Δ^m-Strongly summable sequences spaces in 2-normed spaces defined by ideal convergence and an Orlicz function

Abstract

In this paper, we study certain new difference sequence spaces using ideal convergence and an Orlicz function in 2-normed spaces and we give some relations related to these sequence spaces.

Introduction

Recall in [13] that an Orlicz function M: [0, ∞) → [0, ∞) is continuous, convex, non-decreasing function such that M(0) = 0 and M(x) > 0 for x > 0, and M(x) → ∞ as x → ∞.

Subsequently Orlicz function was used to define sequence spaces by Parashar and Choudhary [16] and others [6], [19], [21].

An Orlicz function M can always be represented in the following integral form: $M(x)={\int }_0^{x}p(t)\mathit{dt}$ where p is the known kernel of M, right differential for t ⩾ 0, p(0) = 0, p(t) > 0 for t > 0, p is non-decreasing and p(t) → ∞ as t → ∞.

If convexity of Orlicz function M is replaced by M(x + y) ⩽ M(x) + M(y) then this function is called Modulus function, which was presented and discussed by Ruckle [18] and Maddox [14].

The notion of ideal convergence was introduced first by Kostyrko et al. [11] as a generalization of statistical convergence. More applications of ideals can be seen in [3], [4], [5], [12].

The concept of 2-normed space was initially introduced by Gähler [7] as an interesting non-linear generalization of a normed linear space which was subsequently studied by many authors (see [8], [17]). Recently a lot of activities have started to study summability, sequence spaces and related topics in these nonlinear spaces (see [20], [9], [10]).

In this article, we define some new sequence spaces in 2-normed spaces by using Orlicz functions, generalized difference sequences and also ideals. In this context it should be noted that though generalized difference sequence spaces have been studied before (see [1], [2], [21]). They have not been studied in non-linear structures like 2-normed spaces and there ideals were not used.

Let (X, ∥·∥) be a normed space. Recall that a sequence $(x_n{)}_{n\in \mathbb{N}}$ of elements of X is called to be statistically convergent to $x\in X$ if the set $A(\epsilon )=\{n\in \mathbb{N}:\Vert x_n-x\Vert ⩾\epsilon \}$ has natural density zero for each ε > 0.

A family $\mathcal{I}\subset 2^Y$ of subsets a nonempty set Y is said to be an ideal in Y if

• $\varnothing \in \mathcal{I}$;

• $A\text{,}B\in \mathcal{I}$ imply $A\cup B\in \mathcal{I}$;

• $A\in \mathcal{I}\text{,}B\subset A$ imply $B\in \mathcal{I}$, while an admissible ideal $\mathcal{I}$ of Y further satisfies $\{x\}\in \mathcal{I}$ for each x ∈ Y, (see [8], [9]).

Given $\mathcal{I}\subset 2^{\mathbb{N}}$ be a nontrivial ideal in $\mathbb{N}$. The sequence $(x_n{)}_{n\in \mathbb{N}}$ in X is said to be $\mathcal{I}$-convergent to $x\in X$, if for each ε > 0 the set $A(\epsilon )=\{n\in \mathbb{N}:\Vert x_n-x\Vert ⩾\epsilon \}$ belongs to $\mathcal{I}$, (see [11], [12]).

Let X be a real vector space of dimension d, where 2 ⩽ d <  ∞. A 2-norm on X is a function $\Vert ·\text{,}·\Vert :X\times X\to \mathbb{R}$ which satisfies;

• ∥x, y∥ = 0 if and only if x and y are linearly dependent;

• ∥x, y∥ = ∥y, x∥;

• $\Vert \alpha x\text{,}y\Vert =|\alpha |\Vert x\text{,}y\Vert \text{,}\alpha \in \mathbb{R}$;

• ∥x, y + z∥ < ∥x, y∥ + ∥x,z∥.

The pair (X, ∥·, ·∥) is then called a 2-normed space [8]. As an example of a 2-normed space we may take $X={\mathbb{R}}^2$ being equipped with the 2-norm ∥x, y∥ ≔ the area of the parallelogram spanned by the vectors x and y, which may be given explicitly by the formula

$$\Vert x_1{x}_2{\Vert }_E=\mathit{abs}\left(\left|\begin{array}{cc}{x}_{11}\hfill & x_{12}\hfill \\ x_{21}\hfill & x_{22}\hfill \end{array}\right|\right)\text{.}$$

Recall that (X, ∥·, ·∥) is a 2-Banach space if every Cauchy sequence in X is convergent to some x in X.