On A Class of New Type Generalized Difference Sequences Related to the P-Normed lp...

AYHAN ESİ

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On A Class of New Type Generalized Difference Sequences Related to the P-Normed lp Space Defined By Orlicz Functions

AYHAN ESİ

Department of Mathematics, Science and Art Faculty in Adiyaman, Adiyaman University, Adiyaman, Turkey

Abstract

The idea of difference sequence spaces were defined by Kizmaz [6] and generalized by Et and Colak [5]. Later Esi et al. [4] introduced the notion of the new difference operator for fixed n,m∈N. In this article we introduce new type generalized difference sequence space using by the Orlicz function. We give various properties and inclusion relations on this new type difference sequence space.

Cite this article:

  • ESİ, AYHAN. "On A Class of New Type Generalized Difference Sequences Related to the P-Normed lp Space Defined By Orlicz Functions." American Journal of Applied Mathematics and Statistics 1.4 (2013): 52-56.
  • ESİ, A. (2013). On A Class of New Type Generalized Difference Sequences Related to the P-Normed lp Space Defined By Orlicz Functions. American Journal of Applied Mathematics and Statistics, 1(4), 52-56.
  • ESİ, AYHAN. "On A Class of New Type Generalized Difference Sequences Related to the P-Normed lp Space Defined By Orlicz Functions." American Journal of Applied Mathematics and Statistics 1, no. 4 (2013): 52-56.

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1. Introduction

Throughout the article w, and denote the spaces all, bounded and p absolutely summable sequences, respectively. The zero sequence is denoted by . The sequence space was introduced by Sargent [11], who studied some of its properties and obtained its relationship with the space . Later on it was investigated from sequence space point of view by Rath [9], Rath and Tripathy [10], Tripathy and Sen [15], Tripathy and Mahanta [14], Esi [2] and others.

An Orlicz function is a function M: [0,∞)→ [0,∞) , which is continuous, non-decreasing and convex with M(0)=0, M(x) > 0 for x > 0 and M(x).

An Orlicz function is said to satisfy Δ2-condition for all values of u, if there exists a constant K > 0, such that M(2u) KM(u), u 0.

Remark. An Orlicz function satisfies the inequality for all .

Lindenstrauss and Tzafriri [7] used the idea of Orlicz function to construct the sequence space

The space with the norm

becomes a Banach space which is called an Orlicz sequence space. The space is closely related to the space which is an Orlicz sequence space with

In the later stage different Orlicz sequence spaces were introduced and studied by Tripathy and Mahanta [14] , Esi [1], Esi and Et [3], Parashar and Choudhary [8], and many others.

Kizmaz [6] defined the difference sequence spaces (Δ), c(Δ) and co(Δ) as follows:

for Z = , c and co, where Δx = = for all kN.

The above spaces are Banach spaces, normed by

Later, the difference sequence spaces were generalized by Et and Çolak [5] as follows: Let be fixed integer, then for , where and so .

They showed that the above spaces are Banach spaces, normed by

After then, the notion new type of difference sequence spaces were further generalized Esi and et.al. [4] as follows:

Let be fixed integers, then

for , where and for all . The new type generalized difference has the following binomial representation:

They showed that the above spaces are Banach spaces, normed by

where, for ; and

2. Definitions and Background

Throughout the article denotes the set of all subsets of N, the set of natural numbers, those do not contain more than s elements. Further will denote a non-decreasing sequence of positive real numbers such that for all nN. The class of all the sequences satisfying this property is denoted by Φ.

The space introduced and studied by Sargent [11] is defined as follows:

Recently Tripathy and Mahanta [13] defined and studied the following sequence space: Let M be an Orlicz function, then

The purpose of this paper is to introduce and study a class of new type generalized difference sequences related to the space using by Orlicz function.

In this article we introduce the following sequence space: Let M be an Orlicz function and p= be bounded sequence of strictly positive real numbers and be fixed integers, then

Taking for all k and m=n=1 i.e., considering only first difference we have the following difference sequence space which were defined and studied by Tripathy and Mahanta [13]

Taking for all k, M(x)=x and m=n=1 i.e., considering only first difference we have the following difference sequence space which were defined and studied by Tripathy [12]

Taking for all k, M(x)=x and n=1, we have the following difference sequence space which were defined and studied by Esi [2]

The space for is defined by Rath [9] as follows:

Let be a sequence, then S ( X ) denotes the set of all permutations of the elements of i.e. , S ( X ) = {: π ( k ) is a permutation on N}. A sequence space E is said to be symmetric if S ( X ) ⊂ E for all xE.

A sequence space E is said to be solid (or normal ) if E , whenever (xk) ∈ E and for all sequences of scalars (αk) with for all kN

A sequence space E is said to be monotone, if it contains the canonical pre-images of its step spaces.

The following inequality will be used throughout the paper

where and are complex numbers, and

3. Main Results

In this section we prove some results involving the sequence space

Theorem 1. Let p= be bounded sequence of strictly positive real numbers.Then the space is a linear space over the complex field C.

Proof: Let , and C. Then there exists positive numbers and such that

and

Let . Since M is non-decreasing and convex

Hence

Theorem 2. Let p= be bounded sequence of strictly positive real numbers and . Then is a linear topological space paranormed by

where for ; for m=0 and r=m for n=0.

Proof: Clearly . Next = implies and such as , therefore =0. It can be easily shown that =0=

Next, let and be such that

and

Let . Then we have

Since the are non-negative, we have

Next, for , without loss of generality, let , then

where

So, the continuity of the scalar multiplication follows from the above inequality.

Theorem 3. if and only if

Proof: Let and Then

for some .

So,

Therefore

Conversely , let . Suppose that .Then there exists a sequence of natural numbers such that . Let . Then there exists such that

Now we have

Therefore . As such we arrive at a contradiction. Hence

The following result is a consequence of Theorem 3.

Corollary 4: Let M be an Orlicz function. Then if and only if and for all s=1,2,3,... .

Theorem 5: Let p= be bounded sequence of strictly positive real numbers and let be Orlicz functions satisfying Δ2-condition. Then

Proof: Let . Then we have

for some

Let and choose with such that for . Let for all m and n and for any , let

where the first summation is over and the second is over . For the first summation above, we can write

(1)

For the second summation, we will make following procedure. For , we have

Since M is non-decreasing and convex, it follows that

Since M satisfies Δ2 condition, we can write

Hence

(2)

By (1) and (2), we have

Taking in Theorem 5, we have the following result.

Corollary 6: Let p= be bounded sequence of strictly positive real numbers and let M be an Orlicz function satisfying Δ2-condition. Then

From Theorem 3 and Corollary 6, we have

Corollary 7: Let p= be bounded sequence of strictly positive real numbers and let M be an Orlicz function satisfying Δ2-condition. Then

if and only if

Corollary 8: The space is not solid and symmetric in general.

Proof: To show this space is not solid and symmetric in general, consider the following examples, respectively.

Example 1. Let m=n=1, and for all k. Consider for all k and M(x)=x. Then but . Hence the space is not solid in general.

Example 2. Let m=n=1, and for all k and M(x)=x. Then the sequence define for all k is in . Consider the sequence , the rearrangement of define as follows

Then . Hence the space is not symmetric in general.

Finally, in this section, we consider that and are any bounded sequences of strictly positive real numbers. We are able to prove below results only under additional conditions.

Corollary 9: a) If for all k, then

b) If for all k, then

c) Let for all k and be bounded, then

Proof: Using the same technique as in Theorem 4 in [1], it is easy to prove the Corollary 9.

Acknowledgement.

The author would like to thank the anonymous reviewer for his/her careaful reading and making some useful comments on earlier version of this paper which improved the presentation and its readability.

References

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