Open Access

On Generalized Paranormed Statistically Convergent Sequence Spaces Defined by Orlicz Function

Journal of Inequalities and Applications20092009:729045

DOI: 10.1155/2009/729045

Received: 8 May 2009

Accepted: 26 August 2009

Published: 27 September 2009

Abstract

We define generalized paranormed sequence spaces , , , and defined over a seminormed sequence space . We establish some inclusion relations between these spaces under some conditions.

1. Introduction

will represent the spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent, and bounded statistically null -valued sequence spaces throughout the paper, where is a seminormed space, seminormed by For the space of complex numbers, these spaces represent the which are the spaces of all, convergent, null, statistically convergent, statistically null, bounded, bounded statistically convergent, and bounded statistically null sequences, respectively. The zero sequence is denoted by , where is the zero element of .
The idea of statistical convergence was introduced by Fast [1] and studied by various authors (see [24]). The notion depends on the density of subsets of the set of natural numbers. A subset of is said to have density if
(1.1)

where is the characteristic function of .

A sequence is said to be statistically convergent to the number (i.e., ) if for every
(1.2)

In this case, we write or stat lim

Let be a mapping of the set of positive integers into itself. A continuous linear functional on , the space of real bounded sequences, is said to be an invariant mean or -mean if and only if

(1) when the sequence has for all ,

(2) , where

(3) for all

The mappings are one to one and such that for all positive integers and where denotes the th iterate of the mapping at . Thus extends the limit functional on , the space of convergent sequences, in the sense that for all . In that case is translation mapping , a -mean is often called a Banach limit, and , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences [5].

If , set It can be shown [6] that

(1.3)

where

Several authors including Schaefer [7], Mursaleen [6], Sava [8], and others have studied invariant convergent sequences.

An Orlicz function is a function , which is continuous, nondecreasing, and convex with for and as . If the convexity of an Orlicz function is replaced by

(1.4)

then this function is called modulus function, introduced and investigated by Nakano [9] and followed by Ruckle [10], Maddox [11], and many others.

Lindenstrauss and Tzafriri [12] used the idea of Orlicz function to construct the sequence space

(1.5)

which is called an Orlicz sequence space.

The space becomes a Banach space with the norm
(1.6)

The space is closely related to the space which is an Orlicz sequence space with for . Orlicz sequence spaces were introduced and studied by Parashar and Choudhary [13], Bhardwaj and Singh [14], and many others.

It is well known that since is a convex function and then for all with .

An Orlicz funtion is said to satisfy -condition for all values of , if there exists constant , such that . The -condition is equivalent to the inequality for all values of and for being satisfied [15].

The notion of paranormed space was introduced by Nakano [16] and Simons [17]. Later on it was investigated by Maddox [18], Lascarides [19], Rath and Tripathy [20], Tripathy and Sen [21], Tripathy [22], and many others.

The following inequality will be used throughout this paper. Let be a sequence of positive real numbers with and let . Then for , the set of complex numbers for all , we have [23]
(1.7)

2. Definitions and Notations

A sequence space is said to be solid (or normal) if , whenever and for all sequences of scalars with for all .

A sequence space is said to be symmetric if implies , where is a permutation of .

A sequence space is said to be monotone if it contains the canonical preimages of its step spaces.

Throughout the paper will represent a sequence of positive real numbers and a seminormed space over the field of complex numbers with the seminorm . We define the following sequence spaces:

(2.1)

We write

(2.2)

If , for each and then these spaces reduce to the spaces

(2.3)

defined by Tripathy and Sen [21].

Firstly, we give some results; those will help in establishing the results of this paper.

Lemma 2.1 ([21]).

For two sequences and one has if and only if , where such that

Lemma 2.2 ([21]).

Let and , then the followings are equivalent:

(i) and

(ii)

Lemma 2.3 ([21]).

Let be an infinite subset of such that Let
(2.4)

Then is uncountable.

Lemma 2.4 ([24]).

If a sequence space is solid then is monotone.

3. Main Results

Theorem 3.1.

, , , are linear spaces.

Proof.

Let . Then there exist , positive real numbers and , such that
(3.1)

Let , be scalars and let . Then by (1.7) we have

(3.2)

Hence is a linear space.

The rest of the cases will follow similarly.

Theorem 3.2.

The spaces and are paranormed spaces, paranormed by
(3.3)

where .

Proof.

We prove the theorem for the space . The proof for the other space can be proved by the same way. Clearly for all and . Let Then we have , such that
(3.4)

Let . Then by the convexity of , we have

(3.5)

Hence from above inequality, we have

(3.6)

For the continuity of scalar multiplication let be any complex number. Then by the definition of we have

(3.7)

where .

Since we have Then

(3.8)

and therefore converges to zero when converges to zero or converges to zero.

Hence the spaces and are paranormed by .

Theorem 3.3.

Let be complete seminormed space, then the spaces and are complete.

Proof.

We prove it for the case and the other case can be established similarly. Let be a Cauchy sequence in for all . Then , as . For a given , let and to be such that . Then there exists a positive integer such that
(3.9)

Using definition of paranorm we get

(3.10)

Hence is a Cauchy sequence in . Therefore for each there exists a positive integer such that

(3.11)

Using continuity of , we find that

(3.12)

Thus

(3.13)

Taking infimum of such s we get

(3.14)

for all and . Since and is continuous, it follows that . This completes the proof of the theorem.

Theorem 3.4.

Let and be two Orlicz functions satisfying -condition. Then

(i)

(ii)

where , , and .

Proof.
  1. (i)
    We prove this part for and the rest of the cases will follow similarly. Let . Then for a given , there exists such that there exists a subset of with , where
    (3.15)
     

If we take then implies that . Hence we have by convexity of

(3.16)

Thus

(3.17)

Hence by (3.15) it follows that for a given , there exists such that

(3.18)
Therefore
  1. (ii)

    We prove this part for the case and the other cases will follow similarly.

     
Let . Then by using (1.7) it can be shown that . Hence
(3.19)

This completes the proof.

Theorem 3.5.

For any sequence of positive real numbers and for any two seminorms and on one has
(3.20)

where , , and .

Proof.

The proof follows from the fact that the zero sequence belongs to each of the classes the sequence spaces involved in the intersection.

The proof of the following result is easy, so omitted.

Proposition 3.6.

Let be an Orlicz function which satisfies condition, and let and be two seminorms on . Then

(i)

(ii)

(iii) where , , , and ,

(iv)if is stronger than , then

(3.21)

where , , and .

Theorem 3.7.

The spaces are not solid, where and .

Proof.

To show that the spaces are not solid in general, consider the following example. Let , for all , , where and for all . Then we have for all . Consider the sequence , where is defined by and for each fixed . Hence for and . Let if is odd and , otherwise. Then for and . Thus is not solid for and .

The proof of the following result is obvious in view of Lemma 2.4.

Proposition 3.8.

The space is solid as well as monotone for and .

Theorem 3.9.

The spaces are not symmetric, where , , and .

Proof.

To show that the spaces are not symmetric, consider the following examples. Let , for all , , where and for all . Then we have for all . We consider the sequence defined by if and otherwise. Then for and . Let be a rearrangement of , which is defined as if is odd and , otherwise. Then for and .

To show for and , let for all odd and for all even. Let and , where . Let and for all . Then we have for all . We consider

(3.22)
Then for and . We consider the rearrengement of as
(3.23)

Then for and . Thus the spaces are not symmetric in general, where , , and .

Proposition 3.10.

For two sequences and one has if and only if , where such that .

Proof.

The proof is obvious in view of Lemma 2.1.

The following result is a consequence of the above result.

Corollary 3.11.

For two sequences and one has if and only if and , where such that .

The following result is obvious in view of Lemma 2.2.

Proposition 3.12.

Let and , then the followings are equivalent:

(i) and

(ii)

Theorem 3.13.

Let be a sequence of nonnegative bounded real numbers such that Then
(3.24)

Proof.

Let . Then for a given , we have
(3.25)

where the vertical bar indicates the number of elements in the enclosed set.

From the above inequality it follows that .

Conversely let . Let such that

(3.26)

For a given , let .

Let .

Since , so , uniformly in , as . There exits a positive integer such that for all . Then for all , we have

(3.27)

Hence .

This completes the proof of the theorem.

The following result is a consequence of the above theorem.

Corollary 3.14.

Let and be two bounded sequences of real numbers such that and Then
(3.28)

Since the inclusion relations and are strict, we have the following result.

Corollary 3.15.

The spaces and are nowhere dense subsets of

The following result is obvious in view of Lemma 2.3.

Proposition 3.16.

The spaces and are not separable.

Declarations

Acknowledgment

The authors would like to express their gratitude to the reviewers for their careful reading and valuable suggestions which improved the presentation of the paper.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science and Arts, Sakarya University

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Copyright

© M. Başarir and S. Altundağ. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.