Some New Double Sequence Spaces Defined by Orlicz Function in -Normed Space

Abstract

The aim of this paper is to introduce and study some new double sequence spaces with respect to an Orlicz function, and also some properties of the resulting sequence spaces were examined.

1. Introduction

We recall that the concept of a 2-normed space was first given in the works of Gähler ([1, 2]) as an interesting nonlinear generalization of a normed linear space which was subsequently studied by many authors (see, [3, 4]). Recently, a lot of activities have started to study summability, sequence spaces, and related topics in these nonlinear spaces (see, e.g., [5–9]). In particular, Savaş [10] combined Orlicz function and ideal convergence to define some sequence spaces using 2-norm.

In this paper, we introduce and study some new double-sequence spaces, whose elements are form -normed spaces, using an Orlicz function, which may be considered as an extension of various sequence spaces to -normed spaces. We begin with recalling some notations and backgrounds.

Recall in [11] that an Orlicz function is continuous, convex, and nondecreasing function such that and for , and as .

Subsequently, Orlicz function was used to define sequence spaces by Parashar and Choudhary [12] and others. An Orlicz function can always be represented in the following integral form: , where is the known kernel of , right differential for , , for , is nondecreasing, and as .

If convexity of Orlicz function is replaced by , then this function is called Modulus function, which was presented and discussed by Ruckle [13] and Maddox [14].

Remark 1.1.

If is a convex function and , then for all with .

Let and be real vector space of dimension , where . An -norm on is a function which satisfies the following four conditions:

(i) if and only if are linearly dependent,

(ii) are invariant under permutation,

(iii), ,

(iv).

The pair is then called an -normed space [3].

Let be equipped with the -norm, then the volume of the -dimensional parallelepiped spanned by the vectors, which may be given explicitly by the formula

(1.1)

where denotes inner product. Let be an -normed space of dimension and a linearly independent set in . Then, the function on is defined by

(1.2)

is defines an norm on with respect to (see, [15]).

Definition 1.2 (see [7]).

A sequence in -normed space is aid to be convergent to an in (in the -norm) if

(1.3)

for every .

Definition 1.3 (see [16]).

Let be a linear space. Then, a map is called a paranorm (on ) if it is satisfies the following conditions for all and scalar:

(i)  ,

(ii),

(iii),

(iv) and imply .

2. Main Results

Let be any -normed space, and let denote -valued sequence spaces. Clearly is a linear space under addition and scalar multiplication.

Definition 2.1.

Let be an Orlicz function and any -normed space. Further, let be a bounded sequence of positive real numbers. Now, we define the following new double sequence space as follows:

(2.1)

for each .

The following inequalities will be used throughout the paper. Let be a double sequence of positive real numbers with , and let . Then, for the factorable sequences and in the complex plane, we have as in Maddox [16]

(2.2)

Theorem 2.2.

sequences space is a linear space.

Proof.

Now, assume that and . Then,

(2.3)

Since is a -norm on , and is an Orlicz function, we get

(2.4)

where

(2.5)

and this completes the proof.

Theorem 2.3.

space is a paranormed space with the paranorm defined by

(2.6)

where , .

Proof.

• (i) Clearly, and (ii) . (iii) Let , then there exists such that

  

  (2.7)

So, we have

(2.8)

and thus

(2.9)

• (iv) Now, let and . Since

  

  (2.10)

This gives us .

Theorem 2.4.

If for each and , then .

Proof.

If , then there exists some such that

(2.11)

This implies that

(2.12)

for sufficiently large values of and . Since is nondecreasing, we are granted

(2.13)

Thus, . This completes the proof.

The following result is a consequence of the above theorem.

Corollary 2.5.

• (i) If for each and , then

  

  (2.14)

• (ii) If for each and , then

  

  (2.15)

Theorem 2.6.

, where is the double space of bounded sequences and .

Proof.

. Then, there exists an such that for each , . We want to show . But

(2.16)

and this completes the proof.

Theorem 2.7.

Let and be Orlicz function. Then, we have

(2.17)

Proof.

We have

(2.18)

Let ; when adding the above inequality from to we get and this completes the proof.

Definition 2.8 (see [10]).

Let be a sequence space. Then, is called solid if whenever for all sequences of scalars with for all .

Definition 2.9.

Let be a sequence space. Then, is called monotone if it contains the canonical preimages of all its step spaces (see, [17]).

Theorem 2.10.

The sequence space is solid.

Proof.

Let ; that is,

(2.19)

Let () be double sequence of scalars such that for all . Then, the result follows from the following inequality:

(2.20)

and this completes the proof.

We have the following result in view of Remark 1.1 and Theorem 2.10.

Corollary 2.11.

The sequence space is monotone.

Definition 2.12 (see [18]).

Let denote a four-dimensional summability method that maps the complex double sequences into the double-sequence , where the th term to is as follows:

(2.21)

Such transformation is said to be nonnegative if is nonnegative for all , and .

Definition 2.13.

Let be a nonnegative matrix. Let be an Orlicz function and a factorable double sequence of strictly positive real numbers. Then, we define the following sequence spaces:

(2.22)

for each . If , then we say is summable to , where .

If we take and for all , then we have

(2.23)

Theorem 2.14.

is linear spaces.

Proof.

This can be proved by using the techniques similar to those used in Theorem 2.2.

Theorem 2.15.

• (1) If , then

  

  (2.24)

• (2) If , then

  

  (2.25)

Proof.

• (1) Let ; since , we have

  

  (2.26)

and hence .

• (2) Let for each and . Let .

Then, for each , there exists a positive integer such that

(2.27)

for all . This implies that

(2.28)

Thus, , and this completes the proof.