Difference Sequence Spaces of Second Order Defined by a Sequence of Moduli

Khalid Ebadullah; Kibreab Gebreselassie

I. INTRODUCTION

Let N, R and C be the sets of all natural, real and complex numbers respectively. We write

the space of all real or complex sequences. Let $\ell_{\infty}$ , c and $c_{0}$ denote the Banach spaces of bounded, convergent and null sequences respectively.

The following subspaces of $\omega$ were first introduced and discussed by Maddox [10-12].

where $p = (p_{k})$ is a sequence of strictly positive real numbers.

The idea of difference sequence sets

where $X = \ell_{\infty}$ , $c$ or $c_0$ was introduced by Kizmaz [6].

Kizmaz [6] defined the following sequence spaces,

where $\triangle x = (x_{k} - x_{k+1})$ . These are Banach spaces with the norm

Ref

Mikail [14] defined the sequence spaces

Where $(\triangle^2 x) = (\triangle^2 x_k) = (\triangle x_k - \triangle x_{k + 1})$

The sequence spaces $\ell_{\infty}(\triangle^{2})$ , $c(\triangle^{2})$ and $c_{0}(\triangle^{2})$ are Banach spaces with the norm

Mikail and Colak [15] defined the sequence spaces

where $m\in N$

and so that

and showed that these are Banach spaces with the norm

Ref

Let U be the set of all sequences $u = (u_k)$ such that $u_k \neq 0$ ( $k = 1,2,3 \ldots$ ). Malkowsky[13] defined the following sequence spaces

where $u \in U$ .

The concept of paranorm (see[12]) is closely related to linear metric spaces. It is a generalization of that of absolute value.

Let X be a linear space. A function $g: X \longrightarrow R$ is called paranorm, if for all $x, y \in X$ ,

(PI) $g(x) = 0$ if x = 0,

(P2) $g(-x) = g(x)$ ,

(P3) $g(x + y) \leq g(x) + g(y)$ ,

(P4) If $(\lambda_{n})$ is a sequence of scalars with $\lambda_{n} \to \lambda (n \to \infty)$ and $x_{n}, a \in X$ with $x_{n} \to a (n \to \infty)$ , in the sense that $g(x_{n} - a) \to 0 (n \to \infty)$ , in the sense that $g(\lambda_{n}x_{n} - \lambda a) \to 0 (n \to \infty)$ .

A paranorm g for which $g(x) = 0$ implies x = 0 is called a total paranorm on X, and the pair $(X, g)$ is called a totally paranormed space.

The idea of modulus was structured by Nakano[16].

A function $f:[0,\infty)\longrightarrow [0,\infty)$ is called a modulus if

(P1) $f(t) = 0$ if and only if $t = 0$ ,

(P2) $f(t+u) \leq f(t) + f(u)$ for all $t, u \geq 0$ ,

(P3) $f$ is increasing, and

(P4) $f$ is continuous from the right at zero.

Ruckle [17-19] used the idea of a modulus function $f$ to construct the sequence space

This space is an FK space. Ruckle[17-19] proved that the intersection of all such $X(f)$ spaces is $\phi$ , the space of all finite sequences.

The space $X(f)$ is closely related to the space $l_{1}$ which is an $X(f)$ space with $f(x) = x$ for all real $x \geq 0$ . Thus Ruckle[17-19] proved that, for any modulus $f$ .

Ref

The space $X(f)$ is a Banach space with respect to the norm

Spaces of the type $X(f)$ are a special case of the spaces structured by Gramsch in[4]. From the point of view of local convexity, spaces of the type $X(f)$ are quite pathological. Symmetric sequence spaces, which are locally convex have been frequently studied by Garling[2-3], Köthe[9] and Ruckle[17-19].

Kolk [7-8] gave an extension of $X(f)$ by considering a sequence of moduli $F = (f_k)$ and defined the sequence space

Khan and Lohani [5] defined the following sequence spaces

If we take $x_{k}$ instead of $\triangle x$ , then we have the following sequence spaces

Asma and Colak[1] defined the following sequence spaces

Ref

where $u \in U$ , $p = (p_k)$ be any sequence of positive reals.

Khan and Lohani [5] defined the following sequence spaces

which are paranormed spaces paranormed with

where $H = \max(1, \sup_{k \geq 0} p_k)$ and $a = f_k(l)$ , $l = \sup_{k \geq 0} (|u_k \triangle x_k|)$ .

II. MAIN RESULTS

In this article we introduce the following class of sequence spaces.

Theorem 2.1. $\ell_{\infty}(u,\triangle^{2},F)$ is a Banach space with norm

where $\alpha = f_k(l)$ and $l = \sup_{k\geq 0}(|u_k\triangle^2 x_k|)$ .

Proof. Let $(x^{i})$ be a cauchy sequence in $\ell_{\infty}(u,\triangle^2,F)$ for each $i\in N$ . Let $r,x_0$ be fixed. Then for each $\frac{\epsilon}{rx_0} >0$ there exists a positive integer $N$ such that

Using the definition of norm, we get

Notes ie,

Hence we can find $r > 0$ with $f_{k}\left(\frac{rx_{0}}{2}\right) \geq \alpha$ such that

This implies that

Since $u_{k} \neq 0$ for all $k$ , we have

| \triangle^ {2} x _ {k} ^ {i} - \triangle^ {2} x _ {k} ^ {j} | \leq \frac {\epsilon}{2} \quad \text { for all i, j\geq N }

Hence $(\triangle^2 x_k^i)$ is a cauchy sequence in R.

Using the continuity of $F = (f_{k})$ we can show that

since $(x^{i})\in \ell_{\infty}(u,\triangle^{2},F)$ and $F = (f_k)$ is continuous it follows that $x\in \ell_{\infty}(u,\triangle^{2},F)$ . Thus $\ell_{\infty}(u,\triangle^{2},F)$ is complete.

Theorem 2.2. $\ell_{\infty}(u,\triangle^{2},F,p)$ is a complete paranormed space with

where $H = \max(1, \sup_{k \geq 0} p_k)$ and $\alpha = f_k(l)$ , $l = \sup_{k \geq 0} (|u_k \triangle^2 x_k|)$ .

Notes

Proof. Let $(x^{i})$ be a cauchy sequence in $\ell_{\infty}(u,\triangle^2,F,p)$ for each $i\in N$ .

Let $r > 0, x_0$ be fixed. Then for each $\frac{\epsilon}{rx_0} > 0$ there exists a positive integer $N$ such that

Q _ {u} (x ^ {i} - x ^ {j}) _ {\triangle^ {2}} < \frac {\epsilon}{r x _ {0}} \quad \text { for all i, j\geq N }

Using the definition of paranorm, we get

\sup _ {k \geq 0} f _ {k} (\frac {| u _ {k} (\triangle^ {2} x _ {k} ^ {i} - \triangle^ {2} x _ {k} ^ {j}) |}{Q _ {u} (x ^ {i} - x ^ {j}) _ {\triangle^ {2}}}) ^ {\frac {p _ {k}}{H}} \leq \alpha , \quad \text { for all i, j\geq N }

ie,

Hence we can find $r > 0$ with $f_{k}\left(\frac{rx_{0}}{2}\right) \geq \alpha$ such that

This implies that

Since $u_{k} \neq 0$ for all $k$ , we have

Hence $(\triangle^2 x_k^i)$ is a cauchy sequence in R.

Using the continuity of $F = (f_{k})$ we can show that

since $(x^{i})\in \ell_{\infty}(u,\triangle^{2},F,p)$ and $F = (f_k)$ is continuous it follows that $x\in \ell_{\infty}(u,\triangle^{2},F,p)$ . Thus $\ell_{\infty}(u,\triangle^{2},F,p)$ is complete.

Theorem 2.3. Let $0 < p_k \leq q_k < \infty$ for each $k$ . Then we have

Notes

Proof. Let $x \in c_0(u, \triangle^2, F, p)$ that is

This implies that

for sufficiently large k, since modulus function is non decreasing.

Hence we get

Therefore $x\in c_0(u,\triangle^2,F,q)$

Theorem 2.4. (a) Let $0 < \inf p_k \leq p_k \leq 1$ . Then we have

(b) Let $1 \leq p_k \leq \sup_k p_k < \infty$ . Then we have

Proof. (a) Let $x \in c_0(u, \triangle^2, F, p)$ , that is

Since $0 < \inf p_k \leq p_k \leq 1$ ,

Hence $x \in c_0(u, \triangle^2, F)$ .

(b) Let $p_k \geq 1$ for each $k$ and $\sup p_k < \infty$ .

Suppose that $x \in c_0(u, \triangle^2, F)$ .

Then for each $\epsilon > 0$ there exists a positive integer $N$ such that

Notes

Since $1 \leq p_{k} \leq \sup_{k} p_{k} < \infty$ , we have

Therefore $x\in c_0(u,\triangle^2,F,p)$