus etc.Also Grossman extended this notion to the other fields in [11,12].Many authors have extensively developed the notion of multiplicative calculus.The complete mathematical description of multiplicative calculus, was given by Bashirov et al. [13].Also some authors have also worked on the classical sequence spaces and related topics by using this type calculus [14,15].Further Kadak et al. [16,17,18] have examined matrix transformations between certain sequence spaces and have generalized Runge-Kutta numerical method.Definition 2.1.[13] Let X be a nonempty set.A multiplicative metric (*metric) is a mapping d * : X × X → R + satisfying the following conditions: (i) d * (x, y) ≥ 1 for all x, y ∈ X and d * (x, y) = 1 if and only if x = y; (ii) d * (x, y) = d * (y, x) for all x, y ∈ X; (iii) d * (x, y) ≤ d * (x, z) • d * (y, z) for all x, y, z ∈ X (multiplicative triangle inequality).
The multiplicative absolute value of x ∈ R + is defined by On the base of this, one can define multiplicative metric spaces as alternative to the ordinary metric spaces.
Definition 2.2.[8] Let X = (X, d * ) be a *metric space.Then, the basic notions are given as follows: (d) A *complete metric space is a *metric space in which every *Cauchy sequence is *convergent.Now, we give the basic concepts *open and *closed sets.
Definition 2.3.Given any point x 0 ∈ X.Then, for a real number r > 0, is a *neighborhood (or *open ball) of centre x 0 and radius r and where the *vectors u = (u 1 , u 2 ), v = (v 1 , v 2 ) ∈ V and the scalar λ ∈ R + .Then the operations must satisfy the following conditions: (a) For all λ ∈ R + and all u, v ∈ V , u ⊕ v and λ ⊙ v are uniquely defined and belong to V .
(d) The set V contains an additive identity element, denoted by θ A = (1, 1), such that for all u ∈ V , u ⊕ θ A = u.
(e) The set V contains an additive inverse element, denoted by (g) The set V contains an element 1 * such that 1 * ⊙ u = u for all u ∈ V .
Definition 2.6.[17] Let X be a *vector space over the field R + and • * be a function from X to R + satisfying the following axioms: For x, y ∈ X and λ ∈ R + , Then, (X, • * ) is said a *normed space.It is trivial that a *norm on X defines a *metric d * on X which is given by d * (x, y) = x/y * ; (x, y ∈ X) and is called the *metric induced by the *norm.Definition 2.7.(Multiplicative linearity) Let V and W be two *linear spaces.An operator T : V → W is said to be multiplicative linear (*linear) if The important point to note here is the notion of *linearity have not the same meaning than in the standard case since the *linear space has NOT an ordinary linear structure with the usual operations.
Definition 2.8.(i) The *limit of a function f , denoted by * lim (ii) A topological *vector(linear) space X is a *vector space over the topological field that endowed with a topology such that *vector addition and scalar multiplication are *continuos functions.
(iii) A topological *vector space is called *normable if the topology of the space can be induced by a *norm.
Definition 2.9.A sequence space λ with a *linear topology is called a *K-space provided each of the maps p i : λ → R + defined by p i (x) = x i is *continuous for all i ∈ N. A *K-space is called a *FK-space provided λ is a complete linear *metric space.A *FK-space whose topology is *normable is called a *BK-space.
(ii) The proof may be obtained by using similar technique with case (i). where where ) are *Cauchy sequences in real field R and X * , respectively.By using the completeness of R and X * , we have that they are *convergent and suppose that for sufficiently large j, i.e. j > 2m.Then (∆ m Proof: It is obvious that X(∆ m * ) is Banach space (see Thm. 3.3).Suppose that

Dual properties
In this section, following [19] we give the α-, β-and γ-duals of a set λ ⊂ ω * which are respectively denoted by λ α , λ β and λ γ , as follows: On multiplicative difference sequence spaces

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A multiplicative infinite matrix A = (a ij ) of positive real numbers is defined by a function A from the set N × N into R + .The addition (⊕) and scalar multiplication (⊙) of the infinite matrices A = (a ij ) and B = (b ij ) are defined by for all i, j ∈ N. Also, the product (A ⊙ B) ij of A = (a ij ) and B = (b ij ) can be interpreted as and provided the infinite product on the right hand side *converge when the *limit exists.Further the product (4.1) may *diverge for some, or all, values of i, j; the product A ⊙ B may not exist.Let µ 1 , µ 2 ⊂ w * and A = (a nk ) be a multiplicative infinite matrix.Then, we say that A defines a matrix mapping from µ 1 into µ 2 , and denote it by writing A : µ 1 → µ 2 , if for every sequence x = (x k ) ∈ µ 1 the sequence A ⊙ x = {(Ax) n }, the multiplicative A-transform of x, exists and is in µ 2 .In this way, we transform the sequence x = (x k ) into the sequence {(Ax) n } defined by for all k, n ∈ N. Thus, A ∈ (µ 1 : µ 2 ) if and only if the infinite product on the right side of (4.2) *converges for each n ∈ N. A sequence z is said to be A-*summable to γ if A ⊙ x *converges to γ ∈ R + which is called as the A- * lim of z.
Conversely suppose that (4.4) holds and iii) The necessary and sufficient conditions can be obtained by taking into account [20, Theorem 10, pp.223-225] and [16, Theorem 23], respectively.✷ Define the matrices B = (b nk ) and C = (c nk ) as follows: (−1) m −m m for each m ∈ N.